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Distribution rules of crystallographic systematic

absences on the Conway topograph and their

application to powder auto-indexinglowast

Ryoko Oishi-Tomiyasudagger

August 16 2018

Abstract

Powder auto-indexing is the crystallographic problem of lattice determinationfrom an average theta series There in addition to all the multiplicities the lengthsof part of lattice vectors cannot be obtained owing to systematic absences As aconsequence solutions are not always unique We develop a new algorithm to enu-merate powder auto-indexing solutions This is a novel application of the reductiontheory of positive-definite quadratic forms to a problem of crystallography Ouralgorithm is proved to be effective for all types of systematic absences using theirnewly obtained common properties The properties are stated as distribution rulesfor lattice vectors corresponding to systematic absences on a topograph Conwaydefined topographs for 2-dimensional lattices as graphs whose edges are associatedwith |l1|

2 |l2|2 |l1 + l2|

2 |l1 minus l2|2 where l1 l2 are lattice vectors In our enumer-

ation algorithm topographs are utilized as a network of lattice vector lengths Asa crystal structure is a lattice of rank 3 the definition of topographs is generalizedto any higher dimensional lattices using Voronoirsquos second reduction theory Theuse of topographs allows us to speed up the algorithm The computation timeis reduced to 1250ndash132 when it is applied to real powder diffraction patternsAnother advantage of our algorithm is its robustness to missing or false elementsin the set of lengths extracted from a powder diffraction pattern Conograph is thepowder indexing software which implements the algorithm We present results ofConograph for 30 diffraction patterns including some very difficult cases

1 Introduction

Lattice determination problems have been the target of much interest in mathematicsIn particular many mathematicians worked on lattice determination from a theta seriesin the second half of the 20th century with the aim of providing higher dimensionalcounterexamples for the isospectral problem proposed by Kac [16] Lattice determina-tion from a theta series was finally resolved in [24] [25] However it is not well known inthe mathematics community that a similar problem was being studied in crystallographyduring this period

The lattice determination problem in crystallography is called powder auto-indexingThere a set of lattice parameters is determined from a powder diffraction pattern Pow-der auto-indexing is equivalent to lattice determination from an average theta seriesas shown in Appendix A At present powder diffraction is the principal technique for

lowastThis research was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki prefecture (J-PARC-23D06)

daggerHigh Energy Accelerator Research Organization Tsukuba Japan (ryokotomiyasukekjp)

1

determining the structure of materials that do not necessarily form large-sized crystalsA highly automated system of analyzing powder diffraction patterns is required bothfor basic scientific research and a wide range of industrial applications not least in thepharmaceutical industry Hence it is necessary to establish a powerful and reliablepowder auto-indexing algorithm and software

In this paper the powder auto-indexing problem is formulated and solved This isregarded as a new application of the reduction theory of positive-definite quadratic formsto crystallography In consideration of its application to crystallography we mainly dis-cuss cases when the lattice has a rank of N = 2 3 Similar to a theta series informationabout the lengths of lattice vectors is extracted from an average theta series (Section2) The most notable difference is that it is very difficult to acquire the multiplicities

of the lengths (ie number of lattice vectors satisfying |l|2 = q for fixed q isin Rgt0) froman average theta series and only a part of the lengths can be obtained owing to thecrystallographic phenomenon of systematic absences

Systematic absences have not been defined sufficiently explicitly as a mathematicalnotion We provide a definition in Section 4 As in the International Tables [14]systematic absences are classified by the pair (GH)

bull isomorphism class of the crystallographic group G sub O(N) ⋉RN

bull conjugate class of the finite subgroup H sub G

For fixed N systematic absences are categorized by finitely many pairs (Proposition41) When a periodic function weierp belongs to the type (GH) the lengths of llowast isinΓext(GH) is not directly extracted from the average theta series of weierp where Γext(GH)is a subset of the reciprocal lattice Llowast of L determined from (GH)

To date powder auto-indexing algorithms have been studied and improved by expertsin crystallography Among existing software packages Ito [10] [29] TREOR (trial-and-error method [34]) and DICVOL (dichotomy method [4]) are widely used McMaille (agrid search based on the Monte Carlo method [18]) and X-cell (dichotomy method [20])were developed comparatively recently With regard to existing powder auto-indexingsoftware several considerable problems have been reported These include

bull existence of more than one solutions

bull systematic absences

bull missing or false elements in the set of extracted lengths

bull observation errors contained in the length values

bull large zero-point shift

bull overlapping peaks

In some software these problems are caused by limitations intended to suppress the com-putation time Therefore both computation time and success rate should be consideredto discuss powder auto-indexing algorithms (For example although the simplest bruteforce grid search may obtain the highest success rate it takes from a few hours to a fewdays in monoclinic and triclinic cases) This paper presents a method that can resolveeach of the problems listed above The first three are explained in more detail in thefollowing discussion The next two are resolved using the estimated error Err[|l|2] of|l|2 (see (A2) in Section 3 and the beginning of Section 7) The final problem is alsoresolved naturally in our algorithm (Section 74)

Our three main results are as follows First we formulate powder auto-indexingas a mathematical problem and summarize mathematical results on the cardinality of

2

solutions in powder auto-indexing This is considered the most important foundationfor the following discussion in general a lattice L is not determined uniquely in N ge 2even if the rank N of L and all the lengths l isin L are given (Appendix B) On the otherhand the cardinality of L2 satisfying ΛL = ΛL2 is always finite in N le 4 thereforeit is possible to enumerate all such L2 algorithmically (Appendix C) However it isnot certain whether the number of solutions is actually finite in powder auto-indexingowing to systematic absences and the finite observational range Fortunately usingboth physical constraints and mathematical theorems we obtain a necessary conditionfor assuming the finiteness of solutions except for events with zero probability (Section3) As a result it is only necessary to enumerate finitely many solutions for powderauto-indexing

Our second result is a new algorithm to enumerate powder auto-indexing solutions(Sections 71 and 72) We prove that it works regardless of the type of systematicabsences Our basic idea is to use the graph whose edges are associated with a set ofsquares of lengths |llowast1 |2 |llowast2|2 |llowast1 + llowast2 | |llowast1 minus llowast2 |2 for some lattice vectors llowast1 l

lowast2 isin Llowast

In Itorsquos method the parallelogram law 2(|llowast1|2 + |llowast2 |2) = |llowast1 + llowast2 |2 + |llowast1 minus llowast2 |2 is usedto obtain Gram matrices of sublattices of rank 2 (called zones) However simple useof the parallelogram law does not always provide successful results as introduced inFact 1 of Section 62 Therefore instead of just enumerating a set of lengths satisfyingthe parallelogram law our algorithm utilizes graphs whose edges are associated with|llowast1 |2 |llowast2 |2 |llowast1 + llowast2| and |llowast1 minus llowast2|2 as a network of lattice vector lengths This provides

a comprehensive method of analyzing how the two sets |llowast1 |2 |llowast2|2 |llowast1 + llowast2|2 |llowast1 minus llowast2|2and |klowast1 |2 |klowast2 |2 |klowast1 + klowast2 |2 |klowast1 minus klowast2 |2 are related to each other

For lattices of rank 2 such a graph has already been defined by Conway who calledit a topograph [6] As explained in section 51 a graph having the required property isconstructed for general N by using Voronoirsquos second reduction theory We also call thisa topograph Basic properties of topographs for lattices of rank N = 2 3 are explainedin Section 52

In Section 6 we shall see how primitive vectors of llowast isin Llowast belonging to Γext(GH)are distributed on a topograph Although the following theorem for the case of N = 2has not been described explicitly it provides a theoretical reason for the parallelogramlaw working appropriately for N = 2

Theorem 1 Let (GH) be a type of systematic absence in N = 2 and let Llowast be thereciprocal lattice of L a lattice consisting of all the translations in G Then for anyprimitive vector llowast of Llowast belonging to Γext(GH) there exists llowast2 isin Llowast such that llowast1 middotllowast2 = 0and llowast1 l

lowast2 make a basis of Llowast

In brief Theorem 1 claims that llowast isin Γext(GH) only appear in the gray area inFigure 1 regardless of the type (GH) With regard to the case of H = 1 we shallintroduce a proof using topographs in Section 61 The case H ) L is not proved heresince it is verified easily by checking lists in [14]

By Theorem 1 a subgraph of a topograph with infinitely many edges is formed by uni-fying substructures as in Figure 2 associated with the lengths |llowast1 |2 |llowast2|2 |llowast1 + llowast2|2 |llowast1 minus llowast2|2of llowast1 l

lowast2 isin Llowast Γext(GH) that are easily extracted from an average theta series

For N = 3 similar statements are proved in Theorems 2 and 3 In this case it isnecessary to use the following formula instead of the parallelogram law

3|llowast1|2 + |llowast1 + 2llowast2|2 = 3|llowast2|2 + |2llowast1 + llowast2 |2 (1)

As another consequence of Theorems 2 and 3 our algorithm enumerates the Grammatrices (llowasti middotllowastj )1leijle3 for multiple bases llowast1 l

lowast2 l

lowast3 of the true solution Llowast This makes the

enumeration procedure robust against missing or false elements in the set of extractedlengths as explained in Section 74

3

0

Figure 1 Reciprocal lattice vectors that are allowed to correspond to systematic ab-sences

Figure 2 Substructure of a topograph corresponding to the parallelogram law

For the third result we provide a method to speed up the enumeration process inSection 73 In order not to reduce the rate to acquire the true solution Theorems 2 and3 are used again here In the test using actual powder diffraction patterns in Section82 it is demonstrated that the improvement makes the enumeration speed 32ndash250 timesfaster As a result the enumeration process is executed in a few minutes at most

The novel algorithm for lattices of rank N = 3 is implemented in the powder auto-indexing software Conograph In section 8 we introduce the default parameters andresults of Conograph The default parameters are selected so that less-experiencedusers of the software can obtain good results without modifying them We prepare 30sets of test data including difficult cases such as samples from Structure Determinationby Powder Diffractometry Round Robin-2 (SDPDRR-2) Results for rather difficultcases are explained in Examples 3ndash6 The total time for powder auto-indexing did notexceed several minutes The Conograph software is scheduled to be distributed in thenear future from httpsourceforgejpprojectsconograph

Notation and symbols

The notation and symbols used in this paper are summarized in this section The innerproduct of the Euclidean space RN is denoted by u middot v and the Euclidean norm u middot uis denoted by |u|2 The standard basis t(0 0

i

1 0 0) of ZN is denoted by ei(1 le i le N)

A lattice L of rank N is a discrete and cocompact subgroup of RN For any latticeL there are linearly independent vectors v1 vN isin RN over R such that v1 vNgenerate L as a Z-module In this case v1 vN are called a basis of L and the matrix(vi middot vj)1leijleN is called a Gram matrix of L The reciprocal lattice Llowast of L is definedas Llowast = llowast isin RN l middot llowast isin Z for all l isin L

If v1 vi sub L (1 le i lt N) is extended to a basis of L it is called a primitiveset of L In particular v isin L is a primitive vector of L if and only if v is a primitiveset of L Pn(L) is the set consisting of all the primitive sets of L of cardinality n

A symmetric matrix (sij)1leijleN is always identified with a quadratic form Q(x1 xN ) =

4

solutions in powder auto-indexing This is considered the most important foundationfor the following discussion in general a lattice L is not determined uniquely in N ge 2even if the rank N of L and all the lengths l isin L are given (Appendix B) On the otherhand the cardinality of L2 satisfying ΛL = ΛL2 is always finite in N le 4 thereforeit is possible to enumerate all such L2 algorithmically (Appendix C) However it isnot certain whether the number of solutions is actually finite in powder auto-indexingowing to systematic absences and the finite observational range Fortunately usingboth physical constraints and mathematical theorems we obtain a necessary conditionfor assuming the finiteness of solutions except for events with zero probability (Section3) As a result it is only necessary to enumerate finitely many solutions for powderauto-indexing

Our second result is a new algorithm to enumerate powder auto-indexing solutions(Sections 71 and 72) We prove that it works regardless of the type of systematicabsences Our basic idea is to use the graph whose edges are associated with a set ofsquares of lengths |llowast1 |2 |llowast2|2 |llowast1 + llowast2 | |llowast1 minus llowast2 |2 for some lattice vectors llowast1 l

lowast2 isin Llowast

In Itorsquos method the parallelogram law 2(|llowast1|2 + |llowast2 |2) = |llowast1 + llowast2 |2 + |llowast1 minus llowast2 |2 is usedto obtain Gram matrices of sublattices of rank 2 (called zones) However simple useof the parallelogram law does not always provide successful results as introduced inFact 1 of Section 62 Therefore instead of just enumerating a set of lengths satisfyingthe parallelogram law our algorithm utilizes graphs whose edges are associated with|llowast1 |2 |llowast2 |2 |llowast1 + llowast2| and |llowast1 minus llowast2|2 as a network of lattice vector lengths This provides

a comprehensive method of analyzing how the two sets |llowast1 |2 |llowast2|2 |llowast1 + llowast2|2 |llowast1 minus llowast2|2and |klowast1 |2 |klowast2 |2 |klowast1 + klowast2 |2 |klowast1 minus klowast2 |2 are related to each other

For lattices of rank 2 such a graph has already been defined by Conway who calledit a topograph [6] As explained in section 51 a graph having the required property isconstructed for general N by using Voronoirsquos second reduction theory We also call thisa topograph Basic properties of topographs for lattices of rank N = 2 3 are explainedin Section 52

In Section 6 we shall see how primitive vectors of llowast isin Llowast belonging to Γext(GH)are distributed on a topograph Although the following theorem for the case of N = 2has not been described explicitly it provides a theoretical reason for the parallelogramlaw working appropriately for N = 2

Theorem 1 Let (GH) be a type of systematic absence in N = 2 and let Llowast be thereciprocal lattice of L a lattice consisting of all the translations in G Then for anyprimitive vector llowast of Llowast belonging to Γext(GH) there exists llowast2 isin Llowast such that llowast1 middotllowast2 = 0and llowast1 l

lowast2 make a basis of Llowast

In brief Theorem 1 claims that llowast isin Γext(GH) only appear in the gray area inFigure 1 regardless of the type (GH) With regard to the case of H = 1 we shallintroduce a proof using topographs in Section 61 The case H ) L is not proved heresince it is verified easily by checking lists in [14]

By Theorem 1 a subgraph of a topograph with infinitely many edges is formed by uni-fying substructures as in Figure 2 associated with the lengths |llowast1 |2 |llowast2|2 |llowast1 + llowast2|2 |llowast1 minus llowast2|2of llowast1 l

lowast2 isin Llowast Γext(GH) that are easily extracted from an average theta series

For N = 3 similar statements are proved in Theorems 2 and 3 In this case it isnecessary to use the following formula instead of the parallelogram law

3|llowast1|2 + |llowast1 + 2llowast2|2 = 3|llowast2|2 + |2llowast1 + llowast2 |2 (1)

As another consequence of Theorems 2 and 3 our algorithm enumerates the Grammatrices (llowasti middotllowastj )1leijle3 for multiple bases llowast1 l

lowast2 l

lowast3 of the true solution Llowast This makes the

enumeration procedure robust against missing or false elements in the set of extractedlengths as explained in Section 74

3

0

Figure 1 Reciprocal lattice vectors that are allowed to correspond to systematic ab-sences

Figure 2 Substructure of a topograph corresponding to the parallelogram law

For the third result we provide a method to speed up the enumeration process inSection 73 In order not to reduce the rate to acquire the true solution Theorems 2 and3 are used again here In the test using actual powder diffraction patterns in Section82 it is demonstrated that the improvement makes the enumeration speed 32ndash250 timesfaster As a result the enumeration process is executed in a few minutes at most

The novel algorithm for lattices of rank N = 3 is implemented in the powder auto-indexing software Conograph In section 8 we introduce the default parameters andresults of Conograph The default parameters are selected so that less-experiencedusers of the software can obtain good results without modifying them We prepare 30sets of test data including difficult cases such as samples from Structure Determinationby Powder Diffractometry Round Robin-2 (SDPDRR-2) Results for rather difficultcases are explained in Examples 3ndash6 The total time for powder auto-indexing did notexceed several minutes The Conograph software is scheduled to be distributed in thenear future from httpsourceforgejpprojectsconograph

Notation and symbols

The notation and symbols used in this paper are summarized in this section The innerproduct of the Euclidean space RN is denoted by u middot v and the Euclidean norm u middot uis denoted by |u|2 The standard basis t(0 0

i

1 0 0) of ZN is denoted by ei(1 le i le N)

A lattice L of rank N is a discrete and cocompact subgroup of RN For any latticeL there are linearly independent vectors v1 vN isin RN over R such that v1 vNgenerate L as a Z-module In this case v1 vN are called a basis of L and the matrix(vi middot vj)1leijleN is called a Gram matrix of L The reciprocal lattice Llowast of L is definedas Llowast = llowast isin RN l middot llowast isin Z for all l isin L

If v1 vi sub L (1 le i lt N) is extended to a basis of L it is called a primitiveset of L In particular v isin L is a primitive vector of L if and only if v is a primitiveset of L Pn(L) is the set consisting of all the primitive sets of L of cardinality n

A symmetric matrix (sij)1leijleN is always identified with a quadratic form Q(x1 xN ) =

4

0

Figure 1 Reciprocal lattice vectors that are allowed to correspond to systematic ab-sences

Figure 2 Substructure of a topograph corresponding to the parallelogram law

For the third result we provide a method to speed up the enumeration process inSection 73 In order not to reduce the rate to acquire the true solution Theorems 2 and3 are used again here In the test using actual powder diffraction patterns in Section82 it is demonstrated that the improvement makes the enumeration speed 32ndash250 timesfaster As a result the enumeration process is executed in a few minutes at most

The novel algorithm for lattices of rank N = 3 is implemented in the powder auto-indexing software Conograph In section 8 we introduce the default parameters andresults of Conograph The default parameters are selected so that less-experiencedusers of the software can obtain good results without modifying them We prepare 30sets of test data including difficult cases such as samples from Structure Determinationby Powder Diffractometry Round Robin-2 (SDPDRR-2) Results for rather difficultcases are explained in Examples 3ndash6 The total time for powder auto-indexing did notexceed several minutes The Conograph software is scheduled to be distributed in thenear future from httpsourceforgejpprojectsconograph

Notation and symbols

The notation and symbols used in this paper are summarized in this section The innerproduct of the Euclidean space RN is denoted by u middot v and the Euclidean norm u middot uis denoted by |u|2 The standard basis t(0 0

i

1 0 0) of ZN is denoted by ei(1 le i le N)

A lattice L of rank N is a discrete and cocompact subgroup of RN For any latticeL there are linearly independent vectors v1 vN isin RN over R such that v1 vNgenerate L as a Z-module In this case v1 vN are called a basis of L and the matrix(vi middot vj)1leijleN is called a Gram matrix of L The reciprocal lattice Llowast of L is definedas Llowast = llowast isin RN l middot llowast isin Z for all l isin L

If v1 vi sub L (1 le i lt N) is extended to a basis of L it is called a primitiveset of L In particular v isin L is a primitive vector of L if and only if v is a primitiveset of L Pn(L) is the set consisting of all the primitive sets of L of cardinality n

A symmetric matrix (sij)1leijleN is always identified with a quadratic form Q(x1 xN ) =

4

Figure 3 Lattice parameters and the unit cell

sumNi=1

sumNj=1 sijxixj The linear space consisting of N timesN symmetric matrices with real

entries is denoted by SN SN≻0 (resp SN

0) is the subset of SN consisting of all the

positive-definite (resp semidefinite) symmetric matrices S1 S2 isin SN are equivalent ifand only if there exists g isin GLN (Z) such that S2 = gS1

tgFor any S isin SN elements of ΛS are called representations of S over Z

ΛS = tvSv 0 6= v isin ZN (2)

