crystal indexing method using a simulated annealing algorithm with particular applications in...

} }< <

Crystal Indexing Method Using aSimulated Annealing Algorithm withParticular Applications inNanocrystal Research

K. F. C. YIUU

Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK

K. Y. TAMPhysical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX13QZ, UK

S. C. TSANG†

The Catalysis Research Centre, Department of Chemistry, University of Reading, Whiteknights,Reading RG6 6AD, UK

Received 27 November 1995; accepted 1 April 1996

ABSTRACT

The development of a crystal indexing computer program using interplanarangles and lattice spacings is very useful, particularly in nanocrystal research bytransmission electron microscopy. However, the indexing involves a largenumber of possible variables, which prohibits the use of simple mathematicaltechniques. This article is concerned with an application of a combinatorialoptimization technique using the simulated annealing algorithm for solving thecrystal indexing problem where traditional descent optimization cannot be used.We show that the program can unambiguously identify the Miller indices usinga set of interplanar angles even for crystals with low symmetry elements.Q 1997 by John Wiley & Sons, Inc.

* Present address: Department of Engineering Science, Ox-ford University, Parks Road, Oxford OX1 3PJ, UK.

† To whom all correspondence should be addressed.

( )Journal of Computational Chemistry, Vol. 18, No. 2, 290]299 1997Q 1997 by John Wiley & Sons CCC 0192-8651 / 97 / 020290-11

CRYSTAL INDEXING METHOD

Introduction

he science of nanoscale clusters, crystallites,T and particles has attracted much attention inrecent years.1 Because the physical properties ofthe materials with diameters in the range of 1]50nm would neither correspond to those of freeatoms or molecules that make up the particles, norto the bulk solids with identical composition, theyhave been shown to give interesting and uniqueelectronic, optical, and magnetic properties. Al-though a variety of techniques have been devel-oped to prepare nanosized materials, there areonly a few techniques available for characteriza-tion. X-ray diffraction is by far the most popularmethod for the determination of crystal structuresand lattice parameters; however, the diffractionlines of nanosized particles are so broad that theyeffectively ‘‘disappear’’ into the background radia-tion.2

At present, transmission electron microscopyŽ . Ž .TEM combined with electron diffraction ED inthe modern electron microscope provides the mostpowerful technique for the characterization ofnanosized materials. Because of the relatively shortwavelengths associated with high energy electronbeams, the image patterns can be used to measurethe interplanar angles and the lattice spacings for

Ž .identifying the Miller indices MIs of differentcrystal phases. This gives useful information aboutthe phase orientation relationships, crystal twin-ning, crystal faulting, and the growth characteris-tics of small nano-sized materials.

However, nanocrystal indexing by TEM is not asimple task. Usually, indexing of selected phases isaccomplished through a comparison of the mea-sured angles and lattice spacings with a series ofcalculated data. Published tables of MIs with d-spacings and angles between crystallographicplanes are available for high crystal symmetry

Ž .systems such as cubic and hexagonal crystals .Docherty et al.3 were among the first to use acomputer program, MORANG, to calculate thepossible planes from the given interplanar angleswith estimated measurement errors. Although thishas greatly reduced the number of possible candi-dates for the matching process, the matching isstill very tedious if more than two interplanarangles are involved.

In fact, the problem of identifying the set of MIsof a crystal from the measured interplanar angles

can be formulated as an integer programmingproblem, with the cost function defined as themismatch between a function of the measuredangles and a function of the calculated angles. Theglobal minimum of the cost function should corre-spond to the correct set of MIs. However, we alsonote that when the number of planes grows, theset of all possible MIs becomes factorially large,which is almost impossible to explore exhaus-tively. Also, traditional descent optimization tech-niques cannot be applied as the cost function andthe solution spaces are not continuous functions.These types of problems are known as NP-com-plete combinatorial optimization problem,4 whichcannot be readily solved.

Recently, Tam and Compton5 introduced acombinatorial optimization technique based on agenetic algorithm for searching the permissibleMIs by best fitting experimental and calculatedinterplanar angles. They showed that the geneticalgorithm can be successfully applied to index alow symmetry triclinic system. However, they re-stricted their permissible MI domain between y1and 1. As the permissible MI space increases expo-nentially with the number of higher indices al-lowed, the inclusions of higher index planesbecome inevitably tedious in their chromosomalrepresentations. In addition, extensive computingtime was required using their investigation tech-niques.

