school's book (15mb, convenient version)

http://www.cryst.ehu.es/crystr2015

Second Balkan School on Fundamental Crystallography and Workshop on Magnetic Symmetry

13-19 July 2015, Energy Institute, Istanbul Technical University Campus,Istanbul / Turkey

CONTENTS

Welcome...............................................................................................................................................3Program................................................................................................................................................7Abstracts.............................................................................................................................................11A Short Practical Course to ITA & BCS with Exercises....................................................................19Tutorial: Use of BCS Tools on the Study of Group-Subgroup Phase Transitions .............................47Tutorial: Use of the BCS program PSEUDO ....................................................................................73Tutorial: Use of the BCS program AMPLIMODES ........................................................................107Tutorial: Use of the BCS program MAXMAGN and Others for Magnetic Structures ...................133List of Participants............................................................................................................................169

WELCOME

The story begins two years ago, in September 2013. We(Sibel, Damla, Cem, Mustafa, Emin & me) wereheaded for Sofia to attended to the 'international schoolon fundamental crystallography' organized by theBulgarian Crystallographic Society1.

Thanks to the excellent hospitality of our organizers,we had a wonderful time there and on our return toTurkey, we decided to “return the favor” by continuingthe tradition. And after many challenges, finally herewe are, welcoming you!

These schools are curious events: unlike conferences or symposiums, it brings out and utilizes theenergy of the young people attending to it. Friendships are made and these friendships furtherevolve into collaborations. What is better than to work with someone you know from sharing thesame classroom once? For me, the crystallography schools are even more personal: I've met Mois,Manu and Massimo in the first crystallography school I had attended all the way back in 2009 2;visited Massimo at his hometown Nancy in another summer school on MathematicalCrystallography in 20103 and met Rosica & Bobby in Sofia, 20131.

This time, we have learned first hand that organization is a very difficult process but thanks to allour contributors and sponsors, we hope that it will be a nice experience with many good memories.I thank to all of our contributors, sponsors and organizers, especially to Prof. Suheyla Ozbey & Dr. Tolga Birkandan without whom this school wouldn't happen in the first place. Many thanks alsogo to ITU Energy Institute & ITAP for making us feel at home.

I wish you all a very good time, filled with friendship, joy and knowledge!

Dr. Emre S. Tasci

1 International school on fundamental crystallography, 30/09 – 05/10 2013, Gulechitza – Bulgaria http://www.bgcryst.com/school2013/ 2 Crystallography Online: International school on the Use and Applications of the Bilbao Crystallographic Server, 21-27/06 2009, Lekeitio – Spain http://www.cryst.ehu.es/html/lekeitio.html 3 Summer School on Mathematical Crystallography, 21/06 - 02/07 2010, Nancy – France http://www.crystallography.fr/mathcryst/nancy2010.php

Emre, Damla, Sibel and Cem after one of theorganization meetings, 2015

Emre, Damla, Mustafa and Sibel, "just tired" in Sofia, 2013

http://www.bgcryst.com/school2013/

http://www.crystallography.fr/mathcryst/nancy2010.php

http://www.cryst.ehu.es/html/lekeitio.html

ORGANIZERS & LECTURERS

Program CommitteeProf. Suheyla Ozbey, Hacettepe University, Ankara, TurkeyProf. Massimo Nespolo, Université de Lorraine, Nancy, FranceProf. Mois I. Aroyo, University of the Basque Country, Bilbao, SpainAssist. Prof. Emre S. Tasci, Hacettepe University, Ankara, Turkey

LecturersProf. Mois I. Aroyo, University of the Basque Country, Bilbao, SpainProf. J. Manuel Perez-Mato, University of the Basque Country, Bilbao, SpainAssoc. Prof. Rosica P. Nicolova, Bulgarian Academy of Sciences, Sofia, BulgariaAssoc. Prof. Boris Lubomirov Chivatchev, Bulgarian Academy of Sciences, Sofia, BulgariaAssist. Prof. Emre S. Tasci, Hacettepe University, Ankara, Turkey

Local OrganizersAssist. Prof. Tolga Birkandan, Istanbul Technical University, Istanbul, TurkeyDr. Ceren Sibel Sayin, Bilkent University, Ankara, TurkeyDamla Kocak, M.S.Cem Lider, M.S., Middle East Technical University, Ankara, TurkeyMustafa Tek, M.S., Middle East Technical University, Ankara, Turkey

Mois, Massimo, Emre and Rosica in Sofia, 2013

PAST SCHOOLS

Crystallography Online: International school on the Use and Applications of the Bilbao Crystallographic Server, 21-27/06 2009, Lekeitio – Spain

International school on fundamental crystallography, 30/09 – 05/10 2013, Gulechitza – Bulgaria

ORGANIZING INSTITUTION

SPONSORS & CONTRIBUTORS

Istanbul Technical University | International Union of Crystallography European Crystallographic Association | MathCryst

Institute of Theoretical and Applied Physics | National Crystallography Association of TurkeyBilbao Crystallographic Server | Atomika Teknik

Hacettepe University

Program

Morning session (4 hours): Lectures: 8:30 – 10:30 / 11:00 – 13:00 Coffee Break: 10:30 – 11:00

Lunch: 12:30 – 14:30

Afternoon session (4 hours): Lectures: 14:30 – 16:30 / 17:00 – 19:00 Coffee Break: 16:30 – 17:00

Dinner: 19:00 – 20:30 Evening session (optional): 20:30 – 21:15

1st Day – 13/07 MondayMorning Session (Pre-school session) [MIA]

» 10:00 – 13:00 «Matrix calculus applied to crystallography.

Isometries and crystallographic symmetry operations. Crystallographic point and space-group symmetry operations. Matrix-column presentation of symmetry operations. Symmetryelements: geometric elements and element sets.

14:10 – 14:30 «Official opening of the school

Afternoon Session [RN + BS]

» 14:30 – 19:00 «Introduction to group theory. Crystal lattice vs. crystal pattern and crystal structure. Symmetrydirections in a lattice .Unit cells: primitive cells, multiple cells, conventional cells in 2D and 3D.Crystal families. Symmetry groups and types of symmetry in direct space: morphologicalsymmetry; symmetry of physical properties; symmetry of lattices; symmetry of the unit cellcontent; symmetry of crystallographic patterns.

2nd Day – 14/07 TuesdayMorning Session [MIA]

» 8:30 – 13:00 «Crystallographic point groups. Stereographic projection and the morphology of crystals. Hermann-Mauguin symbols for point groups.

Basic properties of groups (group axioms, order, multiplication tables, etc.). Subgroups,index, coset decompositions, Lagrange theorem. Concept of isomorphism and homomorphism.Group actions: conjugation and normalizers.

Poster contributions-discussion during the coffee break

Afternoon Session [MIA]

» 14:30 – 19:00 «Group-subgroup relations (general considerations): index, coset decomposition and normalsubgroups; conjugate elements and conjugate subgroups; factor groups and homomorphism.

Wyckoff positions for point group. Relations of Wyckoff positions for a group-subgroup pairof point groups. Supergroups of point groups.

Poster contributions-discussion during the coffee break

Evening Session (optional)

Discussions, questions and comments on the lecture and exercise material of the day

3rd Day – 15/07 WednesdayMorning Session [RN + BS]

» 8:30 – 13:00 «Space groups – general introduction: periodic structure of the crystalline matter. Orthogonalprojections of space groups. Symmetry-elements and general-position space-group diagrams fromVolume A of the International Tables for Crystallography. General and special positions, site-symmetry groups, crystallographic orbits.

Space groups – general introduction I: Exercises on orthogonal projections of space groups. Poster contributions-discussion during the coffee break

Afternoon Session [RN + BS]

» 14:30 – 17:00 «Space groups – general introduction II: Exercises on orthogonal projections of space groups.

» 17:30 – «Excursion

4th Day – 16/07 ThursdayMorning Session [MIA]

» 8:30 – 12:50 «Space groups and their description in International Tables for Crystallography, Vol. A.Crystallographic point and space-group symmetry operations. Geometric interpretation of thematrix-column presentation of the symmetry operations. General and special Wyckoff positions.Sections and projections.

» 12:50 – 13:00 «Sponsor company presentation

Afternoon Session [MIA]

» 14:30 – 15:30 «Computer databases and access tools to crystallographic symmetry data for space groups(International Tables for Crystallography, Volume A: Space-group symmetry

» 15:30 – 18:00 «Transformations of the coordinate systems: change of origin and orientation. Conventional and non-conventional descriptions of space groups; ITA-settings.

» 18:00 – 19:00 «Oral presentation of the poster contributions by the participants



5th Day – 17/07 Friday Morning Session [RN + BS]

» 8:30 – 13:00 «Symmetry in reciprocal space. Diffraction symmetry: Laue classes, Friedel's law, resonantscattering. Integral, zonal and serial reflection conditions and their use in the determination of thespace-group symmetry. Special reflection conditions.

» 11:00 – 13:00 «Structure solution and refinement: introductory strategies.

Afternoon Session

» 14:30 – 16:30 « [MIA]Group-subgroup relations between space groups: Subgroups of space groups: types of subgroups ofspace groups; Hermann theorem; maximal subgroups; series of isomorphic subgroups. Cosetdecomposition. Relations of Wyckoff positions for a group-subgroup pair. Supergroups of spacegroups.

Maximal subgroups data and related computer application in the Bilbao CrystallographicServer.

» 17:00 – 19:00 « [EST + MIA]Crystal-structure descriptions. Descriptions of crystal structures with respect to different ITAsettings of the space groups (the program SETSTRU). Equivalent crystal structure descriptions (theprograms EQUIVSTRU and COMPSTRU). Crystal-structure descriptions compatible withsymmetry reduction (the program TRANSTRU).



6th Day – 18/07 SaturdayMorning Session

» 8:30 – 10:30 « [EST]Crystal-structure relationships. Family trees (Baernighausen trees) of crystal structures: aristotype(basic) and hettotypes (derivative) structures (the program STRUCTURE RELATIONS).

» 11:00 – 13:00 « [JMPM]Structural pseudosymmetry. Pseudosymmetry search for new ferroics. Application in structuralphase transitions (the program PSEUDO).

Afternoon Session [JMPM]

» 14:30 – 19:00 «Group-subgroup relations of space groups and their applications to phase transitions (the programAMPLIMODES).

WORKSHOP ON MAGNETIC SYMMETRY

7th Day – 19/07 SundayMorning Session [JMPM]

» 8:30 – 10:30 «Magnetic symmetry. Magnetic point operations and point groups, connection to magnetic tensorproperties. Magnetic space groups and their presentation in D. Litvin’s tables. OG and BNS settingsof type-IV magnetic space groups

» 11:00 – 13:00 «Representation analysis vs magnetic symmetry description of magnetic structures

Afternoon Session [JMPM]

» 14:30 – 16:30 «Magnetic and its applications in in the Bilbao Crystallographic Server. Magnetic point and magneticspace groups. General and special Wyckoff positions. Magnetic symmetry material tensors.Extinction rules for magnetic space groups . Maximal magnetic space groups for a given apropagation vector and resulting magnetic structural models

» 17:00 – 18:00 «Database of magnetic structures on the Bilbao Crystallographic Server.

» 18:00 – 18:15 «Final remarks

» 20:00 – «Farewell Dinner; end of the school

ABSTRACTS

CRYSTR2015 – Second Balkan School on Fundamental Crystallography / 13-19 July 2015, Turkey

On group-subgroup relations of double Na-Ca carbonates Ca3Na2(CO3)4 and CaNa2(CO3)4

Pavel N. Gavryushkina,b*, Anton F. Shatskiy a,b, Konstantin D. Litasov a,b,

Victor G. Thomas b, Nadezhda B. Bolotinac

a V.S. Sobolev Institute of Geology and Mineralogy, SB RAS, Novosibirsk, Russia b Novosibirsk State University, Novosibirsk, Russia d Shubnikov Institute of Crystallography RAS, Moscow, Russia

Abstract

Double Na-Ca carbonates is relatively rare class of compounds. In addition to the two known compounds nyerereite Ca2Na2(CO3)4 and shortite CaNa2(CO3)4, 3 new compounds CaNa4(CO3)3, Ca3Na2(CO3)4 and Ca4Na2(CO3)5 were synthesized under high pressure. From point of view of symmetry, the noteworthy feature of all Na-Ca carbonates is the absence of the center of symmetry which define this compounds as perspective for non-linear optics.

The main focus of the present work is on the application of group-subgroup analysis to Ca3Na2(CO3)4 crystal structure, which let to make assumption about new phases of this compound. Also, the analysis of modulations of CaNa2(CO3)4 is presented. The length and direction of this modulations is different for different conditions of crystal growth. Thus, investigation of this modulations gives key to the estimation of conditions under which natural crystals of nyerereite grow.

Samples of Ca3Na2(CO3)4 were synthesized under 6 Gpa and 1000 °C in the BARS apparatus, the samples of CaNa2(CO3)4

– by hydrothermal method in the standard steel autoclaves at 450 °C and 1 kbar.

The structures of synthesized single-crystals Ca3Na2(CO3)4 and CaNa2(CO3)4 were solved with charge-flipping algorithm in the P1 space group [1]. The unit cell parameters of the Ca3Na2(CO3)4 is the following P1n1, Z=8, a=31.4421(8) Å, b=8.1960(2)Å, c=7.4360(2) Å, β=89.923(2)°. The complex character of the structure, presence of CO3 group in many different orientations and the low symmetry of the crystal is untypical for carbonates. The

symmetry of the cation array is Pmnb (#62), with halved a-axis. We find two compounds, Ba3Nd2(BO3)4 and

Gd5Si4, which is characterized by the same symmetry and almost the same (in the first case) cell parameters. Oxygen atoms gradually decrease symmetry to Pmn21 (#31) then to P21/n and finally to P1n1 (fig.1). For

determination of full group-subgroup graph, seven additional structures Ca3Na2(CO3)4 I – VII is necessary (fig.1). Probable phase transitions between this new structures were fixed under decompression from 1 Gpa to 1 atm in our in-situ synchrotron experiments [2].

The results on another carbonate CaNa2(CO3)4 will be presented in the conferences.

References: [1] Gavryushkin, P. N, Bakakin, V.V, Bolotina, N.A, Shatskiy, A.F., Seryotkin, Y.,V and Litasov, K.D, Crystal Growth & Design, 14, 4610-4616 (2014)

[2] Shatskiy, A., Gavryushkin, P. N., Litasov, K. D., Koroleva, O. N., Kupriyanov, I. N., Borzdov, Y. M., Sharygin, I. S., Funakoshi, K., Palyanov, Y. N., and Ohtani, E., European Journal of Mineralogy, in press (2015)


Magnetodielectric coupling and potential ferroelectricity in polar antiferromagnetic Er2Cu2O5 and Yb2Cu2O5

Umut Adem1,2, Arramel2, Gwilherm Nenert3 and Thomas T. M. Palstra2

1 Department of Materials Science and Engineering, İzmir Institute of Technology, Urla, 35430 İzmir, Turkey 2Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands 3PANalytical B. V., Lelyweg 1, 7602 EA Almelo, The Netherlands

AbstractMultiferroic materials with simultaneous magnetic and electric ordering received a lot of attention during the past decade due to their potential applications such as multiferroic memories, taking advantage of electrically written magnetic information[1]. While there has been progress towards the realization of applications in the form of thin films[2], fundamental research on understanding the coupling between ferroelectricity and magnetism is still needed. R2Cu2O5 (R is a rare earth ion with smaller ionic radius than Gd, including Sc, In and Y) constitute a family of polar magnetic oxides. These materials crystallize in an orthorhombic space group, Pna21[3]. Magnetic ordering in orthorhombic cuprates is complex, especially in the members with magnetic ions at both A and B sites. Antiferromagnetic ordering temperatures range from 11 to 30 K in the series[4]. We have previously reported on the magnetodielectric coupling in single crystals of Y2Cu2O5 [5]. Here, we report on the magnetodielectric coupling in two other members of polar R2Cu2O5 family: Yb2Cu2O5 and Er2Cu2O5 using magnetization and dielectric measurements. Due to the presence of magnetic rare-earth ions at the A-site, unlike the case in Y2Cu2O5, magnetic exchange interactions become complex, resulting in a number of field-induced magnetic phase transitions. Coupling of dielectric constant to various magnetic phases can yield interesting phenomena. Despite the polar space group they possess, ferroelectric switching could not be demonstrated previously in Y2Cu2O5[4]. Here, we estimate an upper bound magnitude for the spontaneous polarization using the crystal structure information and pseudosymmetry analysis using PSEUDO[6]. We have also analyzed the ferroelectric phase transition using AMPLIMODES[7]. References: [1] M. Bibes and A. Barthlemy, Nat. Mater. 7, 425 (2008). [2] Y. S. Chai, S. Kwon, S. H. Chun, I. Kim, B.-G. Jeon, K. H. Kim, and S. Lee, Nat. Commun. 5, 4208 (2014). [3] J. L. Garcia-Munoz and J. Rodriguez-Carvajal, J. Solid State Chem. 115, 324 (1995). [4] R. Troc, J. Klamut, Z. Bukowski, R. Horyn, and J. Stepien-Damm, Physica B Condens. Matter 154, 189 (1989). [5] U. Adem, G. Nenert, Arramel, N. Mufti, G. R. Blake, and T. T. M. Palstra, EPJ B 71, 393 (2009). [6] C. Capillas, E. Tasci, G. de la Flor, D. Orobengoa, J. Perez-Mato, and M.I. Aroyo, Z. Krist.

226, 186 (2011). [7] D. Orobengoa, C. Capillas, M. I. Aroyo and J. M. Perez-Mato, J. Appl. Crystallogr. 42, 820-833 (2009).


Crystal Structure of Triaqua(4-cyanobenzoato-к2O,O')(nicotinamide-кN1)zinc 4-cyanobenzoate

Gülçin Şefiye Aşkın a, Hacali Necefoğlu b,c, Gamze Yılmaz Nayir b, Raziye Çatak Çelikd and Tuncer Hökeleka*

aDepartment of Physics, Hacettepe University, 06800 Beytepe, Ankara, Turkey, bDepartment

of Chemistry, Kafkas University, 36100 Kars, Turkey, cInternational Scientific Research Centre, Baku State University, 1148 Baku, Azerbaijan, and dScientific and Technological

Application and Research Center, Aksaray University, 68100, Aksaray, Turkey.

Abstract Nicotinamide (NA) is a form of niacin. A deficiency of this vitamin leads to loss of copper from the body, known as pellagra disease.[1] The nicotinic acid derivative N,N-

diethylnicotinamide is an important respiratory stimulant.[2] The asymmetric unit of the crystal structure of the title salt, [Zn(C8H4O2N)(C6H6ON2)(H2O)3](C8H4O2N), is composed of one complex cation and one 4-cyanobenzoate (CNB) counter-anion. The ZnII atom is coordinated by one 4-cyanobenzoate (CNB) anion, one nicotinamide (NA) ligand and three water molecules, the CNB anion and NA ligand coordinating in bidentate and monodentate modes, respectively (Fig. 1). In the cation, the four coordinating atoms (O1, O2, O5 and N2) around the Zn1 atom show a distorted square-planar arrangement, while the considerably distorted octahedral coordination environment of ZnII is completed by two additional water O atoms (O4 and O6) in the axial positions. Intermolecular O-H∙∙∙O hydrogen bonds link two of the coordinating water molecules to two free CNB anions. In the crystal, further hydrogen-bonding interactions are present, namely N-H∙∙∙O, O-H∙∙∙O and C-H∙∙∙O hydrogen bonds that link the molecular components, enclosing R2

2(12), R33(8) and R3

3(9) ring motifs and forming layers parallel to (001).[3] π– π contacts between benzene rings [centroid-to-centroid distances = 3.791(1) and 3.882 (1) Å ] may further stabilize the crystal structure.

References: [1] K. A. V. R. A

Krishnamachari, Am. J. Clin. Nutr. 27, 108–111 (1974).

[2] F. Bigoli, A. Braibanti,

M.A. Pellinghelli, & A. Tiripicchio, Acta Cryst. B28, 962–966 (1972).

[3] J. Bernstein, R.E.

Davis, L. Shimoni, & N.L. Chang, Angew. Chem. Int. Ed. Engl. 34, 1555–1573 (1995).

Figure 1: The molecular entities of the title salt, showing the

atom-numbering scheme. Intermolecular O—H∙∙∙O hydrogen

bonds are shown as dashed lines.


Bridging the gap between crystallographic and “standard” conventional unit cells

Yoyo Hinuma1, Atsushi Togo2, Hiroyuki Hayashi1, and Isao Tanaka1,2,3

1 Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan 2 Elements Strategy Initiative for Structural Materials, Kyoto University, Sakyo, Kyoto 606-8501, Japan

3 Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan

Abstract

Use of “standard” primitive cells as defined by Setyawan and Curtarolo [1] is convenient for band diagram and effective mass calculations because the definitions and coordinates of high symmetry points are provided; however, the basis vectors are taken differently and/or high symmetry point definitions differ from the crystallographic convention in many cases. Therefore, we propose a computer-friendly algorithm to convert crystallographic conventional cells to a “standard” primitive cell by retaking basis vectors.

In our procedure, we determine the conventional cell through a standardized algorithm

to select the choice of basis vectors. We basically follow Parthé et al. [2] but the first choice that appears in Table A1.4.2.7 of the International Tables of Crystallography (ITB) [3] is taken as the standard choice. A list of transformation matrices to convert a crystallographic conventional cell to a “standard” conventional cell is identified. The relevant transformation matrix depends on the Bravais lattice and, when applicable, on criteria that depend on lattice vector lengths and interaxial angles. We find that about 20 matrices are needed to cover all space group types.

The high symmetry point symbols by Setyawan and Curtarolo [1] differ in many cases from established databases, for instance, from those in the Bilbao Crystallographic Server [4]. This is particularly problematic in orthorhombic cells. Crystallographic convention determines basis vectors based on symmetry operations whereas the basis vectors of the “standard” cell is defined solely on the order of basis vector lengths. Therefore, different conversion matrices can exist for the same space group type. This is very awkward, for instance, when compiling a band diagram that shows the valence band maximum (VBM) and conduction band minimum (CBM) and when identifying effective masses along high symmetry lines. Here, having one set of high symmetry site symbol definitions, including sites with “irrational” coordinates, for each space group would be very convenient. We refrain from defining additional high symmetry point symbols at this point to avoid further confusion regarding this problem. However, we are open to discussion to design a scheme that is crystallographic convention friendly and systematically defines Brillouin zone points with “irrational” coordinates. References: [1] W. Setyawan and S. Curtarolo, Comp. Mater. Sci. 49, 299 (2010). [2] E. Parthé, K. Cenzual, and R. E. Gladyshevskii, J. Alloys Compd. 197, 291 (1993). [3] International Union of Crystallography, International Tables of Crystography B 3rd edition (2008). [4] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Acta Crystallographica Section A 62, 115 (2006).


Uranyl stretching frequencies in solid Cs2UO2Cl4

H. H. Osman* , Pilar Pertierra, Miguel A. Salvadó, F. Izquierdo-Ruiz, and J. M. Recio

MALTA-Team and Dpto. de Química Física y Analítica, Univ. de Oviedo, Oviedo(Spain)

Abstract The chemistry of uranium plays an importantrole in the treatment of nuclear waste and in the understanding of migration mechanisms ofthis radioactive species through the groundwater and soil. The linearity of UO2

2+

and its remarkable chemical stability has drawn considerable theoretical and experimental work aimed at understanding thenature of the U-O chemical bonding [1, 2]. At this respect, Cs2UO2Cl4 provides one of the simplest examples of a solid containing the uranyl cation. Cs2UO2Cl4 belongs to the monoclinic, C2/m space group. The U atom is located at a special position with C2h site symmetry, resulting in a strictly linear conformation of theuranyl moiety as shown below [3].

First principle VASP calculations based on the density functional theory (DFT+U) using the projector-augmented wave (PAW) and the formalism by Dudarev et al. were carried out to study the response of Cs2UO2Cl4 crystal to pressure conditions up to 50 Gpa [4]. An effective Hubbard parameter (Ueff) and a cut-off energy 3.49 eV and 250 eV, respectively, were selected. Bond distances and uranyl symmetric and asymmetric stretch frequencies were calculated at selected pressures within this 0-50 GPa range. In addition, bond and interaction force constants were calculated

and the uranyl bond length using Badger's relationship. Badger's rule was found to be valid for all geometries obtained at different pressures. This means that uranyl normal modes are decoupled from those modes involving equatorial ligands (Cl) [5]. Interestingly, unexpected behavior of the uranyl bond length was obtained at low pressure as shown below: a slight increase of 0.006 Å for the U–O distance. Both symmetric(gold) and asymmetric (red) frequencies decrease 4 and 9 cm-1, respectively. This behavior can be explained by studying other bond distances e.g. Cs–O and U–Cl andalso by analyzing how the charge transfer to the uranyl moiety changes as pressure is applied..

[1] D. D. Schnaars and R. E. Wilson, Inorg. Chem. 52, 14138 (2013).[2] V. V. Zhurov, E. A. Zhurova, A. I. Stash and A. A. Pinkerton, J. Phys. Chem. A 115, 13016 (2011). [3] VESTA 3. K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011). [4] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A. P. Sutton, Phys. Rev. B 57, 1505 (1998).[5] V. Vallet, U. Wahlgren, and I. Grenthe, J. Phys. Chem. A 116, 12373 (2012). * [email protected]


Epitaxial growth of ZnO thin films on square symmetry substrate planes

Iuliana Pasuk, Ioana Pintilie, Cristina Chirila, Lucian Pintilie

National Institute of Materials Physics, Atomistilor 105 bis, Magurele 077125, Romania

Abstract It is known that ZnO with hexagonal symmetry (s.g. P63mc) can be epitaxially grown on the (001) plane of cubic perovskites with the lattice parameter around 4 Å, such as SrTiO3 (STO), either with ZnO (001) or with ZnO (110) parallel to the surface, depending on the deposition conditions. In the first case the in-plane symmetry of ZnO is hexagonal, while in the second case the in-plane symmetry is tetragonal. We obtained both types of orientations for thin ZnO films deposited on STO by pulsed laser deposition [1]. The in-plane orientations of the films relative to the square symmetry substrates were analysed by X-ray diffraction performing

azimuth-scans on skew crystal planes with respect to the surface (of tilt angle, τ), using

asymmetric measurement geometry (Fig. 1).

The epitaxy of both ZnO (001) and ZnO (110) films on the square symmetry STO surface is driven by the close values of the periods along ZnO [120] and STO [110], which determines

the parallelism of these directions of the film and substrate. For (001) ZnO the -scan around the ZnO [001] axis shows 12 maxima, twice the order of the rotational symmetry of this axis,

while in the case of (110) ZnO the -scan yields 4 peaks, twice of that of the twofold symmetry of the normal to the (110) ZnO plane. This is due to the equivalence of the

diagonals, [110] and ]011[ , of the square symmetry substrate plane, which induces a twinned

in-plane growth at 90o. The small peaks in Fig. 1.2.c. reveal an unusual secondary in-plane orientation. In the poster we present structural models of the interface planes consistent with all the features of the azimuth scans presented in Fig. 1.2. Reference: [1] I.Pintilie, I.Pasuk, G. A.Ibanescu, R.Negrea, C.Chirila, E.Vasile, L.Pintilie, J.Appl.Phys. 112, 104103 (2012)

Fig.1. The measurement

geometry of the azimuth-

scans (1.1); azimuth-

scans on skew planes of

the single crystal (001)

STO substrate (1.2.a), on

the (001) oriented ZnO

film (1.2.b) and on the

(110) oriented ZnO film

(1.2.c).

=0

n

X-yars

1.1 1.2


A study of high-pressure X-ray diffraction of the Topological Insulator SnBi2Te4

R. Vilaplana b,*, J.A. Sans a, F.J. Manjón a, C. Popescu c, O. Gomis b, A.L.J., P. Rodríguez-Hernández e,

A. Muñoz e a Instituto de Diseño para la Fabricación y Producción Automatizada, MALTA Consolider Team,

Universitat Politècnica de València, Valencia, Spain

b Centro de Tecnologías Físicas, MALTA Consolider Team, Universitat Politècnica de València,

Valencia, Spain

c ALBA-CELLS, Barcelona, Spain d Departamento de Física, Instituto de Materiales y Nanotecnología, MALTA Consolider Team,

Universidad de La Laguna, Tenerife, Spain

Abstract

Topological insulators (TIs) have gained considerable attention in the solid-state community. Distinct from a simple insulator or metal, a typical TI contains exotic metallic surface states, which are protected by the time-reversal symmetry despite having a bulk band gap. This unusual state of matter is distinguished by the nontrivial topological invariant Z2, which is determined by the bulk electronic structure. Such a structure is characterized by the band inversion in which electronic bands with opposite parity around the Fermi level are inverted. The spin orbit coupling (SOC) effect has been considered dominant during the band inversion. Materials containing elements with atomic number greater than 50 are naturally candidates of TIs because of the strong SOC in these heavy elements.

New ternary layered topological insulators (TI) based on well-known binary TIs Bi2Te3, Bi2Se3, and Sb2Te3 [1] is one of the possible ways to obtain the necessary properties of TI for practical application [2–4]. To this respect, calculations of the electronic structure of the layered ternary compound SnBi2Te4 which is composed of sextuple layer (SL) atomic blocks separated by van der Waals spacing indicates that this compound is three-dimensional TI [4]. High-pressure (HP) studies in TI play an important role for understanding the physics of these layered semiconductors [5-7].

We report a study of HP X-ray diffraction of the TI SnBi2Te4 in order to account for the electrical resistivity changes observed in our measurements about 3-4 GPa. We have also carried out Raman measurements in order to shed more light into the nature of the changes occurring at 3-4 GPa, apparently, without a phase transition.

References: [1] H. Zhang, C.X. Liu, X.L. Qi, et al., Nat. Phys. 5, 438 (2009) [2] S. V. Eremeev, G. Landolt, T. V. Menshchikova, et al.,Nature Commun. 3, 635 (2012). [3] T.V. Menshchikova, S.V. Eremeev, E.V. Chulkov, JETP Letters 94, 106 (2011) [4] T.V. Menshchikova, et al., Appl. Surf. Sci. 267, 1 (2013). [5] R. Vilaplana, et al. Phys. Rev. B 84, 184110 (2011). [6] R. Vilaplana, et al. Phys. Rev. B 84, 104112 (2011). [7] O. Gomis, et al. Phys. Rev. B 84, 174305 (2011).