In crystallography a crystal lattice in three-dimensional Euclidean space is repre-sented by a set of lattice parameters a b c α β and γ as in Figure 3 This parame-terization is transformed into a 3 times 3 matrix S = (sij)1leijle3 as follows and S is theGram matrix of the Bravais lattice of the crystal

s11 = a2 s22 = b2 s33 = c2 (3)

s12 = ab cosγ s13 = ac cosβ s23 = bc cosα

In general for any lattice L sub RN its automorphism group is defined by g isinGL(3Z) gS tg = S and L is categorized into its Bravais type by the conjugacyclass of the group in GL(3Z) In N = 3 it is known that there exist 14 Bravaistypes In crystallography selection of the Bravais lattice L2 sub L and its basis l1 l2 l3is standardized according to the Bravais type of L S in (3) is the Gram matrix definedfor the basis of L2 When L belongs the category called a primitive centring L = L2and l1 l2 l3 are chosen by the method called the Niggli reduction which is very similarwith the Minkowski reduction in N = 3 (see [14]) When L belongs to the categorycalled a face-centered (resp body-centered) lattice there exist l1 l2 l3 isin L such that L

is generated byli+lj2 (1 le i lt j le 3) (resp l1+l2+l3

2 minus li (1 le i le 3)) and li middot lj = 0(1 le i lt j le 3) holds Regardless of the choice of l1 l2 l3 their Gram matrix S and L2

generated by these l1 l2 l3 are determined uniquely With regard to the Bravais latticethe explanation above is sufficient in order to understand our following discussions

In the context of powder structure analysis representations of Sminus1 over Z are calledq-values of diffraction peaks

2 Outline of the powder auto-indexing problem

A function weierp on RN is said to be periodic if L = l isin RN weierp(x+ l) = weierp(x) for any x isinRN is a lattice We call L the period lattice of weierp For example an electron (or nucleus)density in a crystal (cf the left figure in Figure 4) is a periodic function of N = 3

The following weierp provides its standard model

weierp(x) =

msum

i=1

disum

k=1

sum

lisinL

pi(x minus xik minus l) (4)

5

where

m number of different elements in a crystal

di number of the ith element in the primitive cell R3L

pi(x) rapidly decreasing function on R3 that represents the electron distribution of respective atoms

According to diffraction theory if a crystal has an electron density weierp the diffrac-tion image of its single-crystal sample equals cfsingle(x

lowastweierp) for some c gt 0 wherefsingle(x

lowastweierp) is a sum of delta functions given by

fsingle(xlowastweierp) =

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast) (5)

weierp(llowast) =

int

RNL

weierp(x)eminus2πradicminus1xmiddotllowastdx (6)

A powder sample is an ensemble of a very large number of randomly oriented crystal-lites As a result its diffraction image is proportional to the integration of fsingle(x

lowastweierp)on a sphere of radius

radicq

fpowder(qweierp) =

int

|xlowast|2=q

fsingle(xlowastweierp)dxlowast = 2

radicqsum

llowastisinLlowast

|weierp(llowast)|2δ(q minus |llowast|2) (7)

The weierp(llowast) is called a structure factor in crystallographyThe right figure in Figure 4 presents an actual powder diffraction pattern It is

obtained by replacing every delta function in (7) with some kind of peak-shape modelfunction g that is close to a Gaussian distribution and satisfies

int

Rg(q)dq = 1

Figure 4 Powder diffraction pattern

Ab-initio powder crystal structure determination retrieves the electron density weierpfrom the powder diffraction pattern under the assumption that the following additionalinformation is available

bull chemical formula (ie the ratio [d1 middot middot middot dm])

bull density of a single crystal

bull rapidly decreasing functions pi (1 le i le m)

Powder auto-indexing is the initial stage of ab-initio powder crystal structure de-termination and aims to find the period lattice L of weierp As the positions of the deltafunctions elements of Λweierp are extracted from a powder diffraction pattern fpowder(qweierp)

Λweierp = |llowast|2 llowast isin Llowast Fweierp(|llowast|2) 6= 0 (8)

6

L is normally determined from elements of Λweierp in powder auto-indexing After L isobtained using the coefficients Fweierp(q) powder crystal structure determination is carriedout

Fweierp(q) =sum

llowastisinLlowast|llowast|2=q

|weierp(llowast)|2 (9)

3 Formulation of the powder auto-indexing problem

In this section we formulate the problem explicitly In powder auto-indexing extractedΛobs is different from the set Λweierp of true |llowast|2 (llowast isin Llowast) owing to observational problemsConsequently the Gram matrix S of the period lattice L of weierp must be retrieved fromΛobs under the following assumptions

(A1) The observed range of a powder diffraction pattern is contained in a finite interval[qmin qmax] sub (0infin) Consequently only information about Λweierp cap [qmin qmax] isavailable

(A2) Every qobs isin Λobs has some observation error It may be assumed that the thresh-old Err[qobs] on the error

∣

∣qobs minus qcal∣

∣ is given (A method to compute Err[qobs] isintroduced in [22])

(A3) Owing to probabilistic mistakes and errors in acquisition of the peak-positions(ie peak-search) Λobs differs from the true Λweierp cap [qmin qmax] There exist smallǫ1 gt 0 and ǫ2 gt 0 satisfying the following conditions

(i) For arbitrarily fixed q isin Λweierp cap [qmin qmax] the following occurs with proba-bility ǫ1

q isin⋃

qobsisinΛobs

[qobs minus Err[qobs] qobs + Err[qobs]] (10)

(ii) For arbitrarily fixed qobs isin Λobs the following occurs with probability ǫ2

Λweierp cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty (11)

(A4) When the distribution weierp is constrained by group symmetry infinitely many weierp(llowast)and Fweierp(q) become zero owing to systematic absences deterministically (Section 4)

As assumed in (3) Λobs extracted from a powder diffraction pattern has missing orfalse elements These are caused by observational problems including background noiseor false peaks due to sample impurity

With regard to (4) sometimes Fweierp(q) = 0 holds due to a special arrangement of atompositions xik rather than systematic absences (cf [12] [27]) However the probabilityis zero if every xik is distributed uniformly in RNL (The only known exceptions aresystematic absences)

For the special arrangement with zero probability we replace Λweierp by Λext(weierp) and

consider (A3) instead of (A3)

Λext(weierp) = Λweierp cup

|llowast|2 llowast isin Llowast Fweierp(q) = 0 owing to reasonsother than systematic absences

(12)

(A3) Assume that

(i) For any q isin Λext(weierp) cap [qmin qmax] q isin ⋃

qobsisinΛobs [qobs minus Err[qobs] qobs +

Err[qobs]] occurs with probability ǫ1

7

(ii) For any qobs isin Λobs Λext(weierp) cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty occurswith probability ǫ2

Under the conditions (A1)ndash(A4) infinitely many solutions may exist in some cases(for example consider the case of very small [qmin qmax]) Hence additional assump-tions are necessary in order to guarantee a finite number of solutions

(A5) Owing to repulsive force between atoms we may assume min|l|2 0 6= l isin L ge d2

for a positive constant d asymp 2A Then by the inequalities on successive minimaof L and its reciprocal lattice Llowast proved by Lagarias et al [17] in N = 2 3the maximum diagonal entry DN of a Minkowski-reduced(defined in Appendix C)Gram matrix S of Llowast satisfies

DN le N + 3

4d2maxγi 1 le i le N (13)

where γi is the Hermite constant

γi = sup

min tvSv 0 6= v isin ZN S isin Si≻0 detS = 1

(14)

In particular D2 le 53d

minus2 and D3 le 3 middot 2minus13dminus2 follow from γ1 = 1 γ22 = 4

3 γ33 = 2 (In N = 2 the estimation is improved up to D2 le 4

3dminus2 easily)

(A6) The length of the intervalradicqmax minus radic

qmin is sufficiently greater thanradicDN that

there exists llowast1 llowast2 l

lowast3 is a basis of Llowast such that Λobs includes |llowast1 plusmn llowast2|2 |llowast1 plusmn llowast3 |2

|plusmnllowast1 + llowast2 + llowast3|2 and at least one of the following for both i = 2 3

(i) |llowast1 |2 |llowasti |2(ii) |llowast1 |2 |llowast1 minus 2llowasti |2(iii) |llowast1 |2 |llowast1 + 2llowasti |2(iv) |llowasti |2 |2llowast1 minus llowasti |2(v) |llowasti |2 |2llowast1 + llowasti |2

In (A5) 2A is selected as the minimum distance between the two closest lattticepoints of L satisfied by any existing crystals (A6) looks rather artificial This assump-tion is necessary for our algorithm in Table 5 By Theorems 2 and 3 (A6) holds exceptfor events with zero probability if a sufficiently large qmax is chosen regardless of thetype of systematic absences Under assumptions (A5) and (A6) the number of solutionsis always finite because Λobs sub [qmin qmax] contains only finite elements and the Grammatrix of Llowast is computed from a combination of elements of Λobs due to assumption(A6)

Here it is still non-trivial how to select qmax At least it is clear qmax ge D3 isrequired otherwise Λext(weierp) cap [qmin qmax] contain only |llowast|2 of llowast isin Llowast

2 in some caseswhere Llowast

2 sub Llowast is a sublattice of rank less than 3 On the other hand as seen in (3)in Section 8 too many q-values are frequently extracted from the interval [qmin D3]when we set d = 2A and D3 = 3 middot 2minus13dminus2 nevertheless powder auto-indexing is veryfrequently successful even with a smaller interval Since the time of our enumerationalgorithm is roughly proportional to the fourth power of the number of elements ofΛobs (cf Section 73) the time can be significantly decreased by minimizing the range[qmin qmax] Considering the current accuracy of diffractometers and the power of per-sonal computers qmax should be chosen empirically to some degree in addition to thetheoretical estimation above See (3) in Section 81 for a more detailed approach to thisissue

8

4 Summary of crystallographic groups and system-

atic absences

We now give some definitions for crystallographic groups and systematic absences Inthe following we represent the elements x isin RNL as a row vector and llowast isin Llowast as acolumn vector Furthermore any group action on RNL (resp Llowast) is represented as aright action (resp left action)

Any congruent transformation of the Euclidean space RN is represented as a compo-sition of the orthogonal group O(N) and a translation if σ is a congruent transformationof RN there exist τ isin O(N) and ν isin RN such that

xσ = xτ + ν for any x isin RN (15)

Such a σ is denoted by τ |ν The group consisting of all congruent transformations ofRN is the semidirect group O(N)⋉RN By expanding the composition (xτ1|ν1)τ2|ν2it is seen that the group multiplication is given by

τ1|ν1 middot τ2|ν2 = τ1τ2|ντ21 + ν2 (16)

Definition 41 A crystallographic group is a discrete and cocompact subgroup of O(N)⋉RN

A crystallographic group is also called a wallpaper group in N = 2 and a space groupin N = 3

For a crystallographic group G two groups RG and L are defined by

RG = τ isin O(N) τ |ν for some ν isin RN (17)

L = ν isin RN 1N |ν isin G (18)

where 1N is the identity of O(N) L is a lattice and RG is a finite subgroup of O(N)consisting of τ that maps any elements of L to L RG is called a point group of G Gis a group extension of L by RG From the definition of L for any τ |ντ isin G theclass ντ + L isin RNL is uniquely determined Furthermore the map RG minusrarr RNL τ 7rarr ντ + L is a 1-cocycle ie it satisfies

νστ equiv ντσ + ντ mod L (19)

We now proceed to the definition of systematic absences Let G be a crystallographicgroup with the point group RG and the translation group L and consider the followingperiodic function weierp

weierp(x) =msum

i=1

sum

σisinG

pi(x minus xσi ) (20)

Note that the density model function weierp in (4) is represented as in (20) for some spacegroup G

Let L2(RN ) be the L2-space ie the set of all measurable functions f on RN with

a finite L2-norm ||f ||2 = (int

RN |f(x)|2dx)12 lt infin In order to compute the same list as[14] let us assume that

(Isotropy condition) pi belongs to L2(RN )RG ie pi(xτ ) = pi(x) holds for any

τ isin RG

9

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

where

m number of different elements in a crystal

di number of the ith element in the primitive cell R3L

pi(x) rapidly decreasing function on R3 that represents the electron distribution of respective atoms

According to diffraction theory if a crystal has an electron density weierp the diffrac-tion image of its single-crystal sample equals cfsingle(x

lowastweierp) for some c gt 0 wherefsingle(x

lowastweierp) is a sum of delta functions given by

fsingle(xlowastweierp) =

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast) (5)

weierp(llowast) =

int

RNL

weierp(x)eminus2πradicminus1xmiddotllowastdx (6)

A powder sample is an ensemble of a very large number of randomly oriented crystal-lites As a result its diffraction image is proportional to the integration of fsingle(x

lowastweierp)on a sphere of radius

radicq

fpowder(qweierp) =

int

|xlowast|2=q

fsingle(xlowastweierp)dxlowast = 2

radicqsum

llowastisinLlowast

|weierp(llowast)|2δ(q minus |llowast|2) (7)

The weierp(llowast) is called a structure factor in crystallographyThe right figure in Figure 4 presents an actual powder diffraction pattern It is

obtained by replacing every delta function in (7) with some kind of peak-shape modelfunction g that is close to a Gaussian distribution and satisfies

int

Rg(q)dq = 1

Figure 4 Powder diffraction pattern

Ab-initio powder crystal structure determination retrieves the electron density weierpfrom the powder diffraction pattern under the assumption that the following additionalinformation is available

bull chemical formula (ie the ratio [d1 middot middot middot dm])

bull density of a single crystal

bull rapidly decreasing functions pi (1 le i le m)

Powder auto-indexing is the initial stage of ab-initio powder crystal structure de-termination and aims to find the period lattice L of weierp As the positions of the deltafunctions elements of Λweierp are extracted from a powder diffraction pattern fpowder(qweierp)

Λweierp = |llowast|2 llowast isin Llowast Fweierp(|llowast|2) 6= 0 (8)

6

L is normally determined from elements of Λweierp in powder auto-indexing After L isobtained using the coefficients Fweierp(q) powder crystal structure determination is carriedout

Fweierp(q) =sum

llowastisinLlowast|llowast|2=q

|weierp(llowast)|2 (9)

3 Formulation of the powder auto-indexing problem

In this section we formulate the problem explicitly In powder auto-indexing extractedΛobs is different from the set Λweierp of true |llowast|2 (llowast isin Llowast) owing to observational problemsConsequently the Gram matrix S of the period lattice L of weierp must be retrieved fromΛobs under the following assumptions

(A1) The observed range of a powder diffraction pattern is contained in a finite interval[qmin qmax] sub (0infin) Consequently only information about Λweierp cap [qmin qmax] isavailable

(A2) Every qobs isin Λobs has some observation error It may be assumed that the thresh-old Err[qobs] on the error

∣

∣qobs minus qcal∣

∣ is given (A method to compute Err[qobs] isintroduced in [22])

(A3) Owing to probabilistic mistakes and errors in acquisition of the peak-positions(ie peak-search) Λobs differs from the true Λweierp cap [qmin qmax] There exist smallǫ1 gt 0 and ǫ2 gt 0 satisfying the following conditions

(i) For arbitrarily fixed q isin Λweierp cap [qmin qmax] the following occurs with proba-bility ǫ1

q isin⋃

qobsisinΛobs

[qobs minus Err[qobs] qobs + Err[qobs]] (10)

(ii) For arbitrarily fixed qobs isin Λobs the following occurs with probability ǫ2

Λweierp cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty (11)

(A4) When the distribution weierp is constrained by group symmetry infinitely many weierp(llowast)and Fweierp(q) become zero owing to systematic absences deterministically (Section 4)

As assumed in (3) Λobs extracted from a powder diffraction pattern has missing orfalse elements These are caused by observational problems including background noiseor false peaks due to sample impurity

With regard to (4) sometimes Fweierp(q) = 0 holds due to a special arrangement of atompositions xik rather than systematic absences (cf [12] [27]) However the probabilityis zero if every xik is distributed uniformly in RNL (The only known exceptions aresystematic absences)

For the special arrangement with zero probability we replace Λweierp by Λext(weierp) and

consider (A3) instead of (A3)

Λext(weierp) = Λweierp cup

|llowast|2 llowast isin Llowast Fweierp(q) = 0 owing to reasonsother than systematic absences

(12)

(A3) Assume that

(i) For any q isin Λext(weierp) cap [qmin qmax] q isin ⋃

qobsisinΛobs [qobs minus Err[qobs] qobs +

Err[qobs]] occurs with probability ǫ1

7

(ii) For any qobs isin Λobs Λext(weierp) cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty occurswith probability ǫ2

Under the conditions (A1)ndash(A4) infinitely many solutions may exist in some cases(for example consider the case of very small [qmin qmax]) Hence additional assump-tions are necessary in order to guarantee a finite number of solutions

(A5) Owing to repulsive force between atoms we may assume min|l|2 0 6= l isin L ge d2

for a positive constant d asymp 2A Then by the inequalities on successive minimaof L and its reciprocal lattice Llowast proved by Lagarias et al [17] in N = 2 3the maximum diagonal entry DN of a Minkowski-reduced(defined in Appendix C)Gram matrix S of Llowast satisfies

DN le N + 3

4d2maxγi 1 le i le N (13)

where γi is the Hermite constant

γi = sup

min tvSv 0 6= v isin ZN S isin Si≻0 detS = 1

(14)

In particular D2 le 53d

minus2 and D3 le 3 middot 2minus13dminus2 follow from γ1 = 1 γ22 = 4

3 γ33 = 2 (In N = 2 the estimation is improved up to D2 le 4

3dminus2 easily)

(A6) The length of the intervalradicqmax minus radic

qmin is sufficiently greater thanradicDN that

there exists llowast1 llowast2 l

lowast3 is a basis of Llowast such that Λobs includes |llowast1 plusmn llowast2|2 |llowast1 plusmn llowast3 |2

|plusmnllowast1 + llowast2 + llowast3|2 and at least one of the following for both i = 2 3

(i) |llowast1 |2 |llowasti |2(ii) |llowast1 |2 |llowast1 minus 2llowasti |2(iii) |llowast1 |2 |llowast1 + 2llowasti |2(iv) |llowasti |2 |2llowast1 minus llowasti |2(v) |llowasti |2 |2llowast1 + llowasti |2

In (A5) 2A is selected as the minimum distance between the two closest lattticepoints of L satisfied by any existing crystals (A6) looks rather artificial This assump-tion is necessary for our algorithm in Table 5 By Theorems 2 and 3 (A6) holds exceptfor events with zero probability if a sufficiently large qmax is chosen regardless of thetype of systematic absences Under assumptions (A5) and (A6) the number of solutionsis always finite because Λobs sub [qmin qmax] contains only finite elements and the Grammatrix of Llowast is computed from a combination of elements of Λobs due to assumption(A6)

Here it is still non-trivial how to select qmax At least it is clear qmax ge D3 isrequired otherwise Λext(weierp) cap [qmin qmax] contain only |llowast|2 of llowast isin Llowast

2 in some caseswhere Llowast

2 sub Llowast is a sublattice of rank less than 3 On the other hand as seen in (3)in Section 8 too many q-values are frequently extracted from the interval [qmin D3]when we set d = 2A and D3 = 3 middot 2minus13dminus2 nevertheless powder auto-indexing is veryfrequently successful even with a smaller interval Since the time of our enumerationalgorithm is roughly proportional to the fourth power of the number of elements ofΛobs (cf Section 73) the time can be significantly decreased by minimizing the range[qmin qmax] Considering the current accuracy of diffractometers and the power of per-sonal computers qmax should be chosen empirically to some degree in addition to thetheoretical estimation above See (3) in Section 81 for a more detailed approach to thisissue

8

4 Summary of crystallographic groups and system-

atic absences

We now give some definitions for crystallographic groups and systematic absences Inthe following we represent the elements x isin RNL as a row vector and llowast isin Llowast as acolumn vector Furthermore any group action on RNL (resp Llowast) is represented as aright action (resp left action)