In this article, we report a simple but effectivecombinatorial optimization algorithm, namely, thesimulated annealing algorithm for the indexingcrystal structures. This algorithm is based on theconcept of the physical annealing process. A briefhistorical description of the technique is also given.We demonstrate that this algorithm can be suc-cessfully applied in the search of MIs for differentcrystal systems. The algorithm can easily takehigher index planes into account and is thereforeregarded as a more suitable and direct algorithmin solving the indexing problem. Indexing MIs ofboth hypothetical and experimental nanocrystalsare also demonstrated in this article. We envisagethat this optimization algorithm could be useful,particularly in nanocrystal research.

Interplanar Angles

It is well known that the interplanar angles arerelated to the Miller indices via some geometricformulas. For a different crystal system, a different

JOURNAL OF COMPUTATIONAL CHEMISTRY 291

YIU, TAM, AND TSANG

set of equations can be derived. These equationshave been fully recorded in International Tables forX-ray Crystallography.6 Lattice constants a, b, c, a,b , and g and the reciprocal lattice constants, aU ,bU , cU , aU , b U , and g U are defined as usual.6 The

Ž .Miller indices of a plane are denoted by h, k, l .The formulas for the triclinic, hexagonal, and mon-oclinic systems are summarized next.

TRICLINIC SYSTEM

The interplanar angle, f, between two planesŽ . Ž X X X.hkl and h k l can be calculated as:

w X U 2 X U 2 X U 2cos f s hh a q kk b q ll c

Ž U X . U U Uq kl q lk b c cos a

Ž X X . U U Uq lh q hl c a cos b

X X U U UŽ . x X X Xq hk q kh a b cos g r Q Q' hk l h k l

Ž .1

where:

Q s h2aU 2 q k 2 bU 2 q l 2cU 2 q 2klbUcUcos aUhk l

U U U U U U Ž .q 2 lhc a cos b q 2hka b cos g 2

HEXAGONAL SYSTEM

The interplanar angle, f, between two planesŽ . Ž X X X.hk.l and h k. l is given as:

X X 1 X X U 2 X U 2w Ž .xhh q kk q hk q kh a q ll c2cos f s

X X X XQ Q' hk l h k . l

Ž .3

where:

Ž 2 2 . U 2 2 U 2 Ž .Q s h q k q hk a q l c 4hk .l

MONOCLINIC SYSTEM

The interplanar angle, f, between two planesŽ . Ž X X X.hkl and h k l can be computed through theformula:

cos f

X U 2 X U 2 X U 2 Ž X X . U U Uhh a qkk b qll c q lh qhl c a cosbs

X X XQ Q' hk l h k l

Ž .5

where

Q s h2aU 2 q k 2 bU 2 q l 2cU 2 q 2 lhcUaUcos b Uhk l

Ž .6

Cost Function

In formulating an optimization problem, one ofthe key steps is to identify a suitable cost functionso that, when it is minimized, its solution is theexpected result. Here, it can be seen from theaforementioned relationships that the interplanarangles for a particular crystal system are a function

Ž . Ž X X X .of hkl and h k l only, provided that the unitcell dimensions and the lattice parameters aregiven. Also, notice that for an angle between 08and 1808, the cosine function is a single-valuedfunction. Therefore, the cost function is defined as:

nd2ob sŽ . Ž . Ž .f I s cos f y cos f 7Ý i i

is1

where nd is the number of interplanar angles mea-sured, I is the set of MIs, and the superscript obsdenotes the measured angles.