International Union of Crystallography

Commission on Mathematical and TheoreticalCrystallography

SECOND BALKAN SCHOOL ON FUNDAMENTALCRYSTALLOGRAPHY AND WORKSHOP ON

MAGNETIC SYMMETRY

Istanbul, Turkey, 13 - 19 July 2015

EXERCISES

INTERNATIONAL TABLES FOR CRYSTALLOGRAPHY

Volume A: Space-group Symmetry

Volume A1: Symmetry Relations between Space Groups

THE BILBAO CRYSTALLOGRAPHIC SERVER

Short Practical Course

Mois I. Aroyo

Departamento Fısica de la Materia CondensadaUniversidad del Paıs Vasco, UPV/EHU, Bilbao, Spain

[email protected]

Chapter 1

Space-group symmetry data inInternational Tables forCrystallography, Volume A andVolume A1: Basic concepts andnotation

1.1 Crystallographic Symmetry Operations

In order to describe the symmetry operations analytically one introduces a coordinate system {O, a, b, c},consisting of a set of basis vectors a, b, c and an origin O. A symmetry operation can be regarded as aninstruction of how to calculate the coordinates x, y, z of the image point X from the coordinates x, y, zof the original point X.The equations are

x = W11 x+W12 y +W13 z + w1

y = W21 x+W22 y +W23 z + w2

z = W31 x+W32 y +W33 z + w3,(1.1.1)

These equations can be written using the matrix formalism:

x = W x + w = (W ,w)x where

the symmetry operations (W ,w) are given in a matrix-column form consisting of a (3×3) matrix (linear)part W and a (3× 1)-column(translation) part w :

(W , w) =

W11 W12 W13 w1

W21 W22 W23 w2

W31 W32 W33 w3

(1.1.2)

Apart from the matrix-column pair presentation of (W ,w) often the so-called short-hand notation forthe symmetry operations is used. It is obtained from the left-hand side of equ. (1.1.1) by omitting theterms with coefficients 0 and writing in one line the three different rows of equ.(1.1.1), separated bycommas.

3

4 CHAPTER 1. SPACE-GROUP SYMMETRY DATA

For example, the matrix-column pair (W ,w) = (

1 1 00 1 00 0 1

,

00

1/2

) is represented in short-

hand notation by the coordinate triplet: x+ y, y, z + 1/2. This is the entry (11) of the General positionsof the space group P6522, No. 179 (cf. the space-group tables of International Tables for Crystallography,Volume A: Space-group symmetry, referred to as ITA).

1.1.1 Crystallographic Symmetry Operations and Their Representations byMatrices

Geometric meaning of matrix-column pairs (W ,w)

The geometric meaning of a matrix-column pair (W ,w) can be determined only if the reference coordi-nate system is known. The following procedure indicates the necessary steps for the complete geometriccharacterization of (W ,w).

Procedure for the geometric interpretation of (W ,w)

1. W -information

(a) Type of isometry: the types 1,2,3,4,6 or 1,2,3,4,6 can be determined by the matrix invariants:det(W ) and tr(W )

det(W ) = +1 det(W ) = −1

tr(W ) 3 2 1 0 −1 −3 −2 −1 0 1

type 1 6 4 3 2 1 6 4 3 2 = m

order 1 6 4 3 2 2 6 4 6 2

.

(b) Direction of u the rotation or rotoinversion axis or the normal of the reflection plane

i. Rotations: Calculate the matrix Y (W ) = W k−1 + W k−2 + . . .+ W + I . The elementsof any non-zero column of Y give the components of the vector u with respect to thereference co-ordinate system.

ii. Rotoinversions: Calculate the matrix Y (−W ). The elements of any non-zero column ofY give the components of the vector u with respect to the reference co-ordinate system.For 2 = m, Y (−W ) = −W + I .

(c) Sense of rotation (for rotations or rotoinversions with k > 2): The sense of rotation is de-termined by the sign of the determinant of the matrix Z , given by Z = [u |x |(detW )Wx ],where u is the vector of 1b and x is a non-parallel vector of u , e.g. one of the basis vectors.

2. w -information

(a) Intrinsic translation part (screw or glide component) t/k

i. Screw rotations

t/k =1

kYw ,whereW k = I (1.1.3)

ii. Glide reflections

t/k =1

2(W + I )w (1.1.4)

1.2. SYMMETRY OPERATIONS AND SYMMETRY ELEMENTS 5

(b) Location of the symmetry elements (fixed points xF )

i. t/k = 0(W ,w)xF = xF . (1.1.5)

ii. t/k 6= 0(W ,w lp)xF = xF . (1.1.6)

The column w lp = w − t/k is the so-called location part as it determines the position of the rotation orscrew-rotation axis or of the reflection or glide-reflection plane in space.

The formulæ of this section enable the user to find the geometric contents of any symmetry operation.In reality, ITA have provided the necessary information for all symmetry operations which are listed inthe plane–group or space–group tables. The entries of the General position are numbered. The geometricmeaning of these entries is listed under the same number in the block Symmetry operations in thetables of ITA. The explanation of the symbols for the symmetry operations is found in Sections 2.9 and11.2 of ITA.

1.2 Symmetry Operations and Symmetry Elements

The definition and detailed discussion on the term symmetry element can be found in the reports of theAd-Hoc Committee on the Nomenclature of Symmetry Wolff, P. M. de et al. Acta Cryst A45, 494 (1989)and Acta Cryst A48, 727 (1992). In short, a symmetry element is defined as a concept with a doublemeaning, namely the combination of a geometric element with the set of symmetry operations having thisgeometric element in common (the so-called element set). The geometric element is defined as a geometricitem that allows the reduced symmetry operation (i.e. after removing any intrinsic translation) to belocated and oriented in space. It is important to note that the element set of a symmetry element canconsist of symmetry operations of the same type (such as the powers of a rotation) or of different types,e.g. by a reflection and a glide reflection through the same plane. Among these operations one choosesthe so-called defining operation (defined as the simplest symmetry operation that suffice to identify thesymmetry element, cf. Wolff, P. M. de et al., 1992), which in a way specifies the name and the symbol ofthe symmetry element.According to the recommendations of the same Ad-Hoc Committee (Wolff, P. M. de et al., 1992) thecharacters appearing after the lattice letter in the (short or full) Hermann-Mauguin (HM) symbol of aspace group which originally were meant to represent generating operations, should be interpreted assymmetry elements. The introduction of the e-glide notation in HM symbols of five oC space groupswas a direct consequence of this decision: the e-glide planes are symmetry elements characterized bythe existence of two glide reflections through the same plane with perpendicular glide vectors with theadditional requirement that at least one glide vector is along a crystal axis.The definitions of symmetry elements, geometric elements and the related element sets of symmetryoperations for crystallographic space groups and point groups are summarised in the following table.


Table 1.2.1 Symmetry elements in point and space groups

Name of Geometric Defining Operationssymmetry element element operation (d.o) in element set

Mirror plane Plane A Reflection in A D.o. and its coplanarequivalents∗

Glide plane Plane A Glide reflection in A; 2ν (not ν) D.o. and its coplanara lattice translation equivalents∗

Rotation axis Line b Rotation around b, angle 2π/n 1st, . . . , (n− 1)th powers of d.o.n = 2, 3 , 4 or 6 and their coaxial equivalents†

Screw axis Line b Screw rotation around b, angle 2π/n, 1st, . . . , (n− 1)th powers of d.o.u = j/n times shortest lattice and their coaxial equivalents†

translation along b, right-hand screw,n = 2, 3 , 4 or 6, j = 1, . . . , (n− 1)

Rotoinversion Line b Rotoinversion: rotation around b, D.o. and its inverseaxis and point angle 2π/n, and inversion

P on b through P , n = 3 , 4 or 6

Center Point P Inversion through P D.o. only

∗ That is, all glide reflections with the same reflection plane, with glide vectors v differing from that ofd.o. (taken to be zero for reflections) by a lattice translation vector. The glide planes a, b, c, d and e aredistinguished.† That is, all rotations and screw rotations with the same axis b, the same angle and sense of rotationand the same screw vector u (zero for rotation) up to a lattice translation vector.

!!Examples

1. Glide planes!!

The element set of a glide plane with a glide vector v consists of infinitely many different glidereflections with glide vectors that are obtained from v by adding any lattice translation parallel tothe glide plane, including centring translations of the centred cells. If, however, among the infinitelymany glide reflections of the element set of the same plane there exists one operation with zero glidevector, then this operation is taken as defining operation, i. e. the symmetry element is a mirrorplane.

(a) The symmetry operation x + 5/2, y − 7/2,−z + 3 is a glide reflection. Its geometric elementis a the plane x, y, 3/2. Its symmetry element is a glide plane in space group Pmmn becausethere is no lattice translation by which the glide vector can be changed to o. If, however, thesame mapping is a symmetry operation of space group Cmmm, then its symmetry element is areflection plane, because the glide vector with components 5/2,−7/2 can be cancelled througha translation (2 + 1/2)a + (−4 + 1/2)b which is a lattice translation in a C lattice. Evidently,the correct specification of the symmetry element is possible only with respect to a specifictranslation lattice.

(b) Similarly, in Cmma with an a-glide reflection x+1/2, y, z also the b-glide reflection x, y+1/2, zoccurs. The geometric element is the plane x, y, 0 and the symmetry element is an e-glide plane.

(c) More general, all vectors (u+ 12 )a+ vb+ 1

2k(a+b), u, v, k integers, are glide vectors of a-glidereflections through the (001) plane of a space group with a C-centered lattice. Among them

1.3. SITE SYMMETRY: GENERAL AND SPECIAL POSITIONS 7

one finds a glide reflection b with a glide vector 12b related to 1

2a by the centring translation;an a-glide reflection and a b-glide reflection share the same plane as a geometric element. Theirsymmetry element is thus an e-glide plane.

2. Screw axes !!

The element set of a screw axis is formed by a screw rotation of angle 2π/n, its (n − 1)th powersand all their co-axial equivalents that include all rotations and screw rotations with the same axis,the same angle and the same screw vector up to lattice translation vector (i. e. any screw vectorobtained by adding a lattice translation vector, parallel to the screw axis). If any of the screwvectors is zero, then the symmetry element is a rotation axis.

(a) 21 ‖ [001] in a primitive cell: The element set is formed by all twofold screw rotations withscrew vectors of the type (u− 1

2 )c, i. e. screw components as 12c, − 1

2c, 32c, etc.

(b) The symmetry operation 4− x, −2− y, z + 5/2 is a screw rotation of space group P2221. Itsgeometric element is the line 2,−1, z and its symmetry element is a screw axis.

(c) The determination of the complete element set of a geometric element is important for thecorrect designation of the corresponding symmetry element. For example, the symmetry ele-ment of a twofold screw rotation with an axis through the origin is a twofold screw axis in thespace group P2221 but a fourfold screw axis in P41.

1.3 Site symmetry: General and Special positions

The concept of Site symmetry, ı.e. the set of symmetry operations that leave a given point fixed, allowsto define General and Special positions for space groups.

Let G be a space group and X a point. The subgroup SX = {(W , w)} of all (W , w) ∈ G that leave Xfixed, i.e. for which (W , w)X = X holds, is called the site symmetry group SX of G for the point X.The group SX < G is of finite order. If SX = {I}, i.e. only the identity operation maps X onto itself, Xis called a point of General position. Otherwise, if SX > {I}, X is called a point of Special position.

Each point Xi of a G-orbit has its site symmetry group Si < G. The site symmetry groups Si and Sj oftwo points Xi and Xj of the same G-orbit are conjugate subgroups of G: if Xj = (W , w)Xi, (W , w) ∈ G,then Sj = (W , w)Si(W , w)−1. For this reason, all points of one special position in ITA are describedby the same site-symmetry symbol.

In ITA the so-called oriented site-symmetry symbols are used to show how the symmetry elements at asite are related to the symmetry elements of the crystal lattice. The oriented site-symmetry symbols ofthe site-symmetry groups display the same sequence of symmetry directions as the space-group symbol.Sets of equivalent symmetry directions that do not contribute any element to the site-symmetry groupare represented by a dot.

1.4 Coordinate Transformations: basic results

Let a coordinate system be given with a basis (a1, a2, a3) and an origin O. The general transformation(affine transformation) of the coordinate system consists of two parts, a linear part and a shift of theorigin. The transformation is uniquely defined by the (3× 3) matrix P of the linear part and the (3× 1)column matrix p containing the components of the shift vector p.


1. The linear part is described by a (3× 3) matrix

P =

P11 P12 P13

P21 P22 P23

P31 P32 P33

i.e. the matrix which relates the new basis (a′1, a

′2, a

′3) to the old basis (a1, a2, a3) according to

(a′1, a′2, a

′3) = (a1, a2, a3) P = (a1, a2, a3)

P11 P12 P13

P21 P22 P23

P31 P32 P33

. (1.4.7)

2. A shift of the origin is defined by the shift vector

p = (p1a1, p2a2, p3a3)

The basis vectors a1, a2, a3 are fixed at the origin O; the new basis vectors (a′1, a′2, a

′3) are fixed

at the new origin O′ that has the coordinates (p1, p2, p3) in the old coordinate system.

The general affine transformation of the coordinates of a point X in direct space (given by the columnx = (x1, x2, x3)) is given by the following formula:

x ′ = (P ,p)−1x = P−1x −P−1p = P−1(x − p). (1.4.8)

The metric tensor G of the unit cell in direct lattice is transformed by the matrix P as follows:

G ′ = P tGP (1.4.9)

where P t is the transposed matrix of P .The volume of the unit cell V changes with the transformation. The volume of the new unit cell V ′ isobtained by

V ′ = det(P)V (1.4.10)

with det(P) being the determinant of the matrix P .Also, the matrix-column pairs of the symmetry operations are changed by a change of the coordinatesystem. If a symmetry operation is described in the “old”(unprimed) coordinate system by the matrix-column pair (W , w) and in the “new”(primed) coordinate system by the pair (W ′, w ′), then therelation between the pairs (W , w) and (W ′, w ′) is given by:

(W ′, w ′) = (P ,p)−1(W ,w)(P ,p) (1.4.11)

The coordinate systems of the space groups used by the programs and database on the Bilbao Crystallo-graphic Server (referred to as standard or default settings) for the presentation of the space-group datacoincide with the conventional space-group descriptions found in ITA. For space groups with more thanone description in ITA, the following settings are chosen as standard: unique axis b setting, cell choice 1for monoclinic groups, hexagonal axes setting for rhombohedral groups, and origin choice 2 (origin in 1)for the centrosymmetric groups listed with respect to two origins in ITA. Optionally certain applicationsallow the usage of the so-called ITA settings which include all conventional settings applied in ITA(e.g. rhombohedral axes setting for rhombohedral groups, and origin choice 1 for the centrosymmetricgroups) and the great variety of about 530 settings of monoclinic and orthorhombic groups listed inTable 4.3.2.1 of ITA. Settings different from the standard ones and the ITA settings are designated asnon-conventional.

1.5. GROUP-SUBGROUP RELATIONS OF SPACE GROUPS 9

1.5 Group-subgroup relations of space groups

1.5.1 Basic definitions

A subset H of elements of a group G is called a subgroup of G, G > H if it fulfills the group postulateswith respect to the law of composition of G. In general, the group G itself is included among the set ofsubgroups of G, i.e. G ≥ H. If G > H is fulfilled, then the subgroup H is called a proper subgroup of G.In a relation G ≥ H or G > H, G is called a supergroup of H. A subgroup H < G is a maximal subgroupif no group Z exists for which H < Z < G holds. If H is a maximal subgroup of G, then G is a minimalsupergroup of H.Let H < G be a subgroup of G of order |H|. Because H is a proper subgroup of G there must beelements gq ∈ G which are not elements of H. Let g2 ∈ G be one of them. Then the set of elementsg2H = {g2 hj | hj ∈ H}1 is a subset of elements of G with the property that all its elements are differentand that the sets H and g2H have no element in common. Thus, also the set g2H contains |H| elementsof G. If there is another element g3 ∈ G which does belong neither to H nor to g2H, one can form anotherset g3H = {g3 hj | hj ∈ H}. All elements of g3H are different and no one occurs already in H or in g2H.This procedure can be continued until each element gr ∈ G belongs to one of these sets. In this way thegroup G can be partitioned, such that each element g ∈ G belongs to exactly one of these sets.The partition just described is called a decomposition (G : H) into left cosets of the group G relative tothe group H.

G = H ∪ g2H ∪ · · · ∪ giH (1.5.12)

The sets gpH, p = 1, . . . , i are called left cosets, because the elements hj ∈ H are multiplied with thenew elements from the left-hand side. The procedure is called a decomposition into right cosets H gs ifthe elements hj ∈ H are multiplied with the new elements gs from the right-hand side.

G = H ∪Hg2 ∪ · · · ∪ Hgi (1.5.13)

The elements gp or gs are called the coset representatives. The number of cosets is called the index[i] = |G : H| of H in G.Two subgroups Hj ,Hk < G are called conjugate if there is an element gq ∈ G such that g−1q Hj gq = Hk

holds. In this way, the subgroups of G are distributed into classes of conjugate subgroups that are alsocalled conjugacy classes of subgroups. Subgroups in the same conjugacy class are isomorphic and thus havethe same order. Different conjugacy classes of subgroups may contain different numbers of subgroups, i.e.have different lengths.A subgroup H of a group G is a normal subgroup H C G if it is identical with all of its conjugates,g−1q H gq = H, for all gq ∈ G, i.e. if its conjugacy class consists of the one subgroup H only.

1.5.2 Subgroups of space groups

The set of all symmetry operations of a three-dimensional crystal pattern forms its symmetry group, whichis the space group of this crystal pattern. An essential feature of a crystal pattern is its periodicity whichindicates that there are translations among its symmetry operations. The infinite number of translationsdetermines the infinite order of any space group. The set of all translations of a space group G forms theinfinite translation subgroup T (G)CG which is a normal subgroup of G of finite index. Consider the rightcoset decomposition of G relative to T (G).

1The formulation g2 H = {g2 hj | hj ∈ H} means: ‘g2 H is the set of the products g2 hj of g2 with all elements hj ∈ H.’


(I ,o) (W 2, w2) ... (Wm, wm) ... (W i, w i)(I , t1) (W 2, w2 + t1) ... (Wm, wm + t1) ... (W i, w i + t1)(I , t2) (W 2, w2 + t2) ... (Wm, wm + t2) ... (W i, w i + t2)

... ... ... ... ... ...(I , tj) (W 2, w2 + tj) ... (Wm, wm + tj) ... (W i, w i + tj)

... ... ... ... ... ...

.

Obviously, the coset representatives of the decomposition (G : T (G)) represent in a clear and compactway the infinite number of elements of the space group G. And this is one of the ways of presenting thespace groups in ITA and also in the Bilbao Crystallographic Server, i.e. by the matrices of the cosetrepresentatives of (G : T (G)) listed in the General position.Each coset in the decomposition (G : T (G)) is characterized by its linear part. One can show that theset of linear parts, represented by the set of matrices W j , forms a group which is called the point groupPG of the space group G. The point groups which can belong to space groups are called crystallographicpoint groups.

The following types of subgroups of space groups are to be distinguished:

A subgroup H of a space group G is called a translationengleiche subgroup or a t-subgroup of G if the setT (G) of translations is retained, i.e. T (H) = T (G), but the number of cosets of the decomposition(G : T (G)), i.e. the order of the point group PG is reduced.

A subgroup H < G of a space group G is called a klassengleiche subgroup or a k-subgroup if the set T (G)of all translations of G is reduced to T (H) < T (G) but all linear parts of G are retained. Then thenumber of cosets of the decompositions (H : T (H)) and (G : T (G)) is the same, i.e. the order ofthe point group PH is the same as that of PG .

A klassengleiche or k-subgroup H < G is called isomorphic or an isomorphic subgroup if it belongs tothe same affine space-group type (isomorphism type) as G does.

A subgroup of a space group is called general or a general subgroup if it is neither a translationengleichenor a klassengleiche subgroup. It has lost translations as well as linear parts, i.e. point-groupsymmetry.

Any subgroup H of a group G is related to a specific subset of elements of G and this subset defines thesubgroup uniquely: different subgroups of G, even those isomorphic to H, correspond to different subsetsof the elements of G. For example, the listing of the maximal t-subgroups of the space groups in ITA isbased on this fact: apart from the space-group type and index, each t-subgroup H is specified by the setof coordinate triplets of the general position of G which are retained in H.

In the Bilbao Crystallographic Server any subgroup H of a space group G is specified by its ITA-number,the index in the group G and the transformation matrix-column pair (P , p) that relates the standardbases (a,b, c)H of H and (a,b, c)G of G:

(a,b, c)H = (a,b, c)GP (1.5.14)

The column p = (p1, p2, p3) of coordinates of the origin OH of H is referred to the coordinate system ofG.The subgroup data listed in the server, i.e. the space-group type of H and the transformation matrix(P , p), are completely sufficient to define the subgroup uniquely: the transformation of the coordinatetriplets of general-position of H (in standard setting) to the coordinate system of G by (P , p)−1 yieldsexactly the subset of elements of G corresponding to H.


A very important result on group-subgroup relations between space groups is given by Hermann’s theorem:For any group–subgroup chain G > H between space groups there exists a uniquely defined space groupM with G ≥M ≥ H, whereM is a translationengleiche subgroup of G andH is a klassengleiche subgroupof M. The decisive point is that any group-subgroup chain between space groups G > H of index [i] canbe split into a translationengleiche subgroup chain between the space groups G and M of index [iP ] anda klassengleiche subgroup chain between the space groups M and H of index [iL] where [i] = [iP ] · [iL].

The first one, also called t-chain GiP> M, is related to the reduction of the point-group symmetry in

the subgroup. The second one MiL> Hj is known also as k -chain and it takes account of the loss of

translations.

It may happen, that either G = M or H = M holds. In particular, one of these equations must holdif H < G is a maximal subgroup of G. In other words, a maximal subgroup of a space group is either atranslationengleiche subgroup or a klassengleiche subgroup, never a general subgroup.

If the maximal subgroups are known for each space group, then in principle each non-maximal subgroupof a space group G with finite index can be obtained from the data on maximal subgroups. A non-maximal subgroup H < G of finite index [i] is connected with the original group G through a chainH = Zk < Zk−1 < · · · < Z1 < Z0 = G, where each group Zj < Zj−1 is a maximal subgroup of Zj−1,

with the index [ij ] = |Zj−1 : Zj |, j = 1, . . . , k. The number k is finite and the relation i =∏k

j=1 ijholds, i.e. the total index [i ] is the product of the indices ij .

In a similar way, one can express the transformation matrix (P , p) for the symmetry reduction G −→ H asa product of the transformation matrices (P ,p)j characterizing each of the intermediate steps Zj−1 > Zj :(P ,p) = (P ,p)1(P ,p)2 · · · (P ,p)k (here the matrices (P ,p)j relate the bases of Zj−1 and Zj , i.e.(a,b, c)j = (a,b, c)j−1Pj).

1.5.3 Generation of Space Groups

A set of generators of a group is a subset of the group elements which by proper combination permitsthe generation of all elements of the group. Different sets of generators are possible.

In International Tables for Crystallography, Vol. A (referred to as ITA in the following), the gener-ators and the generating procedure have been chosen such as to make the entries in the blocks ofGeneral position and Symmetry operations as transparent as possible. Given the set of h generatorsG1,G2, . . . ,Gp, . . .Gh, any space-group operation W is generated by the following algorithm, starting withthe identity and the translations as right-most factors:

W = Gkh

h .Gkh−1

h−1 . . . . .Gkpp . . . . .G

k33 .G

k22 .G1. (1.5.15)

Here, the exponents kp are positive or negative integers, including zero.

The space-group generator G1 is the identity (zero translation). It is chosen first and assures that thegeneral position of G starts with the coordinate triplet x, y, z.The following generatorsG2, G3, G4 arethe translations corresponding to the three basis vectors a, b, c and G5, G6 are the generators for thecentring translations, if present. The rest of the generators G7, G8, . . . give all those symmetry operationsof the space group G which are not pure translations. They have been chosen such that their exponentscan assume only the values 0,1 and 2. Space groups of the same crystal class are generated in the sameway. In ITA, the generators are designated by the numbers in front of the corresponding general-positionco-ordinate triplets.

The coordinate triplets of the General position are obtained by single-sided, (i.e left-sided) multiplication


of the matrices representing the generators until no new matrices are found. Resulting matrices that differonly by a lattice translation are considered as equal, and the translations parts are chosen such so thatthe symmetry operations lie within the unit cell.

The generating procedure used in ITA highlights important subgroups of space groups as much as possible.For example, once the translation subgroup T G of a space group G is generated, the process of generationfollows step-wise procedure via a chain of normal and maximal subgroups

G B H1 B H2 B · · · B T G , (1.5.16)

with indices |Hi : Hi+1| equal to 2 or 3. In other words, each new (non-translational) generator generatesa minimal translationengleiche or t-supergroup Hi of Hi+1 of index 2 or 3.

Chapter 2

Exercises

2.1 Matrix calculus in crystallography (brief revision)

• Exercise 2.1.1. Matrix transposition

1. Construct the transposed matrix of the (3× 1) row matrix A =(

1 3 4).

2. Determine which of the following matrices are symmetric and which are skew-symmetric

A =

(3 00 2

);B =

(3 4−4 1

);C =

(2 −1−1 1

);D =

(0 2−2 0

);E =

(0 01 0

);

F =(

2)

;G =

0 1 −2−1 0 3

2 −3 0

;H =

3 22 11 0

;J =

(0 00 0

).

• Exercise 2.1.2. Matrix addition and subtraction

1. Find 3A-2B , where A =

(1 23 0

)and B =

(1 30 −4

).

2. Show that the sum of any matrix and its transposed is a symmetric matrix, i.e. (A+AT )T =A + AT .

3. Show that the difference of any matrix and its transposed is a skew-symmetric matrix, i.e.(A−AT )T = −(A−AT ).

• Exercise 2.1.3. Matrix multiplication

1. Find the products AB and BA if they exists, where A =

(1 23 −4

)and B =

(3 −2 21 0 −1

).

2. Find the matrix products AB and BA of the row vector A =(

1 2 3)

and the column vector

B =

−241

.

3. Prove that A(BC )=(AB)C where A =

(1 2−1 3

), B =

(1 0 −12 1 0

)and C =

1 −13 22 1

.

• Exercise 2.1.4. Trace and determinant of a matrix

13

14 CHAPTER 2. EXERCISES

1. Find the values of the traces and the determinants of A and B where

A =

(1 2−1 3

)and B =

0 4 24 −2 −15 1 3

.

2. Show that det(AB) = det(A)det(B) where A =

(3 25 1

)and B =

(1 62 9

).

3. Show that det(A) = det(AT ) where A =

1 1 32 2 23 2 3

.

• Exercise 2.1.5. Inverse of a matrix

1. Show that the matrix B = 1/3

11 −9 1−7 9 −2

2 −3 1

is the inverse of A =

1 2 31 3 51 5 12

.

2. Determine the inverses of the matrices A =

−1 0 00 −1 00 0 1

; B =

0 −1 01 0 00 0 −1

; C =

0 0 11 0 00 1 0

;

D =

1 1 0−1 1 0

0 0 1

; E =

−1 1 11 −1 11 1 −1

and F =

0 1 11 0 11 1 0

.

3. Given that A =

1 2 0−1 0 3

2 −1 0

determine A−1.

• Exercise 2.1.6. Matrix-column presentation of symmetry operations

1. Referred to an ‘orthorhombic’ coordinate system (a 6= b 6= c;α = β = γ = 90) two symmetry

operations are represented by the following matrix-column pairs: (W 1,w1) =

1 0 00 1 00 0 1

,

000

and (W 2,w2) =

1 0 00 1 00 0 1

,

1/20

1/2

.

(a) Determine the images Xi of a point X =

0.70.310.95

under the action of the symmetry

operations.

(b) Can you guess what is the ‘geometric nature’ of (W 1,w1) and (W 2,w2)?

(c) Determine the determinant and the trace of W 1.

(d) Determine the sets of fixed points of (W 1,w1) and (W 2,w2).

2. Consider the matrix-column pairs of the two symmetry operations (W 1,w1) =

0 1 01 0 00 0 1

,

000

and (W 2,w2) =

1 0 00 1 00 0 1

,

1/20

1/2

.

(a) Determine and compare the matrix-column pairs of the combined symmetry operations:(W ,w) = (W 1,w1)(W 2,w2) and (W ,w)′ = (W 2,w2)(W 1,w1).

2.2. GROUP THEORY IN CRYSTALLOGRAPHY (BASIC CONCEPTS) 15

(b) Determine the inverse symmetry operations (W 1,w1)−1 and (W 2,w2)−1.

(c) Determine the inverse symmetry operation (W ,w)−1 if (W ,w) = (W 1,w1)(W 2,w2).

3. Consider the matrix-column pairs (A,a) =

0 1 01 0 00 0 1

,

1/21/21/2

and (B , b) =

0 1 00 0 11 0 0

,

000

.

(a) What are the matrix-column pairs resulting from: (A,a)(B , b) = (C , c) and (B , b)(A,a) =(D ,d).

(b) Determine (A,a)−1, (B , b)−1, (C , c)−1 and (D ,d)−1. What is (B , b)−1(A,a)−1?

2.2 Group theory in crystallography (basic concepts)

• Exercise 2.2.1. Symmetry group of the pentacene molecule

Consider the model of the molecule of the organic semiconductor pentacene (C22H14):

1. Determine all symmetry operations and their matrix and the short-hand (x, y) presentation;

2. Draw up the general-position and the symmetry-elements stereographic-projection diagrams;

3. Find a set of generators;

4. Construct the multiplication table of the group of the pentacene molecule;

5. Distribute the elements of the group into classes of conjugate elements.

• Exercise 2.2.2. Symmetry group of the square

b

a

2

4 1

3

-1,-1

1,-1

-

1,1

-1,1

y +

m+

mmm-

mx

my

mx

m

Consider the symmetry group of the square:

1. Determine all symmetry operations and theirmatrix and the short-hand (x, y) presentation;

2. Draw up the general-position and thesymmetry-elements stereographic-projectiondiagrams;


4. Construct the multiplication table of the groupof the square;

5. Distribute the elements of the group intoclasses of conjugate elements.


• Exercise 2.2.3. Symmetry group of the equilateral triangle

Consider the symmetry group of the equilateral tri-angle:

1. Determine all symmetry operations and theirmatrix and the short-hand (x, y) presentation;

2. Draw up the general-position and thesymmetry-elements stereographic-projectiondiagrams;


4. Construct the multiplication table of the groupof the equilateral triangle;

5. Distribute the elements of the group intoclasses of conjugate elements.

• Exercise 2.2.4. Consider the symmetry group of the square 4mm, cf. Exercise 2.2.2.