Any congruent transformation of the Euclidean space RN is represented as a compo-sition of the orthogonal group O(N) and a translation if σ is a congruent transformationof RN there exist τ isin O(N) and ν isin RN such that

xσ = xτ + ν for any x isin RN (15)

Such a σ is denoted by τ |ν The group consisting of all congruent transformations ofRN is the semidirect group O(N)⋉RN By expanding the composition (xτ1|ν1)τ2|ν2it is seen that the group multiplication is given by

τ1|ν1 middot τ2|ν2 = τ1τ2|ντ21 + ν2 (16)

Definition 41 A crystallographic group is a discrete and cocompact subgroup of O(N)⋉RN

A crystallographic group is also called a wallpaper group in N = 2 and a space groupin N = 3

For a crystallographic group G two groups RG and L are defined by

RG = τ isin O(N) τ |ν for some ν isin RN (17)

L = ν isin RN 1N |ν isin G (18)

where 1N is the identity of O(N) L is a lattice and RG is a finite subgroup of O(N)consisting of τ that maps any elements of L to L RG is called a point group of G Gis a group extension of L by RG From the definition of L for any τ |ντ isin G theclass ντ + L isin RNL is uniquely determined Furthermore the map RG minusrarr RNL τ 7rarr ντ + L is a 1-cocycle ie it satisfies

νστ equiv ντσ + ντ mod L (19)

We now proceed to the definition of systematic absences Let G be a crystallographicgroup with the point group RG and the translation group L and consider the followingperiodic function weierp

weierp(x) =msum

i=1

sum

σisinG

pi(x minus xσi ) (20)

Note that the density model function weierp in (4) is represented as in (20) for some spacegroup G

Let L2(RN ) be the L2-space ie the set of all measurable functions f on RN with

a finite L2-norm ||f ||2 = (int

RN |f(x)|2dx)12 lt infin In order to compute the same list as[14] let us assume that

(Isotropy condition) pi belongs to L2(RN )RG ie pi(xτ ) = pi(x) holds for any

τ isin RG

9

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

L is normally determined from elements of Λweierp in powder auto-indexing After L isobtained using the coefficients Fweierp(q) powder crystal structure determination is carriedout

Fweierp(q) =sum

llowastisinLlowast|llowast|2=q

|weierp(llowast)|2 (9)

3 Formulation of the powder auto-indexing problem

In this section we formulate the problem explicitly In powder auto-indexing extractedΛobs is different from the set Λweierp of true |llowast|2 (llowast isin Llowast) owing to observational problemsConsequently the Gram matrix S of the period lattice L of weierp must be retrieved fromΛobs under the following assumptions

(A1) The observed range of a powder diffraction pattern is contained in a finite interval[qmin qmax] sub (0infin) Consequently only information about Λweierp cap [qmin qmax] isavailable

(A2) Every qobs isin Λobs has some observation error It may be assumed that the thresh-old Err[qobs] on the error

∣

∣qobs minus qcal∣

∣ is given (A method to compute Err[qobs] isintroduced in [22])

(A3) Owing to probabilistic mistakes and errors in acquisition of the peak-positions(ie peak-search) Λobs differs from the true Λweierp cap [qmin qmax] There exist smallǫ1 gt 0 and ǫ2 gt 0 satisfying the following conditions

(i) For arbitrarily fixed q isin Λweierp cap [qmin qmax] the following occurs with proba-bility ǫ1

q isin⋃

qobsisinΛobs

[qobs minus Err[qobs] qobs + Err[qobs]] (10)

(ii) For arbitrarily fixed qobs isin Λobs the following occurs with probability ǫ2

Λweierp cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty (11)

(A4) When the distribution weierp is constrained by group symmetry infinitely many weierp(llowast)and Fweierp(q) become zero owing to systematic absences deterministically (Section 4)

As assumed in (3) Λobs extracted from a powder diffraction pattern has missing orfalse elements These are caused by observational problems including background noiseor false peaks due to sample impurity

With regard to (4) sometimes Fweierp(q) = 0 holds due to a special arrangement of atompositions xik rather than systematic absences (cf [12] [27]) However the probabilityis zero if every xik is distributed uniformly in RNL (The only known exceptions aresystematic absences)

For the special arrangement with zero probability we replace Λweierp by Λext(weierp) and

consider (A3) instead of (A3)

Λext(weierp) = Λweierp cup

|llowast|2 llowast isin Llowast Fweierp(q) = 0 owing to reasonsother than systematic absences

(12)

(A3) Assume that

(i) For any q isin Λext(weierp) cap [qmin qmax] q isin ⋃

qobsisinΛobs [qobs minus Err[qobs] qobs +

Err[qobs]] occurs with probability ǫ1

7

(ii) For any qobs isin Λobs Λext(weierp) cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty occurswith probability ǫ2

Under the conditions (A1)ndash(A4) infinitely many solutions may exist in some cases(for example consider the case of very small [qmin qmax]) Hence additional assump-tions are necessary in order to guarantee a finite number of solutions

(A5) Owing to repulsive force between atoms we may assume min|l|2 0 6= l isin L ge d2

for a positive constant d asymp 2A Then by the inequalities on successive minimaof L and its reciprocal lattice Llowast proved by Lagarias et al [17] in N = 2 3the maximum diagonal entry DN of a Minkowski-reduced(defined in Appendix C)Gram matrix S of Llowast satisfies

DN le N + 3

4d2maxγi 1 le i le N (13)

where γi is the Hermite constant

γi = sup

min tvSv 0 6= v isin ZN S isin Si≻0 detS = 1

(14)

In particular D2 le 53d

minus2 and D3 le 3 middot 2minus13dminus2 follow from γ1 = 1 γ22 = 4

3 γ33 = 2 (In N = 2 the estimation is improved up to D2 le 4

3dminus2 easily)

(A6) The length of the intervalradicqmax minus radic

qmin is sufficiently greater thanradicDN that

there exists llowast1 llowast2 l

lowast3 is a basis of Llowast such that Λobs includes |llowast1 plusmn llowast2|2 |llowast1 plusmn llowast3 |2

|plusmnllowast1 + llowast2 + llowast3|2 and at least one of the following for both i = 2 3

(i) |llowast1 |2 |llowasti |2(ii) |llowast1 |2 |llowast1 minus 2llowasti |2(iii) |llowast1 |2 |llowast1 + 2llowasti |2(iv) |llowasti |2 |2llowast1 minus llowasti |2(v) |llowasti |2 |2llowast1 + llowasti |2

In (A5) 2A is selected as the minimum distance between the two closest lattticepoints of L satisfied by any existing crystals (A6) looks rather artificial This assump-tion is necessary for our algorithm in Table 5 By Theorems 2 and 3 (A6) holds exceptfor events with zero probability if a sufficiently large qmax is chosen regardless of thetype of systematic absences Under assumptions (A5) and (A6) the number of solutionsis always finite because Λobs sub [qmin qmax] contains only finite elements and the Grammatrix of Llowast is computed from a combination of elements of Λobs due to assumption(A6)

Here it is still non-trivial how to select qmax At least it is clear qmax ge D3 isrequired otherwise Λext(weierp) cap [qmin qmax] contain only |llowast|2 of llowast isin Llowast

2 in some caseswhere Llowast

2 sub Llowast is a sublattice of rank less than 3 On the other hand as seen in (3)in Section 8 too many q-values are frequently extracted from the interval [qmin D3]when we set d = 2A and D3 = 3 middot 2minus13dminus2 nevertheless powder auto-indexing is veryfrequently successful even with a smaller interval Since the time of our enumerationalgorithm is roughly proportional to the fourth power of the number of elements ofΛobs (cf Section 73) the time can be significantly decreased by minimizing the range[qmin qmax] Considering the current accuracy of diffractometers and the power of per-sonal computers qmax should be chosen empirically to some degree in addition to thetheoretical estimation above See (3) in Section 81 for a more detailed approach to thisissue

8

4 Summary of crystallographic groups and system-

atic absences

We now give some definitions for crystallographic groups and systematic absences Inthe following we represent the elements x isin RNL as a row vector and llowast isin Llowast as acolumn vector Furthermore any group action on RNL (resp Llowast) is represented as aright action (resp left action)

Any congruent transformation of the Euclidean space RN is represented as a compo-sition of the orthogonal group O(N) and a translation if σ is a congruent transformationof RN there exist τ isin O(N) and ν isin RN such that

xσ = xτ + ν for any x isin RN (15)

Such a σ is denoted by τ |ν The group consisting of all congruent transformations ofRN is the semidirect group O(N)⋉RN By expanding the composition (xτ1|ν1)τ2|ν2it is seen that the group multiplication is given by

τ1|ν1 middot τ2|ν2 = τ1τ2|ντ21 + ν2 (16)

Definition 41 A crystallographic group is a discrete and cocompact subgroup of O(N)⋉RN

A crystallographic group is also called a wallpaper group in N = 2 and a space groupin N = 3

For a crystallographic group G two groups RG and L are defined by

RG = τ isin O(N) τ |ν for some ν isin RN (17)

L = ν isin RN 1N |ν isin G (18)

where 1N is the identity of O(N) L is a lattice and RG is a finite subgroup of O(N)consisting of τ that maps any elements of L to L RG is called a point group of G Gis a group extension of L by RG From the definition of L for any τ |ντ isin G theclass ντ + L isin RNL is uniquely determined Furthermore the map RG minusrarr RNL τ 7rarr ντ + L is a 1-cocycle ie it satisfies

νστ equiv ντσ + ντ mod L (19)

We now proceed to the definition of systematic absences Let G be a crystallographicgroup with the point group RG and the translation group L and consider the followingperiodic function weierp

weierp(x) =msum

i=1

sum

σisinG

pi(x minus xσi ) (20)

Note that the density model function weierp in (4) is represented as in (20) for some spacegroup G

Let L2(RN ) be the L2-space ie the set of all measurable functions f on RN with

a finite L2-norm ||f ||2 = (int

RN |f(x)|2dx)12 lt infin In order to compute the same list as[14] let us assume that

(Isotropy condition) pi belongs to L2(RN )RG ie pi(xτ ) = pi(x) holds for any

τ isin RG

9

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

(ii) For any qobs isin Λobs Λext(weierp) cap [qobs minus Err[qobs] qobs + Err[qobs]] = empty occurswith probability ǫ2

Under the conditions (A1)ndash(A4) infinitely many solutions may exist in some cases(for example consider the case of very small [qmin qmax]) Hence additional assump-tions are necessary in order to guarantee a finite number of solutions

(A5) Owing to repulsive force between atoms we may assume min|l|2 0 6= l isin L ge d2

for a positive constant d asymp 2A Then by the inequalities on successive minimaof L and its reciprocal lattice Llowast proved by Lagarias et al [17] in N = 2 3the maximum diagonal entry DN of a Minkowski-reduced(defined in Appendix C)Gram matrix S of Llowast satisfies

DN le N + 3

4d2maxγi 1 le i le N (13)

where γi is the Hermite constant

γi = sup

min tvSv 0 6= v isin ZN S isin Si≻0 detS = 1

(14)

In particular D2 le 53d

minus2 and D3 le 3 middot 2minus13dminus2 follow from γ1 = 1 γ22 = 4

3 γ33 = 2 (In N = 2 the estimation is improved up to D2 le 4

3dminus2 easily)

(A6) The length of the intervalradicqmax minus radic

qmin is sufficiently greater thanradicDN that

there exists llowast1 llowast2 l

lowast3 is a basis of Llowast such that Λobs includes |llowast1 plusmn llowast2|2 |llowast1 plusmn llowast3 |2

|plusmnllowast1 + llowast2 + llowast3|2 and at least one of the following for both i = 2 3

(i) |llowast1 |2 |llowasti |2(ii) |llowast1 |2 |llowast1 minus 2llowasti |2(iii) |llowast1 |2 |llowast1 + 2llowasti |2(iv) |llowasti |2 |2llowast1 minus llowasti |2(v) |llowasti |2 |2llowast1 + llowasti |2

In (A5) 2A is selected as the minimum distance between the two closest lattticepoints of L satisfied by any existing crystals (A6) looks rather artificial This assump-tion is necessary for our algorithm in Table 5 By Theorems 2 and 3 (A6) holds exceptfor events with zero probability if a sufficiently large qmax is chosen regardless of thetype of systematic absences Under assumptions (A5) and (A6) the number of solutionsis always finite because Λobs sub [qmin qmax] contains only finite elements and the Grammatrix of Llowast is computed from a combination of elements of Λobs due to assumption(A6)

Here it is still non-trivial how to select qmax At least it is clear qmax ge D3 isrequired otherwise Λext(weierp) cap [qmin qmax] contain only |llowast|2 of llowast isin Llowast

2 in some caseswhere Llowast

2 sub Llowast is a sublattice of rank less than 3 On the other hand as seen in (3)in Section 8 too many q-values are frequently extracted from the interval [qmin D3]when we set d = 2A and D3 = 3 middot 2minus13dminus2 nevertheless powder auto-indexing is veryfrequently successful even with a smaller interval Since the time of our enumerationalgorithm is roughly proportional to the fourth power of the number of elements ofΛobs (cf Section 73) the time can be significantly decreased by minimizing the range[qmin qmax] Considering the current accuracy of diffractometers and the power of per-sonal computers qmax should be chosen empirically to some degree in addition to thetheoretical estimation above See (3) in Section 81 for a more detailed approach to thisissue

8

4 Summary of crystallographic groups and system-

atic absences

We now give some definitions for crystallographic groups and systematic absences Inthe following we represent the elements x isin RNL as a row vector and llowast isin Llowast as acolumn vector Furthermore any group action on RNL (resp Llowast) is represented as aright action (resp left action)

Any congruent transformation of the Euclidean space RN is represented as a compo-sition of the orthogonal group O(N) and a translation if σ is a congruent transformationof RN there exist τ isin O(N) and ν isin RN such that

xσ = xτ + ν for any x isin RN (15)

Such a σ is denoted by τ |ν The group consisting of all congruent transformations ofRN is the semidirect group O(N)⋉RN By expanding the composition (xτ1|ν1)τ2|ν2it is seen that the group multiplication is given by

τ1|ν1 middot τ2|ν2 = τ1τ2|ντ21 + ν2 (16)

Definition 41 A crystallographic group is a discrete and cocompact subgroup of O(N)⋉RN

A crystallographic group is also called a wallpaper group in N = 2 and a space groupin N = 3

For a crystallographic group G two groups RG and L are defined by

RG = τ isin O(N) τ |ν for some ν isin RN (17)

L = ν isin RN 1N |ν isin G (18)

where 1N is the identity of O(N) L is a lattice and RG is a finite subgroup of O(N)consisting of τ that maps any elements of L to L RG is called a point group of G Gis a group extension of L by RG From the definition of L for any τ |ντ isin G theclass ντ + L isin RNL is uniquely determined Furthermore the map RG minusrarr RNL τ 7rarr ντ + L is a 1-cocycle ie it satisfies

νστ equiv ντσ + ντ mod L (19)

We now proceed to the definition of systematic absences Let G be a crystallographicgroup with the point group RG and the translation group L and consider the followingperiodic function weierp

weierp(x) =msum

i=1

sum

σisinG

pi(x minus xσi ) (20)

Note that the density model function weierp in (4) is represented as in (20) for some spacegroup G

Let L2(RN ) be the L2-space ie the set of all measurable functions f on RN with

a finite L2-norm ||f ||2 = (int

RN |f(x)|2dx)12 lt infin In order to compute the same list as[14] let us assume that

(Isotropy condition) pi belongs to L2(RN )RG ie pi(xτ ) = pi(x) holds for any

τ isin RG

9

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

4 Summary of crystallographic groups and system-

atic absences

We now give some definitions for crystallographic groups and systematic absences Inthe following we represent the elements x isin RNL as a row vector and llowast isin Llowast as acolumn vector Furthermore any group action on RNL (resp Llowast) is represented as aright action (resp left action)

Any congruent transformation of the Euclidean space RN is represented as a compo-sition of the orthogonal group O(N) and a translation if σ is a congruent transformationof RN there exist τ isin O(N) and ν isin RN such that

xσ = xτ + ν for any x isin RN (15)

Such a σ is denoted by τ |ν The group consisting of all congruent transformations ofRN is the semidirect group O(N)⋉RN By expanding the composition (xτ1|ν1)τ2|ν2it is seen that the group multiplication is given by

τ1|ν1 middot τ2|ν2 = τ1τ2|ντ21 + ν2 (16)

Definition 41 A crystallographic group is a discrete and cocompact subgroup of O(N)⋉RN

A crystallographic group is also called a wallpaper group in N = 2 and a space groupin N = 3

For a crystallographic group G two groups RG and L are defined by

RG = τ isin O(N) τ |ν for some ν isin RN (17)

L = ν isin RN 1N |ν isin G (18)

where 1N is the identity of O(N) L is a lattice and RG is a finite subgroup of O(N)consisting of τ that maps any elements of L to L RG is called a point group of G Gis a group extension of L by RG From the definition of L for any τ |ντ isin G theclass ντ + L isin RNL is uniquely determined Furthermore the map RG minusrarr RNL τ 7rarr ντ + L is a 1-cocycle ie it satisfies

νστ equiv ντσ + ντ mod L (19)

We now proceed to the definition of systematic absences Let G be a crystallographicgroup with the point group RG and the translation group L and consider the followingperiodic function weierp

weierp(x) =msum

i=1

sum

σisinG

pi(x minus xσi ) (20)

Note that the density model function weierp in (4) is represented as in (20) for some spacegroup G

Let L2(RN ) be the L2-space ie the set of all measurable functions f on RN with

a finite L2-norm ||f ||2 = (int

RN |f(x)|2dx)12 lt infin In order to compute the same list as[14] let us assume that

(Isotropy condition) pi belongs to L2(RN )RG ie pi(xτ ) = pi(x) holds for any

τ isin RG

9

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Then weierp(xσ) = weierp(x) holds for any σ isin G and weierp has the Fourier coefficient

weierp(llowast) =

msum

i=1

sum

σisinG

int

RNL

pi(xminus xσi )e

minus2πradicminus1xmiddotllowastdx

=msum

i=1

pi(llowast)

sum

σisinGL

eminus2πradicminus1xσ

i middotllowast (21)

where pi(llowast) =

int

RN pi(x)eminus2π

radicminus1xmiddotllowastdx

Hence the probability of weierp(llowast) = 0 depends on the size of WGllowast sub RNL

WGllowast =

x isin RNL sum

σisinGL eminus2πradicminus1xσ middotllowast = 0

(22)

Definition 42 For any finite subgroup H sub G let (RN )H be the subset of RN con-sisting of all the fixed points of H Under this notation Γext(G) and Γext(GH) aredefined by

Γext(G) = llowast isin Llowast WGllowast = RNL (23)

Γext(GH) = llowast isin Llowast (RN )H((RN )H cap L) sub WGllowast (24)

According to the terminology of crystallography we say that llowast isin Llowast corresponds to asystematic absence at general positions (resp special positions) if and only if llowast belongsto Γext(G) (resp Γext(GH)) We shall call (GH) a type of systematic absences

In the following we shall focus on Γext(GH) since we have Γext(G) = Γext(G id)From the definition it is clear that minusllowast and τllowast belong to Γext(GH) for any τ isin RG ifand only if llowast does

For weierp in (20) satisfying the isotropy condition it is not difficult to confirm thatthe following equivalence condition holds when Hi sub G is the stabilizer subgroup ofxi isin RN

weierp(llowast) = 0 holds constantly when(x1 xm p1 pm)

is perturbed in (RN )H1 times middot middot middot times (RN )Hm times (L2(RN )RG)m

lArrrArr llowast isinm⋂

i=1

Γext(GHi) (25)

As proved by Bieberbach [2] [3] a homomorphism ϕ G1 minusrarr G2 is an isomorphismbetween two crystallographic groups G1 and G2 if and only if there is an affine map αof RN such that ϕ(g) = αgαminus1 For such G1 G2 Γext(G1 H) = Γext(G2 ϕ(H)) clearlyholds In general we also have Γext(GH) = Γext(G σHσminus1) for any σ isin G