Simulated Annealing Algorithm

The term ‘‘annealing’’ refers to the process inwhich a solid material is first melted and thenallowed to cool by slowly reducing the tempera-ture. Particles within the system attain a groundstate arrangement which can be considered as aglobal minimum of its potential energy function ateach temperature, T. However, if the system iscooled quickly, it will not reach the ground statebut rather ends up in a polycrystalline or amor-phous state having somewhat higher energy. Thesolidified material may be reheated and cooledslowly with the hope that it will then migrate to alower energy state. In nature, the energy states of asystem follow the so-called Boltzmann probabilitydistribution which expresses the idea that a systemin thermal equilibrium at temperature T has itsenergy probabilistically distributed among all dif-ferent energy states. Metropolis et al.7 were thefirst to incorporate this principle into numericalcalculations by employing the Monte Carlo methodto simulate the evolution to thermal equilibrium ofa solid for a fixed temperature, T. This idea hasbeen extended further to form the basis of thesimulated annealing algorithm, which was first

VOL. 18, NO. 2292


proposed by Kirkpatrick et al.,8 and independentlyby Cerny.9 Specifically, the simulated annealingalgorithm is a stochastic computational techniquethat can be applied for finding global or nearglobal minimum solutions to large optimizationproblems.

Ž . Ž .Here, the cost function, f I , defined by eq. 7 ,can be regarded analogously as the energy state ofa system that follows the Boltzmann probabilitydistribution. A biased random walk in the MI

space is implemented to search for the minimumcost function. During the search, some of the MIsare randomly perturbed and acceptance dependson the Boltzmann probabilistic distribution givenbelow. The perturbation process is repeated untilthe system reaches its allowed final temperature. IfŽ .f I attains a reasonably small value, the assign-

ment may represent the global solution. The struc-ture of the algorithm is given as follows:

InitializationGenerate an initial set of Miller indices, I.Select an initial , « , N , and N. Input N and other parameters.c p

CoolingFor i s 1, . . . , Nc

Iterate s 0.Searching

X. � 4 � 41 Index s random 0, 1 , p s random 1, . . . , N , I s I.p.2 If Index s 1,

X X X� 4 � 4 � 4H s random y4, 4 , k s random y4, 4 , l s random y4, 4 .p p p3 If Index s 0,

X X X � 4randomly accept either h , k or l s random y4, 4 .p p p. Ž .4 If 000 , goto 2.

X. Ž . Ž .5 Calculate d s f I y f I .XIf d - 0, then I s I ;

Xw x Ž .else if random 0, l - T exp ydrT , then I s I ..6 Iterate s Iterate q 1; goto 1 until Iterate ) N.

T s «T.

In the algorithm, the variables T , « , and N repre-csent the current temperature, cooling speed, andnumber of cooling steps, respectively. N indicatespthe number of planes to be indexed and N is thenumber of random perturbations in I per eachcooling step. All uniformly distributed randomnumbers are generated using an established rou-

10 � 4tine. The expression a, b is used to denote aninteger number ranging from a to b, whereas theexpression with the square bracket indicates a real,continuous value.

For each cooling step, N searching steps areexecuted. In each searching step, the MIs of aplane are randomly selected for perturbation. Themodification, which is implemented in a random

Ž .fashion, can be either all indices h, k, and l orŽ .just one being replaced by random number s . If

the perturbed solution IX contains any illegal indexŽ .000 , the searching step is reiterated. Subse-quently, the cost function of IX is evaluated. If thenew function value is less than the previous one,

Ž X.the new move is accepted I s I . Otherwise, theacceptance of the perturbed solution is dependenton a Boltzmann probabilistic distribution chosenby numerical experiences as:

Ž . Ž X .f I y f IŽ .P s T exp 8ž /T

If a random number generated is less than P, thenew move is accepted. Otherwise, control will passto the next search step. The temperature of thesystem is updated after N search steps. The cool-ing process is repeated until the temperature of thesystem reaches the final value. Then, the solution,I, is reported as the assigned MIs.

The program begins with a relatively high tem-perature T to avoid premature entrapment at localminima. As T starts to decrease, the chance ofrejecting an upward move in the cost function willincrease. When the temperature become very low,the efforts to reduce the cost function further will


YIU, TAM, AND TSANG

become very discouraging, which signals to termi-nate the annealing procedure. As the number ofplanes increases, a larger number of searchingsteps, N, may be helpful to converge to the globalsolution. It should be pointed out that there is norigorous rule governing the selection of the initialtemperature, the cooling speed, and the number ofsearching steps, N. These annealing parametersthemselves have to be optimized by experiencewith regard to convergence speed.

Results

In all the test cases reported in what follows, theinitial temperature, T , and the cooling speed werechosen by numerical experience as 0.5 and 0.9,respectively. The maximum cooling step, N , wasctaken to be 30. If the maximum cooling step ex-ceeded without reaching the global minimum, theprocedure was restarted with a higher number ofiterations N per each cooling step. The computertime was estimated using a Sun Sparc workstation.