1. Which are the possible orders of the subgroups of 4mm?

2. With the help of the stereographic projections of 4mm derive all subgroups of indices [2] and[4] of 4mm.

3. Determine the subgroups of 4mm and construct the complete subgroup graph of point group4mm, see Remarks.

4. Which of these subgroups are conjugate (symmetrically equivalent) in 4mm and which arenormal subgroups?

RemarksIn a subgroup diagram each, subgroup is located at a level which is determined by its index (theoriginal group with index [1] on top, subgroups of index [2] next lower level, etc.). Each of thesegroups is connected with its maximal subgroups by straight lines.

• Exercise 2.2.5. Consider the symmetry group of the equilateral triangle 3m and its stereographicprojections, cf. Exercise 2.2.3.


1. Determine the subgroups of 3m;

2. Distribute the subgroups into classes of conjugate subgroups;

3. Construct the graph of maximal subgroups of 3m.

• Exercise 2.2.6. Consider the subgroup 2 = {1, 2} of 4mm.

1. Write down the left and right coset decomposition of the group 4mm with respect to thesubgroup 2 .

2. Are the right and the left coset decompositions equal or different? Why?

3. Show that the cosets of the decomposition 4mm with respect to 2 satisfy the group axiomsand the set of cosets forms a factor group.

4. Construct the multiplication table of the factor group.

5. Indicate a crystallographic point group isomorphic to the factor group.

• Exercise 2.2.7. Show that H C G holds always for |G : H| = 2, i. e. subgroups of index 2 are alwaysnormal subgroups.

• Exercise 2.2.8. Consider the symmetry group of the square 4mm (cf. Exercise 2.2.2) and the pointgroup 422 that is isomorphic to it. Determine the general and special Wyckoff positions for the twogroups.

• Exercise 2.2.9. Consider the general and the special Wyckoff positions of the symmetry group of thesquare 4mm (cf. Exercise 2.2.8) and those of its subgroup mm2 (cf. Exercise 2.2.4). Determine thesplitting schemes of the Wyckoff positions for the pair 4mm > mm2.

• Exercise 2.2.10. Consider the group 4mm and its subgroups of index 4 (cf. Exercise 2.2.4). Determinetheir normalizers in 4mm. Distribute the subgroups of 4mm of index 4 into conjugacy classes withthe help of their normalizers in 4mm.

• Exercise 2.2.11. Show that the rotational symmetries of a crystal pattern with translational symmetryare limited to 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold.

• Exercise 2.2.12. Consider the following 10 figures of the symmetry elements and the general positionsof the plane point groups.

1. Determine the order of the point groups and arrange them vertically by descending point-group orders (i.e. the point group of highest order at the top, and that of lowest order at thebottom).

2. Determine the complete group-subgroup graph for all plane point groups.

3. Consider the point group 2mm. Determine its maximal subgroups, its minimal supergroupsand the corresponding indices.


1

2

3

4

6

m

2mm

3m

4mm

6mm

• Exercise 2.2.13. Point groups isomorphic to 4mm Consider the following three pairs of stereographicprojections. Each of them correspond to a crystallographic point group isomorphic to 4mm:


1. Determine the point groups isomorphic to 4mm by indicating their symbols, symmetry oper-ations and possible sets of generators;

2. Construct the corresponding multiplication tables;

3. For each of the isomorphic point groups indicate the one-to-one correspondence with thesymmetry operations of 4mm;

4. Distribute the elements of the point groups isomorphic to 4mm into classes of conjugateelements.

• Exercise 2.2.14. Determine the symmetry elements and the corresponding point groups for each ofthe following models of molecules:

(a) Benzene C6H6 (b) Acetone C[CH3]2O

(c) Methane CH4 (d) Ethyne C2H2

• Exercise 2.2.15. Explain the terms composition series and solvable groups. Write down the compositionseries for the point groups 4/mmm, m3m and 6/mmm. Generate the symmetry operations of thegroups 4/mmm and 3m following their composition series.


2.3 Space-group symmetry data

• Exercise 2.3.1. The General position of a space group is listed as: (1) x, y, z (2) x, y + 12 , z + 1

2 (3)x, y, z (4) x, y + 1

2 , z + 12 .

1. Construct the matrix-column pairs of these ‘coordinate triplets’. Write down the corresponding(4× 4) matrix representation.

2. Characterize geometrically the matrices if they refer to a monoclinic basis with unique axis b(type of operation, glide (screw) component, fixed points, nature and location of the symmetryelement).

3. Use the program SYMMETRY OPERATIONS for the geometric interpretation of the matrix-columnpairs of the symmetry operations.

• Exercise 2.3.2. The following matrix-column pairs (W , w) are determined with respect to a basis(a,b,c): (1) x, y, z (2) x, y + 1

2 , z + 12 (3) x, y, z (4) x, y + 1

2 , z + 12 .

- Determine the corresponding matrix-column pairs (W ′,w ′) with respect to the basis (a′,b′, c′) =(a,b, c)P , with P= c,a,b.

- The coordinates of a point X =

0.700.310.95

are determined with respect to the basis (a,b,c).

What would be the coordinates X ′ referred to the basis (a′,b′, c′)?

• Exercise 2.3.3.

(a) Consider the General position data given in ITA for the space group Cmm2 (No. 35):

1. Characterize geometrically the matrix-column pairs listed under General position of thespace group Cmm2. Compare the results with the data listed under Symmetry operations.

2. Consider the diagram of the symmetry elements of Cmm2. Try to determine the matrix-column pairs of the symmetry operations whose symmetry elements are indicated on theunit-cell diagram.

3. Compare your results with the results of the program SYMMETRY OPERATIONS for the ge-ometric interpretation of the matrix-column pairs of the symmetry operations consideredin this exercise.

(b) The same for the space group P4mm (No.99).

Attachments: Copies of the ITA pages with the space-group data of Cmm2, No. 35.

Copies of the ITA pages with the space-group data of P4mm, No. 99.

• Exercise 2.3.4. Consider the special Wyckoff positions of the the space group P4mm (No. 99)

1. Determine the site-symmetry groups of Wyckoff positions 1a and 1b. Compare the results withthe listed data of P4mm in ITA .

2. The coordinate triplets (x, 1/2, z) and (1/2, x, z), belong to Wyckoff position 4f . Comparetheir site-symmetry groups.

3. Compare your results with the output of the program WYCKPOS for the space group P4mm.

4. Use the option Non-conventional settings of the program WYCKPOS to determine the coordinatetriplets of the Wyckoff positions of the space group P4mm referred to a non-conventionalsetting with the four-fold rotation axes parallel to a axis.


Attachments: Copies of the ITA pages with the space-group data of P4mm, No. 99.

• Exercise 2.3.5. Consider the Wyckoff-position data given in ITA for the space group P42/mbc (No.135):

1. Determine the site-symmetry groups of the following Wyckoff positions: 4(a); 4(c); 4(d); 8(g).Construct the corresponding oriented site-symmetry symbols and compare them with thoselisted in ITA. Compare your results with the results listed by the program WYCKSPLIT.

2. Characterize geometrically the isometries (3), (8), (12), (15) and (16) as listed under Gen-eral Position. Compare the results with the corresponding geometric descriptions listed underSymmetry operations in ITA.

3. Compare the calculated geometric descriptions of the isometries by the program SYMMETRY

OPERATIONS

Attachments: Copies of the ITA pages with the space-group data of P42/mbc, No. 135.

• Exercise 2.3.6. ITA -conventional settings of space groups

1. Consider the space group P21/c (No. 14). Show that the relation between the General andSpecial position data of P1121/a (setting unique axis c ) can be obtained from the data P121/c1(setting unique axis b ) applying the transformation (a,b, c)c = (a,b, c)bP , with P= c,a,b.

2. Use the retrieval tools GENPOS (generators and general positions) and WYCKPOS (Wyckoff po-sitions for accessing ITA data. Get the data on general and special positions in differentsettings either by specifying transformation matrices to new bases, or by selecting one of the530 settings of the monoclinic and orthorhombic groups listed in ITA (cf. Table 4.3.2.1).

Attachments: Copies of the ITA pages with the space-group data of P21/c (No. 14).

• Exercise 2.3.7. ITA and Non-conventional settings of space groups

1. Use the retrieval tools GENPOS (generators and general positions), WYCKPOS (Wyckoff positionsand HKLCOND (reflection conditions) for accessing ITA data. Get the data on general and specialpositions in different settings either by specifying transformation matrices to new bases, or byselecting one of the 530 settings of the monoclinic and orthorhombic groups listed in ITA (cf.Table 4.3.2.1).

2. Consider the General position data of the space group Im3m (No. 229). Using the optionNon− conventional setting obtain the matrix-column pairs of the symmetry operations withrespect to a primitive basis (ap,bp, cp), applying the transformation ap,bp, cp = 1

2 (−a + b +c), 12 (a− b + c), 12 (a + b− c) (where (a,b, c) is the conventional basis).

2.4 Group-subgroup relations of space groups

• Exercise 2.4.1. Construct the diagram of the t-subgroups of P4mm using the ‘analogy’ with thesubgroup diagram of the group 4mm, cf. Exercise 2.2.4. Give the standard Hermann-Mauguinsymbols of the t -subgroups of P4mm.

• Exercise 2.4.2. The retrieval tool MAXSUB gives an access to the database on maximal subgroups ofspace groups as listed in ITA1. Consider the maximal subgroups of the group P4mm, (No. 99) andcompare them with the maximal subgroups of P4mm derived in Problem 2.17 (ITA Exercises).Comment on the differences, if any.


• Exercise 2.4.3. Study the group–subgroup relations between the groups G = P41212, No. 92, andH = P21, No. 4 using the program SUBGROUPGRAPH. Consider the cases with specified (e.g. [i] = 4)and unspecified index of the group-subgroup pair.

• Exercise 2.4.4. Translationengleiche subgroups of P4mm

(a) Explain the difference between the contracted and complete graphs of the t-subgroups of P4mm(No. 99) obtained by the program SUBGROUPGRAPH. Compare the complete graph with theresults of Problems 2.4 and 2.17 of ITA Exercises.

(b) Explain why the t-subgroup graphs of all 8 space groups from No. 99 (P4mm) to No. 106(P42bc) have the same ‘topology’ (i. e. the same type of ‘family tree’), only the correspondingsubgroup entries differ.

• Exercise 2.4.5. Phase transitions in BaTiO3

The crystal structure of BaTiO3 is of perovskite type. Above 120C BaTiO3 has the ideal paraelectriccubic structure (space group Pm3m ) shown in Figure 2.1. Below 120C BaTiO3 assumes threestructures with slightly deformed unit cells, all three being ferroelectric with different directions ofthe axis of spontaneous polarisation (polar axis). The three ferroelectric polymorphs differ in thedirection of displacement of the Ti-atoms from the centres of the octahedra (and the accompanyinglattice distortion):

(a) No displacement: cubic structure

(b) Displacement parallel to a cube edge: < 100 >, symmetry group P4mm;

(c) Displacement parallel to face diagonal of the cube:< 110 >, symmetry group Amm2;

(d) Displacement parallel to a body diagonal of the cube: < 111 >, symmetry group R3m.

Figure 2.1: (1) Perovskite structure (undistorted); (2) Distorted perovskite structure: Ti displacementsand lattice distortion parallel to a cube edge, and the related dipole generation in BaTiO3

(i) Which subgroup indices do the three space groups of the ferroelectric polymorphs display withrespect to the cubic group Pm3m?

(ii) How many orientation states of the twin domains occur for each polymorph? Which mutualorientation do the domains exhibit for case (b)?

• Exercise 2.4.6. SrTiO3 has the cubic perovskite structure, space group Pm3m. Upon cooling below105K, the coordination octahedra are mutually rotated and the space group is reduced to I 4/mcm;c is doubled and the unit cell is increased by the factor of four. Can we expect twinned crystals ofthe low symmetry form? If so, how many kinds of domains?

2.5. SYMMETRY RELATIONS BETWEEN CRYSTAL STRUCTURES 23

Determine the number and type of domains of the low-symmetry form of SrTiO3 using the computertools of the Bilbao Crystallographic server.

• Exercise 2.4.7. Study the splittings of the Wyckoff positions for the group-subgroup pair P4mm(No. 99)> Cm (No. 4) of index 4 by the program WYCKSPLIT.

• Exercise 2.4.8. Consider the group–supergroup pair H < G with H = P222, No. 16, and the super-group G = P422, No. 89, of index [i] = 2. Using the program MINSUP determine all supergroupsP422 of P222 of index [i] = 2. How does the result depend on the normalizer of the supergroupand/or that of the subgroup.

2.5 Symmetry relations between crystal structures

Structure data for the exercises:

http://www.cryst.ehu.es/resources/Istanbul2015

• Exercise 2.5.1. Structure descriptions for different space-group settings (Wondratschek, 2002)

(a) In R. W. G. Wyckoff, Crystal structures, vol. II, Ch. VIII, one finds the important mineral zir-con ZrSiO4 and a description of its crystal structure. Many rare-earth phosphates, arsenates,and vanadates belong to the same structure type.

Structural data: Space group I41/amd = D194h, No. 141;

lattice constants a = 6.60 A; c = 5.88 A.

The origin choice is not stated explicitly. However, Wyckoff’s Crystal Structures started toappear in 1948, when there was one conventional origin only (the later ORIGIN CHOICE 1,i. e. Origin at 4m2).

Zr : (a) 0, 0, 0; 0, 12 ,14 ; 1

2 , 0,34 ; 1

2 ,12 ,

12 ;

Si : (b) 0, 0, 12 ; 0, 12 ,34 ; 1

2 , 0,14 ; 1

2 ,12 , 0;

O : (h) (0, u, v; 0, u, v; u, 0, v; u, 0, v; 0, 12 + u, 14 − v; 0, 12 − u,14 − v;

u, 12 , v + 14 ; u, 12 , v + 1

4 ; ) [ and the same with ( 12 ,

12 ,

12 )+ ].

The parameters u and v are listed with u = 0.20 and v = 0.34.

(b) In the Structure Reports, vol. 22, (1958), p. 314 one finds:

‘a = 6.6164(5) A, c = 6.0150(5) A’

‘Atomic parameters. Origin at center (2/m) at 0, 14 ,18 from 4m2.’

‘Oxygen: (0, y, z) with y = 0.067, z = 0.198.’

Compare the two structure descriptions and check if they belong to the same structure type. Whichof the structure tools of the Bilbao Crystallographic Server could help you to solve the problem?

Hint:In order to compare the different data, the parameters of Wyckoff’s book are to be transformedto ‘origin at center 2/m’, i. e. ORIGIN CHOICE 2.

• Exercise 2.5.2. Equivalent structure descriptions


(a) CsCl is cubic, space group Pm3m, with the following co-ordinates

Atom Wyckoff Coordinate triplets

position x y z

Cl 1a 0.0 0.0 0.0

Cs 1b 0.5 0.5 0.5

How many equivalent sets of co-ordinates can be used to describe the structure? What aretheir co-ordinates?

Hint : The number of different equivalent descriptions of CsCl is equal to the index of its spacegroup Pm3m(a,b, c) in the Euclidean normalizer Im3m(a,b, c), i.e. [i] = 2. The two differentdescriptions are generated by the coset representatives of the decomposition of the normalizerwith respect to the space group.

(b) P(C6C5)4[MoNCl4] is tetragonal, space group P4/n, with the following co-ordinates:


position x y z

P 2b 0.25 0.75 0

Mo 2c 0.25 0.25 0.121

N 2c 0.25 0.25 −0.093

C1 8g 0.362 0.760 0.141

C2 8g 0.437 0.836 0.117

Cl 8g 0.400 0.347 0.191

(H and C3 to C6 omitted)

How many equivalent sets of co-ordinates can be used to describe the structure? What aretheir co-ordinates?

Hint : The number of different equivalent descriptions of P(C6C5)4[MoNCl4] is equal to theindex of its space group P4/n in the Euclidean normalizer. The different descriptions aregenerated by the coset representatives of the decomposition of the normalizer with respect tothe space group. In the special case of P(C6C5)4[MoNCl4] such equivalent descriptions can begenerated, for example, by the translations t(0, 0, 1/2) and t(1/2, 1/2, 0), and by a reflectionthrough a mirror plane at (x, x, z) represented by the coordinate triplet (y, x, z).

• Exercise 2.5.3. Isoconfigurational structure types (Koch & , Fischer, 2002)

Do the following three structures belong to the same structure type? Try to find analogous coordi-nate descriptions for all three crystal structures.

1. KAsF6 (ICSD: 59413)

Unit Cell 7.348(1) 7.348(1) 7.274(8) 90. 90. 120Space group R-3h


position x y z

K 3b 0.33333 0.66667 0.166667

As 3a 0 0 0

F 18f 0.1292(2) 0.2165(2) 0.1381(2)

2. BaIrF6 (ICSD: 803188)

Unit Cell 7.3965(1) 7.3965(1) 7,2826(1) 90. 90. 120Space group R-3h



position x y z

Ba 3b 0.33333 0.6666 0.166666

Ir 3a 0 0 0

F 18f 0.0729(2) 0.2325(2) 0.1640(2)

3. BaSnF6 (ICSD: 33788)

Unit Cell 7.4279(2) 7.4279(2) 7.418(2) 90. 90. 120Space group R-3h


position x y z

Ba 3a 0 0 0

Sn 3b 0 0 0.5

F 18f 0.2586(3) 0.8262(3) 0.0047(3)

Hint : Consider the Euclidean normalizer of symmetry group R3(hex) of KAsF6. The number ofdifferent equivalent descriptions of KAsF6 is equal to the index of its space group in the Euclideannormalizer. The different descriptions are generated by the coset representatives of the decomposi-tion of the normalizer with respect to the space group. In the special case of KAsF6 such equivalentdescriptions can be generated, for example, by the translation t(0, 0, 1/2), by a reflection througha mirror plane at (x,−x, z) represented by the coordinate triplet (−y,−x, z), etc.

• Exercise 2.5.4. Crystal structure descriptions

In Inorganic Crystal Structure Database can be found several structure data sets of ε-Fe2O3, all ofthem of symmetry Pna21 (No.33). Compare the two structure descriptions listed in the ExerciseData file and check if they belong to the same structure type.

• Exercise 2.5.5. Cristobalite phase transitions

At low temperatures, the space-group symmetry of cristobalite is given by the space group isP41212 (92) with lattice parameters a = 4.9586A, c = 6.9074A. The four silicon atoms are locatedin Wyckoff position 4(a)..2 with the coordinates x, x, 0;−x,−x, 1/2; 1/2 − x, 1/2 + x, 1/4; 1/2 +x, 1/2 − x, 3/4, x = 0.3028. During the phase transition, the tetragonal structure is transformedinto a cubic one with space group Fd3m(227), a = 7.147A. It is listed in the space-group tableswith two different origins.

1. If Origin choice 2 setting is used (with point symmetry 3m at the origin), then the siliconatoms occupy the position 8(a) 43m with the coordinates 1/8, 1/8, 1/8; 7/8, 3/8, 3/8 andthose related by the face - centring translations. Describe the structural distortion from thecubic to the tetragonal phase by the determination of (i) the displacements if the Si atoms inrelative and absolute units, and (ii) the lattice distortion accompanying the transition.

2. Repeat the calculations for the characterization of the phase transition using the Origin-choice1 description of the high-symmetry phase (cf. Exercise Data file for the structure data).

• Exercise 2.5.6. Ferroelastic phase transitions

(a) Lead phosphate Pb3(PO4)2 shows a phase transition from a paraelastic high-temperaturephase with symmetry R3m (No.166) to a ferroelastic phase of symmetry C 2/c (No.15). Using


the structure data given in the ExerciseData file and the tools of the Bilbao CrystallographicServer:

(a) characterize the symmetry reduction between the high- and low-symmetry phases (indexand transformation matrix);

(b) describe the structural distortion from the rhombohedral to the monoclinic phase by theevaluation of the lattice strain and the atomic displacements accompanying the phasetransition.

(b) Lead phosphate-vanadate Pb3(PVO4)2 shows a phase transition from a paraelastic high-temperature phase with symmetry R3m (No.166) to a ferroelastic phase of symmetry P21/c(No.14). Using the structure data given in the ExerciseData file and the tools of the BilbaoCrystallographic Server describe the structural distortion from the rhombohedral to the mono-clinic phase by the evaluation of the lattice strain and the atomic displacements accompanyingthe phase transition.

• Exercise 2.5.7. Order-disorder phase transitions in CuAu alloy

(i) A high-temperature form where Au and Cu are distributed statistically over the sites of anfcc packing: i.e. symmetry Fm3m, Wyckoff position 4a m3m 0, 0, 0 and a lattice parameterac = 3.83A;

(ii) A low temperature phase where Au and Cu are ordered in layers perpendicular to one of thefour-fold axes of Fm3m: the positions 0,0,0 and 1/2,1/2,0 are occupied by Cu, and 1/2,0,1/2and 0,1/2,1/2 by Au. The symmetry is reduced to P4/mmm, with lattice parameters at =2.86A, ct = ac.

(a) characterize the symmetry reduction between the high- and low-symmetry phases (index andtransformation matrix);

(b) describe the structural distortion from the rhombohedral to the monoclinic phase by the eval-uation of the lattice strain and the atomic displacements accompanying the phase transition.

• Exercise 2.5.8. CoU hettotype of the β−brass structure

Show that the crystal structure of CoU maybe interpreted as a slightly distorted CsCl (or β−brass,CuZn)-type structure. Using the structural data in the Exercise Data file, characterize the structuralrelationship between the CoU structure and CsCl structure.

• Exercise 2.5.9. (Muller, 2008): Crystal-structure relationships

1. HT -quartz and LT -quartz

Upon heating above 573 C the LT -quartz transforms to its HT form. Set up the correspondingBarnighausen tree that describes the symmetry relations between the two quartz forms. Whichadditional degree of freedom are present in the lower symmetry form?

The crystal data of LT -quartz are:

Unit Cell: 4.91 4.91 5.41 90. 90. 120Space group: P3221


position x y z

Si 3a 0.470 0 16

O 6c 0.414 0.268 0.286


Unit Cell: 5 5 5.46 90. 90. 120Space group: P6222


position x y z

Si 3c 0.5 0 0.5

O 6j 0.416 0.208 0.66666

2. α-AlPO4

The crystal data of α-AlPO4:

Unit Cell: 4.94 4.94 10.95 90. 90. 120Space group: P3121


position x y z

Al 3a 0.466 0 0.33333

P 3b 0.467 0 56

O1 6c 0.417 0.292 0.398

O2 6c 0.417 0.257 0.883

What is the symmetry relation of α-AlPO4 to quartz?

• Exercise 2.5.10. Hettotypes of the fluorite structure (Hahn & Wondratschek, 1984)

The structure of α-XOF (X=La, Y, and Pu) can be derived from that of cubic CaF2 (fluoritestructure) by splitting the fluorine positions into two: one for oxygen and one for fluorine, and byshifting the metal positions along c. By these changes the space-group symmetry is reduced.

The conventional basis a′, b′, c′ of α-XOF is a′ = 12 (a− b), b′ = 1

2 (a + b), c′ = c where a,b, c isthe basis of CaF2. Moreover, the conventional origin of α-XOF is shifted by p = 1

4 , 0,14 relative to

that of α-XOF (symmetry group P4/nmm (129)).

The coordinates of CaF2 are:

Ca 4a m3m 0, 0, 0 12 ,

12 , 0

12 , 0

12 0, 12 ,

12

F 8c 43m 14 ,

14 ,

14

14 ,

34 ,

34

34 ,

14 ,

34

34 ,

34 ,

14

34 ,

34 ,

34

34 ,

14 ,

14

14 ,

34 ,

34

14 ,

14 ,

34

Questions

(i) Display the relation between the old (a,b, c) and the new (a′,b′, c′) unit cell by means of adrawing.

(ii) Which is the crystal system of the new unit cell? Which is its centring type? (The lattice ofCaF2 is F -centred cubic (fcc), a = b = c, α = β = γ.)

(iii) Construct the transformation matrix P describing the change of the basis.

(iv) What is the volume of the new unit cell compared to that of the old one?

(v) What are the coordinates of the atoms of the CaF2 structure referred to the new coordinatesystem?

(vi) Can the structure of α-LaOF be considered as a hettotype (derivative structure) of the aristo-type (basic) structure of CaF2 ? (structure data of α-LaOF in Exercise Data file)

Tutorial on the application of the tools of the Bilbao Crystallographic Server in the study of group-subgroup phase transitions J.M. Perez-Mato and M.I. Aroyo Depto de Física de la Materia Condensada, Facultad de Ciencia y Tecnologia, Universidad del Pais Vasco, UPV/EHU, 48080 Bilbao, Spain. 1. In a phase transition, not all symmetry-breaks are equally possible! Isotropy Subgroups The Landau theory of symmetry-breaking structural phase transitions is based on the basic postulate that the symmetry break taking place in such transitions is due to the condensation (i.e. the change from zero to a non-zero amplitude) of one or a set of collective degrees of freedom that transform according to a single irreducible representation (irrep) of the space group of the high-symmetry phase (the so-called active irrep). These amplitudes {Qi, i=1,…,n} that become spontaneously non-zero in the low-symmetry phase, constitute the so-called order parameter, and the n-dimensional irrep describing its transformation properties is usually called the active irrep of the transition. Although Landau theory may not reproduce accurately the thermodynamic behaviour close to the transition points, its symmetry aspects and resulting restrictions are of much general validity. Its basic postulate of a single active irrep is fulfilled in most cases by group-subgroup phase transitions, even if they are discontinuous (for which its validity is not ensured). That the symmetry change in a transition is fully determined by a single active irrep, independently of the condensation of other degrees of freedom, is a very restrictive condition, which can be very powerful in the analysis and prediction of the structural and symmetry properties of these systems. Basic problem to solve: we know the high symmetry group and the active irrep, and we want to know the possible symmetries of the low symmetry phase.