Proposition 41 All types of systematic absences are classified by pairs (GH) whereG H range respectively in

(a) isomorphism classes of a crystallographic group G

(b) conjugacy classes of finite subgroups H sub G in G We also assume H = σ isin G xσ = x for some x isin RN

As a result there are only finitely many types of systematic absences for each N gt 0

Proof As proved by Bieberbach [2] [3] there are only finitely many isomorphism classesof crystallographic groups in each N Hence we shall only show that the number of theconjugacy classes is finite For any finite subgroup H sub G the natural epimorphism

10

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

ϕ G GL induces an injective map H rarr GL Thus it is sufficient for the proof ifwe can prove that for any fixed subgroup H sub GL the set S(H) = H sub G ϕ(H) =HH = σ isin G xσ = x(existx isin RN ) contains finitely many conjugacy classes Let

(RNL)H be the subset of RNL consisting of all the fixed points of H and π be the

natural onto map RN minusrarr RNL For any x isin (RNL)H and x isin RN with π(x) = xlet Hx sub G be the subgroup consisting of all σ isin G with xσ = x The conjugacyclass of Hx in G clearly depends only on x It is also straightforward to check that any

elements of S(H) can be represented as Hx for some x isin (RNL)H Furthermore if

x and y are contained in the same connected component of (RNL)H there is a path

w [0 1] rarr (RNL)H such that w(0) = x and w(1) = y and therefore Hx = Hy must

hold since ϕ(Hw(t)) = H holds for any t isin [0 1] from the assumption Since (RNL)H

is compact it contains only finitely many connected components As a result S(H)contains only finitely many conjugacy classes

Γext(GH) is computed using the following proposition

Proposition 42 For fixed llowast isin Llowast the equivalence relation among the right cosetsRHRG is defined by

RHτ1llowastsimRHτ2 lArrrArr

def

sum

τisinRHτ1

τllowast =sum

τisinRHτ2

τllowast (26)

Then for any x isin (RN )H llowast isin Llowast belongs to Γext(GH) if and only if the followingholds

sum

RHτ2llowastsimRHτ1

e2πradicminus1xτ2|ντ2middotllowast = 0 for every RHτ1 isin RHR (27)

Proof We fix x isin RN stabilized by H arbitrarily In this case llowast isin Γext(GH) holds ifand only if

sum

RHτ1isinRHRe2πi((x+δx)τ1+ντ1 )middotl

lowast= 0 for any δx isin RN stabilized by RH (28)

Furthermore

δxτ1 middot llowast = δxτ2 middot llowast for any δx isin RN stabilized by RH

lArrrArrsum

τisinRH

(δxττ1 minus δxττ2) middot llowast = 0 for any δx isin RN

lArrrArrsum

τisinRH

δx middot (ττ1llowast minus ττ2llowast) = 0 for any δx isin RN

lArrrArrsum

τisinRH

τ(τ1llowast minus τ2l

lowast) = 0 lArrrArr RHτ1llowastsimRHτ2 (29)

Hence (28) holds if and only if the following does for any δx stabilized by RH

sum

[τ1]isin(RHR)llowastsim

e2πradicminus1δxτ1 middotllowast

sum

RHτ2llowastsimRHτ1

e2πradicminus1(xτ2+ντ2 )middotl

lowast= 0 (30)

which leads to the statement

11

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Corollary 41 Let M be the order of RG and HGH sub RN be the following union offinite linear subspaces of dimension less than N

HGH =⋃

RHτ1RHτ2isinRHRGsum

τisinRHτ(τ1minusτ2)6=0

xlowast isin RN sum

τisinRHτ(τ1 minus τ2)x

lowast = 0

(31)

There then exists Ω sub LlowastMLlowast such that llowast isin Γext(GH) lArrrArr llowast +MLlowast isin Ω holds forany llowast isin Llowast HGH

Proof If x is stabilized by H ντ equiv x minus xτ mod L holds for any τ isin H Hence [ντ ] isinH1(RGR

NL) is mapped to 0 by the natural map H1(RGRNL) minusrarr H1(RH RNL)

As a result it is also mapped to 0 by H1(RGRNL)

timesMmminusrarr H1(RGRNL) where m is

the order of RH (cf Proposition 6 in Chap VII of Serre (1968)) Thus for any τ isin RGthere exist y isin RN and microτ isin (Mm)minus1L such that ντ equiv y minus yτ + microτ mod L HenceMm (xminusy) equiv M

m (xminusy)σ holds for any σ isin RH Therefore M(xminusy) equivsumσisinRH

Mm (xminusy)σ

If u isin RN with mu = xminus y is fixed we have (x minus y)minussumσisinRHuσ isin Mminus1L hence for

some ξτ isin Mminus1L ντ is represented as follows

ντ equiv xminus xτ minussum

σisinRH

uσ +sum

σisinRH

uστ + ξτ (32)

From Lemma 42 llowast isin Llowast belongs to Γext(GH) if and only if the following holds forany RHτ1 isin RHRG

sum

RHτ2llowastsimRHτ1

e2πi(xτ2+ντ2 )middotl

lowast=

sum

RHτ2llowastsimRHτ1

e2πi(xminus

sumσisinRH

uσ+sum

σisinRHuστ2+ξτ2 )middotl

lowast

= e2πi(xminussum

σisinRHuσ)middotllowast+2πiumiddotsum

σisinRHστ1l

lowast sum

RHτ2llowastsimRHτ1

e2πiξτ2 middotllowast= 0 (33)

This is impossible if llowast belongs to MLlowast

There exists a simple condition equivalent to llowast isin Γext(G)

Corollary 42 For any given crystallographic group G

llowast isin Llowast belongs to Γext(G) lArrrArr existτ isin RG such that τllowast = llowast and ντ middot llowast isin Z (34)

Proof Let RGllowast sub RG be the stabilizer subgroup of llowast Then RLτ2llowastsimRLτ1 if and only

if τ2 isin τ1RGllowast Hence

sum

RLτ2llowastsimRLτ1

e2πradicminus1xτ2|ντ2middotllowast =

sum

τisinRGllowast

e2πradicminus1(xτ1τ+ντ1τ )middotllowast

= e2πradicminus1(xτ1+ντ1 )middotl

lowast sum

τisinRGllowast

e2πradicminus1ντ middotllowast (35)

The statement follows from the fact that τ 7rarr e2πradicminus1ντ middotllowast is a homomorphism on RGllowast

Now that we have defined systematic absences it is possible to formulate Λext(weierp) in(12) precisely for any type (GH) of systematic absences we define

Λext(GH) =

|llowast|2 0 6= llowast isin Llowast Γext(GH)

(36)

Λext(G) = Λext(G id) (37)

12

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

From the equivalence condition (25) for weierp in (20)

Λext(weierp) =m⋃

i=1

Λext(GHi) (38)

As a result it is only necessary to consider the case of Λext(weierp) = Λext(GH) for all thetypes (GH) of systematic absences so as to discuss the problem (A4) of systematicabsences

5 C-type domains and Conwayrsquos topographs

The reduction theory deals with the problem of specifying a domain D sub SN≻0 satisfying

(R1) When we put D[g] = gS tg S isin D for any subset D sub SN≻0 and g isin GLN (Z)

the subgroup of GLN(Z) consisting of all g isin GLN(Z) satisfying D = D[g] hasonly finite elements

(R2) For any g isin GLN (Z) D and D[g] do not share interior points except for g isin H

(R3) SN≻0 is decomposed as follows

SN≻0 =

⋃

gHisinGLN (Z)H

D[g] (39)

In [6] a topograph was defined from the Selling reduction with N = 2 So we shallrecall the Selling reduction first let AN = (aij)1leijleN isin SN

≻0 be the Gram matrix ofthe root lattice AN having entries as follows

aij =

2 if i = j

1 otherwise(40)

Using AN and the inner-product 〈S T 〉 = Trace(ST ) =sumN

i=1

sumNj=1 sijtij on SN

the domain DNSel sub SN

≻0 is defined by

DNSel =

S isin SN≻0 〈SAN 〉 le 〈gS tgAN 〉 for any g isin GLN (Z)

(41)

If the subgroup of all g isin GLN(Z) satisfying tgANg = AN is denoted by H(AN )SN≻0 is partitioned using DN

Sel by the Selling reduction [26]

SN≻0 =

⋃

gisinGLN(Z)H(AN )

DNSel[g] (42)

In order to generalize the definition of topographs for general N wersquod like to notethat the tessellation of (42) coincides with the one given by Voronoirsquos two reductiontheories [30] [31] in N = 2 3 Hence DN

Sel (N = 2 3) is same as the principal domainof the first type defined in the two reduction theories In the first reduction theorythe principal domain is defined as the convex cone expanded by v tv of all the minimalvectors v of AN In the second reduction theory the same domain is represented asfollows

V(Φ) = S isin SN≻0 tvSu le tuSu for any v isin Φ and u isin ZN (43)

ΦN0 =

plusmnsumNk=1 ikek ik = 0 1

(44)

13

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Different from DNSel in the tessellation of SN

≻0 given by the first reduction theoryevery domain is associated explicitly with the set of integral vectors formed by minimalvectors of a perfect form Even in the second reduction theory such an association isprovided by using C-type domains (a union of finite L-type domains) instead of L-typedomains

In the following we choose the association by the second reduction theory becauseof Proposition 51 on the association of facets of primitive C-type domains and theparallelogram law

51 Topographs for lattices of general rank

In this section C-type domains and topographs are defined for the general dimensionN We also refer to [23] for more detailed information about C-type domains andits connection with covering problems In [6] a topograph was defined to explain theSelling reduction with N = 2 In contrast vonorms and conorms were used for the caseof N gt 2

The idea of vonorms and conorms as invariants of a lattice seems to have originatedfrom the Voronoi vectors defined in Voronoirsquos second reduction theory for a fixed S isinSN0 if v isin ZN satisfies tvSv = voS(v + 2ZN ) v is called a Voronoi vector of S The

vonorm map of S is defined as a map from v + 2ZN isin ZN2ZN to the representationby the Voronoi vector corresponding to v + 2ZN

voS(v + 2ZN ) = min twSw w isin v + 2ZN (45)

Conorms are the Fourier transform of vonorms when χ is any character on ZN2ZN the conorm map of S is defined by

coS(χ) = minus 1

2Nminus1

sum

v+2ZNisinZN2ZN

voS(v + 2ZN )χ(v) (46)

In the following we use vonorm maps in the definition of C-type domains for clarityBefore introducing C-type domains we shall recall the definition of L-type domains(also called secondary cones cf [28]) The DirichletndashVoronoi polytope of S is defined by

DV(S) = x isin RN txSx le t(x + l)S(x+ l) for any l isin ZN (47)

From the definition DV(S) is the intersection of half-spaces

DV(S) =⋂

06=visinZN

x isin RN txSx le t(x+ v)S(x + v) (48)

As proved in [31] (cf [7]) v isin ZN is a Voronoi vector if and only if the hyperplaneHSv =

x isin RN txSx = t(x+ v)S(x+ v)

intersects DV(S)A tiling of RN is given by the DirichletndashVoronoi polytopes

RN =⋃

lisinZN

(DV(S) + l) (49)

The Delone subdivision is a dual tiling of (49) If we let PS be the set of extremepoints of DV(S) and denote the set of all Voronoi vectors v satisfying p isin HSv by Ψp forany p isin PS then every v isin Ψp satisfies tpSp = t(p+ v)S(p+ v) Therefore an ellipsoidx isin RN t(x+ p)S(x+ p) = tpSp passes through all the elements of 0 cupΨp sub ZN If Lp is the convex hull of 0 cupΨp then the Delone subdivision of RN is given by

Del(S) =⋃

pisinPS lisinZN

Lp + l (50)

14

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

In this case a L-type domain containing S is defined by S2 isin SN Del(S) =Del(S2)

For any subset D sub SN≻0 and g isin GLN(Z) define D[g] = gS tg S isin D Two

domains D1 D2 sub SN≻0 are said to be equivalent if and only if D1[g] = D2 holds for some

g isin GLN (Z) Voronoi proved that the number of equivalence classes of L-type domains

of dimension N(N+1)2 is finite and provided an algorithm to gain all the equivalence

classes [31] This is the outline of Voronoirsquos second reduction theoryFor fixed S isin SN

≻0 define ΦS = [v] v isin ZN is a Voronoi vector of S where[v] represents the class of v isin ZN when v and minusv are identified A C-type domaincontaining S is defined by V(Φ) = S2 isin SN

≻0 ΦS = ΦS2 ie a set of elements ofSN≻0 having the same Voronoi vectors Then V(Φ) is a union of finite L-type domains

because a set of Voronoi vectors of S is decomposed into⋃

pisinPSΨp in different ways

depending on S From the definition the Voronoi map is a linear function on V(ΦS) forany S isin SN

≻0More generally we shall define C-type domains for any set Φ = [v1] [vn] of

arbitrary size n with v1 vn isin ZN

V(Φ) = S isin SN≻0 tvSv = voS(v + 2ZN ) for any [v] isin Φ (51)

This is well defined because both tvSv and v + 2Z are invariant if v is replaced byminusv From the definition V(Φ) is an intersection of the following half-spaces

V(Φ) =⋂

[v]isinΦ

⋂

u+2ZN=v+2ZN

Hge0(u v) (52)

Hge0(u v) = S isin SN≻0 tuSu ge tvSv (53)

Any S isin SN≻0 is contained in V(ΦS) S is said to be in a general position if ΦS has

exactly 2N elements If S is in a general position V(ΦS) includes an open neighbor ofS Such C-type domains are called primitive Otherwise there exist [v] 6= [u] isin ZN

such that u+ 2ZN = v + 2ZN and S belongs to the following hyperplanes

H(u v) = S isin SN tuSu = tvSv (54)

Even for such S by perturbing the entries of S S in a general position satisfyingΦS sub ΦS is obtained As a result the following partitioning of SN

≻0 is obtained

SN≻0 =

⋃

VisinPN

V (55)

PN =

V(ΦS) S isin SN≻0V(ΦS) is primitive

(56)

This tessellation is coarser than that given by L-type domains Hence similarlywith L-type domains the number of equivalence classes of C-type domains of dimensionN(N+1)

2 is finiteTable 1 lists all the representatives of equivalence classes of C-type domains for

1 le N le 4 Each domain V(Φki ) in Table 1 has a set of extreme rays provided by

M(Φk0) =

v tv v = ei (1 le i le N) ei minus ej (1 le i lt j le N)

(57)

M(Φ41) =

(

M(Φ40) cup D4

)

v tv v = e1 minus e2

(58)

M(Φ42) =

(

M(Φ40) cup

v tv v = e1 + e2 minus e3 minus e4

cup D4 D42)

v tv v = e1 minus e2 e3 minus e4

(59)

where D4 and D42 are the equivalent perfect forms corresponding to the root latticeD4

D4 =

2 1 minus1 minus11 2 minus1 minus1minus1 minus1 2 0minus1 minus1 0 2

D42 =

2 0 minus1 minus10 2 minus1 minus1minus1 minus1 2 1minus1 minus1 1 2

(60)

15

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 1 Equivalence classes of primitive C-type domainsVoronoi vectors Primitive C-type domain

N = 1 Φ10 = [0] [e1] V(Φ1

0) = S1ge0

N = 2 Φ20 = [0] [e1] [e2] [e1 + e2] V(Φ2

0) =⋂

1leiltjle3 Hge0(ei minus ej ei + ej)

N = 3 Φ30 =

[

sum3k=1 ikek

]

ik = 0 or 1

V(Φ30) =

⋂

1leiltjle4 Hge0(ei minus ej ei + ej)

N = 4 Φ40 =

[

sum4k=1 ikek

]

ik = 0 or 1

V(Φ40) =

⋂

1leiltjle5 Hge0(ei minus ej ei + ej)

Φ41 = Φ4

0 cup [e1 minus e2] [e1 + e2] V(Φ41) = Hge0 (e1 + e2 e1 minus e2)

cap(

⋂

3leiltjle5 Hge0 (ei minus ej ei + ej)

)

cap(

⋂

3leiltjle5k=12

Hge0 (ei + ej + 2ek ei + ej))

Φ42 = Φ4

0 cup [e1 minus e2] [e3 minus e4] [e1 + e2] [e3 + e4] V(Φ4

2) = Hge0 (e1 + e2 e1 minus e2) capHge0 (e3 + e4 e3 minus e4)

cap(

⋂

k=34 Hge0 (e1 + e2 + 2ek e1 minus e2)

)

cap(

⋂

k=12 Hge0 (e3 + e4 + 2ek e3 minus e4)

)

cap(

⋂4i=1 H

ge0 (ei minus e5 ei + e5))

cap(

⋂

i=12k=34or i=34k=12

Hge0 (ei + e5 + 2ek ei + e5))

aHere we put eN+1 = minussumNi=1 ei in every N-dimensional case

bAmong the above domains only V(Φ42) is not an L-type domain but is a union of two L-type

domains L-type domains for N = 4 are listed in [28]

The following proposition claims that every facet of a primitive C-type domain isassociated with a set of four vectors satisfying the parallelogram law Although this isthe most important property in our discussion we could not find references mentioningthis explicitly

Proposition 51 Suppose that two primitive C-type domains V(ΦS1) 6= V(ΦS2) have

an (N(N+1)2 minus1)-dimensional cone as their intersection Then ΦS1capΦS2 contains exactly

2Nminus1 elements Hence there exist u v isin ZN such that ΦS1ΦS2 = [u] and ΦS2ΦS1 =[v] V(ΦS1 cupΦS2) sub H(u v) is a common facet of V(ΦS1) and V(ΦS2) Furthermore v+u

2 vminusu2 is a primitive set of ZN and [ v+u

2 ] [ vminusu2 ] are elements of ΦS1 cap ΦS2

Proof From V(ΦS1) 6= V(ΦS2) there exist u1 v1 such that u1 + 2ZN = v1 + 2ZN [u1] isin ΦS1 ΦS2 and [v1] isin ΦS2 ΦS1 Since u1 v1 are Voronoi vectors of any S3 isinV(ΦS1) cap V(ΦS2) we have tu1S3l le tlSl and tv1S3l le tlSl Hence u1+v1

2 and u1minusv12

are also Voronoi vectors of S3 Replacing ui vi with gui gvi (g isin GLN (Z)) if necessarywe may assume u1minusv1

2 = e1 and u1+v12 = me1 + ne2 for some mn isin Z Then u1 v1

and u1plusmnv12 are Voronoi vectors of S4 = ( teiS3ej)1leijle2 isin S2

≻0 Hence S4 belongsto V([u1plusmnv1

2 ] u1) which is equivalent to V(Φ20) in Table 1 As a result there exists

g isin GLN (Z) such that g(u1minusv12 ) = e1 and g(u1+v1

2 ) = e2 Hence it is concluded thatu1+v1

2 u1minusv12 is a primitive set of ZN Suppose that there exists another (u2 v2) 6=

(u1 v1) satisfying u2 + 2ZN = v2 + 2ZN [u2] isin ΦS1 ΦS2 and [v2] isin ΦS2 ΦS1 Fromthe dimension of V(ΦS1) cap V(ΦS2) the following must hold for any S3 isin SN

t(u1 + v1

2)S3(

u1 minus v12

) = 0 lArrrArr t(u2 + v2

2)S3(

u2 minus v22

) = 0 (61)

16

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Figure 5 Representations associated with an edge of a topograph

0

When V(Φ1) V(Φ2) isin PN have a common facet there exists u v isin ZN such that Φ1 Φ2 =plusmn(u+ v) Φ2 Φ1 = plusmn(uminus v) In this case the direction of the edge connecting V(Φ1) and V(Φ2)is defined as in [6]

Figure 6 Direction of edges of a topograph CTNS0

We may now assume u1minusv12 = e1 and u1+v1

2 = e2 Then [u1] [v1] = [u2] [v2] iseasily obtained Therefore ΦS1 ΦS2 and ΦS2 ΦS1 consist of only one element Nowthe remaining statements follow immediately