HYPOTHETICAL HEXAGONAL ANDTRICLINIC CRYSTALS

The algorithm was tested using some hypotheti-cal crystals of hexagonal and triclinic systems. Inthese cases, we deliberately chose a wide range ofparameters and interplanar angles to verify thealgorithm. The lattice parameters used to generatethe corresponding interplanar angles are given inTable I. For each crystal system, the interplanar

Ž .angles up to two decimal places were deliber-ately chosen as shown in Table II. We found thatthe algorithm could produce the global minimavery quickly if the exact angles were used. Forpractical purposes, we believe the angles will bemeasured with a certain degree of error. Therefore,random noise of magnitude up to "0.58 is added

TABLE I.Crystallographic Data Used for theHypothetical Crystals.

System Triclinic Hexagonal

˚( )a A 14.16 14.10˚( )b A 21.32 13.07˚( )c A 13.07 21.90

( )a 8 99.92 90.000( )b 8 92.68 90.000( )g 8 106.15 120.000

TABLE II.Interplanar Angles of the Hypothetical Crystals.

Hexagonal Triclinica obs obs( ) ( )Planes f f

1 n 2 24.46 86.551 n 3 34.36 58.851 n 4 109.50 55.691 n 5 74.09 88.551 n 6 128.53 136.142 n 3 40.07 48.872 n 4 128.77 73.862 n 5 98.29 3.952 n 6 120.89 50.723 n 4 90.12 25.323 n 5 83.52 47.613 n 6 94.32 82.174 n 5 55.16 72.244 n 6 58.86 97.315 n 6 113.96 48.19

( ) ( ) ( ) ( )Expected 1: 010 ; 2: 231 ; 1: 010 ; 2: 312 ;b ( ) ( ) ( ) ( )indices 3: 134 ; 4: 102 ; 3: 134 ;, 4: 123 ;

( ) ( ) ( ) ( )5: 431 ; 6: 123 5: 413 ; 6: 121y4 y 4Expected 1.6330 = 10 2.1441 = 10

( )f I values

a The ‘‘n’’ symbols denote the interception between twocrystal planes.b The numbers are arbitrarily assigned to the expected in-dices for identification purposes.

to the interplanar angles to imitate the error inmeasurements. We found that the computer-pro-gram-generated MIs agreed perfectly with the the-oretical calculations showing the robustness of themethod. Typical convergence histories of the algo-rithms were obtained as shown in Figures 1 and 2.Both figures show that a large fluctuation in costfunction is observed when T was high. As Tdecreased, more steps were rejected and thus thefluctuation decreased significantly. Also, it can beseen that the cost function decreased gradually asthe cooling step increased. These are typical behav-iors of a simulated annealing algorithm. Here, inboth cases, N was chosen to be 40,000. The cputime for the hexagonal crystal was about 4 min-utes, whereas for the triclinic crystal it took about13 minutes to finish the plane indexing.

PRACTICAL HEXAGONAL, TRICLINIC, ANDMONOCLINIC SYSTEMS

The algorithm was tested using some realisticcrystals of hexagonal, triclinic, and monoclinic sys-tems as given in the study by Bishop.11 The latticeparameters are given in Table III. For each crystal

VOL. 18, NO. 2294


FIGURE 1. The convergence history of the algorithmfor the hypothetical triclinic crystal.

Žsystem, the interplanar angles up to two decimal.places are shown in Table IV; also, the shapes and

the MIs for the chosen hexagonal, tricliniccliniccrystals are shown in Figures 3, 4, and 5, respec-tively. In all cases, N was chosen to be 200,000.The cpu time for the hexagonal crystal was about36 minutes and the procedure converged in the19th cooling step. For the triclinic crystal, the cputime taken was about 39 minutes, which con-

FIGURE 2. The convergence history of the algorithmfor the hypothetical hexagonal crystal.

TABLE III.Crystallographic Data Used for the Crystals.