Let us call G the space group of the high-symmetry phase, and τ the (n-dimensional) active irrep for a certain low-symmetry phase, and let be Q=(Q1,…,Qn) the n-dimensional order parameter transforming according to the irrep τ. By definition, if an operation g of the space group G is applied to the low-symmetry distorted structure, the amplitudes Q=( Q1,…,Qn) will change/transform according to the irrep τ:

Tτ(g) Q = Q’ (1) where the Tτ(g) is the nxn matrix associated by the irrep τ to the operation g. The system will keep the operation g among its symmetry operations, i.e. g will belong to the low-symmetry distorted phase, if the system is undistinguishable after the operation is applied, i.e. Q’=Q. Hence, all possible space groups, H (subgroups of G), for the low-symmetry phase, resulting from an active irrep τ, can be obtained from the condition that the space group operations h belonging to H should fulfill: Tτ(h) Q = Q (2) The possible subgroups of G, H, that can satisfy this invariance equation, and therefore be the symmetry of a distorted phase with τ as active irrep, are usually called isotropy subgroups (for the irrep τ). The “lost” operations g, such that: Tτ(g) Q = Q’ ≠ Q (3) transform the structure into configurations which are distinguishable from the original one, but energetically equivalent, and represent domain-related structures. In the case of a 1-dim irrep, eq. (2) is rather trivial, and only a single isotropy subgroup exists. It is formed by all operations of G for which Tτ(g)=1. The symmetry of the low-symmetry phase is therefore fully determined by the 1-dim irrep, and very simple to derive. Let us consider for instance the example of a material having I4/mmm symmetry and consider the possible phase transitions/symmetry breaks that the material can suffer, without changing its unit cell, i.e. without loosing any lattice translation. By definition, the matrix Tτ({E,l}) associated to a lattice translation {E,l} by an irrep τ is a diagonal matrix of the form:

where {k1,…ks} are the wave vectors of the star of the irrep. Therefore, according to eq. (2), if we want all lattice translations {E,l} maintained in the isotropy subgroup, the

wave vectors ki should be zero, i.e. the active irrep should have null wave vector, or in other words correspond to the point Γ of the Brillouin zone. The irreps of a space group at the Brillouin zone center are equivalent to those of the corresponding point group. The lattice translations have associated identity operators. We get then to the irreps of I4/mmm at the Γ point by looking at the irreps of the point group 4/mmm in the program POINT of this server: Table 1: Character table of the irreps of the point group 4/mmm obtained with the server tool POINT

Character Table

D4h(4/mmm) # 1 2 4 2h 2h' -1 mz -4 mv md functions

Mult. - 1 1 2 2 2 1 1 2 2 2 _

A1g Γ1+ 1 1 1 1 1 1 1 1 1 1 x2+y2,z2

A2g Γ2+ 1 1 1 -1 -1 1 1 1 -1 -1 Jz

B1g Γ3+ 1 1 -1 1 -1 1 1 -1 1 -1 x2-y2

B2g Γ4+ 1 1 -1 -1 1 1 1 -1 -1 1 xy

Eg Γ5+ 2 -2 0 0 0 2 -2 0 0 0 (xz,yz),(Jx,Jy)

A1u Γ1- 1 1 1 1 1 -1 -1 -1 -1 -1 _

A2u Γ2- 1 1 1 -1 -1 -1 -1 -1 1 1 z

B1u Γ3- 1 1 -1 1 -1 -1 -1 1 -1 1 _

B2u Γ4- 1 1 -1 -1 1 -1 -1 1 1 -1 _

Eu Γ5- 2 -2 0 0 0 -2 2 0 0 0 (x,y)

To determine the space groups resulting from each 1-dim irrep acting as active irrep, we have just to keep the operations with character 1: Table 2: Isotropy subgroups of I4/mmm for onedimensional irreps with k=0 irrep Operations conserved Isotropy subgroup A1g all operations I4/mmm A2g 1, 2z, 4z, -1, mz, -4 I4/m B1g 1, 2z, 2h (2x, 2y), -1, mz, mv (mx, my) Immm B2g 1, 2z, 2h’ (2xy, 2x-y), -1, mz, md (mxy, mx-y) Fmmm A2u 1, 2z, 4z, mh(mx, my), md(mxy,mx-y) I4mm B1u 1, 2z, 2h (2x, 2y), -4, md(mxy, mx-y) I-42m B2u 1, 2z, 2h’(2xy, 2x-y), -4, mv (mx, my) I-4m2

The irrep A1g is trivial and does not break the symmetry. We have taken into account that the columns in Table 1 correspond to equivalence classes with several operations. The lattice is labelled as F in the case of the irrep B2g, despite that the lattice is not changed, because in this case the operations of the orthorhombic group mmm are defined along the diagonal directions on the plane xy of the tetragonal I lattice. The I lattice described in a unit cell with (110) and (-110) as basic translations becomes an F lattice. Table1 contains a final column, headed with “functions”, which contains useful additional information. It lists functions of the coordinates x,y,z of a polar vector, or the coordinates Jx, Jy and Jz, of an axial vector, that transform according to the corresponding irrep. For instance, the product xy transforms according to the irrep B2g. But a shear strain εxy of the crystal transforms as a function xy. Therefore a shear strain εxy is a crystal magnitude of the I4/mmm crystal transforming according to B2g, and could be the order parameter for a phase transition I4/mmm --- Fmmm, i.e. a proper ferroelastic transition, with a switchable spontaneous strain. In general irreps of dimension n > 1 have more than one isotropy subgroup depending on the direction taken by the order parameter Q within its n-dimensional space. In our example, we have two 2-dim irreps, Eg and Eu. Let us consider the irrep Eg, whose matrices can be chosen in the following form (it straightforward to derive them from the fact that they should correspond to the transformation properties of the pair of functions xz, yz): Table 3: matrices of the irrep Eg of the point group 4/mmm, for a certain choice of the basis in the irrep space. 1,-1 2z, mz 4z+, -4z+ 4z-, -4z- 2x, mx 2y, my 2xy, mxy 2x-y, mx-y Eg 1 0

0 1 -1 0 0 -1

0 -1 1 0

0 1 -1 0

-1 0 0 1

1 0 0 -1

0 -1 -1 0

0 1 1 0

As the inversion -1 has associated the identity matrix, operations g and -1.g have the same matrix. Identity and inversion will always fullfill eq. (2) and operations 2z, mz, 4z+, -4z+, 4z- and -4z- will never fullfill it, independently of the direction of Q. The conservation of the other operations depends on the direction of Q. For instance if we consider that the Eg order parameter is of the form (0,Q), i.e. takes the direction (0,1) in the irrep space, 2x and mx will be conserved in the distorted structure, and the resulting isotropy subgroup is P2/m11. It is easy to check, considering the matrices above that the whole set of possible isotropy subgroups for Eg, depending on the order parameter direction, are: Table 4: Isotropy subgroups of the irrep Eg of I4/mmm, depending on the direction within the irrep space direction operations space group (0,1) 1, 2x, -1, mx C2/m (-b-c, a, c ; 0 0 0) (1,0) 1, 2y, -1, my C2/m (-a-c, b, c; 0 0 0) (1,1) 1, 2xy, -1, mxy C2/m (a-b+c, a+b, c; 0 0 0) (-1,1) 1, 2x-y, -1, mx-y C2/m ( a+b+c, -a+b, c; 0 0 0 arbitrary 1,-1 P-1 (a, b, a/2+b/2+c/2; 0 0 0)

The I centred unit cell must be transformed to a C centred cell for the monoclinic space groups and to a primitive unit cell for the tricilinic one, if expressed in a conventional setting. The b-axis is the monoclinic unique axis in the conventional setting used. The lattice does not change, it is only expressed in a conventional unit cell consistent with the direction of the monoclinic axis. The directions (0,1) and (1,0) for the order parameter are equivalent: they correspond to domain related directions. The four equivalent order parameter values obtained by applying the matrices of the lost operations are {(Q,0), (0,Q), (-Q,0), (0,-Q)}. This means mathematically that the isotropy subgroups corresponding to the order parameter directions (0,1) and (1,0) belong to the same conjugacy class. In general the number of domain-related configurations is equal to the index of the subgroup (4 in this case). Similarly for the solutions of type (1,1), there are four equivalent order parameter directions {(a,a), (-a,a), (-a,-a), (a,-a)}. We have then three possible isotropy subgroups for the irrep Eγ, which are not equivalent: C2/m (with monoclinic axis along x or y), C2/m (with monoclinic axis along the oblique directions (110) or (1-10) ) and P-1. With a a similar analysis for Eu we can complete the set of isotropy subgroups of I4/mmm for irreps with k=0: Table 5: Non-equivalent isotropy subgroups of I4/mmm for irreps with k=0

irrep Isotropy subgroup A1g I4/mmm A2g I4/m B1g Immm B2g Fmmm A2u I4mm B1u I-42m B2u I-4m2 Eg C2/m

C2/m P-1

Eu Imm2 Fmm2 Cm

It is interesting to compare this table with the set of all subgroups of I4/mmm that maintain the lattice, which can be obtained with the program CELLSUB of the Bilbao Crystallographic Server. The program CELLSUB: CELLSUB lists all possible subgroups of a given space group for a given k-index (or for a k-index smaller than a certain value). The program lists first the space group types, and then, for each space group type, it can distinguish the different conjugacy classes. The k-index ik is the “klassengleich” index, and indicates the multiplication factor relating the volume of the primitive cell of the subgroup with

respect to the primitive cell of the original structure. Thus, ik=1 implies conservation of the primitive unit cell, and therefore of the whole lattice. For ik= 1, CELLSUB lists for I4/mmm the following subgroups: Table 6: Subgroups of I4/mmm with ik=1

Clicking over each of them we can see the non-equivalent subgroups of each type. Thus for C2/m, we obtain: Table 7: Conjugacy classes of subgroups of I4/mmm of type C2/m with index ik=1

There are therefore 3 conjugacy classes of space groups of type C2/m. Classes 2 and 3 in the list correspond to the two non-equivalent isotropy subgroups discussed above. On the other hand, class 1 has the monoclinic axis along the z direction of the tetragonal setting, and it is not an isotropy subgroup. (Warning: the ordering and numbering of the classes done by the program is not fixed. Different runs can order the classes differently!) Inspecting the classes for all subgroup types, the number of non-equivalent subgroups with ik=1 is then 26, while the number of non-equivalent isotropy subgroups is only 13. Therefore, not all subgroups of I4/mmm compatible with its lattice can be reached through a phase transition with a single active irrep, as demanded by the Landau postulate. The Landau postulate restricts the number of possible symmetries to 13 of the 26 possible subgroups. The restriction of the possible symmetry break to only isotropy subgroups is very demanding, and can provide very valuable information. For instance, if a symmetry change between two phases does not comply with this condition, it indicates that the distorted phase contains at least two active irreps, i.e. the symmetry break can

only be explained considering at least two different order parameters transforming according to different irreps. It will therefore be highly probable that a sequence of at least two phase transitions will exist, due to the separate condensation of the two order parameters, and an intermediate phase is most likely (see below for an example). Some predictive power about the phase diagram of a compound or that of similar compounds can be achieved from the knowledge of the isotropy subgroups of a certain irrep. If a mutidimensional irrep is known to be active in a certain material, it may exhibit several phases with different symmetries corresponding to different isotropy subgroups of this active irrep, depending on the thermodynamic stabilization of the order parameter along different directions. Similarly, if the same irrep is active in a family of isomorphous compounds, the distorted symmetries observed in different compounds may be different, but will correspond to different isotropy subgroups of the same irrep. Let us consider for instance the example of the perovskites. Perovskites are known to have an intrinsic instability due to the softness of three degenerate rigid unit modes (modes tilting the octahedral framework and keeping the BX6 octahedra approximately rigid), which transform according to the 3-dim irrep R4+ of the parent symmetry Pm-3m. The isotropy subgroups of Pm-3m for irrep R4+ are the following: Table 8: Isotropy subgroups of Pm-3m for irrep R4+

In this list, the last triad for each space group indicates the restricted direction of the order parameter in its 3-dim space. For each of the possible isotropy subgroups, we have listed one perovskite compound that is known to have this symmetry in one particular phase. One can see that any of the possible symmetries is realized in some perovskite compound, except the lowest one, which corresponds to an arbitrary direction of the order parameter. There are perovskites, as CeAlO3, where the following phase transition sequence takes place as temperature is increased: I4/mcm --- Imma ---- R-3c ----- Pm3m Thus, the system is changing the direction of its R4+ order parameter from one phase to the next, through first-order phase transitions, yielding symmetries given by different isotropy subgroups of the list above.

The determination by hand of the isotropy subgroups for a given irrep is time consuming and tedious. The Bilbao Crystallographic Server does not have a tool to do this job directly, but the isotropy subgroup are listed (with some restrictions) in the book: H. T. Stokes and D. M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups (World Scientific, Singapore, 1988). or they can be obtained automatically using the program ISOTROPY (H. T. Stokes, D. M. Hatch, and B. J. Campbell, (2007), stokes.byu.edu/isotropy.html) , or some accompanying programs, all available in http://stokes.byu.edu/isotropy.html The use of ISOTROPY requires getting familiar with some specific program commands, which are necessary for running the program. A more direct form to get the information about the isotropy subgroups is to run the associated programs INVARIANTS or ISODISPLACE, which are more user-friendly, and can be executed by filling some self-evident menus. These two programs yield the list of isotropy subgroups of a chosen irrep as a preliminary result. The information is more complete in ISODISPLACE, but running this program, requires the ntroduction of some structure for the high-symmetry phase (which can be a fake one, since for deriving the list of isotropy subgroups the structure is not used at all). Some variations on the same problem: In many practical cases, we ignore the active irrep of the transition, but we may know some restrictive conditions on the symmetry of the distorted phase, which can be sufficient for deriving a restricted set of possible space groups H, and possible active irreps. Let us consider as an example a real case. Recently crystals that include both molecules of fullerene and cubane have been synthetized. They are known to crystallize at high temperatures in the Fm-3m space group with the disordered fullerenes centred at the site 4a (0 0 0) and disordered cubane molecules at 4b (½ ½ ½) (Nature Mat. 4, 764 (2005)). At low temperature the system exhibits a couple of phase transitions, as the molecules become ordered. From powder diffraction experiments, the symmetry of the final phase has been reported to be a primitive orthorhombic structure, with its lattice parameters satisfying the approximate relations a and b ≈ ac/√2, while c ≈ 2c, but its space group, and therefore its structure could not be determined (J. Phys. Chem B 113 2042 (2009)). Obviously, if we could restrict the symmetry of this phase to a minimal set of possible/probable space groups, we could have a better chance for succeeding in the interpretation and analysis of the diffraction diagram of this phase. We can start by determining all possible primitive orthorhombic subgroups of Fm-3m, fulfilling the unit cell relation that has been observed. We can use for that CELLSUB, provided that we know the k-index of the subgroup we are searching. This index can be derived in a straightforward manner comparing the volumes of the primitive unit cell of the two space groups: Fm3m: ac

3/4 Orthorhombic phase: ac

3

This means, the orthorhombic space group keeps only ¼ of the lattice translations, or in other words the primitive orthorhombic unit cell contains 4 formula units, while the Fm-3m structure only one. Hence ik= 4. The point group of the searched orthorhombic space groups can be 222, mm2 or mmm. To simplify the example, we are going to assume that this point group symmetry is the maximal mmm (but we could proceed similarly with the other two possible point groups). Restricted for ik=4, point group mmm, and no centring, CELLSUB lists a quite long list of subgroups:

Table 9: Subgroups (types) of Fm-3m with point group mmm, ik=4 and no centring.

N HM Symbol ITA index t-index k-index More info

1 Pnma 062 24 6 4 show...

2 Pbca 061 24 6 4 show...

3 Pbcn 060 24 6 4 show...

4 Pmmn 059 24 6 4 show...

5 Pnnm 058 24 6 4 show...

6 Pbcm 057 24 6 4 show...

7 Pccn 056 24 6 4 show...

8 Pbam 055 24 6 4 show...

9 Pcca 054 24 6 4 show...

10 Pmna 053 24 6 4 show...

11 Pnna 052 24 6 4 show...

12 Pmma 051 24 6 4 show...

13 Pban 050 24 6 4 show...

14 Pccm 049 24 6 4 show...

15 Pnnn 048 24 6 4 show...

16 Pmmm 047 24 6 4 show...

But not all of them will have unit cell parameters consistent with the experimental values. By clicking on “show” we get the different non-equivalent classes of subgroups of this type. For instance, for Pbca, there is a single class as shown by the output of CELLSUB (warning: the ordering of the classes is not fixed, and different runs of the program will yield in general different orderings):

Table 10: conjugacy classes of subgroups of Fm-3m of type Pbca Classification of the subgroups of type Pbca(61) of group Fm-3m(225) with index 24

For the group G = Fm-3m there are 2 different subgroups Hj isomorphic to H = Pbca of index 24. These subgroups are distributed in 1 class of conjugate subgroups with respect to the group G. In the tables below each table corresponds to one class. For each class are given the chains corresponding to the different subgroups in the class, and the obtained transformation matrices. The list with the chains and transformation matrices that give identical subgroups can be seen by clicking on the button in the column Identical of the table.

Class 1

Check Chain [indices] Chain with HM symbols Transformation Identical

1 225 139 069 064 061 [3 2 2 2]

Fm-3m > I4/mmm > Fmmm > Cmce > Pbca

0 0 1 0 -1 0 0 0 0 -1 0 0

2 225 139 069 064 061 [3 2 2 2]

Fm-3m > I4/mmm > Fmmm > Cmce > Pbca

0 1 0 0 -1 0 0 0 0 0 1 0

The transformation matrix relating the two groups is then (-b,-c,a; 0 0 0) or equivalent ones, with the unit cell parameters coinciding with those of the cubic cell. Therefore this space group can be discarded.

On the other hand, if we consider subgroups of type Pnma, there are four classes of Pnma subgroups, and one of these classes is listed as:

Table 11: One of the conjugacy classes of subgroups of Fm-3m of type Pnma

Check Chain [indices] Chain with HM symbols Transformation Identical

31 225 139 137 059 062 [3 2 2 2]

Fm-3m > I4/mmm > P42/nmc > Pmmn > Pnma

0 1/2 1/2 1/4 2 0 0 3/4 0 1/2 -1/2 1/2

32 225 139 071 059 062 [3 2 2 2]

Fm-3m > I4/mmm > Immm > Pmmn > Pnma

0 1/2 1/2 1/4 0 -1/2 1/2 0 2 0 0 3/4

33 225 139 137 059 062 [3 2 2 2]


0 1/2 -1/2 1/2 0 1/2 1/2 1/4 2 0 0 1/4

34 225 139 137 059 062 [3 2 2 2]


0 1/2 -1/2 1/2 0 1/2 1/2 1/4 2 0 0 3/4

35 225 139 137 059 062 [3 2 2 2]


0 1/2 1/2 1/2 0 -1/2 1/2 1/4 2 0 0 1/4

36 225 139 129 059 062 [3 2 2 2]

Fm-3m > I4/mmm > P4/nmm > Pmmn > Pnma

2 0 0 3/4 0 1/2 -1/2 1/4 0 1/2 1/2 0

37 225 139 071 059 062 [3 2 2 2]


2 0 0 3/4 0 1/2 1/2 1/4 0 -1/2 1/2 0

38 225 139 137 059 062 [3 2 2 2]


2 0 0 1/4 0 1/2 -1/2 1/2 0 1/2 1/2 1/4

39 225 139 137 059 062 [3 2 2 2]


2 0 0 1/4 0 1/2 1/2 1/2 0 -1/2 1/2 1/4

40 225 139 129 059 062 [3 2 2 2]


0 1/2 1/2 0 2 0 0 1/4 0 1/2 -1/2 1/4

41 225 139 129 059 062 [3 2 2 2]


0 -1/2 1/2 0 2 0 0 1/4 0 1/2 1/2 1/4

42 225 139 071 059 062 [3 2 2 2]


0 -1/2 1/2 0 2 0 0 3/4 0 1/2 1/2 1/4

The parameters of the Pnma orthorhombic cell for this class are therefore a=2ac, b= c=ac/√2, which are consistent with the orthorhombic cell observed, if a permutation of the parameter labels is done.

Inspecting the classes for all the subgroups in the list, it is straightforward to conclude that only six symmetries Pnma, Pmmn, Pccn, Pmma, Pccm and Pmmm with transformations matrices listed below (or equivalent), are consistent with the experimental observation.

Table 12: Possible centrosymmetric orthorhombic subgroups of Fm-3m fulfilling the lattice metrics observed in the orthorhombic phase of fullerene-cubane crystals.

Pnma (2a, b/2+c/2, -b/2+c/2; ¾, ¼ , 0) Pmmn (a/2-b/2, a/2+b/2, 2c; ¼ , 0, ¾) Pccn (a/2-b/2, a/2+b/2, 2c; ¼ , 0, ¾) Pmma (2a, b/2+c/2, -b/2+c/2; ½ , 0 , 0) Pccm (a/2-b/2, a/2+b/2, 2c; 0, 0, 1/2)

Pmmm (a/2-b/2, a/2+b/2, 2c; 0, 0, 0)

This list could be further reduced if we assume that the searched symmetry should correspond to an isotropy subgroup (see exercise 2 below), i.e. an active irrep can explain the corresponding symmetry break, but this lead us into a new type of problem, which we treat in the following section.

Exercise 1: A structure has symmetry Pnma. At lower temperatures, a phase transition happens, and diffraction experiments show that superstructure reflections at points (h, k, l+ ½) appear, indicating the duplication of the c parameter, while keepin an orthorhombic lattice. Assuming a group-subgroup related transition and using CELLSUB, predict the only possible space group of this low-temperature phase, and the transformation matrix relating it with the parent space group Pnma.

2. The inverse Landau problem: The problem: We know the symmetry break and we want to identify the active irrep

From the discussion above, it should be clear that the most important information required for the characterization of a certain phase transition is the identification of its active irrep. In most cases it is sufficient to know the high and low space groups to identify it, but for some symmetry breaks this may not be sufficient. The ISOTROPY website (stokes.byu.edu/isotropy.html) includes the program COPL, which does this identification, and also provides the irreps of all additional spontaneous secondary variables. In the Bilbao Crystallographic Server, the program SYMMODES can also give this information as a by-product. Secondary spontaneous variables: In the low symmetry phase, apart from the order parameter and quantities transforming according to the active irrep, variables or degrees of freedom transforming according to other irreps can also condense or become spontaneous. The only requirement is that they are compatible with the low-symmetry space group. This is the realization of the Von Neumann principle: any variable/degree of freedom compatible with the symmetry of the crystal is allowed and will therefore in general have a non-zero value. Using the concept of isotropy subgroup, we can say that any quantity with transformation properties given by an irrep having an isotropy subgroup containing the group H of the distorted phase will be spontaneous in the transition (i.e. it will change from zero to non-zero values in the distorted phase). Therefore, while the active irrep

must have H as an isotropy subgroup, the irreps associated to spontaneous secondary variables have in general an isotropy subgroup which is a supergroup H. Let us consider the following case: I4/mmm -------- C2/m (-b-c, a, c ; 0 0 0) From the example discussed above, we know that the active irrep for this symmetry change is Eg (k=0). If introduce this symmetry change, including the transformation matrix, in the program COPL (stokes.byu.edu/isotropy.html), the following result is obtained: Table 13: Output of COPL for the symmetry break I4/mmm --- C2/m (-b-c,a,c; 000) ------------------------------------------------------------------------------------ COPL, Version 1.0, August 2001 Written by Harold T. Stokes and Dorian M. Hatch Brigham Young University Parent: 139 D4h-17, I4/mmm, I4/m2/m2/m Subgroup: 12 C2h-3, C2/m, C12/m1, unique axis b, cell choice 1 Lattice vectors: 0 -1 -1 1 0 0 0 0 1 origin: 0 0 0 Irrep Dir Subgroup Size GM1+ (a) 139 I4/mmm 1 GM2+ (a) 71 Immm 1 GM5+ (a,0) 12 C2/m 1 ----------------------------------------------------------------------------------- The program lists a set of irreps, a “direction” or subspace within the irrep space and their isotropy subgroup. The final column headed “size” indicates the multiplication of the primitiva unit cell of the corresponding isotropy subgroup relative to the parent high-symmetry space group (i.e. the klassengleiche index ik). We have then the irrep GM5+ as the active irrep with the actual observed symmetry as isotropy subgroup, and then a secondary irrep GM2+ with an isotropy subgroup Immm, which necessarily must be a supergroup of C2/m. In addition, the trivial irrep GM1+ always appear as possible symmetry of secondary variables that are allowed to have non-zero values both at the distorted phase, and also at the parent phase. The label GM is being used to indicate that they are irreps at the Brillouin zone centre (Gamma point)), as the translational symmetry is not broken. GM5+ must be the irrep labelled Eg in the Table above, while GM2+ should be the irrep labelled above as B1g.

The reason for the existence of secondary spontaneous quantities transforming according to the irrep B1g becomes obvious if we use SUBGROUPGRAPH or CELLSUB to construct the graph of minimal subgroups connecting the two end space groups: Figure 1: Graph of minimal subgroups connecting the space group I4/mmm and its subgroup C2/m (-b-c, a, c ; 0 0 0) (or equivalent), obtained with SUBGROUPGRAPH. (to obtain this graph one has to choose the correct conjugacy class among the three classes given by the program)

The group Immm, which is the isotropy subgroup of B1g, as shown in the previous section, is indeed a supergroup of C2/m. Therefore, B1g variables are compatible with the symmetry of the distorted phase, and therefore can be non-zero in this phase. They will be secondary spontaneous variables. Table 1 indicates that x2-y2 and (xz, yz) transform according to B1g and Eg, respectively. This means that the shear strain components (εxz, εyz) have the symmetry properties of the active irrep, and could in principle be identified with the order parameter of the transition, while the strain difference εxx-εyy transforms according to B1g, being zero in the parent phase, and becoming spontaneous, as a secondary variable, in the C2/m phase. We have then a proper or pseudoproper ferroelastic transition (i.e. some strain component(s) have the symmetry of the order parameter), and we expect the softness at the transition of the stiffness coefficients corresponding to εxz and εyz, i.e. the elastic constants C55 and C66. The shear strain εyz (direction (0,1) in Eg) should become spontaneous in the C2/m phase, deviating the monoclinic angle from 90º, as a primary order parameter effect. [Equivalently we could consider the domain related distortion with εxz (direction (1,0) in Eg)]. On the other hand, the quantity εxx-εyy is zero in the parent phase (cell parameters a and b are equal), while in the C2/m phase εxx-εyy ≠ 0 (cell parameter a and b become different), but as a secondary (weaker) effect. As the index of C2/m is 4, four equivalent domain-related structures must exist. These domains should be distinguishable by the value of the order parameter (see eq. (3) above). From the structure having a spontaneous shear strain yz εο and zero shear strain xz: (0,εο), we obtain the values of the spontaneous shear strains εxz and εyz in the other domains applying lost symmetry operations according to the matrices listed in

Table 3: {2z|000} → (0,-εο), {4z-|000} → (εο, 0), {4z+|000}→ (-εο, 0). Therefore, the primary spontaneous monoclinic strain can have opposite values in different domains, and can also have two different orientations with respect to the parent tetragonal phase, corresponding to the monoclinic axis being along x or y, as listed in Table 4 above. Note: One must be take into account that the computer-adapted irrep labels used by COPL and ISOTROPY differ from those used in some programs of the Bilbao server, but they coincide with those employed by the programs SYMMODES and AMPLIMODES of the Bilbao server, as both of them use an adapted version of COPL kindly provided by H. Stokes. The program SYMMODES (J. Appl. Cryst. (2003). 36, 953): SYMMODES is designed to provide a basis of symmetry-adapted displacive modes for describing any displacive distortion relating a group-subgroup-related pair of phases. These modes, which are allowed to condense in the distorted phase are both primary (with their symmetry given by the active irrep) and secondary (with their symmetry given by secondary irreps having isotropy subgroups containing the observed space group). The program has been developed in collaboration with H. Stokes (Brigham Young University), and uses the program COPL. For each allowed irrep (either active/primary or secondary), and restricted to the necessary irrep subspace, SYMMODES lists a complete set of symmetry-adapted displacive modes for the Wyckoff orbit types chosen by the user. In contrast with COPL, the program only requires as initial input the space group types of the high and low symmetry phase, G and H, and their index. The program then determines all the different classes of subgroups H of G, and the user has to choose the one relevant for the problem. A direct link to SUBGROUPGRAPH to obtain the graph of minimal subgroups is also available. Once the relevant class of subgroups H is decided, the user chooses the set of Wyckoff positions, for which the basis of symmetry-adapted modes is desired. Note that the choice of unit cell for the H-group done by the program may not coincide with the one wished, but an equivalent one should appear on the list. The program also allows to introduce a specific desired matrix transformation (If the matrix is not consistent with the subgroup, the program will write a warning and will stop). For our I4/mmm example, we use this option to introduce the specific matrix transformation we have been considering above for C2/m, and the following menu appears:

Table 14: First Menu of SYMMODES for the pair I4/mmm --- C2/m (-b-c, a, c ; 0 0 0)

We choose the Wyckoff position 8i to see the displacive modes that will be triggered by the transition for atoms on sites of this type. The output is the following: Table 15: Output of SYMMODES for the pair I4/mmm -- C2/m (-b-c,a,c;000) and the Wyckoff positions 8i.

This ouput gives first a “Symmetry Modes Summary” listing the irreps corresponding to the displacive modes allowed for the chosen Wyckoff position(s), and highlighting in bold letters the active irrep, i.e. the one having C2/m as isotropy subgroup. We can see that modes of symmetry GM1+, GM2+ and GM5+ are allowed for 8i sites in the distorted structures, in agreement with Table 13. The output then lists for each irrep the corresponding isotropy subgroup with its transformation matrix, the eventual restriction on the direction within the irrep space (this is the data headed as “order parameter”), and a set of linearly independent modes having this irrep as symmetry.

The form of the modes are given as a column of triads indicating the correlated displacements of the atoms of the Wyckoff orbit extended to the primitive unit cell of the distorted phase. As the distorted phase in our example has the same primitive unit cell (ik = 1) than the parent phase, the set of listed atomic positions is limited to 4. The other atoms in the structure belonging to the same orbit are related by lattice translations to those listed in the output and their mode displacements will have the same displacements. The components of the atomic displacements describing the modes are given in the setting of the parent structure. Modes are not normalized and, in the case of existing several modes for the same irrep, they are not orthogonalized. We can see in Table 15 that for 8i atoms, there is only a single mode with the symmetry of the active irrep. It involves displacements along the z-axis of only 2 of the 4 atoms within a primitive unit cell. Only atoms with non-zero y-component have non-zero displacements. Atoms with opposite y-components have opposite displacements along z. Modes GM1+ and GM2+ are restricted to the xy plane. As it should be, GM1+ displaces the atoms in such a way that their components will maintain the I4/mmm symmetry relation described in the first column. In contrast, the mode GM2+ mode breaks this relationship: (x,0,0) and (0,x,0) sites have (0,1,0) and (0,-1,0) displacements respectively. From this output one expects that the atoms 8i of type (0,x,0) in the I4/mmm structure and their related ones by the I centring will suffer displacements along the c axis much stronger than along the y axis, the only two directions allowed (always working in the tetragonal setting), because the z-displacements correspond to the primary mode with the symmetry of the active irrep, and will be in principle related with the transitions mechanism. The atoms with positions (x,0,0) and the related ones by the I centring, on the other hand, will only have secondary displacements along the x-direction, that maintain the symmetry relation among them. SYMMODES also allows to examine the splitting of the Wyckoff orbit due to the symmetry decrease, by means of a link to the program WYCKSPLIT of the server:

Table 16 : Splitting of orbits 8i for the symmetry break I4/mmm -- C2/m (-b-c,a,c;000)

Wyckoff position(s) Group Subgroup More...

1 8i 4g 4i

Table 17: Detailed splitting of orbits 8i for the symmetry break I4/mmm -- C2/m (-b-c,a,c;000)

The last column in Table 17 describes the generic positions of the two splitted orbits in the subgroup basis, while the third column indicates the corresponding position in the higher I4/mmm structure. One can see that the first 4 positions forming the 4g1 orbit do not really break the tetragonal symmetry; their only allowed variation of the y component (in the subgroup basis) is also compatible with the tetragonal space group. On the other hand, the other four positions forming the 4i1 orbit introduce an additional degree of freedom by breaking the correlation between the x and z components (in the subgroup basis). The number of symmetry free components in the two orbits is 3. This should coincide with the number of distinct modes listed in Table 15. In general, the number of modes should be equal to the number of degrees of freedom for the 8i atomic positions in the subgroup, and this is the number of free parameters in the resulting Wyckoff orbits in the subgroup symmetry. Even if we are not interested in the displacive modes present in the distorted phase, SYMMODES can still be used to obtain important information on the active and secondary irreps relevant for a given symmetry break. The information is more complete than in COPL as the transformation matrices for each isotropy subgroup are also listed. But a caution note should be added: SYMMODES only lists the compatible irreps which can be associated to displacive modes of the Wyckoff positions chosen. If we are interested in all the irreps compatible with the symmetry reduction, it is convenient to choose a general Wyckoff orbit, in order to have as many symmetry allowed displacive modes as possible. In most cases this will be sufficient to get all compatible irreps, but in some cases some compatible irreps may not be involved in any atomic displacements, and then it will not be listed by SYMMODES. Exercise 2: Using SYMMODES show that only two of the six space groups listed in Table 12 are isotropy subgroups of Fm-3m, and therefore only for these two subgroups a single active irrep can be identified. These two space groups should be therefore the two first obvious choices for the symmetry of the orthorhombic phase of the fullerene-cubane crystals.