By Proposition 51 it is proved that the decomposition (55) is a facet-to-facet tes-sellation Hence (55) is also a face-to-face tessellation by a theorem of Gruber andRyshkov [13]

Using the V -partitioning (55) a topograph is defined for general N

Definition 51 Define CTN as the graph which has PN EN as its sets of nodes andedges respectively

PN =

V(ΦS) S isin SN≻0 ΦS is primitive

(62)

EN =

eV(Φ1)V(Φ2) V(Φ1) 6= V(Φ2) isin PN share a facet

(63)

where eV(Φ1)V(Φ2) connects two nodes V(Φ1) V(Φ2) isin PN When S0 isin SN is fixedarbitrarily edges in EN are associated with two representations of S0 over Z by themap

fS0(eV(Φ1)V(Φ2)) = tuS0utvS0v (64)

where u v isin ZN are taken so that [u+v] = ΦS1 ΦS2 and [uminusv] = ΦS2 ΦS1 Suchan edge is represented in Figure 5 Furthermore the direction of the edge is defined as inFigure 6 Assuming every edge is oriented by this we call the pair CTNS0 = (CTN fS0)a topograph of S0

52 Topographs for low-dimensional lattices

In this section the structures of topographs for lattices of rank N = 2 3 are explainedSince the same topic is also discussed in [6] we mention only basic facts necessary in thefollowing sections In N = 2 3 the structures are also determined from the partitioning(42) of the Selling reduction In particular the set of nodes is provided by

PN =

DNSel[g] = V( tgminus1ΦN

0 ) g isin GLN (Z)

(65)

17

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

By Voronoirsquos second reduction theory every node DNSel[g] is associated with tgminus1ΦN

0 In the following a lattice L of rank N a basis b1 bN of L are fixed and a matrix

(

b1 middot middot middot bN)

is denoted by B In order to clarify tBB is the Gram matrix of (LB)we utilize the following notation instead of f tBB CTN tBB

f(LB)(eV(Φ1)V(Φ2)) = f tBB(eV(Φ1)V(Φ2)) = |Bu|2 |Bv|2 (66)

CTN(LB) = CTN tBB (67)

where u v isin ZN are chosen as in Definition 51Basic properties of CT2(LB) and CT3(LB) are explained in the following examples

Example 1 Case of CT2(LB) D2Sel = V(Φ2

0) is a polyhedral cone surrounded by thethree hyperplanes in Table 1 Hence a node of CT2 is adjacent to three nodes as inFigure 7

Let e3 = minuse1 minus e2 and τ(2)ij be the 2times 2 matrix satisfying

τ(2)ij ei = minusej τ

(2)ij ej = ei (68)

When we put (l1 l2) = (b1 b2) tgminus1 and l3 = minusl1minusl2 for the fixed basis b1 b2 of L and g isin GL2(Z) everynode D2

Sel[g] = V( tgminus1Φ2

0) is an end point of three edges eV( tgminus1Φ2

0)V( tgminus1τ(2)ij

Φ20)

(1 le i lt j le 3) that

are associated with |li|2 |lj |2 (In this figure any edge between two domains labeled |k1|2 and |k2|2is considered to be associated with |k1|2 |k2|2 Such a labeling of domains is achieved by embeddingthe topograph in the upper half plane as in Figure 8 and labeling every domain D sub H surrounded bytopograph edges with |l|2 of the minimal vector l of some S isin ιminus1(D) This is well-defined because lis the only minimal vector of all S isin ιminus1(D))

Figure 7 Local structure of topograph CT2(L(b1 b2))

It can be proved that CT2 is a tree as in Figure 8 The tree is embedded in S2≻0Rgt0 ≃

H by mapping a node V( tgminus1Φ20) to the perfect form gAminus1

2tg and an edge between

V( tgminus11 Φ2

0) and V( tgminus12 Φ2

0) to the geodesic connecting g1Aminus12

tg1 g2Aminus12

tg2

Example 2 Case of CT3(LB) D3Sel = V(Φ3

0) is a polyhedral cone surrounded by thesix hyperplanes in Table 1 Hence a node of CT3 is adjacent to six nodes as in Figure

9 All the adjacent nodes of V( tgminus1Φ20) are given as V( tgminus1τ

(3)ij Φ2

0) (1 le i lt j le 4)

where e4 = minussum3i=1 ei and τ

(3)ij is the 3times 3 matrix satisfying

τ(3)ij ei = minusei τ

(3)ij ej = ej k 6= i j =rArr τ

(3)ij ek = ei + ek (71)

As explained in Figure 9 CT3 contains two kinds of circuits of length 3 and 6 whichcorrespond to the following fundamental relations of GL3(Z)

τ(3)ij τ

(3)kmτ

(3)ij = σikjm isin H(A3)

(

τ(3)ij τ

(3)ik τ

(3)im

)2

= 1 (72)

18

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

0 112

A one-to-one correspondence between the points of the upper half-plane H = z isin C Im(z) gt 0and S2

≻0Rgt0 is given by the map

ι x+radicminus1y 7rarr

[

x2 + y2 xx 1

]

(69)

Under this identification of H and S2≻0Rgt0 the action of g isin GL2(Z) on the latter set by S 7rarr gS tg

coincides with the action of GL2(Z) on H by

(

a bc d

)

middot z =

az+bcz+d

if adminus bc = 1az+bcz+d

if adminus bc = minus1(70)

where z is the complex conjugate of z

Figure 8 Embedding of a topograph in the upper half plane

where i j km are integers satisfying i j km = 1 2 3 4 and σikjm isin H(A3) isthe 3times 3 matrix satisfying

σikjmep = minuseq for any (p q) = (i k) (k i) (jm) (m j) (73)

6 Main results for the distribution rules of systematic

absences

We now discuss on the distribution rules of systematic absences on a topograph Thoseused in our algorithm are described as theorems whereas other properties important forpowder auto-indexing algorithms are mentioned as facts As proved in Proposition 41there are only a finite number of types of systematic absences and they are classifiedby the types (GH)

It is not difficult to prove our theorems if Γext(GH) is contained in HGH in (31)(This always holds for lattices of rank 2) Because too many case-by-case considerationsare required otherwise the most difficult part of the theorems is confirmed by directcomputation using the International Tables (N = 2) and executing a program (N = 3)For confirmation of the case N = 3 we verify that the program outputs exactly thesame list as the International Tables

The most important property of Γext(GH) is that Llowast is generated by elements ofLlowastΓext(GH) under the assumption that L is the period lattice of the periodic functionweierp Therefore if Llowast Γext(GH) holds the type (GH) may be regarded to be invalidand removed from the following consideration

61 Cases of rank 2

According to the International Tables there are 17 wallpaper groups and 72 types ofsystematic absences including invalid ones By direct computation (or by seeing tablesin [14]) it is verified that Γext(GH) = Γext(G id) holds in all the valid cases

19

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

When we put (l1 l2 l3) = (b1 b2 b3) tgminus1 and l4 = minussum3

i=1 li for the fixed basis b1 b2 b3 andg isin GL3(Z) each node D3

Sel[g] = V( tgminus1Φ3

0) is an end point of six edges associated with |li|2 |lj |2(1 le i lt j le 4) For any i j km = 1 2 3 4 two edges associated with |li|2 |lj |2 |lk|2 |lm|2are contained in a circuit of length 3 On the other hand two edges associated with |li|2 |lj |2|li|2 |lk|2 are contained in a circuit of length 6

Figure 9 Local structure of a topograph CT3(L(b1 b2 b3))

In the following we introduce a short proof of Theorem 1 in the case H = L usinga topograph As a result Theorem 1 follows for general cases

Lemma 61 Let Llowast be a lattice of rank 2 and R sub O(N) be the automorphism groupof Llowast If τllowast1 = llowast1 holds for some 1 6= τ isin R and primitive vector of llowast1 isin Llowast either ofthe following holds

(a) there exists llowast2 isin Llowast such that llowast1 middot llowast2 = 0 and llowast1 llowast2 is a basis of Llowast or

(b) τ2 = 1 and there exists llowast2 isin Llowast such that llowast1 = llowast2 + τ1llowast2 and llowast2 τ1l

lowast2 is a basis of Llowast

Proof Fix a Gram matrix S0 of Llowast Let llowast2 isin Llowast be the vector satisfying llowast1 llowast2 isinP2(L

lowast) llowast1 middot llowast2 ge 0 and |llowast2 |2 = min|llowast3|2 llowast1 llowast3 isin P2(Llowast) The edges of CT2S0 then

have the direction presented in Figure 10 The left figure corresponds to the case (a)

0

Figure 10 Directions of edges of a topograph associated with |llowast1 |2

In the case of the right figure τ fixes the node surrounded by |l1|2 |l2|2 and |l1 minus l2|2If we recall that the stabilizer of V(Φ3

0) is given by H(A3) the action of τ or minusτ onllowast1minusllowast2minusllowast1 + llowast2 coincides with a permutation of llowast1 minusllowast2 minusllowast1 + llowast2 From τ 6= 1 andτllowast1 = llowast1 we obtain τllowast2 = llowast1 minus llowast2 This is equivalent to the case (b)

Proof of Theorem 1 in the case of H = L From Corollary 42 for any primitive vectorllowast1 isin Γext(G) there exists τ isin R with τllowast1 = llowast1 such that ντ middot llowast1 isin Z If (b) of Lemma 61holds then

ντ middot llowast1 = ν1+ττ middot llowast2 = ντ2 middot llowast2 = ν1 middot llowast2 isin Z (74)

20

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Hence the statement of the theorem follows

62 Cases of rank 3

For N = 3 we consider the following cases separately

(i) Γext(GH) cap P1(Llowast) is contained in HGH in (31)

(ii) Γext(GH) cap P1(Llowast) is not contained in HGH

Table 2 lists all valid types of systematic absences corresponding to the latter caseFor each type Ω sub LlowastMLlowast in Corollary 41 is given in Table 3

Table 2 Types of systematic absences having Γext(GH) 6sub HGH Space group G (Noa) Hb (x y z)c

A (Face-centered lattice) B (Body-centered lattice) P 4 3 n (218) C2 (x 0 12)

F d d 2 (43) C2 (0 0 z) I 41a (88) Ci (0 14 18) P m 3 n (223) C2 ( 1

4 y y + 1

2)

F d d d (70) C2 (x 0 0) I 41a (88) Ci ( 14 0 3

8) P m 3 n (223) C2v (x 1

2 0)

F d d d (70) D2 (0 0 0) I 41a m d (141) C2h (0 14 18) P m 3 n (223) C2v (x 0 1

2)

F d d d (70) D2 ( 12 12 12) I 41a m d (141) C2h (0 1

4 58) F (Body-centered)

F d 3 (203) C2 (x 0 0) C I 4 3 d (220) C3 (x x x)F d 3 (203) T (0 0 0) I 41a m d (141) C2 (x 1

4 18) I a 3 d (230) C3 (x x x)

F d 3 (203) T ( 12 12 12) I 41a c d (142) C2 ( 1

4 y 1

8) G

F 41 3 2 (210) C2 (x 0 0) D P 42 3 2 (208) D2 ( 14 0 1

2)

F 41 3 2 (210) T (0 0 0) P 3 1 c (159) C3 ( 13 23 z) P 42 3 2 (208) D2 ( 1

4 12 0)

F 41 3 2 (210) T ( 12 12 12) P 3 1 c (163) C3 ( 1

3 23 z) P 4 3 n (218) S4 ( 1

4 0 1

2)

F d 3 m (227) C2v (x 0 0) P 3 1 c (163) D3 ( 23 13 14) P 4 3 n (218) S4 ( 1

4 12 0)

F d 3 m (227) Td (0 0 0) P 3 1 c (163) D3 ( 13 23 14) P m 3 n (223) D2d ( 1

4 0 1

2)

F d 3 m (227) Td ( 12 12 12) P 63 (173) C3 ( 1

3 23 z) P m 3 n (223) D2d ( 1

4 12 0)

A (Body-centered lattice) P 63m (176) C3 ( 13 23 z) H

I 41 (80) C2 (0 0 z) P 63m (176) C3h ( 23 13 14) P 43 3 2 (212) D3 ( 1

8 18 18)

I 41a (88) C2 (0 0 z) P 63m (176) C3h ( 13 23 14) P 43 3 2 (212) D3 ( 5

8 58 58)

I 41a (88) S4 (0 0 0) P 63 2 2 (182) C3 ( 13 23 z) P 41 3 2 (213) D3 ( 3

8 38 38)

I 41a (88) S4 (0 0 12) P 63 2 2 (182) D3 ( 2

3 13 14) P 41 3 2 (213) D3 ( 7

8 78 78)

I 41 2 2 (98) C2 (0 0 z) P 63 2 2 (182) D3 ( 13 23 14) I

I 41 2 2 (98) D2 (0 0 0) P 63 m c (186) C3v ( 13 23 z) I 41 3 2 (214) D3 ( 1

8 18 18)

I 41 2 2 (98) D2 (0 0 12) P 6 2 c (190) C3 ( 1

3 23 z) I 41 3 2 (214) D3 ( 7

8 78 78)

I 41 m d (109) C2v (0 0 z) P 6 2 c (190) C3h ( 23 13 14) J1

I 4 2 d (122) C2 (0 0 z) P 6 2 c (190) C3h ( 13 23 14) I 41 3 2 (214) D2 ( 1

8 0 1

4)

I 4 2 d (122) S4 (0 0 0) P 63m m c (194) C3v ( 13 23 z) I 41 3 2 (214) D2 ( 5

8 0 1

4)

I 4 2 d (122) S4 (0 0 12) P 63m m c (194) D3h ( 2

3 13 14) J2

I 41a m d (141) C2 (x x 0) P 63m m c (194) D3h ( 13 23 14) I 4 3 d (220) S4 ( 7

8 0 1

4)

I 41a m d (141) C2v (0 0 z) E I 4 3 d (220) S4 ( 38 0 1

4)

I 41a m d (141) D2d (0 0 0) P 62 (171) C2 ( 12 12 z) K

I 41a m d (141) D2d (0 0 12) P 64 (172) C2 ( 1

2 12 z) I 41 3 2 (214) C2 (x 0 1

4)

I 41a c d (142) C2 (x x 14) P 62 2 2 (180) C2 ( 1

2 0 z) I 4 3 d (220) C2 (x 0 1

4)

B (Face-centered lattice) P 62 2 2 (180) D2 ( 12 0 0) I a 3 d (230) C2 ( 1

8 yminusy+ 1

4)

F d d d (70) Ci ( 18 18 18) P 62 2 2 (180) D2 ( 1

2 0 1

2) L

F d d d (70) Ci ( 58 58 58) P 64 2 2 (181) C2 ( 1

2 0 z) I a 3 d (230) C2 (x 0 1

4)

F d 3 (203) C3i ( 18 18 18) P 64 2 2 (181) D2 ( 1

2 0 0) M

F d 3 (203) C3i ( 58 58 58) P 64 2 2 (181) D2 ( 1

2 0 1

2) I a 3 d (230) D2 ( 1

8 0 1

4)

F 41 3 2 (210) D3 ( 18 18 18) F (Primitive) I a 3 d (230) S4 ( 3

8 0 1

4)

F 41 3 2 (210) D3 ( 58 58 58) P 42 3 2 (208) C2 (x 1

2 0) N

F d 3 m (227) D3d ( 18 18 18) P 42 3 2 (208) C2 (x 0 1

2) I a 3 d (230) D3 ( 1

8 18 18)

F d 3 m (227) D3d ( 58 58 58) P 4 3 n (218) C2 (x 1

2 0)

aNumber assigned to every space group in [14]bThe isomorphism class of Hc(x y z) isin RNL corresponding to all the elements of (R3)H

21

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 3 Necessary and sufficient conditiona forsum3

i=1 uillowasti isin HGH to belong to

Γext(GH)Type Necessary and sufficient condition

Aface-centered =rArr u1 + u2 + u3 equiv 2 mod 4body-centered =rArr 2u2 + u3 equiv 2 mod 4

Bface-centered =rArr u1 u2 u3 equiv 0 0 2 0 2 2 mod 4body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4 or (u1 u2 u3) equiv (1 1 0) mod 2

C (u1 u2 u3) equiv (1 1 0) mod 2D u1 equiv u2 mod 3 u3 equiv 1 mod 2E u1 u2 equiv 0 mod 2 u3 6equiv 0 mod 3

Fprimitive =rArr u1 u2 u3 equiv 1 1 1 mod 2body-centered =rArr u1 u2 u3 equiv 0 0 2 2 2 2 mod 4

G u1 u2 u3

equiv 0 0 1 1 1 1 mod 2 and

6equiv 0 1 2 0 2 3 mod 4

H u1 u2 u3

equiv 0 0 0 0 0 1 mod 2 and

6equiv 0 1 2 0 2 3 0 0 0 2 2 2 mod 4

I u1 u2 u3 equiv 0 0 2 0 2 2 mod 4

J1 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 1 1 1 4 0 3 3 3 3 4 0 1 7 1 4 7 0 3 5 3 4 5 mod 8

J2 u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 0 1 3 1 3 4 0 1 5 1 4 5 0 3 7 3 4 7 0 5 7 4 5 7 mod 8

K u1 u2 u3 equiv 2 2 2 mod 4L u1 u2 u3 equiv 0 1 1 0 1 3 0 3 3 2 2 2 mod 4

M u1 u2 u3

6equiv 0 0 0 0 2 2 1 1 2 1 2 3 2 3 3 mod 4 and

6equiv 0 2 4 0 4 6 mod 8

N u1 u2 u3 6equiv 0 0 0 1 1 2 1 2 3 2 3 3 mod 4

a〈llowast1 llowast2 llowast3〉 is the reciprocal basis of the basis l1 l2 l3 of R3 taken as in the footnote of Table 2

First we shall explain some more details of Itorsquos method and determine why it doesnot work appropriately for some types of systematic absences It was Ito [15] who firstproposed using the parallelogram law for powder auto-indexing if the observed q-valuesq1 q2 q3 q4 isin Λobs satisfy 2(q1 + q2) = q3 + q4 the method assumes that there existllowast1 l

lowast2 isin Llowast satisfying the following

q1 = |llowast1|2 q2 = |llowast2 |2 q3 = |llowast1 + llowast2|2 q4 = |llowast1 minus llowast2 |2 (75)

If (75) is true the Gram matrix of the sublattice expanded by llowast1 llowast2 is determined In

order to obtain candidate solutions for Llowast it is necessary to construct a 3 times 3 Grammatrix from combinations of these sublattices In order to simplify the combinationprocedure it is very desirable that llowast1 llowast2 in (75) is a primitive set of Llowast Otherwiseit is necessary to check whether Llowast has rank 3 sublattices that are more plausible as asolution once Llowast is obtained The program of Visser [29] which adopted Itorsquos methodalso implicitly requires llowast1 llowast2 isin P2(L

lowast) [10]However according to the following fact in some types of systematic absences

llowast1 llowast2 sub Llowast is never a primitive set of Llowast if llowast1 llowast2 satisfies (75)

Fact 1 If (GH) is of the category B or N there exists no primitive set llowast1 llowast2 of Llowast

such that none of llowast1 llowast2 l

lowast1 plusmn llowast2 belong to Γext(GH)

To remove adverse effects of systematic absences from Itorsquos algorithm it has beenproposed that a formula other than the parallelogram law should be used [9]

|llowast1 +mllowast2 |2 minus |llowast1 minusmllowast2 |2 = m(|llowast1 + llowast2 |2 minus |llowast1 minus llowast2|2) (76)

However it has not been ascertained whether the equation works appropriately for alltypes of systematic absences As another example the following was proposed to obtain