Triclinic Hexagonal Monoclinica b c( ) ( ) ( )System Rhodonite Vanadinite Scolecite

( )Space group P1 P6 3 / m P2˚( )a A 7.699 10.331 9.850˚( )b A 12.220 10.331 18.980˚( )c A 6.702 7.343 6.520

( )a 8 93.975 90.000 90.000( )b 8 93.067 90.000 110.100( )g 8 68.200 120.000 90.000

a Mn Si O , PDF 13-138.2 2 6b ( )Pb VO Cl, PDF 13-585.5 4 3c CaAl Si O ? 3H O, PDF 26-1048.2 3 10 2

( )FIGURE 3. The triclinic crystal Rhodonite .

( )FIGURE 4. The hexagonal crystal Vanadinite .


YIU, TAM, AND TSANG

TABLE IV.Interplanar Angles of the Crystals.

Triclinic Hexagonal Monclinica obs obs obs( ) ( ) ( )Planes Rhodonite f Vanadinite f Scolecite f

1 n 2 90.15 65.27 45.061 n 3 88.28 65.27 43.041 n 4 86.22 90.00 45.061 n 5 111.52 90.00 28.931 n 6 116.05 114.73 28.931 n 7 } 114.73 90.001 n 8 } } 90.761 n 9 } } 90.762 n 3 25.43 54.02 14.892 n 4 62.81 30.88 29.782 n 5 21.37 46.64 42.062 n 6 65.62 49.45 60.402 n 7 } 76.26 104.892 n 8 } } 127.422 n 9 } } 134.903 n 4 37.38 80.12 14.893 n 5 33.98 30.88 50.233 n 6 45.53 76.26 50.233 n 7 } 49.45 90.003 n 8 } } 132.813 n 9 } } 132.814 n 5 66.35 60.00 60.404 n 6 29.83 30.88 42.064 n 7 } 80.12 75.114 n 8 } } 134.904 n 9 } } 127.415 n 6 57.04 46.64 57.855 n 7 } 30.88 118.935 n 8 } } 85.365 n 9 } } 95.976 n 7 } 54.02 61.076 n 8 } } 95.976 n 9 } } 85.367 n 8 } } 101.027 n 9 } } 78.988 n 9 } } 22.04

( ) ( ) ( ) ( ) ( ) ( )Expected 1: 001 ; 2: 110 ; 1: 0001 2: 2131 ; 1: 100 ; 2: 111 ;b ( ) ( ) ( ) ( ) ( ) ( )indices 3: 100 ; 4: 110 ; 3: 1321 4: 1010 ; 3: 101 ; 4: 111 ;

( ) ( ) ( ) ( ) ( ) ( )5: 221 ; 6: 221 5: 0110 ; 6: 2131 ; 5: 110 ; 6: 110 ;( ) ( ) ( )7: 1321 7: 010 ; 8: 112 ;

( )9: 112y4 y 4 y 4Expected 1.8608 = 10 2.7696 = 10 4.1902 = 10

( )f I values

a The ‘‘n’’ symbols denote the interception between two crystal planes.b The numbers are assigned to the expected indices arbitrarily for identification purposes.

VOL. 18, NO. 2296


( )FIGURE 5. The monoclinic crystal Scolecite .

verged in the 18th cooling step. Finally, for themonoclinic crystal, the cpu time used was about 83minutes and it converged in the 14th step. We notethat, for these three cases, the highest index neverreached 4 or y4; therefore, a lot of unnecessarycpu time was used in searching the high indexingplanes. Thus, we modified the searched indicesbetween y2 and 2 in the monoclinic system andfound that the cpu time reduced to 17 minuteswith the convergency in the third step. For thehexagonal and triclinic crystals, the searched in-dexes fell to between y3 and 3, and the cpu timeswere found to be 14 and 6 minutes with theconvergency in the 18th and 10th steps, respec-tively. Note that, in these two cases, N was chosento be 80,000 for the hexagonal crystal and 50,000for the triclinic crystal.