Representative Subgroup Wyckoff position No group basis subgroup basis name[n] representative

1 (x, 0, 0 ) (0, x, 0 ) (0, y1, 0 ) 2 (-x, 0, 0 ) (0, -x, 0 ) (0, -y1, 0 ) 3 (x+1/2, -1/2, -1/2 ) (1/2, x+1/2, 0 ) (1/2, y1+1/2, 0 ) 4 (-x+1/2, -1/2, -1/2 ) (1/2, -x+1/2, 0 )

4g1

(1/2, -y1+1/2, 0 ) 5 (0, x, 0 ) (-x, 0, -x ) (x2, 0, z2 ) 6 (0, -x, 0 ) (x, 0, x ) (-x2, 0, -z2 ) 7 (1/2, x-1/2, -1/2 ) (-x+1/2, 1/2, -x ) (x2+1/2, 1/2, z2 ) 8 (1/2, -x-1/2, -1/2 ) (x+1/2, 1/2, x )

4i1

(-x2+1/2, 1/2, -z2 )

Exercise 3: As a continuation of exercise 1, check using SYMMODES that the space group you have determined in exercise 1 is an isotropy subgroup. Identify the wave vector and the label of the active irrep of the transition, and check by hand that indeed this irrep yields the superlattice that has been observed. Exercise 4: A compound with parent symmetry Pmmm exhibits a sequence of phase transitions into two phases with superstructures having their parameter b multiplied by 4 and 3, respectively. The two phases are due to a distortion which varies its wave vector along b*, but keeps the same rotational symmetry given by the same small irrep (only the modulus of the wave vector changes). The space group of the phase with b-parameter = 4b has been identified as Pmmb keeping the same setting (or Pmma in conventional setting). The experiments indicate that the phase with the b-parameter = 3b is also orthorhombic. Using SYMMODES and CELLSUB predict the space group of this second phase. Show that it will be ferroelectric with the polar axis along the z axis (in the original setting). Exercise 5: Monoclinic phase of the system PbZr1-xTixO3 Consider the perovskite-like ferroelectric system PbZr1-xTixO3 (PZT). Some measurements have revealed a monoclinic phase (with no cell multiplication) between the previously established tetragonal (P4mm) and rhombohedral (R3m) regions in its phase diagram as a function of x. Both phases, P4mm and R3m, are ferroelectric distorted phases of the perovskite, due to the condensation of a polar mode of symmetry at k=0. The perfect perovskite structure PbBO3 is cubic Pm-3m (Z=1) with positions: pb 1b, B 1a, O 3d. (i) If you have difficulties to derive directly the index of the subgroups P4mm and R3m with respect to their parent space group Pm-3m, use POINT to obtain the order of the point groups m-3m, 4mm and 3m, and from them, obtain the t-index. This t-index multiplied by the relevant k-index will give you the necessary index to use in SYMMODES. (ii) Using SYMMODES obtain a valid transtormation matrix for the pairs Pm-3m --P4mm, and Pm-3m -- R3m, and check that indeed the two phases, P4mm and R3m, can be assigned to the same active irrep, for two different directions of the order parameter. Take notice of the active irrep and these directions. (iii) A reasonable assumption about the detected monoclinic structure is that it must be some bridging phase with the order parameter changing between the two special directions obtained in (ii). Its symmetry would then be given by a common subgroup of the tetragonal and rhomboedral space groups. Use COMMONSUBS to predict under this assumption the space group of the monoclinic phase. Take notice of the transformation matrix relating it with the space group P4mm. (v) From the transformation matrices for the pairs Pm-3m --- P4mm and P4mm --- monoclinic space, obtain the transformation matrix relating Pm-3m and the monoclinic space group. Using SYMMODES again demonstrate that the active irrep of the postulated monoclinic space group is indeed the same as for the other two phases. Compare the order parameter direction with those obtained in (ii) (vi) Use TRANSTRU to derive a starting structural model of the monoclinic phase (with a single mixed site for the Zr/Ti atoms), which you could use as the starting point for a refinement of the structure.

3. Symmetry breaks with several active irreps As already mentioned above, there may be some symmetry breaks that do not fulfill the Landau postulate of having a single active irrep. In other words, the symmetry reduction cannot be explained by a single irrep, as the symmetry of the distorted phase is not an isotropy subgroup of the parent space group. If the transition is discontinuous or first order, there are indeed mechanisms that can explain the breaking of the Landau postulate and predict the simultaneous condensation of two irreps. However, these situations are rare, and the most plausible explanation for such cases is that the condensation of the two active irreps is in fact stepwise and an intermediate phase has been overlooked Independently of the existence or not of an intermediate phase, the existence of two active irreps implies a scenario, where several degrees of freedom of the structure are independently unstable in the high-symmetry configuration. This is bound to produce rich phase diagrams. Let us consider the example of the Aurivillius compound SrBi2Ta2O9. The compound is known to have a tetragonal I4/mmm phase at high-temperatures with Z=2, and at room temperature crystallizes in the subgroup Cmc21 (c, a-b,a+b; ¼,¼,0) with Z=4. Using SYMMODES we can obtain the following information about the intermediate subgroups relating this group-subgroup pair (note that there are two Cmc21 classes with the same type of cell transformation, and only distinguishable by the origin shift. So one has to take care of choosing the correct one) Figure 2: Graph of minimal subgroups connecting the space group I4/mmm and its subgroup Cmc21 (c, a-b,a+b; ¼,¼,0) (or equivalent), obtained with SUBGROUPGRAPH.

(Note: The group Cmca (N. 64) is labelled Cmce in the new convention of the last version of the International Tables of Crystallography) For each of the subgroups, we have indicated in Figure 2 the corresponding active irrep, if existing. The graph shows then that all subgroups are isotropy subgroups for some irrep, except for the actual observed symmetry. The room temperature phase requires therefore several active irrep. How many? Just two. The effective symmetry resulting from the presence of two of the distortions is the intersection of the two corresponding isotropy subgroups, and this is given in the graph by their first common subgroup. Hence, it is sufficient to consider any pair of the distortions Γ5

-, X3- and X2

+, to reach the symmetry Cmc21. Only the irrep Γ4+ cannot be relevant for producing the observed symmetry Cmc21. i.e. its addition to any of the other ones, does not decrease further the symmetry. We have therefore three possible irreps among which two must be the active ones. Symmetry considerations cannot go further. Only a quantitative analysis of the structure, ab-initio calculations, or the thermal behaviour of the system can indicate which of the three irreps are really the active irreps in this specific case. The distortions having the symmetry of the active irrep are expected to have larger amplitudes, and be the ones that are unstable in the high-symmetry configuration. For instance, a decomposition of the Cmc21 structure of SrBi2Ta2O9 in terms of symmetry-modes, done with AMPLIMODES, shows that the distortions of symmetry Γ5

- and X3- have much

larger amplitude than the one for the irrep X2+. Furthermore, the X3

- distortion is significantly larger than the Γ5

-. Therefore we can identify X3- and Γ5 as the active

irreps. The additional X2+ spontaneous distortion present in the experimental structure

can be considered a secondary effect, essentially induced by the presence of the other two primary distortion modes As the X3

- distortion is the largest one we expect it to be the one that thermalizes to zero value at higher temperatures, and therefore we can infer that an intermediate phase with symmetry Cmcm is highly probable, corresponding to the presence of only this active irrep. Indeed this intermediate phase has been observed! The other mode Γ5

- is the irrep Eu in the notation of the Tables of the program POINT. It corresponds to the symmetry properties of a vector on the plane xy, and therefore is a polar instability, that can produce a spontaneous polarization (its isotropy subgroup Fmm2 is polar along the (1,1,0) direction in the tetragonal setting). This Γ5

- polar distortion is then the fundamental mechanism causing the lost of the inversion centre in the room-temperature structure, and the ferroelectric properties of the room temperature phase Exercise 6: A compound has Pnma symmetry at high temperatures and has space group P1211 at low temperatures, keeping essentially the same lattice, except for some strain. Using SYMMODES obtain the graph of minimal subgroups relating both symmetries. Check that at least two irreps must be active to explain the symmetry of the distorted structure. Indicate the possible pairs of active irreps in the distorted phase. Indicate the possible (alternative) symmetries of a probable intermediate phase. Exercise 7: The multiferroic BiFeO3 has symmetry R3c with Z=6 at room-temperature, having at high temperatures the cubic perovskite configuration (Fe: 1a, Bi: 1b, O: 3d). (i) Using SYMMODES, show that the room temperature phase of BiFeO3 has two active irreps, one being the usual R4+ mentioned above, and the other one is the one discussed in exercise 5, present in BaTiO3.

(ii) Some publications have reported an intermediate phase with symmetry I4/mcm. Crosscheck the consistency of this intermediate symmetry with your previous results. (iii) With TRANSTRU produce a starting structural model to refine the structures I4/mcm and R3c of BiFeO3, indicating the refinable coordinates of the constructed asymmetric unit.

Hands-on exercises on the use of some tools of the Bilbao Crystallographic Server for the analysis of Structural Phase Transitions.

1. Tilting of octahedra in SrTiO3The perovskite SrTiO3 has a tetragonal phase with space group I4/mcm with therelation (a+b, -a+b, 2c; 1/2, 1/2, 1/2) with respect to its parent structure of symmetryPm-3m (Sr: 1a, Ti: 1b, O: 3c). (i) Explore the possible symmetries that would be possible for this change of latticeusing SUBGROUPS. Show that the space group realized is one of the two maximaltetragonal symmetries.(ii) Using the provided cif files of the I4/mcm and parent phases of SrTiO3, obtain withSTRUCTURE RELATIONS the transformation relating both space groups. Do the sameusing PSEUDO and compare the results.(iii) Using SYMMODES, identify the wave vector of the order parameter of the I4/mcmphase and its irrep.(iv) Using SUBGROUPS, derive all possible symmetries that can be realized by thisorder parameter. (files required: icsd_23076_SrTiO3_parent.cif and icsd_56718_SrTiO3_50K.cif)

2. Limiting the possible symmetries of a phaseA structure has symmetry Pnma. At lower temperatures, a phase transition happens,and diffraction experiments show that superstructure reflections at points (h, k, l + 1/2) appear, indicating the duplication of the c parameter, while keeping an orthorhombiclattice.(i). Assuming a group-subgroup related transition and using SUBGROUPS, predict theonly two possible space groups of this low-temperature phase, and the transformationmatrix relating it with the parent space group Pnma. (ii). What is the wave vector associated with the order parameter of this transition?(iii) Using SYMMODES determine if the space groups determined in (i) are isotropysubgroups of an irrep, and in each case, identify the label of the active primary irrepof the transition. (iv) From the output of SYMMODES, in both cases identify the irrep associated with asecondary polar distortion mode(v) Determine using SUBGROUPS all the possibles symmetries that could happen in aphase transition with this wave vector, if it fulfills the Landau assumption, and theorder parameter transforms according to a single irrep.

3. Orthorhombic phase of fullerene-cubane crystalsFullerene-cubane crystals have been reported to crystallize at high temperatures in theFm-3m space group with the disordered fullerene molecules centred at the site 4a (0 00) and disordered cubane molecules at 4b (1/2 1/2 1/2) (Nature Mat. 4, 764 (2005)).At low temperature the system exhibits a couple of phase transitions, as the moleculesbecome ordered. From powder diffraction experiments, the symmetry of the finalphase has been reported to be a primitive orthorhombic structure, with its latticeparameters satisfying the approximate relations a and b ≈ ac/√2, while c ≈ 2ac, but itsspace group, and therefore its structure could not be determined (J. Phys. Chem B 1132042 (2009)). Obviously, if we could restrict the symmetry of this phase to a minimalset of possible/probable space groups, we could have a better chance for thisdetermination.

(i) Assuming that the searched orthorhombic structure is centrosymmetric and usingSUBGROUPS obtain the six possible space groups for this phase. (ii) Show that the wave vector of the order parameter must be (0,0,1/2) or equivalentby the cubic symmetry.(iii) Using SYMMODES show that only two of the six space groups are isotropysubgroups of Fm_3m, and therefore only for these two subgroups a single active irrepcan be identifed. These two space groups should be therefore the two first obviouschoices for the symmetry of the orthorhombic phase of these fullerene- cubanecrystals. Notice that the active irrep for these two space groups is the same one.

4. Monoclinic phase of the system PbZr1-xTixO3

Consider the perovskite-like ferroelectric system PbZr1-xTixO3 (PZT). Measurements haverevealed a monoclinic phase (with no cell multiplication) between the previouslyestablished tetragonal (P4mm) and rhombohedral (R3m) regions in its phase diagramas a function of x. Both phases, P4mm and R3m, are ferroelectric distorted phases ofthe perovskite, due to the condensation of a polar mode at k=0. The perfect perovskitestructure PbBO3 is cubic Pm_3m (Z=1) with positions: Pb 1b, B 1a, O 3d.(i) Using SYMMODES obtain a valid transtormation matrix for the pairs Pm_3m>P4mm,and Pm_3m>R3m, and check that indeed the two phases, P4mm and R3m, can beassigned to the same active irrep, for two different directions of the order parameter.Take notice of the active irrep and these directions.(ii) A reasonable assumption about the detected monoclinic structure is that it must besome bridging phase with the order parameter changing between the two specialdirections obtained in (a). Its symmetry would then be given by a common subgroupof the tetragonal and rhombohedral space groups, corresponding to the same orderparameter, i.e. the same irrep, but for a more general direction. Use SUBGROUPS topredict under this assumption the space group of this monoclinic phase. (iii) Use TRANSTRU to derive a starting structural model of the monoclinic phase (witha single mixed site for the Zr/Ti atoms), which you could use as the starting point for arefinement of the structure.

5. BiFeO3 : A phase with two order parametersThe multiferroic BiFeO3 has symmetry R3c with Z=6 at room-temperature, having athigh temperatures the cubic perovskite configuration (Fe: 1a, Bi: 1b, O: 3d).(i) Using SYMMODES, show that the room temperature phase of BiFeO3 has two activeirreps.(ii) Some publications have reported an intermediate phase with symmetry I4/mcm.Using SUBGROUPS crosscheck the plausibility of this intermediate symmetry asoriginating from the same order parameters (one or both) which are active in the R3cphase.(iii) With TRANSTRU produce a starting structural model to refine the structuresI4/mcm and R3c of BiFeO3, indicating the refinable coordinates of the constructedasymmetric unit.

Tutorial on the use of the program PSEUDO of the Bilbao Crystallographic Server (www.cryst.ehu.es) J. Manuel Perez-Mato, C. Capillas, Mois I. Aroyo and D. Orobengoa, Dept. de Fisica de la Materia Condensada, Facultad de Ciencia y Tecnologia, Universidad del Pais Vasco, Apto. 644, 48080 Bilbao, Spain. (see also E. Kroumova, M. I. Aroyo, J. M. Perez-Mato, S. Ivantchev, J. M. Igartua and H. Wondratschek, J. Appl. Cryst. (2001) 34, 783) If a structure, with space group H, is such that all its atomic positions ri can be described as ri

o+ ui, with ui being small displacements, while the virtual atomic positions rio have

a higher symmetry described by a supergroup G > H, we say that the structure has pseudosymmetry G, or is pseudosymmetric for the space group G. The detection of pseudosymmetry can be very useful for several purposes:

- Prediction of symmetry and structure of some other phase of the material - Prediction of phase transitions. - Identification of ferroic materials: ferroelectrics, ferroelastics,… - Determination of an optimized virtual parent structure - Detection of false symmetry assignments (overlooked symmetry) - Identification of the space group symmetry of a theoretical structure calculated

without symmetry restrictions (ab-initio calculations). The Bilbao Crystallographic server provides three tools for pseudosymmetry search in a given structure: PSEUDO, DOPE and BPLOT. PSEUDO is a new version of a previous program (J. Appl. Cryst. (2001). 34, 783), which examines the possible pseudosymmetry for a given supergroup G of H by applying on the structure the coset representatives of the left coset decomposition of G with respect to the observed symmetry H. The program then checks the approximate coincidence of the transformed structures with the original one through an identification of the atomic displacements relating the two configurations. If these displacements are smaller than a tolerance, the structure is flagged as pseudosymmetric. DOPE works similarly as PSEUDO, but it does the quantitative comparison of the original and transformed structures by checking the degree of superposition of their virtual electronic densities, obtained from the standard atomic diffusion factors assigned to each atom. Results can be more difficult to interpret than those of PSEUDO, since superposition of atoms of different type may contribute to the calculated degree of pseudosymmetry. BPLOT is a modified version (adapted for the Bilbao server with the collaboration of its author) of the program KPLOT (Hundt, R. (1979). KPLOT. A Program for Plotting and Investigating Crystal Structures). BPLOT is intended for searching and identifying symmetry rather than pseudosymmetry. It transforms the input structure to a structure with space group P1 and does not assume a priori any rotational symmetry. In contrast

with PSEUDO, generally BPLOT only works properly if the pseudosymmetry is very high, with displacements of the order of 0.1A or smaller. The program PSEUDO PSEUDO is designed to detect pseudosymmetry in a given structure, and derive a virtual parent high-symmetry structure. The program is intended for the detection, analysis and characterization of displacively distorted structures. It can be used as a preliminary step before the use of AMPLIMODES for a full symmetry mode analysis (if we lack a structural model for the parent high-symmetry structure, required as part of the input in AMPLIMODES). The program is expected to work successfully and detect pseudosymmetry if the maximal atomic displacements relating the input structure with a high symmetry configuration are not larger than about 1 Ǻ. For much larger displacements, the program will not be capable in many cases to detect the pseudosymmetry although it may exist if large distortions are considered. The program is not in principle applicable to structures with order-disorder features in their distortion. However, in many cases some tricks can be done to analyse the displacive component of these distortions, as using average positions of the disordered positions, or treating as distinct atom types the atomic sites with different occupation probabilities. PSEUDO only requires as input the structure to be investigated in the conventional setting of the International Tables for Crystallography (for space groups with two conventional settings the default normally used in this server should be used). A CIF file can be used to introduce the structure. If G is a supergroup of the space group H of the structure to be checked as possible pseudosymmetry, the program first uses the program COSETS (also available in this server as an independent program) to do the left coset decomposition of G with respect to H, choosing a set of coset representatives {1,g2, …,gn}: G = H + g2H + ….. + gnH The operations {1,g2, …,gn} are representatives of the operations of G not belonging to H. All operations belonging to a given coset giH, i.e. all operations of type gih, with h belonging to H, transform the H-symmetric structure in the same form. If we call S the input structure, the structures giS obtained by transforming the structure S by the action of the left coset representatives gi are calculated by the program and compared with the original structure S. If the structures giS differ from S below a given tolerance for the atomic distances, the space group G is then flagged as pseudosymmetry. This tolerance Δmax is the maximum value allowed for the distances Δ between any atomic site of the structure S and the atomic sites that would supposedly coincide with it in the transformed structures giS if the operation gi were actually a symmetry operation. For supergroups of index 2, this distance Δ for each atom is twice the atomic displacement between the hypothetical high symmetry position and the

observed one. The tolerance value Δmax is by default 1 Ǻ, and generally it should not exceed 2 Ǻ. For larger values, the atomic displacements allowed are so large, in many cases comparable to the unit cell parameters, that the comparison of the transformed structures, if successful, may include nonsensical associations between the atoms in both structures. The pseudosymmetry search is done by default checking the minimal supergroups of the actual space group of the structure. Monoclinic and triclinic structures, due to their freedom in the choice of unit cell, require some special additional processing, and this is not yet fully available in the present version of the program. Several options are available when using PSEUDO: Option 1: search of maximal pseudosymmetry stepwise, climbing up through minimal supergroups. Let us consider the following rather simple example of the orthorhombic Pnma structure of Pb2MgWO6 [Acta Cryst. (1995). B51, 668-673]: 62 11.4059 7.9440 5.6866 90.00 90.00 90.00 8 Pb 1 8d 0.1422 0.0032 0.7804 Mg 1 4c 0.3772 0.25 0.7519 W 1 4c 0.1161 0.25 0.2577 O 1 8d 0.1314 0.4907 0.2365 O 2 4c 0.0027 0.25 0.0133 O 3 4c 0.0103 0.25 0.4991 O 4 4c 0.237 0.25 -0.0153 O 5 4c 0.2491 0.25 0.4745

The structure is introduced in a quite obvious format. In consecutive rows the following information is listed: space group number, unit cell parameters, number of atoms in the asymmetric unit, and one row for each atom with atom type, label number, Wyckoff label (can be left unfilled, but substituted by any character) and relative coordinates. This is also the format for the description of a structure in other programs of the Bilbao Crystallographic Server. Also, a CIF structure file can be used as input.

We choose first the default option, i.e. option1: Minimal supergroups. We leave the tolerance in the default value of 1Ǻ, and run the program: List of distinct minimal supergroups: A list of minimal supergroups of Pnma will appear on the screen (by definition, a minimal supergroup of a group H has no subgroup that is also supergroup of H). This list contains all distinct minimal supergroups of Pnma of a type different than Pnma, and a subset of the infinite series of minimal isomorphic supergroups (i.e. minimal supergroups of type Pnma). This subset of minimal isomorphic supergroups includes those with the smallest indices (for most space groups those with index up to 7 are listed).

If a structure of space group H is pseudosymmetric for a supergroup G >H, then the structure will also be pseudosymmetric for any intermediate space group H’ between G and H, and therefore it will be pseudosymmetric for at least one of the minimal supergroups of H, which will necessarily be in a chain of minimal supergroups G> … > H’>…>H, connecting G and H. We can therefore assess the pseudosymmetry of a structure by checking the pseudosymmetry for the minimal supergroups, and if successful for one of them, repeat the process anew for its minimal supergroups, and so on, until the maximal pseudosymmetry is reached.

Note that the program procedure may fail to detect the pseudosymmetry if all possible chains of minimal supergroups connecting the two space groups include an isomorphic supergroup of an index larger than 7, which will not appear in the list of minimal supergroups proposed by the program for checking. In these cases, a more elaborated process using the option 3 of the program can be used (see below). From the list of minimal supergroups provided by the program, many of them can be discarded beforehand, avoiding the subsequent lengthy check, which is bound to be unsuccessful or give non-sensical results. The data provided in the list of minimal supergroups helps the user for choosing the actual minimal supergroups that can make sense. The list gives the index between the two groups and the k-index which indicates the multiplication of the primitive unit cell of the group with respect to the one of the supergroup, the transformation relating both conventional cells, and the actual cell parameters of the hypothetical supergroups. In many cases, the cell parameters of the supergroup will be absurd or unrealistic, or very far from fulfilling the symmetry conditions imposed by the crystalline class of the supergroup. Furthermore, the number of molecules in the primitive unit cell of the structure should be divisible by the k-index, so that the number of molecules in the supergroup primitive unit cell is an integer. In the present example, using these considerations, you can discard many of the listed supergroups. In many cases, the isomorphic supergroups can be discarded because they imply unrealistic divisions of the input unit cell. The program then lists the supergroups that have been checked and the ones for which pseudosymmetry has been detected under the required tolerance. In our example, only the minimal supergroup Pmmn (2c,b,a;0,0,0) gives a positive result:

You can inspect in detail the rest of the output for this detected pseudosymmetry. The program derives a Pmmn structure as close as possible to the input structure, and lists

the atomic displacements of the asymmetric unit of the Pnma structure with respect to this idealized Pmmn configuration. The maximum atomic displacement between the two structures is of the order of 0.17A:

This symmetrized structure is given first in the subgroup Pnma setting, as above, but also in the Pmmn supergroup setting: 059 5.6866 7.9440 5.7030 90.00 90.00 90.00 6 Pb 1 - 0.2500 0.0032 0.2844 Mg 1 - 0.2500 0.2500 0.7544 W 1 - 0.7500 0.2500 0.2322 O 1 - 0.7500 0.4907 0.2628 O 2 - 0.9929 0.2500 0.0130 O 4 - 0.9949 0.2500 0.4861 The number of atoms in the asymmetric unit has been reduced, due to the higher symmetry. Due to the reduction of the unit cell, the Wyckoff orbits has smaller multiplicities for those atoms with no orbit-merging, We can check this point using the program WPASSIGN of the Bilbao server: 59 5.6866 7.9440 5.7030 90.00 90.00 90.00 6 Pb 1 4e 0.2500 0.0032 0.2844 Mg 1 2a 0.2500 0.2500 0.7544 W 1 2b 0.7500 0.2500 0.2322 O 1 4e 0.7500 0.4907 0.2628 O 2 4f 0.9929 0.2500 0.0130 O 4 4f 0.9949 0.2500 0.4861

A more detailed report of the pseudosymmetry assessment is also available with a click at the end of the summary provided by the program. This report contains a full comparison of the original structure with the transformed ones obtained by the application of the coset representatives of the supergroup with respect to the subgroup.

Doing copy-paste, we can now use the symmetrized Pmmn structure as a new input for PSEUDO, so that we can continue the search for pseudosymmetry among the minimal supergroups of this symmetry.

We can discard again beforehand many of the minimal supergroups provided by the program. From the rest, only the space group Immm (a,b,c ; 1/4,1/4,1/4) will be flagged as acceptable with maximal displacements of the order of 0.2A.

Continue the process introducing the symmetrized Immm structure as input in PSEUDO, following the same steps. Now, I4/mmm (b,c,a; 0,1/2,0) will be flagged a pseudosymmetry space group with a maximal atomic displacement smaller than 0.03A, and a symmetrized structure given by:

139 5.7030 5.6866 7.9440 90.00 90.00 90.00 5 Pb 1 - 0.5000 0.0000 0.2500 Mg 1 - 0.0000 0.0000 0.5000 W 1 - 0.5000 0.5000 0.5000 O 1 - 0.5000 0.5000 0.7407 O 2 - 0.2598 0.7402 0.5000 It is important to take into account that the cell parameters given by the program for the supergroup are in general not symmetrized, and correspond exactly to the result of the transformation relating the supergroup-group unit cells, taking as data the cell parameters of the subgroup. In the case that the supergroup belongs to a different crystalline class, the resulting unit cell for the supergroup will include in general some symmetry breaking strain, which should be small, if the pseudosymmetry attribution makes sense. In the present case, for instance, the parameters a and b are not exactly equal, as demanded by the tetragonal symmetry. If this idealized structure is to be used in other contexts, obviously the cell parameters provided by PSEUDO will require a hand-made symmetrization. For instance, in this example, the tetragonal a parameter would be (a+b)/2, with a and b being the ones provided by PSEUDO. However, if we are going to use the structure for a new step further up in the search of maximal pseudosymmetry, it is much better to keep the unsymmetrized unit cell provided by PSEUDO, so that the real lattice of the experimental structure is maintained up to the last step of the process, and the symmetry breaking strain in the final maximal pseudosymmetry space group can be assessed. PSEUDO can work consistently even if a non-tetragonal unit cell is provided, for the tetragonal lattice, because the action of the symmetry operation is calculated on the atomic relative coordinates. The unit cell parameters are only used for producing the transformed unit cells. We use then again this symmetrized I4/mmmm structure as input, and check reasonable minimal supergroups, with the result:

Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P4/mmm (123) 2 2 a-b,a+b,2c ; 0,0,0 >tol -

2 Fm-3m (225) 3 1 1/2a-1/2b,1/2a+1/2b,c ; 0,0,0 0.0040 0.0026

We arrive then to a cubic pseudosymmetry. Note that in this case the maximal atomic displacement is not half the maximal Δ parameter, because the index of the supergroup/group relation is 3. The resulting symmetrized Fm-3m configuration provided by PSEUDO will be (Wyckoff labels added using WPASSIGN): 225 8.0537 8.0537 7.9440 90.00 90.00 90.17 4 Pb 1 8c 0.2500 0.7500 0.2500 Mg 1 4b 0.0000 0.0000 0.5000 W 1 4a 0.5000 0.0000 0.5000 O 1 24e 0.5000 0.0000 0.7404

Note the slight deformation of the unit cell with respect to the required conditions within the cubic crystalline class. The actual symmetrized cell parameter will be the mean value of three a,b,c values, while the angle γ has to be corrected to 90º. The symmetrized Fm-3m structure can be further checked for pseudosymmetry with respect to its only minimal cubic supergroup Pm-3m proposed by PSEUDO, with negative results. In fact this supergroup could be discarded directly, since from the multiplicity of the atomic positions, one can see that the primitive unit cell of the Fm-3m configuration already contains a single formula unit, and therefore, no higher symmetry through smaller primitive cells is possible, while the point-group symmetry is already maximal. ----------------------------------------------- Additional Note: This example compound is in fact a double-perovskite where the composition ABO3 is changed to A2BB’O3. The Fm-3m configuration corresponds to a perovskite cell duplicated along the three directions due to the ordering of the B and B’ cations within the O6 octahedra. This is a simple quite trivial example of a Bärninghausen relation between the space groups Pm-3m --- Fm-3m (2a,2b,2c ; 0,0,0) through the ordering of the Mg and W atoms in the B site of the Pm-3m perovskite. This can be checked with PSEUDO if we introduce as input the above Fm-3m structure with the atoms Mg and W artificially substituted by the same atom type: 225 8.0537 8.0537 7.9440 90.00 90.00 90.17 4 Pb 1 8c 0.2500 0.7500 0.2500 A 1 4b 0.0000 0.0000 0.5000 A 2 4a 0.5000 0.0000 0.5000 O 1 24e 0.5000 0.0000 0.7404

Now the search for pseudosymmetry with respect to the Pm-3m space group will be successful:


1 Pm-3m (221) 2 2 2a,2b,2c ; 0,0,0 0.1546 0.0773

Atom Idealized Coordinates ux uy uz |u|

Pb1 (0.2500, 0.7500, 0.2500) 0.000000 0.000000 0.000000 0.0000

A1 (0.0000, 0.0000, 0.5000) 0.000000 0.000000 0.000000 0.0000

A2 (0.5000, 0.0000, 0.5000) 0.000000 0.000000 0.000000 0.0000

O1 (0.5000, 0.0000, 0.7500) 0.000000 0.000000 -0.009600 0.0763

with the obtained symmetrized structure being of course the cubic perovskite: 221 4.0269 4.0269 3.9720 90.00 90.00 90.17 3 Pb 1 1b 0.5000 0.5000 0.5000 A 1 1a 0.0000 0.0000 0.0000 O 1 3d 0.0000 0.0000 0.5000 where again a small strain in the unit cell is to be corrected by hand. Disregarding the ordering of the atoms Mg and W in the B-site, the distortion of the Fm-3m configuration with respect to a Pm-3m perovskite structure is minimal, with the oxygens displacements below 0.1Ǻ. ----------------- Coming back to the main result: for the actual observed structure with ordering of the Mg and W atoms within the B-sites of the perovskite, the maximal pseudosymmetry is Fm-3m, with the parent structure given above. The process has been a bit long, but we have arrived to an end in a repetitive process of pseudosymmetry check along the following chain of minimal supergroups:

Expressing each of the transformations (P,p) in the chain as a 4-dim matrix (see International Tables for Crystallography), we can obtain the global transformation relating the space groups Fm-3m and Pnma, by multiplying the four matrices:

2c, b, -a; 0 0 0

a, b, c; ¼ ¼ ¼

b, c, a; 0 ½ 0

1/2a-1/2b,1/2a+1/2b,c ; 0,0,0

1/2 1/2 0 0 -1/2 1/2 0 0 0 0 1 0 0 0 0 1 with the following result: 1 0 -1/2 1/2 -1 0 -1/2 1/4 0 1 0 1/4 0 0 0 1 This means the global transformation relating the two groups is: a-b,c,-1/2a-1/2b ; 1/2,1/4,1/4 Using SUBGROUPGRAPH, we could also obtain this transformation or an equivalent one. We only need to introduce the two end space groups Fm-3m and Pnma and their index, which is 24. SUBGROUPGRAPH will provide all equivalent classes of subgroups of Fm-3m, of type Pnma. One of these classes corresponds to our specific case. The class contains all equivalent subgroups corresponding to equivalent domain-related structures. The transformations obtained above does not necessarily appears in the list, but an equivalent one will be listed. The figure above describing the lattice of minimal supergroups connecting the two symmetries has been produced with this program. Note that in this example there is a single chain of minimal supergroups connecting the two space groups. In general, the graph of minimal supergroups connecting the pseudosymmetry space group and observed symmetry can be more complex with several different chains of minimal supergroups connecting both symmetries. In these cases, pseudosymmetry will be detected for several minimal supergroups, and one can choose in principle any of them, for continuing the process and proceed to the next step up to the maximal pseudosymmetry. Generally it is convenient to choose the minimal supergroup with minimal displacements. Let us consider a second example where this happens. Let us take a hypothetical structure of symmetry C2221 (N. 20): 20 5.4435 9.4122 9.0630 90 90 90 7 A 1 4a 0.0790 0 0 A 2 4b 0 0.385 0.25 B 1 8c -0.0323 0.3562 0.6231 C 1 4a 0.5412 0 0 C 2 8c 0.7812 0.222 0.0385 C 3 4b 0 -0.1998 0.25 C 4 8c 0.2596 0.0999 0.2312

0 0 1 0 0 1 0 0 2 0 0 0 0 0 0 1

1 0 0 1/4 0 1 0 1/4 0 0 1 1/4 0 0 0 1

0 0 1 0 1 0 0 1/2 0 1 0 0 0 0 0 1

We apply then PSEUDO with option 1 and tolerance 2 Ǻ, discarding from the list of minimal supergroups those with inconsistent cells or impossible k-indices: No. # Select HM Symb. IT Numb. Index Index ik Transformation (P,p) Transformed Cell

1

P2221 017 2 2 2a,2b,c ; 0,0,0 2.7218 4.7061 9.0630 90.00 90.00 90.00

2

C2221 020 3 3 3a,b,c ; 0,0,0 1.8145 9.4122 9.0630 90.00 90.00 90.00

3

C2221 020 3 3 -b,-3a,-c ; 0,0,3/4 3.1374 5.4435 9.0630 90.00 90.00 90.00

4

C2221 020 3 3 a,b,3c ; 0,0,0 5.4435 9.4122 3.0210 90.00 90.00 90.00

5

C2221 020 5 5 5a,b,c ; 0,0,0 1.0887 9.4122 9.0630 90.00 90.00 90.00

6

C2221 020 5 5 -b,-5a,-c ; 0,0,3/4 1.8824 5.4435 9.0630 90.00 90.00 90.00

7

C2221 020 5 5 a,b,5c ; 0,0,0 5.4435 9.4122 1.8126 90.00 90.00 90.00

8

C2221 020 7 7 7a,b,c ; 0,0,0 0.7776 9.4122 9.0630 90.00 90.00 90.00

9

C2221 020 7 7 -b,-7a,-c ; 0,0,3/4 1.3446 5.4435 9.0630 90.00 90.00 90.00

10

C2221 020 7 7 a,b,7c ; 0,0,0 5.4435 9.4122 1.2947 90.00 90.00 90.00

11

C222 021 2 2 a,b,2c ; 0,0,1/2 5.4435 9.4122 4.5315 90.00 90.00 90.00

12

F222 022 2 2 b,c,a ; ¼,0,1/4 9.0630 5.4435 9.4122 90.00 90.00 90.00

13

Cmcm 063 2 1 a,b,c ; 0,0,0 5.4435 9.4122 9.0630 90.00 90.00 90.00

14

Cmcm 063 2 1 -b,-a,-c ; 0,0,3/4 9.4122 5.4435 9.0630 90.00 90.00 90.00

15

Cmca 064 2 1 a,b,c ; ¼,0,0 5.4435 9.4122 9.0630 90.00 90.00 90.00

16

Cmca 064 2 1 -b,-a,-c ; ¼,0,3/4 9.4122 5.4435 9.0630 90.00 90.00 90.00

17

P4122 091 2 1 a-b,a+b,c ; 0,0,1/8 5.4365 5.4365 9.0630 90.00 90.00 60.09

18

P4122 091 2 1 -a+b,-a-b,-c ; 0,0,1/8 5.4365 5.4365 9.0630 90.00 90.00 60.09

19

P41212 092 2 1 a-b,a+b,c ; 0,0,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

20

P41212 092 2 1 -a+b,-a-b,-c ; 0,0,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

21

P4322 095 2 1 a-b,a+b,c ; 0,0,3/8 5.4365 5.4365 9.0630 90.00 90.00 60.09

22

P4322 095 2 1 -a+b,-a-b,-c ; 0,0,3/8 5.4365 5.4365 9.0630 90.00 90.00 60.09

23

P43212 096 2 1 a-b,a+b,c ; 0,0,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

24

P43212 096 2 1 -a+b,-a-b,-c ; 0,0,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

25

P6122 178 3 1 -a+b,-a-b,c ; 0,0,5/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

26

P6122 178 3 1 -a+b,-a-b,c ; -1/2,1/2,5/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

27

P6122 178 3 1 a-b,a+b,-c ; 0,0,5/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

28

P6122 178 3 1 a-b,a+b,-c ; -1/2,1/2,5/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

29

P6122 178 3 1 -a-b,-a+b,c ; 0,0,2/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

30

P6122 178 3 1 -a-b,-a+b,c ; -1/2,-1/2,2/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

31

P6122 178 3 1 a+b,a-b,-c ; 0,0,2/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

32

P6122 178 3 1 a+b,a-b,-c ; -1/2,-1/2,2/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

33

P6522 179 3 1 -a+b,-a-b,c ; 0,0,1/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

34

P6522 179 3 1 -a+b,-a-b,c ; -1/2,1/2,1/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

35

P6522 179 3 1 a-b,a+b,-c ; 0,0,1/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

36

P6522 179 3 1 a-b,a+b,-c ; -1/2,1/2,1/12 5.4365 5.4365 9.0630 90.00 90.00 60.09

37

P6522 179 3 1 -a-b,-a+b,c ; 0,0,1/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

38

P6522 179 3 1 -a-b,-a+b,c ; -1/2,-1/2,1/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

39

P6522 179 3 1 a+b,a-b,-c ; 0,0,1/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

40

P6522 179 3 1 a+b,a-b,-c ; -1/2,-1/2,1/3 5.4365 5.4365 9.0630 90.00 90.00 119.91

41

P6322 182 3 1 -a+b,-a-b,c ; 0,0,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

42

P6322 182 3 1 -a+b,-a-b,c ; -1/2,1/2,1/4 5.4365 5.4365 9.0630 90.00 90.00 60.09

43

P6322 182 3 1 a+b,a-b,-c ; 0,0,1 5.4365 5.4365 9.0630 90.00 90.00 119.91

44

P6322 182 3 1 a+b,a-b,-c ; -1/2,-1/2,1 5.4365 5.4365 9.0630 90.00 90.00 119.91

with the following results: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P2221 (017) 2 2 2a,2b,c ; 0,0,0 >tol -

2 C222 (021) 2 2 a,b,2c ; 0,0,1/2 >tol -

3 F222 (022) 2 2 b,c,a ; 1/4,0,1/4 >tol -

4 Cmcm (063) 2 1 a,b,c ; 0,0,0 0.9382 0.4691

5 Cmcm (063) 2 1 -b,-a,-c ; 0,0,3/4 >tol -

6 Cmca (064) 2 1 a,b,c ; 1/4,0,0 >tol -

7 Cmca (064) 2 1 -b,-a,-c ; 1/4,0,3/4 >tol -

8 P6122 (178) 3 1 -a-b,-a+b,c ; 0,0,2/3 >tol -

9 P6122 (178) 3 1 -a-b,-a+b,c ; -1/2,-1/2,2/3 >tol -

10 P6122 (178) 3 1 a+b,a-b,-c ; 0,0,2/3 >tol -

11 P6122 (178) 3 1 a+b,a-b,-c ; -1/2,-1/2,2/3 >tol -

12 P6522 (179) 3 1 -a-b,-a+b,c ; 0,0,1/3 >tol -

13 P6522 (179) 3 1 -a-b,-a+b,c ; -1/2,-1/2,1/3 >tol -

14 P6522 (179) 3 1 a+b,a-b,-c ; 0,0,1/3 >tol -

15 P6522 (179) 3 1 a+b,a-b,-c ; -1/2,-1/2,1/3 >tol -

16 P6322 (182) 3 1 a+b,a-b,-c ; 0,0,1 0.8427 0.4863

17 P6322 (182) 3 1 a+b,a-b,-c ; -1/2,-1/2,1 >tol -

Two minimal supergroups have been flagged. We take for the next step upwards the symmetrized structure with smallest atomic displacements, i.e. the structure Cmcm (a,b,c ; 0,0,0): 063 5.4435 9.4122 9.0630 90.00 90.00 90.00 7 A 1 - 0.0000 0.0000 0.0000 A 2 - 0.0000 0.3850 0.2500 B 1 - 0.0000 0.3562 0.6231 C 1 - 0.5000 0.0000 0.0000 C 2 - 0.7500 0.2500 0.0000 C 3 - 0.0000 0.8002 0.2500 C 4 - 0.2596 0.0999 0.2500

with the following result: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 Pmma (051) 2 2 2b,2c,a ; 0,0,0 >tol -

2 Cmmm (065) 2 2 -b,a,2c ; 0,0,0 >tol -

3 Fmmm (069) 2 2 c,b,-a ; 1/4,1/4,0 >tol -

4 P63/mmc (194) 3 1 -a-b,a-b,c ; 0,0,0 0.8427 0.4863

5 P63/mmc (194) 3 1 -a-b,a-b,c ; -1/2,-1/2,0 >tol -

Taking again the symmetrized structure P63/mmc with the smallest distortion for further pseudosymmetry check (Wyckoff labels added with WPASSIGN): 194 5.4365 5.4365 9.0630 90.00 90.00 119.91 5 A 1 2a 0.0000 0.0000 0.0000 A 2 2c 0.333333 0.666667 0.2500 B 1 4f 0.333333 0.666667 0.6231 C 1 6g 0.5000 0.5000 0.0000 C 3 6h 0.8136 0.1864 0.2500

Important: Note that we have modified the output of PSEUDO including 6 decimal digits for the special coordinates 1/3 and 2/3. To use PSEUDO and other programs of

this server, it is important that special coordinates 1/3, 2/3, etc… are expressed including a maximum number of digits in its numerical expression, so that the programs recognise them as exact simple fractions. Applying PSEUDO again to this structure, the result is negative for all consistent minimal supergroups and tolerance 2 Ǻ. P63/mmc is therefore the pseudosymmetry of the structure, with the structural model listed above as P63/mmc reference structure (The cell has some strain to be corrected by hand). The graph of maximal subgrousp connecting this space group with the actual observed symmetry C2221 is the following (obtained with SUBGROUPGRAPH):

One can see now why both supergroups of type Cmcm and P6322 were flagged by PSEUDO, as they are both intermediate symmetries with respect to P63/mmc. If we had taken the symmetrized P6322 structure in the second step, the result would have been the same. Option 3: Search of pseudosymmetry for a specific supergroup defined by the transformation (P,p) Retaking the first example of the Pnma structure of Pb2MgWO6, we can use the option 3 of the program, in which a given supergroup specified by its transformation matrix can be checked, to confirm that this Pnma structure is indeed pseudosymmetric with respect to the symmetry Fm-3m, as obtained step by step checking a chain of minimal supergroups.


1 Fm-3m (225) 24 4 a-b,c,-1/2a-1/2b ; 1/2,1/4,1/4 0.5243 0.2627


Pb1 (0.1250, 0.0000, 0.7500) 0.017200 0.003200 0.030400 0.2627

Mg1 (0.3750, 0.2500, 0.7500) 0.002200 0.000000 0.001900 0.0273

W1 (0.1250, 0.2500, 0.2500) -0.008900 -0.000000 0.007700 0.1106

O1 (0.1250, 0.4904, 0.2500) 0.006400 0.000317 -0.013500 0.1060

O2 (0.0048, 0.2500, 0.0096) -0.002108 0.000000 0.003683 0.0319

O3 (0.0048, 0.2500, 0.4904) 0.005492 0.000000 0.008717 0.0799

O4 (0.2452, 0.2500, 0.0096) -0.008192 0.000000 -0.024917 0.1697

O5 (0.2452, 0.2500, 0.4904) 0.003908 0.000000 -0.015883 0.1007 We can see now directly the atomic displacements relating the actual Pnma structure with the proposed ideal Fm-3m configuration. These displacements are smaller than 0.27A. Option 3 is in principle intended for obtaining a symmetrized structural model of a structure for which we know its pseudosymmetry, and the transformation matrix relating both space groups. But it can also be used to start the determination of some unknown pseudosymmetry in structures which due to their large unit cell may have in their chains of minimal pseudosymmetric supergroups isomorphic space groups with indices larger than those considered in option 1. Example of combined application of option 3 and 1: The apparently complex phase of the phase Ga-II of Ga under pressure: Let us consider for instance the phase of Ga under pressure called Ga-II (Phys. Rev. Lett. 93, 205502 (2004)): 20 5.976 8.576 35.758 90 90 90 14 Ga 1 4b 0.5000 0.1802 0.2500 Ga 2 8c 0.6956 0.4684 0.2716 Ga 3 8c 0.5804 0.7858 0.2861 Ga 4 8c 0.2772 0.5622 0.3081 Ga 5 8c -0.0341 0.7809 0.3292 Ga 6 8c 0.8482 0.4567 0.3430 Ga 7 8c 0.5632 0.6919 0.3666 Ga 8 8c 0.2324 0.4838 0.3851 Ga 9 8c 0.6129 0.2914 0.4003 Ga 10 8c 0.8276 0.5660 0.4250 Ga 11 8c -0.0030 0.2613 0.4435 Ga 12 4a 0.2460 0.0 0.5000 Ga 13 8c 0.1052 0.3090 0.5170 Ga 14 8c 0.3574 0.5518 0.5409

This orthorhombic structure, with space group C2221, has 104 Ga atoms in the conventional centred unit cell. Its unit cell is much elongated along the c axis. One can suspect that the system may be a superstructure of a more proportioned cell, with a division of the cell parameter along the c axis. The observed cell would be then a multiple of a smaller cell, the multiplication factor being necessarily a divisor of 104. Divisors giving reasonable values for the c parameter may be 13, 8, 4, 2. If the structure is a superstructure with the unit cell multiplied along c by a factor 13, the option 1 would not detect it, since this cell multiplication would correspond to an isomorphic supergroup with index 13, above the limit considered in this option. But we can try this possibility by introducing the specific isomorphic supergroup with option 3. Using the program SERIES we can look to the whole series of isomorphic supergroups of C2221, check the existence of one with a division of the cell by 13, and get the corresponding origin shift, if any. In this case the transformation matrix is rather trivial, with no origin shift needed, and introducing a tolerance of 2 Ǻ, PSEUDO reports a positive result, with maximal atomic displacements smaller than 0.8 A: Case # Supergroup G Index i Index ik (P,p) Tr. Matrix Δmax umax

1 C2221 (20) 13 13 a,b,13c ; 0,0,0 [ 1 0 0 ] [ 0] [ 0 1 0 ] [ 0] [ 0 0 13 ] [ 0]

1.6288 0.7794

The resulting symmetrized structure is : 20 5.9760 8.5760 2.7506 90.00 90.00 90.00 2 Ga 1 - 0.5000 0.2484 0.2500 Ga 10 - 0.7504 0.5000 0.5000

Hence, to obtain this very regular structure with a much smaller cell (with c = 2.751 Å) the atomic displacements that have been required are smaller than 0.8 Å. This symmetrized structure can be now checked further for pseudosymmetry using option 1, checking the minimal supergroups. Decreasing now the tolerance to 1Ǻ, and dropping the tetragonal and hexagonal minimal supergroups because they have unit cells very far from the symmetry requirements, and some of the orthorhombic ones because of their inconsistent unit cell, we obtain that the structure is only pseudosymmetric for the minimal supergroup F222 (-b,-c,a ; 3/4,0,1/4) with the following symmetrized reference structure, that requires atomic displacements smaller than 0.01 Å: Atom Idealized Coordinates ux uy uz |u|

Ga1 (0.5000, 0.2500, 0.2500) 0.000000 -0.001600 0.000000 0.0137

Ga10 (0.7500, 0.5000, 0.5000) 0.000400 0.000000 0.000000 0.0024

022 2.7506 5.9760 8.5760 90.00 90.00 90.00 2 Ga 1 - 0.0000 0.5000 0.0000 Ga 10 - 0.2500 0.2500 0.7500 This structure can be further checked for pseudosymmetry with option 1, considering consistent minimal supergroups, and in fact, the structure is detected as having Fddd symmetry, with no additional displacement required:


1 P222 (016) 2 2 -2a,-2b,2c ; 0,0,0 >tol -

2 F222 (022) 3 3 b,c,3a ; 0,0,0 >tol -

3 F222 (022) 3 3 c,3a,b ; 0,0,0 >tol -

4 Fmmm (069) 2 1 -a,-b,c ; 0,0,0 >tol -

5 Fmmm (069) 2 1 a,b,c ; 1/4,1/4,1/4 >tol -

6 Fddd (070) 2 1 -a,-b,c ; 1/8,1/8,1/8 >tol -

7 Fddd (070) 2 1 a,b,c ; 3/8,3/8,3/8 0 0.0000

070 2.7506 5.9760 8.5760 90.00 90.00 90.00 1 Ga 1 - 0.3750 0.8750 0.3750

So, in fact, phase GaII is a distorted Fddd structure with a single atom in the asymmetric unit, with its position fully fixed by symmetry. We can obtain the actual distortion relating both structures, using again PSEUDO with option 3, introducing the supergroup Fddd with the adequate transformation, multiplying, as in the previous case, the three 4x4 matrices corresponding to the three steps done along the chain minimal supergroups: (-b,-c,13a; 1/8, 3/8, 5/8). A summary of the result is: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 Fddd (70) 52 26 -b,-c,13a ; 1/8,3/8,5/8 1.6288 0.7729


Ga1 (0.5000, 0.2500, 0.2500) -0.000000 -0.069800 0.000000 0.5986

Ga2 (0.7500, 0.5000, 0.2692) -0.054400 -0.031600 0.002369 0.4316

Ga3 (0.5000, 0.7500, 0.2885) 0.080400 0.035800 -0.002361 0.5764

Ga4 (0.2500, 0.5000, 0.3077) 0.027200 0.062200 0.000408 0.5578

Ga5 (0.0000, 0.7500, 0.3269) -0.034100 0.030900 0.002277 0.3441

Ga6 (0.7500, 0.5000, 0.3462) 0.098200 -0.043300 -0.003154 0.7036

Ga7 (0.5000, 0.7500, 0.3654) 0.063200 -0.058100 0.001216 0.6267

Ga8 (0.2500, 0.5000, 0.3846) -0.017600 -0.016200 0.000485 0.1751

Ga9 (0.5000, 0.2500, 0.4038) 0.112900 0.041400 -0.003546 0.7729

Ga10 (0.7500, 0.5000, 0.4231) 0.077600 0.066000 0.001923 0.7350

Ga11 (0.0000, 0.2500, 0.4423) -0.003000 0.011300 0.001192 0.1074

Ga12 (0.2500, 0.0000, 0.5000) -0.004000 -0.000000 0.000000 0.0239

Ga13 (0.0206, 0.2685, 0.5211) 0.084617 0.040527 -0.004079 0.6307

Ga14 (0.2679, 0.5200, 0.5366) 0.089521 0.031812 0.004287 0.6198

Therefore, phase GaII is in fact a modulated structure of a simple Fddd structure, with maximal atomic displacements below 0.78A (Phys. Rev. Lett. 97, 115501 (2006)). Option 2: Search among supergroups with a fixed k-index Sometimes, we may be sure that the system is pseudosymmetric with respect to a supergroup from which we know the multiplication factor of its primitive unit cell (k-index). We may be interested for instance to search the pseudosymmetry among supergroups with the no change of lattice, except for some strain (k-index=1), or with a primitive unit cell containing half the number of formula unit than the actual observed one (k-index=2). In this case, option 2 can be more direct. It provides the supergroups for a given k-index (up to 4), and one can choose one by one the desired supergroups. If we want to check a specific supergroup this option spares the user to have to know beforehand (as it happens in option 3) the transformation matrix relating the checked supergroup with the actual space group (only the k-index must be known). Exercises 1: Using PSEUDO, find the pseudosymmetry of the following structure and the symmetrized reference structure. 1 7.0000 4.0000 4.5000 95.00 100.00 82.00 1 A 1 - 0.2800 0.0500 0.09500 A 2 - 0.7500 0.9800 0.9700 This is a case that can be easily deduced from inspection of the structure (you can use VISUALIZE in the Bilbao crystallographic server to visualize the structure with Jmol). Exercise 2: GeF2, having the P212121 (N. 19) structure given below, is reported to have at higher temperature an unknown tetragonal phase, with the primitive unit cell volume being essentially maintained. Using PSEUDO, with the option 2, which allows to check supergroups with a fixed k-index (multiplication of the primitive unit cell) postulate a probable space group or groups and a starting structural model for this high-temperature phase. 19 4.682 5.158 8.312 90 90 90 3 Ge 1 4a 0.2340 0.0083 0.1311 F 1 4a 0.029 0.083 -0.018 F 2 4a 0.067 0.246 0.279

Polar structures: If the structure to be checked for pseudosymmetry is polar, its origin along the polar direction(s) has in general been chosen arbitrarily. A proper search of non-polar pseudosymmetry requires then to optimize somehow the origin of the non-polar (or less polar) supergroup with respect to the input polar structure (formally, the number of distinct non-polar supergroups is infinite due to the possible distinct choices of origin). This optimization is done in PSEUDO by making the pseudosymmetry check for a grid of origin choices for the polar structure whose density is controlled by the user, while the transformation matrix (P,p) defining the origin of the non-polar supergroup is not varied. Let us consider as an example the case of the P63 room-temperature structure of NaSb3F10 (J. Appl. Cryst. (2009). 42, 58–62): 173 8.285 8.285 7.600 90 90 120 6 Sb 1 6c -0.1163 0.2243 0.55 Na 1 2b 0.333333 0.666667 0.467 F 1 6c 0.204 0.393 0.294 F 2 6c 0.111 0.229 0.640 F 3 6c 0.035 0.491 0.581 F 4 2b 0.666667 0.333333 0.545

(we have introduced an arbitrary shift of the origin to the structure reported in the reference above, to simulate a more general case, with unknown pseudosymmetry)

Two projections of the experimental P63 structure of NaSb3F10 This compound has been predicted to be ferroelectric due to its small deviation from a non-polar configuration. The symmetries P6322 and P63/mmc have been proposed for two successive non-polar phases at higher temperatures. The first output of PSEUDO with the list of minimal supergroups includes now a note indicating the polarity of the structure, and allowing the user to introduce the desired maximal distance among the points of the grid to be tried for the origin choice, the default being 0.5 Ǻ (the maximum number of points is however internally limited to 40 along any polar direction). One should take into account that very dense grids can increase computer times to unacceptable values. It is therefore advisable to keep this grid parameter as large as possible.

No. # Select HM Symb. IT Numb. Index Index ik Transformation (P,p) Transformed Cell

1 P6 168 2 2 a,b,2c ; 0,0,2t 8.2850 8.2850 3.8000 90.00 90.00 120.00

2 P63 173 3 3 a,b,3c ; 0,0,3t 8.2850 8.2850 2.5333 90.00 90.00 120.00

3 P63 173 3 3 a-b,a+2b,c ; 0,0,t 4.7833 4.7833 7.6000 90.00 90.00 120.00

4 P63 173 4 4 2a,2b,c ; 0,0,t 4.1425 4.1425 7.6000 90.00 90.00 120.00

5 P63 173 5 5 a,b,5c ; 0,0,5t 8.2850 8.2850 1.5200 90.00 90.00 120.00

6 P63 173 7 7 a,b,7c ; 0,0,7t 8.2850 8.2850 1.0857 90.00 90.00 120.00

7 P63 173 7 7 a-2b,2a+3b,c ; 0,0,t 3.1314 3.1314 7.6000 90.00 90.00 120.00

8 P63 173 7 7 -2a-3b,-a+2b,-c ; 0,0,-t 3.1314 3.1314 7.6000 90.00 90.00 120.00

9 P63/m 176 2 1 a,b,c ; 0,0,t 8.2850 8.2850 7.6000 90.00 90.00 120.00

10 P6322 182 2 1 a,b,c ; 0,0,t 8.2850 8.2850 7.6000 90.00 90.00 120.00

11 P63cm 185 2 1 a,b,c ; 0,0,t 8.2850 8.2850 7.6000 90.00 90.00 120.00

12 P63mc 186 2 1 a,b,c ; 0,0,t 8.2850 8.2850 7.6000 90.00 90.00 120.00

HINT: The initial structure is polar, which means that, in general, an origin shift will be necessary to minimize the displacements between the initial polar structure and the hypothetical idealized parent one. Please, insert a minimum grid for the optimization (in Angstroms) Note the continuous parameter t appearing in the transformation matrix of the listed minimal supergroups, indicating the possible arbitrary choice of its origin along the z-axis. Instead of varying this parameter, the program sets it to zero, while the origin of the input structure is varied according to the defined grid. All isomorphic supergroups can be discarded beforehand because the structure contains only 2 formula unit per primitive cell, and therefore the cell divisions implied by these supergroups (inverse of their k-index) are incompatible. For the rest, with a tolerance of 2 A, the summary of the pseudosymmetry check is the following: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P6 (168) 2 2 a,b,2c ; 0,0,0 >tol -

2 P63/m (176) 2 1 a,b,c ; 0,0,0 1.4010 0.7005

3 P6322 (182) 2 1 a,b,c ; 0,0,0 1.3984 0.6992

4 P63cm (185) 2 1 a,b,c ; 0,0,0 >tol -

5 P63mc (186) 2 1 a,b,c ; 0,0,0 0.8948 0.4474

Three minimal supergroups are therefore detected. The first two are non-polar, while the third one is also polar along z. For the non-polar ones the program indicates an optimal origin shift within the resolution given by the grid used and lists besides the symmetrized structure, the original one with the optimized origin. For instance for the first flagged supergroup: 2# Supergroup P63/m (176): a,b,c ; 0,0,0 and index 2 Displacements:


Sb1 (0.8837, 0.2243, 0.2500) 0.000000 0.000000 0.033333 0.2533

Na1 (0.3333, 0.6667, 0.2500) 0.000000 0.000000 -0.049667 0.3775

F1 (0.1575, 0.3110, 0.0770) 0.046500 0.082000 -0.049667 0.7005

F2 (0.1575, 0.3110, 0.4230) -0.046500 -0.082000 -0.049667 0.7005

F3 (0.0350, 0.4910, 0.2500) 0.000000 0.000000 0.064333 0.4889

F4 (0.6667, 0.3333, 0.2500) 0.000000 0.000000 0.028333 0.2153

NOTE: ux, uy and uz are given in relative units. |u| is the absolute displacement given in Å.