22

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Figure 11 A subgraph of a topograph corresponding to the equation 3|llowast1 |2+|llowast1 + 2llowast2|2 =

|2llowast1 + llowast2|2 + 3|llowast2|2 and a chain formed from such subgraphs

a rank 3 solution directly

|llowast1|2 + |llowast2 |2 + |llowast3 |2 + |llowast1 + llowast2 + llowast3|2 = |llowast1 + llowast2|2 + |llowast1 + llowast3 |2 + |llowast2 + llowast3 |2 (77)

The above equation has a similar property to the parallelogram law

Fact 2 If (GH) is of the category B C F G or N there exists no basis 〈llowast1 llowast2 llowast3〉 ofLlowast such that none of llowast1 l

lowast2 l

lowast3 l

lowast1 + llowast2 l

lowast1 + llowast3 l

lowast2 + llowast3 l

lowast1 + llowast2 + llowast3 belong to Γext(GH)

The following two theorems are intended to resolve any this kind of problems causedby systematic absences There it is proved that there exist ldquoinfinitely manyrdquo elementsof P2(L

lowast) or P3(Llowast) satisfying some common properties and connected subgraphs of

a topograph containing infinitely many nodes are formed from the lengths of latticevectors belonging to such elements Existence of infinitely many elements is requirednot to fail to gain the true solution because only a finite subset Λobs sub Γext(GH)is available in computation From the latter claim it is guaranteed that rather largesubgraph is formed for the true solution even from finite elements belonging to ΛobsThis is useful to select better candidate solutions and reduce computation time

We shall explain the meaning of Theorem 2 before describing the statements in shortit claims that the equation 3|llowast1 |2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 + 3|llowast2 |2 works appropriatelyregardless of the type of systematic absences When (Llowast Blowast) represents the reciprocallattice of the lattice to obtain as a solution and a matrix whose columns are a basisof Llowast this new equation corresponds to the subgraph of CT2(LlowastBlowast) consisting of threenodes and two edges in the left-hand of Figure 11 A subgraph of CT2(LlowastBlowast) containinginfinitely many nodes as the right-hand of Figure 11 is formed by linking such a subgraphobtained from qr qt qs qu isin Λext(GH) satisfying 3qr + qt = 3qs + qu

Theorem 2 For any crystallographic group G with the translation group L and the typeof systematic absences (GH) the following subset of P2(L

lowast) includes infinitely manyelements

P2(Llowast) = llowast1 llowast2 isin P2(L

lowast) mllowast1 + (mminus 1)llowast2 isin Γext(GH) for any m isin Z (78)

More precisely for any open convex cone U sub RN satisfying UcapHGH = empty the followingholds

(1) There exists llowast1 llowast2 isin P2(Llowast) such that mllowast1 + nllowast2 mn isin Zge0 sub U If

Γext(GH) sub HGH such llowast1 llowast2 isin P2(Llowast) belongs to P2(L

lowast)

(2) Assume that (GH) is one of the types in Table 2 and llowast1 llowast2 isin P2(Llowast) satisfy

(a) llowast1 llowast2 sub U and

23

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

(b) llowast1 llowast2 2l

lowast1 + llowast2 l

lowast1 +2llowast2 2l

lowast1 + 3llowast2 3l

lowast1 +2llowast2 3l

lowast1 + 4llowast2 (ie all the vectors other

than llowast1 + llowast2 in the right-hand of Figure 11) do not belong to Γext(GH)

Then llowast1 llowast2 is an element of P2(Llowast) Furthermore llowast1 llowast2 isin P2(L

lowast) satisfyingthe two properties exists

Furthermore there exist infinitely many 2-rank sublattices Llowast2 sub Llowast such that Llowast

2 isexpanded by a primitive set in P2(L

lowast)

Proof If P2(Llowast) is not empty it includes infinitely many elements because for any

n isin Z minusnllowast1 minus (n minus 1)llowast2 (n + 1)llowast1 + nllowast2 isin P2(Llowast) belongs to P2(L

lowast) if and only ifllowast1 llowast2 isin P2(L

lowast) Hence the first statement follows immediately from (1) and (2)(1) is almost straightforward The existence of llowast1 llowast2 llowast3 isin P2(L

lowast) satisfying llowast1 llowast2 llowast3 subU is proved by Lemma 62 Thus

sum3i=1 mil

lowasti isin U holds for any integers mi ge 0 In this

case llowast1 llowast2 + kllowast3 isin P2(Llowast) is a subset of U for any integer k ge 0 and their expanding

sublattices are different from each other From the assumption Γext(GH) sub HGH

plusmn(m1llowast1 + m2(l

lowast2 + kllowast3)) isin Γext(GH) follows As a result llowast1 llowast2 + kllowast3 isin P2(L

lowast) isobtained (Here llowast isin Γext hArr minusllowast isin Γext was used)

In order to prove (2) let M be the order of RG and Ω be the following set definedfor the type (GH) in Corollary 41

Ω = llowast +MLlowast llowast isin Γext(GH) HGH (79)

The following set is defined using Ω

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast llowast1 llowast2 isin P2(Llowast) (80)

P2M (Llowast) = llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω for any m isin Z (81)

By direct computation it is verified that P2M (Llowast) equals the following set

llowast1 +MLlowast llowast2 +MLlowast isin P2M (Llowast) mllowast1 + (mminus 1)llowast2 +MLlowast isin Ω (k le m le k + 6) for some k isin Z

(82)

Furthermore P2M (Llowast) 6= empty holds for any type of systematic absences presented in Table2 (see Table 4) Consequently some llowast1 llowast2 llowast3 isin P3(L

lowast) satisfies the assumptions (a)and (b) as a result of Lemma 62 In this case llowast1 llowast2 + kMllowast3 isin P2(L

lowast) holds for anyk ge 0 because of llowast1 mod M llowast2 mod M isin P2M (Llowast) and U cap HGH = empty The latticesexpanded by llowast1 llowast2+kMllowast3 are different depending on k ge 0 Hence all the statementswere shown

The following lemma is used in the proof of Theorem 2 Although it is straightfor-ward we provide a proof

Lemma 62 Let L sub RN be a lattice of rank N and 1 le m le N be an integer Thenany open cone U sub RN contains a primitive set l1 lm isin Pm(L) Furthermorewhen M gt 0 is a positive integer and k1 km isin Pm(L) U contains infinitely manyl1 lm isin Pm(L) satisfying li minus ki isin ML for any 1 le i le m

Proof We prove the first statement by induction Since al 0 6= a isin Q l isin L =al a isin Q l isin P1(L) is dense in RN there exist 0 6= a isin Q and l isin P1(L) such thatal isin U Hence l isin U is obtained Next suppose that m lt N and there is T isin Pm

contained in U Then there exists l isin L such that T cup l isin Pm+1 For any arbitrarily

fixed l2 isin T there is ǫ gt 0 such that U2 = x isin RN (1 minus ǫ)|x|2|l2|2 le (x middot l2)2is contained in U In this case l + sl2 isin U2 holds for sufficiently large integer s gt 0As a result T cup l + sl2 is a subset of U and primitive In order to prove the secondstatement it is sufficient if some l1 lm isin Pm(L) satisfies the desired property We

24

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

fix a basis l1 lN isin U of L and g isin GL(Z) satisfying ki = gli for any 1 le i le mWhen the subgroup of GL(Z) with positive entries is denoted by GL+(Z) the naturalmap GL+(Z) minusrarr G = g isin GL(ZMZ) det g = plusmn1 mod M is an epimorphismLet g0 isin GL+(Z) be an element belonging to the inverse image of g mod M Theng0l1 g0lN are all contained in U and satisfy g0li minus ki isin ML (1 le i le m)

Table 4 Density of Llowast Γext(GH) in LlowastType P2M (Llowast) in P2M (Llowast) P3M (Llowast) in P3M (Llowast)A 0321 0286B 0286 0143C 0476 0190D 0341 0209E 0736 0604F 0714 0571G 0214 0027H 0429 0058I 0714 0571J1 amp J2 0071 0022K 0857 0786L 0107 0004M 0036 0004N 0036 0004

aM is the cardinality of RG The densities are computed by dividing the cardinality of PiM (Llowast) bythat of PiM (Llowast) (i = 2 3) where PiM (Llowast) is defined by (80) and (83)

bThe densities of G H J L M N which consist of only primitive and body-centered cubic latticesare rather small We were not able to find another combination of q-values with larger densities As apractical measure against small densities the parallelogram law is used in the enumeration procedure

of Table 6 in addition to 3∣

∣llowast1∣

∣

2+

∣

∣llowast1 + 2llowast2∣

∣

2=

∣

∣2llowast1 + llowast2∣

∣

2+ 3

∣

∣llowast2∣

∣

2 (This is effective except for the

category N owing to Fact 1) Furthermore the condition (b) of Theorem 3 is not required to hold forinfinitely many m in the actual algorithm and subgraphs with relatively many edges are given priority

Theorem 3 claims that a solution of rank 3 is obtained from the combination of twosubgraphs of a topograph for lattices expanded by llowast1 and llowasti (i = 2 3) as in the right-

hand of Figure 11 and |plusmnllowast1 + llowast2 + llowast3|2 isin Λext(GH) By using such subgraphs withinfinitely many edges for the purpose the ldquosort criterion for zonesrdquo proposed in Section73 is provided a theoretical foundation

Theorem 3 Using the same notation as Theorem 2 let P3(Llowast) be the set of llowast1 llowast2 llowast3 isin

P3(Llowast) satisfying (a) and (b) with both i = 2 3

(a) plusmnllowast1 + llowast2 + llowast3 isin Γext(GH)

(b) mllowast1 + (mminus 1)(minusllowast1 + llowasti ) isin Γext(GH) for any m isin Z or mllowasti + (m minus 1)(llowast1 minus llowasti ) isinΓext(GH) for any m isin Zge0

P3(Llowast) then contains infinitely many elements regardless of the type of systematic ab-

sences

Remark 1 This theorem may be considered to refer the vectors associated with theedges in the circuit of length 6 in Figure 9 the 6 edges are associated with one of thefour vectors in the following parallelogram law

(a) 2(|l1|2 + |l2 + l3|2) = |l1 + l2 + l3|2 + |minusl1 + l2 + l3|2

25

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Figure 12 Connected subgraph of a topograph consisting of edges associated withlimllowast1 + (mminus 1)(minusl1 + llowasti ) m isin Z or l1mllowasti + (mminus 1)(llowast1 minus llowasti ) m isin Zge0

(b) 2(|l1|2 + |li|2) = |l1 + li|2 + |minusl1 + li|2 (i = 2 or 3)

Proof By Lemma 62 any open convex cone U sub RN HGH contains some llowast1minusllowast1 +llowast2 minusllowast1+ llowast3 isin P3(L

lowast) In this case U also includes plusmnllowast1+ llowast2+ llowast3 mllowast1+(mminus1)(minusl1+ llowasti )mllowast1 +(m+1)(minusl1+ llowasti ) and mllowasti +(mminus 1)(llowast1 minus llowasti ) = mllowast1 +(minusl1 + llowasti ) for any m isin Zge0Consequently if Γext(GH) is contained inHGH the statement is obtained immediately

If (GH) is one of the types in Table 2 let M be the order of RG and Ω and

P2M (Llowast) be the sets defined in the proof of Theorem 2 Furthermore we define

P3M (Llowast) = (llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) llowast1 llowast2 l

lowast3 isin P3(L

lowast) (83)

P3M (Llowast) =

(llowast1 +MLlowast l

lowast2 +ML

lowast l

lowast3 +ML

lowast) isin P3M (Llowast)

plusmnllowast1 + llowast2 + llowast3 +MLlowast isin Ωllowast1 +MLlowastminusllowast1 minus llowasti +MLlowastor llowasti +MLlowastminusllowast1 minus llowasti +MLlowast

belongs to P2M (Llowast) for both i = 2 3

(84)

By direct calculation it is verified that P3M (Llowast) 6= empty regardless of the type of system-atic absences (See Table 4) From Lemma 62 there exist infinitely many llowast1 llowast2 llowast3 isinP3(L

lowast) such that llowast1 + MLlowast llowast2 + MLlowast llowast3 + MLlowast isin P3M (Llowast) holds and llowast1minusllowast1 +

llowast2 minusllowast1 + llowast3 is a subset of C In this case llowast1 llowast2 llowast3 isin P3(Llowast) also holds

7 Algorithm for powder auto-indexing

In order to elaborate our new algorithm we first define a data structure that is assumedto be implemented in the program The set Λobs of observed lattice vector lengths isinput as an array 〈(qi[0] qi[1]) 1 le i le M〉 consisting of pairs of a q-value qi[0] and itsestimated error qi[1] = Err[qi[0]] In our algorithm every candidate for a Gram matrix

of Llowast is provided as a matrix with entriessumM

i=1 niqi[0] having coefficients ni isin 14Z At

various stages of powder auto-indexing the propagated errors of the entries are usefulfor making statistical judgments and strengthening the algorithm against observation

errors in the q-values For this purpose a data structure for formal sumssumNpeak

i=1 ciqiof elements of Λobs with coefficients ci isin 1

4NZ is implemented in Conograph (N is afixed positive integer) The data structure is equipped with the order lt and functions

26

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 5 Procedures for enumeration of four q-values satisfying the parallelogram lawvoid enumerateItoSolutions(Λobs c Ans)

(Input) Λobs array of Npeak pairs of a q-value qi[0] and its approximated error qi[1]c gt 0 parameter setting error tolerance level

(Output) Ans array of a sequence (qr qs qt qu) where qr qs qt qu are elements of Λobs sat-isfying

|Val(2qr + 2qs minus qt minus qu)| le c min2Err(qr + qs)Err(qt + qu) (Parallelogram law)(

radic

qr[0]minusradic

qs[0])2

le qt[0] qu[0] le(

radic

qr[0] +radic

qs[0])2

(Positive-definite condition)

1 (Start) Set a sorted sequence S = 〈qi + qj 1 le i le j le Npeak〉 of formal sums2 for i = 1 to 1

2Npeak(Npeak + 1) do3 Let 1 le Jmin Jmax le Npeak be integers satisfying4 Jmin le j le Jmax lArrrArr |Val(2S[i]minus S[j])| le 2cErr(S[i])5 for j = Jmin to Jmax do6 if |Val(2S[i]minus S[j])| le cErr(S[j]) then7 qr qs = getTerms(S[i])8 qt qu = getTerms(S[j])

9 if (radic

qr[0]minusradic

qs[0])2 le qt[0] qu[0] le (

radic

qr[0] +radic

qs[0])2 then

10 insert (qr qs qt qu) in Ans11 end if12 end if13 end for14 end for

getTerms Val Err as follows

Msum

i=1

aiqi lt

Msum

i=1

biqi hArrdef

Msum

i=1

aiqi[0] lt

Msum

i=1

biqi[0] (85)

getTerms

(

Msum

i=1

niqi

)

= qi 1 le i le Mni 6= 0 (86)

Val

(

Msum

i=1

niqi

)

=Msum

i=1

niqi[0] (87)

Err

(

Msum

i=1

niqi

)

=

(

Msum

i=1

n2i (qi[1])

2

)12

(88)

In particular if Val and Err are called with the argumentsumM

i=1 niqi they return the

value and the propagated error ofsumM

i=1 niqi[0] respectively

71 Algorithm for N = 2

In this case according to Theorem 1 the method using the parallelogram law workssufficiently

On output of the procedure in Table 5 each entry (qr qs qt qu) in Ans satisfiesthe parallelogram law 2(qr + qs) = qt+ qu and corresponds to the 2times 2 positive-definite

27

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

symmetric matrix

(qr qs qt qu) 7rarr(

Val(qr)12Val(qt minus qr minus qs)

12Val(qt minus qr minus qs) Val(qs)

)

(89)

Here we used the assumption that there exist llowast1 llowast2 isin Llowast such that qr = |llowast1|2 qs = |llowast2 |2

qt = |llowast1 + llowast2 |2

72 Algorithm for N = 3

The theorems in Section 62 state how to construct the Gram matrices of lattices ofrank 3 from elements of Λobs satisfying the equation 3|llowast1|2 + |llowast1 + 2llowast2|2 = |2llowast1 + llowast2 |2 +3|llowast2|2 Considering the case in which powder diffraction patterns contain only a smallnumber of peaks it is better to also use q-values satisfying the parallelogram law in theenumeration algorithm Hence the following two computational assumptions are usedin the algorithm of Table 6

(C1) If qr qs qt qu isin Λobs satisfy 2(qr + qs) = qt + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + llowast2 |2 qu = |llowast1 minus llowast2 |2

(C2) If qr qs qt qu isin Λobs satisfy 3qr + qt = 3qs + qu then there are llowast1 llowast2 isin Llowast such

that qr = |llowast1 |2 qs = |llowast2 |2 qt = |llowast1 + 2llowast2|2 qu = |2llowast1 + llowast2 |2

The computation time of the procedure in Table 6 is roughly proportional to N2zone

where Nzone is the size of A2 immediately after (2) This is estimated as follows Thesize of A3 in (3) is approximately 4Nzone if the cases Q1 = Q2 or Q3 = Q4 are ignoredWhen Npeak is the cardinality of Λobs the average number of (R1 R2 R3 R4) isin A2

satisfying (a) or (b) with regard to fixed (Q1 Q2 Q3 Q4) isin A2 is approximated as4Nzone

Npeak Hence the number of combinations of (Q1 Q2 Q3 Q4) (R1 R2 R3 R4) isin A2

and qk isin Λobs is roughly equal to 4Nzone middot 4Nzone

NpeakmiddotNpeak = 16N2

zone Steps (1)ndash(3) take

much less time than (4) Hence it is concluded that the time is proportional to N2zone

The enumeration is completed by calling the procedure in Table 6 after which thefollowing procedures are required before outputting solutions

1 Transform every enumerated Gram matrix into a Niggli-reduced form [21] (whichis standard in crystallography)

2 Bravais lattice determination

3 Sort solutions by some figures of merit (eg de Wolff figure of merit widely usedin powder auto-indexing [11])

4 Remove duplicate solutions

73 Speed-up method using topographs

As described in Section 72 the computation time of the algorithm in Table 6 is pro-portional to the square of the size Nzone of A2 immediately after (2) Thus an effectiveway to speed up the algorithm is to reduce the size of A2

When reducing the size of A2 it is necessary to retain elements obtained fromqr qs qt qu isin Λobs for which the assumption (C1) or (C2) is true because they areessential to obtain the true solution A new criterion to sort the elements of A2 isdefined for the purpose

Every entry (R1 R2 R3 R4) of A2 consists of four formal sums satisfying theparallelogram law 2(R1 + R2) = R3 + R4 Hence it corresponds to the subgraph

28

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 6 Enumeration algorithm for three-dimensional lattices constructed from q-values

void enumerateThreeDimLattices(Λobs c detSmin detSmax Ans)(Input) Λobs c same as in Table 5

detSmin detSmax lower and upper thresholds on determinants of outputmatrices

(Output) Ans array of 3times 3 positive-definite symmetric matrices

(1) By the method in Table 5 enumerate qr qs qt qu of Λobs satisfying 2(qr + qs) =qt + qu and insert (qr qs qt qu) in A2 (Here A2 is an array of four formalsums (Q1 Q2 Q3 Q4))

(2) Enumerate qr qs qt qu of Λobs satisfying the equation 3qr + qt = 3qs + qu Thisis done by a similar method as in Table 5 Using a new formal sum qminus1 =minus2qr+qs+qu

2 = qrminus2qs+qt2 two sets of q-values satisfying the parallelogram law are

generated

2(qminus1 + qr) = qs + qu 2(qminus1 + qs) = qr + qt (90)

Check whether A2 contains (qw qr qs qu) or (qw qs qr qt) for some 1 lew le Npeak If not this suggests that qminus1 = minus2qr+qs+qu

2 = qrminus2qs+qt2 equals

|llowast|2 for some llowast isin Llowast undetected owing to some observational reason Insert(minus2qr+qs+qu