NANOCRYSTAL

In the final example, we demonstrate the use ofthe simulated annealing algorithm for the indexingof a nanocrystal. Figure 6 shows a high resolutionTEM micrograph of a nanosized lathanium-con-taining crystal inside a carbon nanotube. This wasmade by filling the opened nanotubes with latha-nium nitrate solution and then heated to 5008Cunder a flowing stream of oxygen. From our previ-ous study we found that by filling the openednanotubes with metal nitrate solution, followed bysubsequent heat treatment in air, nanocrystals ofmetal oxides can be made. Generally, the thermo-dynamically most stable phase of metal oxide is

made under these conditions.12 The nanocavity ofthe tubes provided templates for the deposition ofmetal nitrate which was thought to decompose tothe corresponding lanthanium oxide at the ele-vated temperature. The crystal was too small tocause detectable X-ray diffraction. Interplanar dis-

˚ ˚ ˚tances of 3.60 A, 2.75 A, and 2.75 A with theinterplanar angles of 1 n 2 s 658, 1 n 3 s 508, and2 n 3 s 658 associated with the crystal were mea-sured from the projected images in the TEM mi-crograph. We employed the stable monoclinic lan-thanum oxide phase in the program where thelattice parameters were obtained from the stan-

Ždard PDF files the unit cell parameters of a s˚ ˚ ˚14.60 A, b s 3.717 A, c s 9.278 A, a s 908, b s

.99.858, g s 908 . The interplanar distances and theangles obtained from the calculation agreed verywell with the experimental results as the measure-ments were fed into the program. Thus, Miller’sindex plane can therefore be fixed from the corre-sponding lattice distancesrangles. On the otherhand, the results did not agree with the experi-mental measurements when the parameters of thecubic or other phases were used. The first Miller

Ž .index plane was fixed to be 202 and was verifiedby the corresponding lattice distance. Upon in-putting the measured angles into the program, the

Ž .Miller indices of the second plane of 112 and theŽ .third plane of 303 were found. These matched

with the d-spacings mentioned previously by onlytaking 2.5 seconds of cpu time and the algorithmwas converged in the first cooling step. We havefound that the same indexing planes were gener-ated, even when feeding the error deviations ofinterplanar angle and spacing of "0.58 and "0.05A, respectively, into the program.

Conclusions

A crystal indexing program using the stimu-lated annealing algorithm has been developed. Us-ing experimentally measured interplanar angles,the program is able to index crystal planes andspends reasonably little computer time withoutinvolving the tedious manual matching work. Theprogram is also particularly useful for applicationin the area of nanocrystal research using the TEMtechnique in which the interplanar distances andangles can be obtained by electron crystallography.

The programs used herein and example datafilescan be obtained from K.F.C.Y. upon request.


YIU, TAM, AND TSANG

(FIGURE 6. A high-resolution TEM micrograph of a lathanium oxide nanocrystal inside a carbon nanotube the scale˚)bar represents 50 A .

VOL. 18, NO. 2298


Acknowledgments

ŽWe thank Dr. Thomas Y. C. Leung Chinese.University of Hong Kong for the stimulating dis-

cussion. S. C. T. kindly acknowledges receipt ofThe Royal Society Research Fellowship.

References

1. R. P. Anders, R. S. Averback, and W. L. Brown, J. Mater.Ž .Res., 4, 704 1989 .

2. A. K. Cheetham and P. Day, Solid State Chemistry Tech-niques, Oxford Science Publications, Oxford, UK, 1987.

3. R. Docherty, K. J., Roberts, and E. Dowty, Comput. Phys.Ž .Commun., 51, 423 1988 .

4. C. H. Papadimitriou and K. Steiglitz, Combinatorial Opti-mization: Algorithms and Complexity, Prentice-Hall, Engle-wood Cliffs, NJ, 1982.

5. K. Y. Tam and R. G. Compton, J. Appl. Cryst., 28, 640Ž .1995 .

6. International Tables for X-ray Crystallography, Vol. 2, KynochPress, Birmingham, UK, 1973.

7. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, andŽ .E. Teller, J. Chem. Phys., 21, 1087 1953 .

8. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science, 220,Ž .671 1983 .

Ž .9. V. Cerny, J. Opt. Theory Appl., 45, 41 1985 .10. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.

Vetterling, Numerical Recipes, Cambridge University Press,Cambridge, UK, 1986.

11. A. C. Bishop, An Outline of Crystal Morphology, HutchinsonPress, London, 1967.

12. S. C. Tsang, Y. K. Chen, P. J. F. Harris, and M. L. H. Green,Ž .Nature, 372, 159 1994 .


crystal indexing method using a simulated annealing algorithm with particular applications in...

Documents