Optimized polar structure: # Origin shifted with t = (0.000000, 0.000000, 0.733333) 173 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.883700 0.224300 0.283333 Na 1 - 0.333333 0.666667 0.200333 F 1 - 0.204000 0.393000 0.027333 F 2 - 0.111000 0.229000 0.373333 F 3 - 0.035000 0.491000 0.314333 F 4 - 0.666667 0.333333 0.278333 Idealized structure (subgroup setting): 173 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.8837 0.2243 0.2500 Na 1 - 0.3333 0.6667 0.2500 F 1 - 0.1575 0.3110 0.0770 F 2 - 0.1575 0.3110 0.4230 F 3 - 0.0350 0.4910 0.2500 F 4 - 0.6667 0.3333 0.2500 Idealized structure (supergroup setting): 176 8.2850 8.2850 7.6000 90.00 90.00 120.00 5 Sb 1 - 0.8837 0.2243 0.2500 Na 1 - 0.3333 0.6667 0.2500 F 1 - 0.1575 0.3110 0.0770 #F 2 - 0.1575 0.3110 0.4230 F 3 - 0.0350 0.4910 0.2500 F 4 - 0.6667 0.3333 0.2500 Notes: * Idealized structure with space group 176 related with the given by the transformation a,b,c ; 0,0,0 and index 2 * Cell parameters have not been symmetrized. They may include in general some symmetry breaking strain, to be removed by hand. * A commented atom means a redundant atom, due to the merging of the Wyckoff orbit with another one in the supergroup

The origin shift done by the program is therefore (0,0,0.733), and it is the same for the other non-polar supergroup P6322. If we increase the density of grid points, the origin choice can be further optimized decreasing the maximum atomic displacements between the symmetrized and the input structure. For instance, if we put for the grid parameter 0.1 A, the pseudosymmetry check gives the following results: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P6 (168) 2 2 a,b,2c ; 0,0,0 >tol -

2 P63/m (176) 2 1 a,b,c ; 0,0,0 1.2824 0.6412

3 P6322 (182) 2 1 a,b,c ; 0,0,0 1.3072 0.6536

4 P63cm (185) 2 1 a,b,c ; 0,0,0 >tol -

5 P63mc (186) 2 1 a,b,c ; 0,0,0 0.8948 0.4474

The corresponding results for the supergroup P63/m are now: Optimized polar structure: # Origin shifted with t = (0.000000, 0.000000, 0.250000) 173 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.883700 0.224300 0.800000 Na 1 - 0.333333 0.666667 0.717000 F 1 - 0.204000 0.393000 0.544000 F 2 - 0.111000 0.229000 0.890000 F 3 - 0.035000 0.491000 0.831000 F 4 - 0.666667 0.333333 0.795000 Idealized structure (subgroup setting): 173 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.8837 0.2243 0.7500 Na 1 - 0.3333 0.6667 0.7500 F 1 - 0.1575 0.3110 0.5770 F 2 - 0.1575 0.3110 0.9230 F 3 - 0.0350 0.4910 0.7500 F 4 - 0.6667 0.3333 0.7500 Idealized structure (supergroup setting): 176 8.2850 8.2850 7.6000 90.00 90.00 120.00 5 Sb 1 - 0.8837 0.2243 0.7500 Na 1 - 0.3333 0.6667 0.7500 F 1 - 0.1575 0.3110 0.5770 #F 2 - 0.1575 0.3110 0.9230 F 3 - 0.0350 0.4910 0.7500 F 4 - 0.6667 0.3333 0.7500 Note that the origin shift is quite different from the one proposed for the 0.5 Å grid. However, the proposed symmetrized P63/m reference structure is the same. In fact, it is an equivalent description of the same structure (the origin is shifted (0 0 ½), which is an operation of the normalizer of P63/m- see program NORMALIZER in the server-). The same happens for the other flagged non-polar supergroup Important: once the program has detected pseudosymmetry for a given non-polar symmetry, it is in principle not necessary to try to optimize further the origin choice

with PSEUDO by minimizing the maximum atomic displacements. In most cases, the best or more sensible origin choice does not correspond to this minimization. For instance, it can be more convenient to take the origin that cancels any global translation when the two structures are compared, so that its geometric centre remains unmoved by the distortion. Once the reference symmetrized structure is known and independently of the magnitude of the atomic displacements, the calculation of this optimal origin choice is straightforward, as done by AMPLIMODES in this server. Continuing with the example, we take now the (polar) symmetrized P63mc structure: 186 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.8297 0.1703 0.5500 Na 1 - 0.3333 0.6667 0.4670 F 1 - 0.1965 0.3930 0.2940 F 2 - 0.1145 0.2290 0.6400 F 3 - 0.0350 0.5175 0.5810 F 4 - 0.6667 0.3333 0.5450

which is the one with the smallest distortion. We use it for a further step up, checking with PSEUDO its pseudosymmetry. To be noticed is the fact that for this symmetry no origin shift has been done by the program, as this supergroups is also polar along z, and the magnitude of the atomic displacements between the distorted and the symmetrized structures do not depend on the origin choice along z. Keeping the grid parameter in 0.1 Å, pseudosymmetry for P63/mmc is detected, with the following values: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P6mm (183) 2 2 a,b,2c ; 0,0,0 >tol -

2 P63/mmc (194) 2 1 a,b,c ; 0,0,0 1.2792 0.6396

Optimized polar structure: # Origin shifted with t = (0.000000, 0.000000, 0.250000) 186 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.829700 0.170300 0.800000 Na 1 - 0.333300 0.666700 0.717000 F 1 - 0.196500 0.393000 0.544000 F 2 - 0.114500 0.229000 0.890000 F 3 - 0.035000 0.517500 0.831000 F 4 - 0.666700 0.333300 0.795000 Idealized structure (subgroup setting): 186 8.2850 8.2850 7.6000 90.00 90.00 120.00 6 Sb 1 - 0.8297 0.1703 0.7500 Na 1 - 0.3333 0.6667 0.7500 F 1 - 0.1555 0.3110 0.5770 F 2 - 0.1555 0.3110 0.9230

F 3 - 0.0350 0.5175 0.7500 F 4 - 0.6667 0.3333 0.7500 Idealized structure (supergroup setting): 194 8.2850 8.2850 7.6000 90.00 90.00 120.00 5 Sb 1 - 0.8297 0.1703 0.7500 Na 1 - 0.3333 0.6667 0.7500 F 1 - 0.1555 0.3110 0.5770 #F 2 - 0.1555 0.3110 0.9230 F 3 - 0.0350 0.5175 0.7500 F 4 - 0.6667 0.3333 0.7500 The optimized origin shift is the same as obtained in the previous step for the non-polar supergroups. The reference symmetrized P63/mmc structure with Wyckoff labels is: 194 8.2850 8.2850 7.6000 90.00 90.00 120.00 5 Sb 1 6h 0.8297 0.1703 0.7500 Na 1 2d 0.333333 0.666667 0.7500 F 1 12k 0.1555 0.3110 0.5770 F 3 6h 0.0350 0.5175 0.7500 F 4 2c 0.666667 0.333333 0.7500

Two projections of the P63/mmc structure obtained with PSEUDO as ideal symmetrized configuration of NaSb3F10. The maximal atomic displacement of the displacive distortion relating this structure with the experimental one (see previous figure (a)) is of the order of 0.65 Å.

Although the rotational symmetry is already maximal, as there are still two formula units per primitive unit cell, we can still check the pseudosymmetry of this structure for supergroups with k-index=2. There is only a supergroup of this type, but the pseudosymmetry check is negative:


1 P6/mmm (191) 2 2 a,b,2c ; 0,0,0 >tol -

In fact if we increase sufficiently the tolerance, this supergroup P6/mmm is flagged by the program with Δmax=3.8 Ǻ (umax= 1.9 Ǻ), but this is due to the fact that c/4 =1.9 A, and therefore with this enormous tolerance atomic sites can change their z-coordinate from 0.75 to 0.0, and similar jumps, changing completely the structure.

Summarizing, and considering the two steps done with PSEUDO, the NaSb3F10 structure has P63/mmc pseudosymmetry with a transformation matrix (a, b, c; 0, 0, 0) relating its setting with the observed P63 space group. We can then use the option 3 of PSEUDO, which also allows a origin optimization for polar cases, for a direct check of this pseudosymmetry and a direct comparison of the two structures: Case # Supergroup G Index i Index ik (P,p) Δmax umax

1 P63/mmc (194) 4 1 a,b,c ; 0,0,0 1.3072 0.6536

Idealized structures

1# Supergroup P63/mmc (194): a,b,c ; 0,0,0 and index 4

Displacements:


Sb1 (0.8297, 0.1703, 0.7500) 0.054000 0.054000 0.050000 0.5870

Na1 (0.3333, 0.6667, 0.7500) 0.000000 0.000000 -0.033000 0.2508

F1 (0.1555, 0.3110, 0.5770) 0.048500 0.082000 -0.033000 0.6426

F2 (0.1555, 0.3110, 0.9230) -0.044500 -0.082000 -0.033000 0.6402

F3 (0.0350, 0.5175, 0.7500) 0.000000 -0.026500 0.081000 0.6536

F4 (0.6667, 0.3333, 0.7500) 0.000000 0.000000 0.045000 0.3420

NOTE: ux, uy and uz are given in relative units. |u| is the absolute displacement given in Å Idealized structure (supergroup setting): 194 8.2850 8.2850 7.6000 90.00 90.00 120.00 5 Sb 1 - 0.8297 0.1703 0.7500 Na 1 - 0.3333 0.6667 0.7500 F 1 - 0.1555 0.3110 0.5770 #F 2 - 0.1555 0.3110 0.9230 F 3 - 0.0350 0.5175 0.7500 F 4 - 0.6667 0.3333 0.7500

The symmetrized structure is the same as the one obtained stepwise through a chain of minimal supergroups. If we search with SUBGROUPGRAPH (also in the server) the possible subgroups P63 of P63/mmc of index 4 (the index of the one here), we can observe that there is a single class and a single subgroup P63, with the following lattice of minimal subgroups relating both space groups:

It is then clear that the pseudosymmetry for P63/mmc makes the system pseudosymmetric for the three intermediate space groups, and this is the reason why the three supergroups were detected by PSEUDO in the first step up the chain of minimal supergroups. Obviously, for the second step we could have used any of the three symmetrized structures to reach the ultimate global supergroup. In (J. Appl. Cryst. (2009). 42, 58–62) only pseudosymmetry for the space groups P6322 and P63/mmc was detected. The other two intermediate subgroups seem to have been overlooked, and the symmetry P6322 was proposed for a possible intermediate phase inferred from experimental results. However, the other two intermediate may be more appropiate candidates for an intermediate phase, since the distortion in the structure with P6322 symmetry, quantified with AMPLIMODES, is marginal compared with those associated with the symmetries P63/m and P63mc. Exercise 3: The compound Nd4GeO8 is reported to have the following structure with Pmc21 symmetry (Doklady Akademii Nauk SSSR (1978) 241, 353-356): 26 7.475 5.727 17.927 90 90 90 20 Nd 1 2a 0.000000 0.265100 0.000000 Nd 2 2b 0.500000 0.231400 0.973700 Nd 3 2a 0.000000 0.292500 0.205700 Nd 4 2b 0.500000 0.787400 0.270500 Nd 5 4c 0.241600 0.768900 0.090200 Nd 6 4c 0.261000 0.274100 0.383800 Ge 1 2b 0.500000 0.258100 0.180900 Ge 2 2a 0.000000 0.755800 0.298600 O 1 2a 0.000000 0.531400 0.104500 O 2 2b 0.500000 0.063000 0.100800 O 3 2a 0.000000 0.040900 0.114200 O 4 2b 0.500000 0.545400 0.143500 O 5 4c 0.186400 0.668100 0.248000 O 6 4c 0.312100 0.192400 0.236700 O 7 2a 0.000000 0.055500 0.319800 O 8 2a 0.000000 0.539900 0.366300 O 9 2b 0.500000 0.004300 0.370900 O 10 2b 0.500000 0.512400 0.362800 O 11 4c 0.215300 0.009700 0.486700 O 12 4c 0.258000 0.492400 0.497300 Show using PSEUDO (option 1) that this structure can be considered a small distortion of a Cmcm structure. Using SUBGROUPGRAPH show the graph of maximal subgroups connecting the two symmetries. Using again PSEUDO (option 3) obtain the atomic displacements relating the two structures. Calculate the theoretical value for the spontaneous polarization of the compound using nominal charges for the ions.

Exercise 4: The ICSD database contains the following structural model for Ca2Ge7O16 (Doklady Akademii Nauk SSSR (1979) 245, 110-113): 32 11.340 11.340 4.6400 90 90 90 13 Ge 1 2a 0 0 0 Ge 2 4c 0.1335 0.1336 0.4996 Ge 3 4c 0.0666 0.3122 -.0107 Ge 4 4c 0.3123 0.0667 0.0095 Ca 1 4c 0.3350 0.3348 0.4970 O 1 4c 0.0012 0.1167 0.7467 O 2 4c 0.0288 0.2197 0.2695 O 3 4c 0.1686 0.2658 0.7246 O 4 4c 0.2657 0.1697 0.2645 O 5 4c 0.1171 0.0006 0.2411 O 6 4c 0.2198 0.0282 0.7180 O 7 4c 0.1398 0.4316 0.1548 O 8 4c 0.4327 0.1403 0.8306 Despite the 4/mmm Laue symmetry of the diffraction diagram the authors of this publication were unable to find an appropiate tetragonal structural model, and refined this Pba2 structure Using PSEUDO, demonstrate that this structure differs from a tetragonal one with space group P-4b2, by atomic displacements which are practically negligible or within experimental accuracy, so that in fact this structure file should be considered incorrect, being a case of “overlooked symmetry”. The extreme pseudosymmetry of this structure was reported in (Acta Cryst. B (2002) 58, 921) and the compound has been recently confirmed to have P-4b2 symmetry by a new study of the structure (Acta Cryst. C (2007) 63, i47) Option 4: Monoclinic and triclinic structures Monoclinic and triclinic structures have a great deal of freedom in the choice of their conventional unit cell, and therefore the checking of their pseudosymmetry with respect to space groups of a higher symmetry class requires in general a specific analysis of the pseudosymmetry of its Bravais lattice. This is provided by option 4 of PSEUDO, which uses a tool of the CCTBX library (http://cci.lbl.gov/cctbx/). This option is still under construction.

Hands-on exercises on the use of PSEUDO

P1. Structural pseudosymmetry of a C2221 structure Using the program PSEUDO, analyse the structural pseudosymmetry of the hypothetical C2221 (No. 20) structure described below. Use option 1, i.e. climbing via the minimal supergroups. Compare the results of different minimal-‐supergroup paths are followed. (file: C2221.cif) 20 5.4435 9.4122 9.0630 90 90 90 7 A 1 4a 0.0790 0 0 A 2 4b 0 0.385 0.25 B 1 8c -‐0.0323 0.3562 0.6231 C 1 4a 0.5412 0 0 C 2 8c 0.7812 0.222 0.0385 C 3 4b 0 -‐0.1998 0.25 C 4 8c 0.2596 0.0999 0.2312

P2. Parent phase of β-NbO2. The symmetry of the polar phase of β-‐NbO2 is given by the space group I41 and its structure data is below (Z. Naturforsch. B 37 (1982) 1361). The structure has 8 formula units per primitve unit cell. The structure is reported to be a deformed rutile structure. The space group of the rutile structure is P42/mnm with 2 formula units per primitive unit cell. i) Using option 2 of PSEUDO, quantify the deviations of the structure with respect to a perfect rutile structure, and obtain this idealized high-‐symmetry configuration. ii) Obtain with SUBGROUPS the graph of maximal subgroups relating the two symmetries. Using option 1 of PSEUDO re-‐obtain the results of i) climbing through a path of minimal supergroups between I41 and P42/mnm. (file: beta-NiO2.cif) 80 9.693 9.693 5.985 90 90 90 6 Nb 1 8b 0.262100 0.507600 0.029900 Nb 2 8b 0.245500 0.490900 0.480400 O 1 8b 0.397400 0.149700 0.007700 O 2 8b 0.110200 0.364200 _0.008900 O 3 8b 0.608600 0.360900 0.506400 O 4 8b 0.650000 0.103800 0.248400 P3. Non-polar phases of NaSb3F10 The compound NaSb3F10 whose room-‐temperatute phase is polar, space group P63 (see structure below), has been predicted to be ferroelectric (J. Appl. Cryst. (2009). 42, 58–62). The symmetries P6322 and P63/mmc had been proposed for two successive non-‐polar phases at high temperature. Applying the pseudosymmetry approach confirm the predictions for the non-‐polar phases of NaSb3F10. Show that apart from a P6322 phase, there are two more appropriate candidates for the intermediate phases between the polar phase P63 and the non-‐polar one of maximal symmetry, P63/mmc. (file: NaSb3F10.cif)

173 8.285 8.285 7.600 90 90 120 6 Sb 1 6c -‐0.1163 0.2243 0.55 Na 1 2b 0.333333 0.666667 0.467 F 1 6c 0.204 0.393 0.294 F 2 6c 0.111 0.229 0.640 F 3 6c 0.035 0.491 0.581 F 4 2b 0.666667 0.333333 0.545 (an arbitrary origin shift alon c has been added to the published structure to simulate a more general case, with unknown symmetry)

P4. Apparently complex phase Ga-II of Ga under pressure Analyse the structural pseudosymmetry of the orthorhombic phase Ga-‐II of Ga under pressure (structure below) using the program PSEUDO. Hint: As a first step check using option 3 the structural pseudosymmetry with respect to an isomorphic supergroup of index 13 (can you guess why?), specified by the transformation matrix: a,b,13c. Then proceed upwards through the chains of minimal supergroups using the default option 1. (file: Ga-II.cif) 20 5.976 8.576 35.758 90 90 90 14 Ga 1 4b 0.5000 0.1802 0.2500 Ga 2 8c 0.6956 0.4684 0.2716 Ga 3 8c 0.5804 0.7858 0.2861 Ga 4 8c 0.2772 0.5622 0.3081 Ga 5 8c -‐0.0341 0.7809 0.3292 Ga 6 8c 0.8482 0.4567 0.3430 Ga 7 8c 0.5632 0.6919 0.3666 Ga 8 8c 0.2324 0.4838 0.3851 Ga 9 8c 0.6129 0.2914 0.4003 Ga 10 8c 0.8276 0.5660 0.4250 Ga 11 8c -‐0.0030 0.2613 0.4435 Ga 12 4a 0.2460 0.0 0.5000 Ga 13 8c 0.1052 0.3090 0.5170 Ga 14 8c 0.3574 0.5518 0.5409 P5. Structural pseudosymmetry of Nd4GeO8 The compound Nd4GeO8 is reported to have polar Pmc21 symmetry (Doklady Akademii Nauk SSSR (1978) 241, 353-‐356) (see structure below). (i) Using PSEUDO (Option 1), show that this structure can be considered a small distortion of a Cmcm structure. (ii) Using SUBGROUPGRAPH or SUBGROUPS show the graph of maximal subgroups connecting the two symmetries. (iii) Using again PSEUDO (Option 3) obtain the atomic displacements relating the two structures. (file: Nd4GeO8.cif) 26 7.475 5.727 17.927 90 90 90

20 Nd 1 2a 0.000000 0.265100 0.000000 Nd 2 2b 0.500000 0.231400 0.973700 Nd 3 2a 0.000000 0.292500 0.205700 Nd 4 2b 0.500000 0.787400 0.270500 Nd 5 4c 0.241600 0.768900 0.090200 Nd 6 4c 0.261000 0.274100 0.383800 Ge 1 2b 0.500000 0.258100 0.180900 Ge 2 2a 0.000000 0.755800 0.298600 O 1 2a 0.000000 0.531400 0.104500 O 2 2b 0.500000 0.063000 0.100800 O 3 2a 0.000000 0.040900 0.114200 O 4 2b 0.500000 0.545400 0.143500 O 5 4c 0.186400 0.668100 0.248000 O 6 4c 0.312100 0.192400 0.236700 O 7 2a 0.000000 0.055500 0.319800 O 8 2a 0.000000 0.539900 0.366300 O 9 2b 0.500000 0.004300 0.370900 O 10 2b 0.500000 0.512400 0.362800 O 11 4c 0.215300 0.009700 0.486700 O 12 4c 0.258000 0.492400 0.497300 P6. Overlooked symmetry of Ca2Ge7O16 According to a structural model for Ca2Ge7O16 published in Doklady Akademii Nauk SSSR (1979) 245, 110-‐113, the symmetry group of the compound is Pba2. Despite the 4/mmm Laue symmetry of the di_raction diagram the authors of this publication were unable to find an appropriate tetragonal structural model, and refined the compound in Pba2 symmetry. Using PSEUDO, demonstrate that this structure differs from a tetragonal one with space group P-‐4b2, by atomic displacements which are practically negligible or within experimental accuracy, so that in fact this structure _le should be considered incorrect, being a case of overlooked symmetry. The pseudosymmetry of the structure was reported in (Acta Cryst. B (2002) 58, 921) and the compound has been recently confirmed to have P-‐4b2 symmetry by a new study of the structure (Acta Cryst. C (2007) 63, i47). (file: Ca2Ge7O16.cif) 32 11.340 11.340 4.6400 90 90 90 13 Ge 1 2a 0 0 0 Ge 2 4c 0.1335 0.1336 0.4996 Ge 3 4c 0.0666 0.3122 -‐.0107 Ge 4 4c 0.3123 0.0667 0.0095 Ca 1 4c 0.3350 0.3348 0.4970 O 1 4c 0.0012 0.1167 0.7467 O 2 4c 0.0288 0.2197 0.2695 O 3 4c 0.1686 0.2658 0.7246 O 4 4c 0.2657 0.1697 0.2645 O 5 4c 0.1171 0.0006 0.2411 O 6 4c 0.2198 0.0282 0.7180 O 7 4c 0.1398 0.4316 0.1548 O 8 4c 0.4327 0.1403 0.8306

P7. High temperature tetragonal phase of Tetragonal phase of GeF2 The compound GeF2, of symmetry P212121 (structure below) is reported to have at high temperatures an unknown tetragonal phase, with the primitive unit cell volume being essentially maintained. Using PSEUDO, postulate a probable space group or groups and a starting structural model for this high-‐temperature phase. (file: GeF2.cif) 19 4.682 5.158 8.312 90 90 90 3 Ge 1 4a 0.2340 0.0083 0.1311 F 1 4a 0.029 0.083 -‐0.018 F 2 4a 0.067 0.246 0.279

Tutorial on the use of the program AMPLIMODES of the Bilbao Crystallographic Server (www.cryst.ehu.es). J. Manuel Perez-Mato, D. Orobengoa, Mois I. Aroyo and C. Capillas Dept. de Fisica de la Materia Condensada, Facultad de Ciencia y Tecnologia, Universidad del Pais Vasco, Apto. 644, 48080 Bilbao, Spain. (see also D. Orobengoa, C. Capillas, M.I. Aroyo & J. M. Perez-Mato. J. Appl. Cryst. (2009), A42, 820-833 ) We know since the works of Landau that the natural language to deal with the static frozen distortions present in displacively distorted structures is the one of modes. Modes are collective correlated atomic displacements fulfilling certain symmetry properties. Structural distortions in distorted structures can be decomposed into contributions of modes with different symmetries given by irreducible representations of the parent space group. One can then distinguish primary and secondary (induced) distortions, which will have in general quite different weights in the structure, and will respond differently to external perturbations. In general, the use of symmetry-adapted modes in the description of distorted structures introduces a natural physical hierarchy among the structural parameters. This can be useful not only for investigating the physical mechanisms that stabilize these phases, but also for pure crystallographic purposes. The set of structural parameters used in a mode description of a distorted phase will in general be better adapted for a controlled refinement of the structure, or for instance for comparative studies between different materials or for ab-initio calculations. AMPLIMODES is a computer program available on the Bilbao Crystallographic Server that can perform the symmetry-mode analysis of any distorted structure of displacive type. The analysis consists in decomposing the symmetry-breaking distortion present in the distorted structure into contributions from different symmetry-adapted modes. Given the high- and the low-symmetry structures, AMPLIMODES determines the atomic displacements that relate them, defines a basis of symmetry-adapted modes, and calculates the amplitudes and polarization vectors of the distortion modes of different symmetry frozen in the structure. The program uses a mode parameterization that is as close as possible to the crystallographic conventions, using an asymmetric unit of the low-symmetry structure for describing modes and distortions. AMPLIMODES uses internally SYMMODES to produce the basis of symmetry-adapted modes to be used in the decomposition of the structural distortion, but instead of describing the modes in the high-symmetry setting, as SYMMODES, works in the setting of the subgroup. Let us consider as an example the Amm2 structure of the ferroelectric distorted perovskite BaTiO3 at 190K (Kwei et al. (1993). The only data needed by AMPLIMODES are the parent structure and the distorted structure to be analyzed (both structures described in standard settings). This, together with the transformation relating the space groups of the two structures, is sufficient for running the program:

Figure 1: Structure of the Amm2 phase of BaTiO3 This orthorhombic is one the three ferroelectric phases of BaTiO3, caused by the condensation at low temperatures of an unstable polar distortion. A three-fold degenerate polar instability associated with a single active 3-dim irrep produces three successive ferroelectric phases, by changing its direction within the 3-dim irrep space. Reference Structure: The program first transforms the input high-symmetry parent structure into the setting of the low symmetry phase (using TRANSTRU). This structure is the so-called REFERENCE STRUCTURE. This structure, and in particular the specific asymmetric unit chosen by the program, will be used as the reference configuration for the atomic displacements describing the low-symmetry structure:

Reference Structure

This listing describes the parent structure, but expressed in the low-symmetry space group. The number of atoms in the asymmetric unit increases in general with respect to the description in the high-symmetry group. Some Wyckoff orbits may split. In the present case, the number of independent atoms increases from 3 to 4, because the orbit of O1 splits into two. Note that the cell parameters listed in the reference structure above correspond to the transformation of the unit cell associated with the parent structure. Thus, they fullfill exactly b = c = √2a, according to the group-subgroup transformation. If the mode analysis demanded to the program really makes sense, this unit cell should not differ much from the one of the input low-symmetry structure, and this should be checked by the user. These idealized cell parameters of the reference structure will be used by the program for computing (when needed) the absolute values of the atomic displacements. In other words, the calculation of these displacements (in Å) is done disregarding the strain of the lattice of the low-symmetry structure with respect to the parent one. Atom Mapping: Once the asymmetric unit of the reference structure has been defined, the program does an “atom mapping” or “pairing” identifying the atoms in the low symmetry structure that correspond to those listed in the asymmetric unit of the reference structure. From the comparison of these pairs of atomic positions in the high and low symmetry structures, the atomic displacements are calculated:

The pairing is only done if the magnitude of the atomic displacements (for all atoms) is smaller than a given tolerance. This tolerance, which by default is 1 Å, can be increased by the user, but one should consider that a sucessful pairing with atomic displacements much larger than 1 Å may be nonsensical from a stereochemical viewpoint, and should be crosschecked carefully. The displacements u for the atoms in the asymmetric unit of the reference structure, listed in the last table above defines completely the displacive distortion relating the two structures. The space group of the low symmetry structure forces that any atom not present in this asymmetric unit, but related by a space group operation (R,t) with one atom in it having a displacement u, must have a displacement given by the rotation or roto-inversion operation R: Ru. Apart from the maximal atomic displacement, the output yields the amplitude of the total distortion, which is given by the square root of the sum of the square of all atomic displacements within a primitive unit cell of the reference structure. Polar structures: The Amm2 structure is polar along the z direction. This means that the displacive distortion relating both structures may include some global translation of the structure as a whole, due to the arbitrary choice of the origin along z in the Amm2 structure. For polar structures, the program shifts its origin to eliminate this global displacement. If this shift is not desired, the user can put (0,0,0) as the polar direction in a second input menu, only appearing for polar structures. If we introduce the correct direction (0,0,1), the program does the mentioned origin shift, yielding the new displacive distortion without global displacement along z:

Note that the origin shift done has slightly increased the maximal atomic displacement, but has decreased the total amplitude of the distortion. One should take into account that in the case of polar structures, its origin choice, relative to the one of the high-symmetry configuration, may include such a large

global displacement of the structure that AMPLIMODES can be unsuccessful to find an appropiate pairing of the two structures. In this case, one has two possibilities:

i) Try AMPLIMODES with a series of equivalent low-symmetry structures with the origin shifted systematically (this shifted structures can be constructed automatically with TRANSTRU) until the program achieves a correct pairing or ii) Apply PSEUDO to the low-symmetry structure to get a first choice for the origin shift minimizing somehow the atomic displacements.