2 qr qs qu) and ( qrminus2qs+qt2 qs qr qt) in A2

(3) For every entry (Q1 Q2 Q3 Q4) isin A2 insert (Q1 Q2 Q3 Q4)(Q1 Q2 Q4 Q3) (Q2 Q1 Q3 Q4) (Q2 Q1 Q4 Q3) in a new array A3

(4) For every (Q1 Q2 Q3 Q4) isin A3 search (R1 R2 R3 R4) isin A3 satisfying either ofthe following

(a) Q1 = R1 isin Λobs

(b) Q1 R1 isin Λobs and |Val(Q1 minusR1)| le cErr(Q1 minusR1)

In addition for every qk isin Λobs assume that there exists llowast1 llowast2 l

lowast3 satisfyinga

Q1 asymp R1 = |l1|2 Q2 = |l2|2 Q3 = |l1 + l2|2R2 = |l3|2 R3 = |l1 + l3|2 qk = |l1 + l2 + l3|2 (91)

Then the Gram matrix S = (li middot lj)1leijle3 is obtained as the following 3 times 3symmetric matrix

Q112 (Q3 minusQ1 minusQ2)

12 (R3 minusQ1 minusR2)

12 (Q3 minusQ1 minusQ2) Q2

12 (Q1 minusQ3 minusR3 + qk)

12 (R3 minusQ1 minusR3)

12 (Q1 minusQ3 minusR3 + qk) R2

(92)

Using Val and Err the values and propagated errors of the entries are computableIf detSmin le detS le detSmax insert S in Ans

29

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

aEvery three graphs on the left-hand side have a common node which is an end point of three edgesassociated with R1 R2 R1 R4 and R2 R4 This figure illustrates how these graphs are unified

Figure 13 Extension of a topograph (12)

aThe left-hand two subgraphs contain a subgraph as in Figure 2 commonly They are unified asabove

Figure 14 Extension of a topograph (22)

of a topograph as in Figure 2 Table 7 explains how a subgraph T of a topographis constructed from these substructures The elements of A2 utilized to obtain thesubgraph are output in A2

In the procedure of Table 7 the subgraph T is expanded to only one side Thewhole subgraph composed of e = (R1 R2 R3 R4) isin A2 and entries of A2 is ob-tained by calling the recursive procedure twice setting e = (R1 R2 R3 R4) and(R1 R2 R4 R3) respectively in the second argument Let C(e) be the cardinality ofthe union of the two A2 output by calling the recursive procedure twice as above ThisC(e) is considered to quantify the size of the subgraph finally obtained

If either (C1) or (C2) is wrong for e isin A2 C(e) will result in a small numberbecause the new q-values required to extend a new edge (ie q isin Λobs in line 2 of Table7) would rarely be found in Λobs This suggests C(e) provides an effective sort criterionfor elements of A2 Part (b) of Theorem 2 indicates that this criterion remains effectiveunder the influence of systematic absences

Now any e = (R1 R2 R3 R4) isin A2 corresponds to a 2times 2 Gram matrix S(e) isinS2 by the following map

S((R1 R2 R3 R4)) 7rarr(

Val(R1)12Val(R3 minusR1 minusR2)

12Val(R3 minusR1 minusR2) Val(R2)

)

(93)

When C(e) is used as a sort criterion e isin A2 with smaller detS(e) is prioritized asa result because Λobs sub [qmin qmax] contains a smaller number of tuS(e)u (u isin Z2) ifdetS(e) has a larger value This property of C(e) is also desirable because (4) in Table6 (and Theorem 3) requires llowast1 llowasti isin P2(L

lowast) (i = 2 3) to obtain the true solution Llowast

30

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 7 Recursive procedure to form a subgraph of a topograph from e and elementsof A2

void expandSubtopograph(A2 e A2 T )(Input) A2 array of four formal sums (Q1 Q2 Q3 Q4) with Qi =

sumNpeak

i=1 nijqj Furthermore it is assumed that every entry sat-isfies

Val(2Q1+2Q2minusQ3minusQ4) le c min2Err(Q1+Q2)Err(Q3+Q4)

for some fixed constant c and Q3 Q4 and either of Q1 Q2 belongto Λobs (ie there is q isin Λobs such that Qi = q)

e = (R1 R2 R3 R4) satisfying (R1 R2 R3 R4) isin A2

(Output) A2 subset of A2T subgrapha of a topograph composed of substructures corre-

sponding to (Q1 Q2 Q3 Q4) isin A2 (If such subgraphs arenot unique A2 containing larger number of entries is prioritized)

1 (Start) For (i j) = (1 2) (2 1) let Sij be the set defined by

2 Sij =

(Ri R4 Rj q) isin A2 q isin Λobs

if Rj isin Λobs

empty otherwise

3 T12 = empty A12 = empty T21 = empty A21 = empty4 for i = 1 to 2 do5 Take j isin 1 2 i6 for e2 isin Sij do7 Call expandSubtopograph(A2 e2 Tij Aij)

8 if∣

∣

∣Aij

∣

∣

∣ lt |Aij | then9 Tij = Tij

10 Aij = Aij 11 end if12 end for13 end for

14 Set A2 = (R1 R2 R3 R4) cup A12 cup A21

15 Construct T by unifying T12 T21 and (R1 R2 R3 R4) as in Figure 13

aIn the actual computation it is not necessary to construct a subgraph T because a sort criterion forelements of A2 is defined using only A2 However as T has a data structure of binary trees consistingof finite nodes and edges it is not difficult to implement T in a program

31

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

By using C(e) a sublattice Llowasti expanded by llowast1 llowasti isin P2(L

lowast) is prioritized over anyother sublattices contained in Llowast

i To summarize the above discussion it is only necessary to insert the following pro-

cedures between (2) and (3) in Table 6 in order to reduce the computation time

(2i) For each element e = (R1 R2 R3 R4) isin A2 compute C(e) by calling theprocedure in Table 6 (In this case any e2 isin A2 satisfies C(e) = C(e2) Hencethe number of calls of the procedure is less than the size of A2)

(2ii) Sort A2 in descending order of C(e) For e1 e2 isin A2 satisfying C(e1) = C(e2)they may be sorted by e1 lt e2 lArrrArr

defdetS(e1) lt detS(e2)

(2iii) Remove the (Nzone + 1)th-to-last elements of A2 using a fixed integer Nzone gt0 (The default value of Nzone used in Conograph is given in (97) of section81 When the default value is used the computation time is approximatelyproportional to Npeak

4)

In Section 82 we shall see how the computation time is decreased in practice bythis improvement

74 Problems with the quality of powder diffraction patterns

Among the problems described in (A1)ndash(A6) and (A3) of Section 3 we have not yetclarified how to handle (A3) In this section we explain how missing and false elementsin Λobs influence the powder auto-indexing results This issue is related to the qualityof powder diffraction patterns as observational data

Using a peak search program equipped with Conograph it is not difficult to obtainthe positions of all the peak heights above a given threshold automatically and ratheruniformly Although the ability to decompose overlapping peaks depends on the peaksearch software this is not a great problem in our algorithm because an almost identicalsolution will be obtained even if q1 isin Λobs is replaced with q2 very close to q1 (to somedegree Err[q1] and Err[q2] will absorb the difference between q1 and q2)

By the algorithm in Table 6 normally multiple Gram matrices of the reciprocallattice Llowast of the correct solution L are generated from Λobs This is because a latticeLlowast has as infinitely many Gram matrices as bases of Llowast Note that Gram matrices ofdifferent bases are computed from different q-values basically Therefore when theseGram matrices are transformed into a reduced form they are a bit different matricesowing to observational errors but rather close to each other

In order to reduce the influence of ǫ1 gt 0 of (A3) the existence of such duplicatesolutions is very useful Suppose that m Gram matrices of Llowast are generated from theset of q-values Λext(weierp) cap [qmin qmax] containing no observational errors by applyingour enumeration method Then when the same method is applied to Λobs containingobservational errors the enumeration process fails to obtain the correct solution onlywhen none of approximate solutions of the m Gram matrices them are generated Thefailure rate becomes very small as m increases regardless of the magnitude of ǫ1 Ifthe size of Λobs is augmented or equivalently the range [qmin qmax] is magnified mincreases naturally As a result the enumeration success rate increases monotonicallyas the size of Λobs increases However it should be noted that the time for enumerationof solutions is roughly proportional to the fourth power of the size of Λobs if the defaultparameters of Conograph are used In addition the success probability will not increasemuch once qmax reaches some observational limit because larger q-values have largererrors owing to peak overlap and observational accuracy

32

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Next we discuss the influence of ǫ2 in (A3) In this case the success rate in obtainingthe true solution remains same if Λobs is replaced with Λobs cup qobs which contains afalse element qobs (Of course such qobs increases the time for enumeration)

In conclusion the enumeration process is considered to be robust to missing and falseelements in Λobs although such elements increase the computation time Indeed theprocedure to sort candidate solutions executed after the enumeration is more sensitiveto missing and false elements because figures of merit are severely affected by theseelements In general for a poor quality powder diffraction pattern it is very difficult tojudge which solution is correct even if the q-values of respective solutions are manuallycompared to actual peak-positions (see Example 6 in Section 82)

Example 5 shows the case of a two-phase sample Both lattice parameters are ac-quired by Conograph However it seems to be almost impossible to judge which is thecorrect second phase parameter in this case

8 Implementation and results of Conograph

Before introducing the input parameters and results of Conograph we explain the cir-cumstances in which the program was verified The Conograph source code is writtenin C++ and OpenMP and compiled with Mingw (GNU Compiler Collection for Win-dows) The computer used for the test has an Intel i7-2620 (270 GHz) processor and 12GB RAM The processor can execute parallel computing with eight hyper-threads allof which were used during the test Additionally we confirmed that a computer with 4GB RAM was able to carry out the same test

81 Input parameters of Conograph

The input parameters used in the enumeration process are listed in Table 8 ldquoAUTOrdquois used to set parameters that are considered to depend on respective powder diffractionpatterns The parameters required after the enumeration are not given here they willbe introduced at another time

Table 8 Default parameters of ConographSymbol Meaning Default∆2θ Zero-point shift (degree) 0c Tolerance level for errors in q-values

Characteristic X-rays or neutron reactor sources 15Synchrotron X-rays or neutron spallation sources 1

Npeak Number of q-values used AUTONzone Threshold for the maximum number of (Q1 Q2 Q3 Q4) AUTONsol Threshold for the maximum number of solutions

First trial AUTOWhen the first trial failed because of too many solutionsa 64000timesm (m isin Zgt0)

Volmin Threshold for the minimum volume of R3L (A3) AUTOVolmax Threshold for the maximum volume of R3L (A3) AUTO

aNsol is set to 64000 at most when AUTO is used in order to prevent memory allocation errors inmemoryless computers When the number of enumerated solutions exceeds Nsol Conograph removesthose with a smaller unit-cell volume As a result the true solution is sometimes also removed Thesimplest method of resolving this is to reduce Volmax or increase Nsol when unsatisfactory results areobtained This time we chose the latter in order to measure computation time However such a trial-and-error approach is not necessary because an alternative method is adopted in Conograph This willbe introduced at another time

33

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

In the following we provide an explanation of the parameters and formulas neededto compute AUTO

(1) Zero-point shift ∆2θ Depending on the type of diffractometers the x-axis of apowder diffraction pattern is represents a diffraction angle 2θ or a time-of-flight tgiven by the following function of q-values

2θ = 2 sinminus1 λradicq

2+ ∆2θ (94)

t =

nsum

i=1

ciqminusi2 (95)

Using these equations x-coordinates are transformed to q-values before executingpowder auto-indexing Among λ ∆2θ and ci (1 le i le n) only the value of ∆2θis unknown Conograph users are recommended to set ∆2θ = 0 normally becausesuccessful results were obtained even in cases of very large zero-point shift such as∆2θ asymp 02 degree After auto-indexing it is possible to refine the lattice parametersand zero-point shift simultaneously using a nonlinear least-squares method

(2) Tolerance level c for errors in q-values Table 5 provides a usage example of cBy setting a larger value of c a wider range of combinations of q-values is searched

(3) Number of q-values used This parameter determines the size of Λobs an array ofinput q-values After sorting the q-values in Λobs into ascending order the (Npeak +1)th-to-last parameters are removed before the powder auto-indexing commencesAs explained in Section 74 both the computation time and success rate increase asNpeak is magnified The default value of Npeak is calculated by

Npeak = min

♯q lt 10d2 q isin Λobs 48

(96)

where ♯T is the cardinality of the set T and d is the lower threshold for the distancebetween two lattice points In Conograph d is set to 20A

In (96) Npeak le 48 is forced because it is frequently meaningless to use large q-values owing to severe peak overlap However 48 is still much larger than the 20ndash30that are normally adopted for powder auto-indexing Conograph uses such manyq-values for another reason in (A5) and the paragraphs following (A6) we explainedthat qmax gt DN = 3 middot2minus13dminus2 asymp 06Aminus2 is required at least theoretically (In (96)qmax gt 4DN is adopted) Figure 15 presents the range of the first Npeak q-values inthe test data of Table 15 Owing to the threshold of 48 the interval [qmin qmax] isfrequently much smaller than 06Aminus2 although powder auto-indexing succeeded inall the test data We suppose 48 q-values might be insufficient in some exceptionalcases Users are recommended to increase Npeak manually if results obtained withthe default parameters are unsatisfactory

(4) Threshold for the maximum size of (Q1 Q2 Q3 Q4)

Nzone =1

3Npeak(Npeak + 1) (97)

(5) Threshold for the maximum number of candidate solutions

Nsol = min64000 N2zone (98)

34

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Synchrotron

Characteristic X-rays

Spallation neutron src(Back scattering bank)

Reactor

06

q (Å-2)

aIf qmax gt 06Aminus2 is imposed the interval [qmin qmax] often includes more than several tens of q-values of diffraction peaks We should not increase qmin because smaller q-values have better accuracyAs a result [qmin qmax] set by the default parameters often does not satisfy the theoretical requirementqmax gt 06Aminus2 Nevertheless powder auto-indexing succeeded in all the test data This is consideredto be because all our test data satisfy d ge 27A (and many also satisfy d ge 4A) and because the formula(13) of Lagarias et al provides an overestimation of D3

Figure 15 Range [qmin qmax] of Npeak q-values used in powder auto-indexing

(6) Threshold for the minimum and maximum of the volume of R3L

Volmin = max5 vminus120 (99)

Volmax = 30Volmin (100)

where 5A3 is chosen as the lower threshold for the volumes of existing crystals andv20 is the upper bound of Vol(R3Llowast) estimated using the 20 smallest elements ofΛobs = q1 q2 qNPeak

by

vj =2π

3

q32j minus q

321

j minus 1 (101)

Equation (101) is based on the following formula which holds for any 0 lt r lt R

2π(R3 minus r3)

3

∣

∣

∣r2 le |llowast|2 le R2 llowast isin Llowast∣

∣

∣

minus1

ge 4π(R3 minus r3)

3

∣

∣

∣llowast isin Llowast r2 le |llowast|2 le R2

∣

∣

∣

minus1

rarr Vol(R3Llowast) as R rarr infin (102)

Note that 20 and 30 are chosen empirically and vNPeakis used instead of v20 if

NPeak lt 20 We have found no cases when this [VolminVolmax] fails to contain thecorrect volume

82 Results

We prepared 26 + 4 powder diffraction patterns as test data A summary of the first26 test data is presented in Table 9 The remaining four are presented in Examples 3ndash6to illustrate some rather special cases

We now evaluate how the method in Section 73 improved enumeration times InSection 72 we explained that the time is roughly proportional to N2

zone From theformula (97) the computation time of the algorithm in Table 6 is approximately pro-portional to Npeak

4 Figure 16 illustrates the relation between the following pairs inpractical tests

35

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 9 Summary of 26 test dataa

Diffractometer number of patternsSynchrotron 11Characteristic X-rays 7Spallation neutron sources (time-of-flight) 6Reactor neutron sources 2Symmetry of latticeTriclinic 7Monoclinic (P) 4Monoclinic (B) 1Orthorhombic (P) 5Tetragonal (P) 2Tetragonal (I) 1Rhombohedral 1Hexagonal 1Cubic (I) 1Cubic (F) 3

Distribution of absolute zero-point shiftsb

ndash 01 1701 ndash 015 1015 ndash 2

aAmong eight samples distributed in SDPDRR-2 samples 1ndash4 8 are included herein The resultof sample 7 is used in Example 6 Samples 5 6 are excluded this time because we could not obtainsolutions with de Wolff figures of meritM20 gt 10 by using all the peaks with sufficiently large intensitiesThese patterns seem to contain a considerable number of false peaks

bThe zero-point shifts were computed after execution of powder auto-indexing by non-linear leastsquares method Time-of-flight data are excluded here

36

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tim

e fo

r en

umer

atio

n (s

ec)

102

103

104

105

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

300

240

180

120

60

0

Tot

al T

ime

(sec

)10

210

310

410

5

Number of enumerated solutions

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aTotal time includes a) Enumeration of solutions b) Transformation of every solution to a reducedform c) Computation of figures of merit d) Refinement of solutions using linear optimization e) Error-stable Bravais lattice determination f) Removal of solutions having a figure of merit less than a user-input threshold g) Removal of duplicate solutions h) Sorting solutions with a selected figure of meritOf these a) and e) are the most time-consuming The time required for a) consumed approximatelyhalf of the total time We have already contributed to reducing the time for e) in [22]

Figure 16 Computation time of Conograph

bull number of q-values Npeak and time for enumeration

bull number of q-values Npeak and total time for powder auto-indexing

bull number of enumerated solutions and time for enumeration

bull number of enumerated solutions and total time for powder auto-indexing

Figure 17 shows the rate of decrease in the size of A2 as a result of applying themethod described in Section 73

Even in the following difficult cases solutions were obtained without special parame-ter settings By Conograph reliable powder auto-indexing results will become availableeven for less-experienced users

Example 3 Non-unique solutions (Figure 18 (a)) For any fixed C gt 0 thefollowing lattice parameters have exactly the same q-values (cf 3 in Appendix B)

Cubic(P) a = b = c = Cα = β = γ = 90 (103)

Tetragonal(P) radic2a =

radic2b = c = Cα = β = γ = 90 (104)

Conograph succeeded in finding both of these

Example 4 Small q-values are lost (Figure 18 (b)) When the size of the unitcell is large many q-values are frequently lost because they are smaller than qmin Wehave confirmed that Conograph is very robust to such a loss This example presents thecase in which the 19 smallest non-zero q-values are not included in the observed range[qmin qmax]

37

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

56

01

2

3

4

56

1 N

zone

Si

ze o

f A

2

50403020100

Number of q-values Npeak

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

56

01

2

3

4

56

1

Npe

ak

Size

of

A2

150100500

Time for enumeration (sec)

Cubic Hexagonal Rhombohedral Tetragonal Orthogonal Monoclinic Triclinic

aExcept for cases of small Npeak all the rates are in the range 006ndash018 Under the assumptionthat the time for enumeration is proportional to N2

zone the enumeration is about 32ndash250 times fasteras a consequence The right-hand figure indicates that a larger improvement was made in more time-consuming cases

Figure 17 Rates of elements of A2 used in enumeration algorithm

Example 5 Two-phase data (Figure 18 (c)) This powder diffraction pattern is atwo-phase sample with the mass ratio 58 42 The lattice parameter of the second phasewas also enumerated

Example 6 Poor quality powder diffraction pattern (Figure 18 (d)) This issample 7 distributed in SDPDRR-2 According to a recent personal report from Le Bailcrystal structures other than sample 7 have been determined Conograph obtained threereasonable solutions

9 Conclusion

Powder auto-indexing is divided into two main stages enumeration and sort of solu-tions We contributed mainly to the stage of enumeration by providing a quick andstrongly reliable algorithm For the purpose Conwayrsquos definition of topographs wasgeneralized to lattices of any rank N using Voronoirsquos second reduction theory holdingthe association of edges and four lengths |llowast1| |llowast2 | |llowast1 + llowast2| |llowast1 minus llowast2| (llowast1 llowast2 isin Llowast) Byusing common properties of systematic absences proved in Theorems 1 2 and 3 the al-gorithm is shown to work regardless of the type of systematic absences This propertiesare stated as distribution rules for reciprocal lattice vectors corresponding to systematicabsences on a topograph Such rules have not been known so far and will be also use-ful in other problems of crystallography Our enumeration algorithm was implementedin Conograph Conograph obtained successful results even for difficult cases Theseexamples proved that the new enumeration method is robust against missing and falseelements in the set of lattice vector lengths extracted from a powder diffraction patternTopographs were also utilized to speed up our enumeration algorithm In practical testswe found that the improvement reduced the enumeration time to 1250ndash132

Acknowledgments The author would like to extend her gratitude to Professor TOda of the University of Tokyo for his daily encouragements to Visible Information Incfor their cooperation in implementing the Conograph GUI and to Professor E Hitzerof International Christian University for proposing the impressive name ldquoConographrdquoI would also like to thank Dr K Fujii and Professors H Uekusa T Ozeki of theTokyo Institute of Technology Dr S Torii Dr J Zhang Dr M Ping and ProfessorsM Yonemura T Kamiyama of KEK and Professors A Hoshikawa and T Ishigaki ofIbaraki University for their valuable comments and for offering test data This research

38

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

(a) Case of more than one solution

5

4

3

2

1

0

2x104 3 4 5 6 7

Time of flight

Cubic(P) Tetragonal(P)

(b) The 19 smallest 0 6= q isin ΛLlowast are lost20

15

10

5

0

2x105 3 4 5

Time of flight

(c) Two-phase data4

3

2

1

0

2x104 3 4 5

Time of flight

(d) Poor quality powder pattern2000

1500

1000

500

0

12108642

2q (deg)

aTriangles indicate peak positions detected by a peak-search program (corresponding to q-values inΛobs) Tick marks represent the peak positions corresponding to q isin ΛLlowast = |llowast|2 llowast isin Llowast It is seenthat two different lattices have exactly the same q-values Both parameters are detected by Conograph

bThis example presents the case in which the 19 smallest non-zero q-values are not in the observedrange It is confirmed that Conograph is very robust to such a loss The lattice parameters are a = 882b = 976 c = 978 α = γ = 90 β = 104 and a diffraction pattern from a back-scattering bank ofneutron sources is used

cThis powder diffraction pattern is a case of a two-phase sample with the mass ratio 58 42

Upper a = b = c = 120 (Cubic(I)M20 = 70) (105)

Lower a = b = 48 c = 130 (RhombohedralM20 = 14) (106)

The unit cell of the first phase is much larger than that of the second phase As a result the latticeparameter of the first phase gained the best de Wolff figure of merit among all the enumerated solutionsregardless of peaks derived from the second phase The lattice parameter of the second phase was alsoenumerated by Conograph This supports our claim in Section 74 that the enumeration process isrobust to missing and false elements of Λobs However the figure of merit of the second phase wasconsiderably small owing to the first-phase peaks Therefore it is almost impossible to judge whichone is the correct second-phase lattice

dThis is sample 7 distributed in SDPDRR-2 held in 2002 The tick marks are peak positions of thefollowing solutions output by Conograph

Upper a = 399 b = 115 c = 171 α = 779 β = 846 γ = 804 (M20 = 66) (107)

Middle a = 407 b = 225 c = 257 α = 888 β = 874 γ = 848 (M20 = 77) (108)

Lower a = 395 b = 171 c = 228 α = 784 β = 866 γ = 841 (M20 = 129) (109)

The upper lattice parameter is the solution proposed by participants in SDPDRR-2 This regards thetwo leftmost peaks to be false The others are newly found by Conograph The second solution assumesthat some peaks are embedded in background noise and the third is intermediate This result provesthat Conograph makes the stage of enumeration much more reliable Although the correct solution isprovided by a larger M20 normally this is not always right In particular it becomes very difficult toselect the best solution from peak positions alone when powder patterns are assumed to have a numberof diffraction peaks embedded in background noise or false peaks as intense as diffraction peaks

Figure 18 Results of Conograph

39

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

was partly supported by a Grant-in-Aid for Young Scientists (B) (No 22740077) andthe Ibaraki Prefecture (J-PARC-23D06)

A On equivalence between powder diffraction pat-

terns and average theta series

For any periodic function weierp with the period lattice L satisfyingint

RNL |weierp(x)|dx lt infin an

average theta series Θweierp(z) is defined by

Θweierp(z) =1

vol(RNL)

sum

lisinL

int

(RNL)2weierp(x)weierp(y)eπ

radicminus1z|xminusy+l|2dxdy (110)

where vol(RNL) is the volume of RNL (Θweierp(z) is invariant if a sublattice L2 sub6= L isregarded as the period of the same weierp This definition is a generalization of the averagetheta series defined in 23 Chapter2 of [8]) The infinite sum converges uniformly andabsolutely in any compact subset of z isin C Im(z) gt 0

Using the Poisson summation formula the functional equation for Θweierp(z) is obtained

Θweierp(z) =

(radicminus1

z

)N2sum

llowastisinLlowast

eminusπradicminus1z

|llowast|2 |weierp(llowast)|2 (111)

where weierp(llowast) = vol(RNL)minus1int

RNLweierp(x)eminus2π

radicminus1xmiddotllowastdx

Assuming that fpowder(qweierp) in (7) equals 0 for any q lt 0 the Fourier transform of(2radicq)minus1fpowder(qweierp) is an average theta series

int

R

fpowder(qweierp)e2π

radicminus1qz dq

2radicq

=

int

R

(

int

|xlowast|2=q

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)dxlowast)

e2πradicminus1qz dq

2radicq

=

int

RN

sum

llowastisinLlowast

|weierp(llowast)|2δ(xlowast minus llowast)e2πradicminus1|xlowast|2zdxlowast

=sum

llowastisinLlowast

e2πradicminus1|llowast|2z|weierp(llowast)|2 = (minus2

radicminus1z)minusN2Θweierp

(

minus 1

2z

)

(112)

Therefore information obtained from a powder diffraction pattern is theoreticallyequivalent to that from an average theta series

B Theorems on the cardinality of solutions

It is well known that the equivalence class of S0 isin SN≻0 is not uniquely determined even

if all elements of ΛS0 = tvS0v 0 6= v isin ZN are provided However for N le 4 itis possible to obtain a finite set containing all the equivalence classes of S isin SN

≻0 withΛS = ΛS0 (cf Appendix C)

In this section several known theorems about the cardinality of solutions are sum-marized for reference

1 Case N = 1 In this case Λweierp in (8) generates Llowast over Z (Otherwise let Llowast2 ( Llowast

be the lattice generated by Λweierp Then the reciprocal lattice L2 of Llowast2 is the period

lattice of weierp since weierp(x) =sum

llowastisinLlowast2weierp(llowast)e2π

radicminus1xmiddotllowast holds This is a contradiction)

Therefore the determination of L is straightforward even from Λweierp

40

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

2 Case N = 2 It was shown by Delone (and independently by Watson [32] [33])that up to a factor the following is the only pair of inequivalent positive-definitesymmetric matrices that have the same representations over Z

(

2 11 2

)

(

2 00 6

)

(113)

3 Case N = 3 From the case of N = 2 an infinite family of inequivalent pairs thathave the same representations over Z is obtained

2 1 01 2 00 0 c

2 0 00 6 00 0 c

(114)

The following is another example (see [19] for the proof)

1 0 00 1 00 0 1

1 0 00 2 00 0 2

(115)

A powder auto-indexing result of Conograph for this case is presented in Example3 of Section 82

4 Case N = 4 S isin SN≻0 is called universal if ΛS equals Zgt0 the set of all positive

integers It was confirmed by Bhargava and Hanke that the number of equivalenceclasses of universal S isin S4

≻0 equals 6436 [1]

5 Case N ge 5 From the existence of universal quadratic forms in N = 4 there areinfinitely many S isin S5

≻0 such that ΛS = Zgt0

C Lattice determination from a complete set of lat-

tice vector lengths

In this section for any N le 4 and a given ΛS0 sub Rgt0 of some S0 isin SN≻0 an algorithm

to enumerate all the equivalence classes of S isin SN≻0 satisfying ΛS0 = ΛS is introduced

First we recall that S = (sij)1leijleN isin SN≻ is Minkowski-reduced if and only if S

satisfies

sii = min tviSvi e1 eiminus1 vi is a primitive set of ZN (116)

When S is Minkowski-reduced the entries of S satisfy

0 lt s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N) (117)

Table 10 gives a recursive procedure to generate all the candidates for S from Λobs =ΛS0 cap (0 c) when 0 lt c le infin is large enough If the recursive procedure is startedwith arguments m = n = I = J = 1 all positive-definite symmetric matrices S =(sij)1leijleN satisfying the followings are enumerated in an array Ans

s11 le middot middot middot le sNN 2|sij | le sii (1 le i lt j le N)

sii+1 le 0 (1 le i lt N)

sii 1 le i le N cup sii + sjj + 2sij 1 le i lt j le N sub Λobs

(118)

41

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Table 10 Recursive procedure for lattice determination from a perfect set of represen-tations

void enumerateLattice(NΛobsm n S I J Ans)(Input) 1 le N le 4 number of rows and columns of S0 isin SN

≻0Λobs = 〈q1 qM 〉 6= empty a sorted sequence of all the elements of ΛS0 that belong to

the interval (0 c)mn integers 1 le m le n le N indicating the (mn)-entry of S

S = (sij) N timesN symmetric matrix that fulfills0 lt s11 le middot middot middot le snn 2|sij | le sii (1 le i lt j lt n) sii+1 le 0(1 le i le n)

I J integers indicating

snn isin qi I le i le J if m = n

snn = qI and smm + snn + 2smn isin qi I le i le J otherwise

(Output) Ans array of positive-definite N timesN symmetric matrices

1 (start) for l = I to J do2 if m = n then3 snn = ql4 else5 smn = snm = 1

2 (ql minus smm minus snn)6 end if7 if m = 1 then8 if det(sij)1leijlen gt 0 then9 if n ge N then10 Insert S in Ans11 else

12 m2a = max

1 le i le M q1 qiminus1 are representations ofthe submatrix (sij)1leijlen over Z

13 if m = n then14 Call searchLattice(NΛobs n+ 1 n+ 1 S lm2 Ans)15 else16 Call searchLattice(NΛobs n+ 1 n+ 1 S Im2 Ans)17 end if18 end if19 end if20 else21 if m = n then22 m2 = max1 le i le M qi le snminus1nminus1 + snn23 Call searchLattice(NΛobsmminus 1 n S lm2 Ans)24 else25 m2 = max1 le i le M qi le 2smminus1mminus1 + snn26 Call searchLattice(NΛobsmminus 1 n S Im2 Ans)27 end if28 end if29 end for

aProposition C1 claims that there always exists m2 lt infin even if the sequence Λobs is replaced byΛS0

virtually

42

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

Consequently any Minkowski-reduced S satisfying (118) and 3sNN le qM is outputin Ans As a result of Proposition C1 the recursive procedure is always completed ina finite number of steps even if we set c = infin ie if ΛS0 is used instead of Λobs Thisindicates all the equivalence classes of S isin SN satisfying ΛS = ΛS0 are enumerated bythe recursive procedure if sufficiently large c is selected Consequently the number ofequivalence classes of S isin SN

≻0 satisfying ΛS = ΛS0 is finite for any ΛS0 sub Rgt0In the remainder of this section we give a proof of Proposition C1

Proposition C1 Suppose that S isin SN≻0 and S2 isin SN2

≻0 and 0 lt N2 lt N le 4 ThenΛS2 6sup ΛS

Lemma C1 is utilized in the proof

Lemma C1 Any S isin SN≻0 is represented as a finite sum

sum

k λkSk such that every Sk

is a positive-definite symmetric matrix with rational entries and λk isin Rgt0 are linearlyindependent over Q

Proof For any S = (sij)1leijleN isin SN let vS be the vector t(s11 s12 sij sNN )

of length N(N+1)2 Then SN is identified in N(N+1)

2 -dimensional vector space by themap S 7rarr vS Define a set P by

P = I sub sij 1 le i le j le N sij isin I are linearly independent over Q(119)

Then P is not empty Let t1 tm be one of the maximal elements of P underinclusive order When vectors t(t1 tm) and t(1 1) of length m are denoted by t

and 1m respectively there exists an N(N+1)2 timesm rational matrix C such that vS = Ct

Furthermore there exists ǫ gt 0 such that for any U = (uij) isin GLm(R) with entries|uij | lt ǫ every column of C(t t1m minus U) is the image of a positive-definite symmetricmatrix by the map S 7rarr vS

Let 1m be the identity matrix of size m If t1mUminus1t 6= 1 we have equations

(t t1m minus U)minus1 = Uminus1(( t1mUminus1tminus 1)minus1t t1mUminus1 minus 1m) (120)

(t t1m minus U)minus1t = ( t1mUminus1tminus 1)minus1Uminus1t (121)

If all the entries of Uminus1t are negative we have that t1mUminus1t lt 0 and every entry of(t t1mminusU)minus1t is positive Clearly there exists U = (uij) isin GLm(R) such that |uij | lt ǫevery entry of Uminus1t is negative and the matrix t t1m minus U is rational Fix such a U

Let Sk (1 le k le m) be a positive-definite symmetric matrix satisfying C(t t1m minusU) = (vS1 vSm

) S is then represented as a linear sum of rational Sk with positivecoefficients as follows

vS = Ct = (vS1 vSm)(t t1m minus U)minus1t (122)

Hence the statement is proved

For any ring R2 sube R and a symmetric matrix S with entries in R2 let ΛS(R) be theset tvSv 0 6= v isin RN consisting of representations of S over R If 0 isin ΛS(R) S issaid to be isotropic over R Otherwise S is anisotropic over R

Proof of Proposition C1 The statement holds if it is true when N2 + 1 = N = 4 ByLemma C1 it is sufficient if the statement is proved in the case that S S2 are rationalAny non-singular quadratic form over Qp of rank 4 satisfies ΛS(Qp) sup Qtimes

p for any p Onthe other hand any anisotropic quadratic form over Qp of rank 3 there exists a finiteprime p such that ΛS2(Qp) 6sup Qtimes

p (cf Corollary 2 of Theorem 41 in Chapter 6 [5]) IfΛS2 sup ΛS ΛS2(Q) sup ΛS(Q) therefore ΛS2(Qp) sup ΛS(Qp) is required for any p This isa contradiction

43

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

References

[1] M Bhargava and J Hanke Universal quadratic forms and the 290 theorem InventMath 2005

[2] L Bieberbach Uber die bewegungsgruppen der euklidischen raume MathematischeAnnalen 70(3)297ndash336 1911

[3] L Bieberbach Uber die bewegungsgruppen der euklidischen raume (zweite ab-handlung) die gruppen mit einem endlichen fundamentalbereich MathematischeAnnalen 72(3)400ndash412 1912

[4] A Boultif and D Louer Powder pattern indexing with the dichotomy method JAppl Cryst 37724ndash731 2004

[5] J W S Cassels Rational Quadratic Forms Academic Press LondonNew York1978

[6] J H Conway The sensual (quadratic) form Carus Mathematical Monographs 26Mathematical Association of America 1997

[7] J H Conway and N J A Sloane Low-dimensional lattices vi Voronoi reductionof three-dimensional lattices Proc Royal Soc London Series A 43655ndash68 1992

[8] J H Conway and N J A Sloane Sphere Packings Lattices and Groups (3rd ed)volume 290 of Grundlehren der mathematischen Wissenschaften Springer 1998

[9] P M de Wolff On the determination of unit-cell dimensions from powder diffractionpatterns Acta Cryst 10590ndash595 1957

[10] P M de Wolff Detection of simultaneous zone relations among powder diffractionlines Acta Cryst 11664ndash665 1958

[11] P M de Wolff A simplified criterion for the reliability of a powder pattern indexingJ Appl Cryst 1108ndash113 1968

[12] P Engel T Matsumoto G Steinmann and H Wondratschek The non-characteristic orbits of the space groups Z Kristallogr Supplement Issue No1 1984

[13] P M Gruber and S S Ryshkov Facet-to-facet implies face-to-face EuropeanJournal of Combinatorics 1083ndash84 1989

[14] T Hahn International tables for Crystallography volume A DordrechtKluwer1983

[15] T Ito A general powder x-ray photography Nature 164755ndash756 1949

[16] M Kac Can one hear the shape of a drum American Mathematical Monthly73(4)1ndash23 1966

[17] J C Lagarias H W Lenstra JR and C P Schnorr Korkin-zolotarev basesand successive minima of a lattice and its reciprocal lattice COMBINATORICA10(4)333ndash348 1990

[18] A Le Bail Monte carlo indexing with McMaille Powder Diffraction 19249ndash2542004

44

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths

[19] Y S Moon Universal quadratic forms and the 15-theorem and 290-theorem Thesis(Stanford University) 2008

[20] M A Neumann X-cell a novel indexing algorithm for routine tasks and difficultcases J Appl Cryst 36(4)356ndash365 2003

[21] P Niggli Kristallographische und strukturtheoretische Grundbegriffe Handbuch derExperimentalphysik volume 7 Leipzig Akademische Verlagsgesellschaft 1928

[22] R Oishi-Tomiyasu Rapid bravais-lattice determination algorithm for lattice pa-rameters containing large observation errors Acta Cryst A 68525ndash535 2012

[23] S S Ryshkov and E P Baranovskii C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)Proceedings of the Steklov Institute of Mathematics 137 1976

[24] A Schiemann Ein beispiel positiv definiter quadratischer formen der dimension 4mit gleichen darstellungszahlen Archiv der Mathematik 54372ndash375 1990

[25] A Schiemann Ternary positive definite quadratic forms are determined by theirtheta series Mathematische Annalen 308507ndash517 1997

[26] E Selling Uber die binaren und ternaren quadratischen formen J Reine AngewMath 77143ndash229 1874

[27] D H Templeton Systematic absences corresponding to false symmetry ActaCryst 9199ndash200 1956

[28] F Vallentin Sphere coverings lattices and tilings (in low dimensions) PhD thesisTechnical University Munich Germany 2003

[29] J W Visser A fully automatic program for finding the unit cell from powder dataJ Appl Crystallogr 289ndash95 1969

[30] G F Voronoi Sur quelques proprietes des formes quadratiques positives parfaitesJ Reine Angew Math 13397ndash178 1907

[31] G F Voronoi Nouvelles applications des parametres continus a la theorie desformes quadratiques J Reine Angew Math 134198ndash287 1908

[32] G L Watson Determination of binary quadratic form by its values at integerpoints MATHEMATIKA 2672ndash75 1979

[33] G L Watson Determination of binary quadratic form by its values at integerpoints Acknowledgement MATHEMATIKA 27188 1980

[34] P-E Werner L Eriksson and M Westdahl Treor a semi-exhaustive trial-and-error powder indexing program for all symmetries J Appl Cryst 18367ndash3701985

45

• 1 Introduction
• 2 Outline of the powder auto-indexing problem
• 3 Formulation of the powder auto-indexing problem
• 4 Summary of crystallographic groups and systematic absences
• 5 C-type domains and Conways topographs
• • 51 Topographs for lattices of general rank
  • 52 Topographs for low-dimensional lattices
  • • 6 Main results for the distribution rules of systematic absences
    • • 61 Cases of rank 2
      • 62 Cases of rank 3
      • • 7 Algorithm for powder auto-indexing
        • • 71 Algorithm for N = 2
          • 72 Algorithm for N = 3
          • 73 Speed-up method using topographs
          • 74 Problems with the quality of powder diffraction patterns
          • • 8 Implementation and results of Conograph
            • • 81 Input parameters of Conograph
              • 82 Results
              • • 9 Conclusion
                • A On equivalence between powder diffraction patterns and average theta series
                • B Theorems on the cardinality of solutions
                • C Lattice determination from a complete set of lattice vector lengths