Summary Output: The program then calls the program SYMMODES to obtain a basis of symmetry-adapted modes for describing the displacive distortion, makes the mode decomposition of the distortion with this basis, and lists the results. First, a summary appears:

The basis of symmetry-adapted modes are chosen such that each mode is restricted to displacements within a single Wyckoff orbit of the high temperature structure. The first Table in this summary lists the number of basis symmetry modes and their irreps, that describe the atomic displacements of the atoms in each occupied Wyckoff orbit of the high symmetry phase. For each Wyckoff type, the table lists the relevant irreps and in parenthesis the number of basis modes corresponding to this irrep and this Wyckoff orbit. This information only depends on the type of Wyckoff orbit, therefore all atoms with the same type of orbit are included in a single row. In our example, there are a total of five modes, four corresponding to irrep GM4- and one to irrep GM5-. The single mode GM5- only involves displacements of the high-symmetry orbit of atoms represented by O1. The total number of modes must coincide with the total number of free structural parameters that are necessary to describe the structure. Indeed, if we inspect the original asymmetric unit of the Amm2 structure, it can be seen that there are five atomic components that are “free” by symmetry, although one of them has been arbitrarily chosen zero due to the polar character of the structure. The second Table in this summary is the most important one. It lists the two irreps present in the distortion and the absolute amplitudes of these two symmetry components of the global distortion. The table gives for each irrep its wave vector, the restricted direction within the irrep space that the modes fulfill, their isotropy subgroup, and the dimension of its subspace. This dimension is the number of basis symmetry-adapted modes of this symmetry. The GM4- subspace is four-dimensional. In order to

describe a GM4- distortion four components must be given corresponding to the four GM4- (orthonormal) basis modes, one for the Ba atoms, one for the Ti atoms, and two for the oxygens. The GM5- subspace is one dimensional as only one basis mode of this symmetry exists. The four dimensional character of the GM4- subspace is however misleading, as it includes the global translation of the structure. Fixing the origin, as in a standard crystallographic description, reduces the number of degrees of freedom within this type of distortions to three. It is remarkable that the isotropy subgroup of the two irreps is Amm2. Therefore any of the two could the active irrep. However, the amplitudes obtained for the two symmetry-adapted distortions are very different. The amplitude of GM5- distortion is more than 20 times smaller. This is an extreme case of what we expect in distorted structures resulting (according to Landau theory) from the instability of an order parameter with symmetry properties given, by a single active irrep. In our example, the GM4- distortion corresponds approximately to a specific combination of the three-fold degenerate unstable polar normal modes which cause the successive ferroelectric phases in BaTiO3. Its amplitude can be identified with the Landau order parameter connecting this phase with the cubic perovskite. The GM4- distortion is therefore at the origin of this ferroelectric phase, while the GM5- distortion is only a secondary effect allowed by symmetry, but marginal in the phase stabilization. Therefore, the strong difference of amplitudes of the two frozen distortions is the signature of the underlying lattice dynamics mechanism causing this phase. Note that the amplitude of the total distortion, 0.1650 Å is again given in this summary. It is in fact a cross check of the calculation done previously summing the square of the atomic displacements of the total distortion. Here, the calculation is done taking the square root of the sum of the square of the amplitudes listed in the Table for all irreps (The Pythagoras theorem is fulfilled by the amplitudes: as we are in a vector space!). In this example: 0.1650= [(0.1649)2+ (0.0056)2]½ . Basis of symmetry-adapted modes: The subsequent detailed output includes first a list of the basis of symmetry-adapted modes used in the decomposition:

The five basis symmetry-adapted modes used by the program are shown. They are listed separately for each irrep. They are labelled using the atom label of the representative of the corresponding Wyckoff orbit, plus a numeric index that enumerates the modes in case that more than one exists. The basis modes are given listing the displacements of only the atoms of the relevant Wyckoff orbit that are present in the asymmetric unit of the reference structure. These displacements are expressed in relative units with respect to the unit cell parameters of the subgroup cell. If transformed into absolute values (in Å) using the reference unit cell, the square of these displacements, when summed for all atoms within a primitive unit cell of the low symmetry space group, must yield 1 Å2: Σi u

2(i) = 1 Å2 This means that the basis modes are normalized to 1 Å. We can check that for instance for mode (O1 ,1): 2x[(0.062406 x 5.665339)2 + (0.062406 x 5.665339)2] + 1x[(0.124813 x 5.665339)2 = 0.999995 where we have used the fact that b=c= 5.665339Å in the reference structure, and that the multiplicity of the splitted orbits of the oxygens O1 and O1_2, reduced to a primitive unit cell, are 2 and 1, respectively. Note that the inclusion in the sum of the atoms not listed in the Table can be done simply by multiplying by the respective multiplicities (for a primitive cell) the contribution of the atoms in the Table, because, due to the symmetry condition mentioned above, the modules of their rotated displacements will be the equal to those of the corresponding atom in the asymmetric unit. The basis modes are also orthogonal: Σi umode1(i). umode2(i) = 0 Let us check this condition for the modes (O1, 1) and (O1, 2): 2x[(0, 0.062406 x 5.665339, 0.062406 x 5.6653399) . (0, - 0.088256 x 5.665339, 0.088256 x 5.6653399)] + 1x[(0, 0, 0.124813 x 5.665339) . (0, 0, 0)] = 1.12331 x 10-7

where again the sum for the full orbit can be done by multiplying the dot products of the atom representatives by their multiplicities. We can observe in the above table of basis modes the symmetry contraints associated with each type of symmetry. For instance, the displacements of the O1 and O1_2 atoms in the mode GM5- are such that the displacement of the O1_2 atom is restricted to the z-direction (subgroup setting), and is twice the value of the equal displacement of the O1 atom along the y and z directions. O1 and O1_2 are independent atomic positions in the Amm2 group, but the GM5- mode correlates their displacements. It is the free combination of the GM5- and GM4- modes for the two atoms, both allowed in the Amm2 space group, that makes the two atomic sites independent. Irrep Distortions. Amplitudes and polarization vectors: Once the basis of symmetry-adapted modes is defined, the output gives the details of the mode decomposition. For each irrep, the parameters describing the distortion of this symmetry (irrep distortion) present in the low-symmetry structure are listed:

After a detailed information on the symmetry of the irrep distortion: wave vector, restricted direction, isotropy subgroup, and transformation matrix, the output

lists the amplitude of this irrep distortion and its normalized so-called polarization vector. The polarization vector of the irrep distortion is the set of normalized correlated atomic displacements of all atoms that multiplied by the given amplitude yields the actual distortion of this symmetry, which is present in the low-symmetry structure. The polarization vector is expressed in two equivalent forms, one algebraic, and one crystallographic. In our example, the space of GM4- distortions is 4-dimensional, and four different basis modes of this symmetry were listed above. Any GM4- distortion can be expressed as a vector in this 4-dim space, by means of its four components with respect to this basis. The output gives these four components: (0.1745, 0.7585, -0.2536, -0.5744) indicating the basis mode corresponding to each of them: (Ba1,1), (Ti1,1), (O1,1) and (O1,2), respectively. The vector is normalized: 0.17452 + 0.75852+ (-0.2536)2 +(-0.5744)2 = 1.00002 The program ist just saying that the GM4- distortion present in the structure can be obtained by combining the four basis modes in the form given by this 4-dim normalized vector and multiplying the resulting distortion by an amplitude of 0.1649. This algebraic form of expressing the irrep distortion can be very useful to compare different distorted structures, obtained for instance at different temperatures, for similar compounds, or to compare experimental and theoretical (ab-initio) structures. In general, one expects the polarization vector of the active/primary irrep distortion to be rather invariant to temperature variations, modelizations, or composition, while its amplitude may vary considerably. We can compare an irrep distortion in two different isomorphic structures by comparing separately their amplitudes, and their polarization vectors. The amplitude just compares the magnitude of the distortion, while the polarization vector compares the type of distortion within a certain symmetry restriction. The polarization vector defines a direction in the 4-dimensional vector space of the GM4- distortions. Two distortions would be of same type, independently of their magnitude, if the dot product of their 4-component vectors is one. The closer to one this dot product is, the more similar are the two distortions. For instance, the Amm2 structure was also determined at 250K (Kwei et al. (1993), and the polarization vector of the GM4- distortion for this structure determined with AMPLIMODES is: (0.2084, 0.7073, -0.1889, -0.6486) The similarity of this vector with the one corresponding to the structure at 190K discussed until now, can be clearly seen by comparing each component. But a more quantitative comparison is just given by their dot product, which is 0.9933.

The output on each irrep distortion also includes a crystallographic description of its polarization vector. For the asymmetric unit of the reference structure, a table is given with the correlated atomic displacements (in relative units), which describe the normalized polarization vector of the irrep distortion. This Table is just obtained by multiplying the components of the polarization vector by the corresponding basis modes. Thus, for instance, the atomic displacements of the oxygens O1 and O1_2 in the table above can be obtained by just combining the two GM4- basis modes for these atoms, each multiplied by the corresponding component in the four-dimensional vector:

For the GM5- distortion the output is somehow simpler, since it is a 1-dim subspace:

For each irrep, the program can provide a virtual structure with only the distortion of this symmetry, which has been calculated and is present in the input distorted structure. The output of the program for our example can be summarized as follows. The Amm2 structure can be understood as the sum of two distortions with symmetry properties given by two different irreps. The amplitudes of the two irrep distortions are

very different. The correlated atomic displacements corresponding to each them, are given by the listed polarization vectors. Schematically we can say that the observed Amm2 structure can be described in the following form:

We can represent graphically this mode decomposition as:

In the figures we have represented the effect of each of the two irrep distortions in an exaggerated way by increasing arbitrarily their amplitudes. Note that the displacements corresponding to the GM4- symmetry include a subtle hidden correlation, namely the displacements of the oxygens O1 and O1_2 fulfill δyO1+δzO1+δzO1_2 = 0. This implies in the resulting structure a non-crystallographic symmetry constraint: yO1 + zO1 + zO1_2 = 0. As the GM5- distortion, that breaks this relation, is a secondary marginal distortion with a very small amplitude, this non-crystallographic relation is approximately maintained by the atomic coordinates of the experimental structure. Finally, it is important to emphasize that for doing a meaningful symmetry mode analysis of a distorted structure with AMPLIMODES is NOT necessary to know a specific real parent structure. Starting from the distorted phase one can construct an

ideal parent structure whose symmetry group is determined by the structural pseudosymmetry of the low-symmetry phase, either by hand, from previous knowledge of similar compounds, or using computer tools as PSEUDO, also available on the Bilbao Crystallographic Server. If the atomic coordinates of the parent high-symmetry structure include some values not forced by symmetry, it is sufficient to give them reasonable approximate values. The structural differences between the ideal parent structure, constructed in such a way, and any other possible parent structure of this symmetry are only due to contributions of symmetry modes compatible with the high space groups, i.e. the so called totally symmetric modes transforming according to the identity irrep. The contributions of the much more important symmetry-breaking distortion modes do not depend on the choice of the variable atomic coordinates of the parent structure. Exercise 1: S2Sn2P6 has a monoclinic P21/c phase at high temperatures (J. Sol. State Chem. (1992) 96, 294) and a ferroelectric non-centrosymmetric phase of symmetry Pc at low temperatures (Z. Naturf. B. (1974) 29, 312-317 ) : 14 6.5500 7.4930 11.3157 90.00 124.19 90.00 5 Sn 1 4e -0.2842 0.3692 -0.2431 P 1 4e 0.3723 0.3914 -0.0671 S 1 4e 0.1362 0.4975 -0.2629 S 2 4e 0.21 0.309 0.0328 S 3 4e 0.5988 0.1976 -0.057 7 6.513000 7.488000 11.309900 90.000000 124.000000 90.000000 10 Sn 1 2a 0.304600 0.385600 0.277000 Sn 2 2a 0.740900 0.124500 0.777900 P 1 2a 0.372800 0.860800 0.433600 P 2 2a 0.626600 0.644700 0.567400 S 1 2a 0.140700 0.751100 0.239800 S 2 2a 0.203800 0.946200 0.528800 S 3 2a 0.602100 0.051700 0.444200 S 4 2a 0.865200 0.743700 0.764900 S 5 2a 0.785900 0.563300 0.464700 S 6 2a 0.402500 0.448000 0.555000 i) Obtain with AMPLIMODES that the ferroelectric structure has two dirtortion modes, a primary one that yields the Pc space group, i.e. the order parameter distortion, and a secondary one compatible with the parent symmetry. ii) Change arbitrarily some of the coordinates of P21/c, but displacing the positions only a small amount (below 1 Å). Check that AMPLIMODES gives the same results (amplitude and polarization vector) for the symmetry-breaking mode, changing only the form of the fully symmetric mode GM1. iii) Which atoms are moving more strongly in the transition? Derive the atomic positions of the Sn atoms in two virtual Pc structures having only the experimental primary ferroelectric mode with an amplitude of 0.1Å and of 0.2 Å. One could use such virtual structures in an ab-initio calculation to characterize the energy variation of the system as a function of the ferroelectric mode.

Exercise 2. “Complex” phase of Ga under pressure Ga under pressure exhibits a phase with symmetry C2221 and 52 atoms per primitive unit cell (see below). It has been reported as a “complex” structure and described in terms of some sequence of atomic layers along the c axis (PRL 93, 205502 (2004). i) Using the program AMPLIMODES show that the structure can be understood as a distorted structure of a simple Fddd structure, with its unit cell having the same a and b parameters, and c being 13 times smaller, and with only a single independent Ga atom at position 8a (1/8,1/8,1/8). (origin shift of the transformation between the two space groups: -1/8 1/8 -3/8). ii) Identify the Ga atom in the C2221 structure with the largest displacement with respect to the ideal Fddd position. iii) From the output identify the prevailing primary mode and its wave vector. iv) Check that the two next dominant modes have wave vectors corresponding to a second and third harmonic of the primary mode. v) If the structure is described with these three first harmonics that have much larger amplitudes, how many parameters are required? Compare this number with the number of parameters required in a conventional crystallographic description. GaII structure: # space group C2221 20 5.976 8.576 35.758 90 90 90 14 Ga 1 4b 0.5000 0.1802 0.2500 Ga 2 8c 0.6956 0.4684 0.2716 Ga 3 8c 0.5804 0.7858 0.2861 Ga 4 8c 0.2772 0.5622 0.3081 Ga 5 8c -0.0341 0.7809 0.3292 Ga 6 8c 0.8482 0.4567 0.3430 Ga 7 8c 0.5632 0.6919 0.3666 Ga 8 8c 0.2324 0.4838 0.3851 Ga 9 8c 0.6129 0.2914 0.4003 Ga 10 8c 0.8276 0.5660 0.4250 Ga 11 8c -0.0030 0.2613 0.4435 Ga 12 4a 0.2460 0.0 0.5000 Ga 13 8c 0.1052 0.3090 0.5170 Ga 14 8c 0.3574 0.5518 0.5409 SYSTEMATIC WEAKNESS OF SECONDARY DISTORTIONS The case of the Amm2 structure of BaTiO3 is an extreme example of the systematic hierarchy that usually happen between primary distortions, associated with the active irrep of the symmetry break, and the secondary ones. Let us consider as a second example the room-temperature P63cm phase of KNiCl3: 185 11.795 11.795 5.926 90 90 120 5

K 1 6c 0.3353 0.3353 0.3294 Ni 1 2a 0 0 0 Ni 2 4b 0.333333 0.666667 0.1230 Cl 1 6c 0.1598 0 0.2604 Cl 2 12d 0.5056 0.1732 0.3852 This structure has the following virtual P63/mmc parent structure: 194 6.80985 6.80985 5.926 90 90 120 3 Ni 1 2a 0.0 0.0 0.0 K 1 2d 0.333333 0.666667 0.75 Cl 1 6h 0.16 0.32 0.25 This virtual structure can be obtained for instance using the program PSEUDO of the server, or just by looking at the P63/mmc structure of similar compounds. Note that the special coordinates 1/3, 2/3 are given with 6 digits. This is important, as in other programs of the server, for the program to recognise the special value of the coordinate. From the comparison of the multiplicities of the atomic positions, it is straightforward to see that the number of fomula unit per primitive unit cell is Z=2 for the parent phase, while Z=6 in the non-centrosymmetric phase. This means a triplication of the unit cell (ik=3). Using CELLSUB or SUBGROUPGRAPH it is then inmediate to obtain the transformation matrix relating both space groups: (a+2b, -2a-b, c; 0 0 0). This transformation is only one of a set of possible ones, which are fully equivalent. In fact the cell parameters of the virtual parent structure have been chosen to fullfill the cell transformation exactly. We use then AMPLIMODES to perform the mode decomposition of the P63cm structure with the following basic result:

It is convenient to analyse this result, having in mind the graph of minimal subgroups connecting the high and low symmetries:

Figure 2: Graph of maximal subgroups connecting the parent and low-symmetry space groups of KNiCl3

One can see that there is a primary active irrep with wave vector (1/3,1/3,0) and label K3 and two secondary active irreps, associated with two intermediate isotropy subgroups. The K1 distortion also corresponds to a wave vector (1/3,1/3,0), so that it produces the same cell multiplication as K3, but maintains the point group of the parent phase, while the second distortion, GM2- at the Brillouin zone centre, keeps the parent lattice and is a polar mode, which should be responsible of some spontaneous polarization in the distorted phase. The mode decomposition provided by AMPLIMODES shows the extreme prevailing role of the primary distortion corresponding the irrep K3. From a total amplitude of 1.72 Å of the total distortion, the contribution of the K3 component is 1.70 Å. This K3 distortion involves only 2 of the 9 structural degrees of freedom present in the P63cm structure. The much larger amplitude of the K3 distortion is a clear indication that it can be identified with the primary order parameter of this phase, and it can be further inferred that K1 and GM2- are induced secondary effects. The material is then a ferroelectric of improper character, the spontaneous polarization (of symmetry GM2-) being an induced secondary effect. The polarization vector of the K3 dominant distortion is given by AMPLIMODES as:

This table only adquires a full meaning when combined with the Table defining the reference structure and the asymmetric unit chosen by the program:

Reference Structure: 185 11.795007 11.795007 5.926000 90.000000 90.000000 120.000000 5 Ni 1 2a 0.000000 0.000000 0.000000 Ni 1_2 4b 0.666667 0.333333 0.000000 K 1 6c 0.333334 0.000000 0.750000 Cl 1 6c 0.160000 0.000000 0.250000 Cl 1_2 12d 0.826667 0.333333 0.250000

This list defines the atoms in the Table that describes the polarization vector of the K3 mode. The splitted Ni1 and Ni1_2 atoms within the asymmetric unit of the distorted structure move in opposite directions along z, with a ½ relation among its displacements. The same relation exists among the displacements of the two Cl sites. These correlations are forced by the K3 symmetry. But the table shows an additional correlation between the displacements of the Cl and Ni sites, namely the displacements of Ni1 and Cl1 are practically the same, and this is not forced by symmetry. Combining AMPLIMODES with FULLPROF (J. Rodriguez-Carvajal, http://www.ill.eu/sites/fullprof/) one can obtain in a straighforward manner a graphical scheme of this K3 distortion (see below for a more detailed explanation of the combined use of AMPLIMODES and FULLPROF):

Figure 3: Scheme of the K3 distortion present in KNiCl3

The correlation of the Ni and Cl displacements implies that the K3 distortion involves global displacements of the NiCl3 columns as rigid units. This feature of the primary K3 distortion is forced neither by symmetry, nor by a strong rigidity of the Ni positions within the octahedra. In the observed structure in fact the Ni atoms clearly displace relatively to their surrounding Cl6 octahedra, and approach along the c axis one of the two Cl3 triangles forming the octahedron. But these Ni displacements are not part of the K3 distortion; they follow a pattern according to the GM2- symmetry. The -½ relation between the displacements of the Cl1_2,Ni1_2 atoms and the Cl1 and Ni1 atoms (this relation indeed forced by the K3 symmetry) yields the antiphase shift of the NiCl3 octahedral columns within the P63cm unit cell, with respect to those on the vertices, causing the triplication of the cell. It important to note that the secondary modes break in general these strict K3-correlations of the displacements between atomic

sites, which are crystallographically independent. For instance the GM2- mode introduces displacements with completely different correlations:

Figure 4: Scheme of the GM2- distortion present in KNiCl3

But the prevailing role of the K3 distortion makes that its atomic correlations are also approximately fulfilled by the total distortion, despite the presence of the secondary distortions. Hence the structure has approximate non-crystallographic constraints coming from the physical mechanism, which estabilizes the phase. Exercise 3: Constrained structural model of SrMnO3

A C2221 structural model for SrMnO3 has been recently proposed (Phys. Rev. B 75 (2007) 104417). The structure is listed below. i) Decompose this structure into modes with respect to the structure of higher symmetry, P63/mmc, observed at 350K (also listed below) and check that the structure has an active irrep and a strong primary distortion with this symmetry, and that two secondary distortions have amplitudes, which are practically negligible. ii) Comparing the atomic coordinates of the model with the general form of the corresponding Wyckoff positions, check that several refinable coordinates have been restricted a priori in the model. How many degrees of freedom have been in practice used in the refinement compared with a full conventional refinement under the C2221 space group? How many degrees of freedom would have been used in the refinement if the secondary modes would have been set a priori to zero. iii) By checking the form of the polarization vector of the primary mode, derive that this model has also included an unjustified restriction on the form of the primary distortion, which is not forced by symmetry considerations SrMnO3 at 350K: 194 5.461 5.461 9.093 90 90 120 5

Sr 1 2a 0 0 0 Sr 2 2c 0.333333 0.666667 0.25 Mn 1 4f 0.3333333 0.666667 0.61264 O 1 6g 0.5 0 0 O 2 6h -0.81858 0.81858 0.75

SrMnO3 at 100K: 20 5.4435 9.4122 9.0630 7 Sr 1 4a 0.0096 0 0 Sr 2 4b 0 0.333333 0.25 Mn 1 8c -0.0123 0.3333333 0.6131 O 1 4a 0.5212 0 0 O 2 8c 0.7712 0.25 0.0085 O 3 4b 0 -0.1798 0.25 O 4 8c 0.2696 0.0899 0.2412 Exercise 4: Hexagonal BaMnO3 BaMnO3 has been determined at 1.7K and at 80K (Chem. Mater. 12 (2000) 831-838) with space group P63cm (see structures below). These structures are supposed to be distorted structures from an ideal P63/mmc ABX3 structure with atoms at positions A 2a, B 2d and O 6h (x, 2x, 0.25), with x about 0.15. (It is a similar situation to the one of the example of KNiCl3 discussed above). i) Draw with SUBGROUPGRAPH the graph of maximal subgroups relating both symmetries. ii) Descompose with AMPLIMODES the two structures into symmetry-adapted modes with respect to the ideal parent phase. Identify the primary dominant distortion. Try to understand and characterize this dominant mode. iii) How many positional atomic parameters are necessary to refine these structures and how many are required to describe the dominant primary distortion? iv) Compare the mode decomposition of the two structures corresponding to different temperatures. First, compare the total amplitudes for each irrep distortion mode, and check if the result is plausible. Second compare the polarization vectors of each distortion mode, by inspection, and by calculating their dot products. Detect that there is some inconsistency in the models for one of the secondary modes (the atomic displacements are more or less opposite in the two models, relative to those of the primary mode). So probably one of the structures is wrong (corresponding to a false minimum in the refinement). v) Compare the mode decomposition with the one of the example KNiCl3 discussed above. BaMnO3 at 1.7K: 185 9.8467 9.8467 4.8075 90.00 90.00 120.00 5 Ba 1 6c 0.339 0.339 0.230 Mn 1 2a 0 0 0 Mn 2 4b 0.666667 0.333333 0.952 O 1 6c 0 0.1492 0.248 O 2 12d 0.6644 0.4824 0.200 BaMnO3 at 80K: 185 9.8467 9.8467 4.8075 90.00 90.00 120.00 5

Ba 1 6c 0.332 0.332 0.238 Mn 1 2a 0 0 0 Mn 2 4b 0.666667 0.333333 0.963 O 1 6c 0 0.150 0.250 O 2 12d 0.667 0.484 0.212

AMPLIMODES COMBINED WITH FULLPROF A symmetry mode decomposition can be done not only a posteriori, i.e. once the low symmetry structure is known, but may be very useful for the actual process of determining the structure. The expected hierarchy among the distortions of different symmetry and the essential invariance of their polarization vectors can make very advantageous a direct refinement of the amplitudes of a basis of symmetry modes, as collective coordinates, instead of the usual individual atomic coordinates. In collaboration with J. Rodriguez-Carvajal we have included in AMPLIMODES an option, which creates a special output to be directly used by FullProf (J. Rodriguez-Carvajal, http://www.ill.eu/sites/fullprof/), for doing such type of refinement. By this means, FullProf can now use in the refinement process the amplitudes of the basis of symmetry modes defined by AMPLIMODES as refinable positional parameters, alternative to the atomic coordinates. This option can also be used to generate, using FullProf, figures of the distortion modes, and of the basis of symmetry-adapted modes, in general. The FullProf option can be chosen in the starting menu, and the program completely changes its output with respect to what has been explained above. Concerning the input, the inclusion of a specific low-symmetry distorted structure becomes optional. In this option, AMPLIMODES creates a so-called pcr file to be used as input for FullProf. This pcr file, by default, is intended to be an input file for simulating with FullProf a neutron powder diffraction diagram. If FullProf is executed using this pcr file, not only the simulated diffraction diagram, but also .fst files are created that can be read by FullProf Studio to represent graphically the different irrep distortions present in the distorted structure. The pcr file created by AMPLIMODES can be easily changed by the user to become the starting pcr file for a refinement of diffraction data, with the amplitudes of the basis of symmetry-adapted modes defined by AMPLIMODES as refinable parameters, instead of the conventional atomic coordinates. Let us take again the example of the Amm2 phase of BaTiO3 discussed above. AMPLIMODES under the FullProf option gives two alternative outputs. One is a set of text lines which can be introduced by copy/paste in a working pcr file of the user, to allow the direct refinement of the amplitudes of basis of symmetry-modes defined by AMPLIMODES:

The first lines define the reference structure. Its atomic coordinates are set constant, and will not be refined. Then, the set of 5 symmetry modes defined by AMPLIMODES are listed. The modes are numbered from 1 to 5, and the first column of each line indicates to which mode belongs the triad of displacements listed in this line. For instance, the mode GM4- labelled (O1 1) in the previous output is here mode 3, and the displacements for atoms O1 and O1_2 corresponding to this mode are given in two consecutive rows headed by the number 3. The number of lines necessary to define all basis modes using this format is 8, which is the variable V_MODES. A_MODES is the number of basis symmetry modes (5 in our example). It should coincide with the number of refinable positional parameters in a conventional treatment. The digit 2 in the same row controls the output of .fst graphic files (consult the webpage of Fullprof for other graphical options). The last lines introduce, as starting model for the eventual refinement, the amplitudes of the 5 basis symmetry modes that have calculated by AMPLIMODES by decomposing the input low-symmetry structure. If no low-symmetry structure is included in the input, these amplitudes would appear with null values, and the starting model would be the reference structure. The value 1.00 after the 5 amplitudes sets them by default as refinable parameters. Compared with the output of AMPLIMODES in its normal option, the amplitudes for the basis modes listed here are obtained by multiplying the corresponding component of the irrep polarization vector by the amplitude of this irrep distortion. But one should take into account that in the FullProf option no origin shift is introduced for polar structures. Therefore, for a comparison, the origin shift should be avoided in the normal option by introducing (0,0,0) as polar direction. Polar Structures: In the FullProf option, AMPLIMODES does not introduce an origin shift of the low symmetry polar structure, since the amplitudes of the modes are only

intended as starting values for a refinement. This is the reason why the amplitude of the GM4- mode for Ba1 is zero, since in the input structure its z-coordinate is fixed to cero. Similarly as it happens with a conventional refinement, in a direct refinement of the mode amplitudes, the amplitude of either mode 1 or mode 2, which involve only atomic displacements along z, should be fixed, while the other amplitudes are refined. As an alternative, AMPLIMODES can create a full pcr file for the simulation of a neutron powder diffraction diagram:

Running FullProf with this pcr file, apart from creating the simulated diffraction diagram, produces a .fst file for each irrep distortion present in the structure according to the mode amplitudes given in the .pcr file. These .fst files can be read by FullProf Studio to represent graphically the distortion modes. Note that the cell parameters in this .pcr file are those of the reference high symmetry structure in the low symmetry setting. For FullProf refinements you will have to change by hand these cell parameters to introduce the experimental values of the low-symmetry phase. Note however that, in this case, the symmetry and orthogonal properties of the modes will in general be broken in a degree that depends on the magnitude of the strain between the two cells. This will be not a problem for refinements (except for extreme unrealistic strains), but a proper visualization with FullProf Studio and a correct interpretation of the symmetry modes will require the use of the reference unit cell provided in the .pcr file provided by AMPLIMODES. A final caution remark: If the reference unit cell is very large the default values used for the pcr file may fail and FullProf may crash. This problem can usually be palliated by changing the U, V, … peak profile parameters, making the peaks thinner. Exercise 5: i) Run the default example of AMPLIMODES for the Amm2 structure of BaTiO3, but making the program not to shift the origin. Inspecting the output of this run, check that the values of the amplitudes for the basis modes in the pcr file above are correct. ii) If your are familiar with FullProf, create the pcr file for this default example, and use it to obtain with FullProf Studio a graphical representation of the GM4- and GM5- distortions in the Amm2 phase of BaTiO3.

Hands-on exercises on the use of AMPLIMODES A1 Secondary modes can be negligible reducing the number of degrees of freedom. NbS3 is triclinic with space group P-1 (N. 2), but belongs to a family where most of the compounds have P21/m (N. 11) symmetry and their cell parameter along b is approximately half the value. NbS3 has a small distortion with respect to this monoclinic structural model common to the family. A virtual parent structure of P21/m can be easily postulated for NbS3 from this knowledge [it can also be derived using the program PSEUDO of the Bilbao crystallographic server that detects pseudosymmetry and builds-up the configuration of higher symmetry closer to the observed one]: 11 4.963 3.365 9.144 90. 97.17 90. 4 Nb 1 2e 0.715500 0.75000 0.348650 S 1 2e 0.763400 0.25000 0.553550 S 2 2e 0.878850 0.25000 0.169450 S 3 2e 0.466950 0.25000 0.174150 The experimental P-1 structure is in the CIF file provided. The transformation matrix between the settings of the two groups must include the duplication of the b parameter, but does not require an origin shift. (file required: icsd_NbS3.cif) i) Apply AMPLIMODES to obtain the mode decomposition. Observe that while the structure has 24 free parameters, the distortion is dominated by an irrep distortion that is determined by only 12 parameters. Check the amplitude of the full symmetric GM1+ distortion (why is it zero?). Compare the polarization vectors of the two irrep distortions. ii) Using one of the different possibilities available, obtain a graphical representation of the irrep distortion modes in NbS3. A2 Polar modes depend on the origin choice. Knowledge of a parent phase is not necessary S2Sn2P6 has a monoclinic P21/c phase at high temperatures (J. Sol. State Chem. (1992) 96, 294) and a ferroelectric non-centrosymmetric phase of symmetry Pc at low temperatures (Z. Naturf. B. (1974) 29, 312-317 ) : 14 6.5500 7.4930 11.3157 90.00 124.19 90.00 5 Sn 1 4e -0.2842 0.3692 -0.2431 P 1 4e 0.3723 0.3914 -0.0671 S 1 4e 0.1362 0.4975 -0.2629 S 2 4e 0.21 0.309 0.0328 S 3 4e 0.5988 0.1976 -0.057 7 6.513000 7.488000 11.309900 90.000000 124.000000 90.000000 10 Sn 1 2a 0.304600 0.385600 0.277000 Sn 2 2a 0.740900 0.124500 0.777900 P 1 2a 0.372800 0.860800 0.433600

P 2 2a 0.626600 0.644700 0.567400 S 1 2a 0.140700 0.751100 0.239800 S 2 2a 0.203800 0.946200 0.528800 S 3 2a 0.602100 0.051700 0.444200 S 4 2a 0.865200 0.743700 0.764900 S 5 2a 0.785900 0.563300 0.464700 S 6 2a 0.402500 0.448000 0.555000

i) Obtain with AMPLIMODES that the ferroelectric structure has two dirtortion modes, a primary one that yields the Pc space group, i.e. the order parameter distortion, and a secondary one compatible with the parent symmetry. (Choose that AMPLIMODES shifts the origin of the polar distorted phase). Which atoms are moving more strongly in the transition? [The transformation matrix between the settings of the two groups includes an origin shift of (0 ¼ 0). This can be obtained for instance using the program SUBGROUPGRAPH ] ii) Let us change now arbitrarily some of the coordinates of the parent P21/c structure displacing the positions by some small amount (below 1 Å): #”Wrong Parent Structure” 14 6.5500 7.4930 11.3157 90.00 124.19 90.00 5 Sn 1 4e -0.3 0.38 -0.27 P 1 4e 0.33 0.36 0.07 S 1 4e 0.1 0.52 -0.23 S 2 4e 0.25 0.33 0.1 S 3 4e 0.6 0.15 -0.1 Apply AMPLIMODES, increasing now the default tolerance D to 2 Å. Check the result choosing that AMPLIMODES shifts the origin, and then without shifting. Check that if the shift is done, the primary polar distortion does not change, while the secondary one, compatible with the parent symmetry, changes drastically. (Conclusion: symmetry mode analysis can be done even without knowing the structure of the parent phase…) A3 Comparison of polarization vectors: detection of a false minimum BaMnO3 has been determined at 1.7K and at 80K (Chem. Mater. 12 (2000) 831-838) with space group P63cm. The structure at room temperature is also known and has a higher symmetry: P63/mmc. The CIF files of the three structures are attached (the icsd CIF files have been modified introducing 6 digits for 1/3 and 2/3 values!). The relation between the settings of the two symmetries can be chosen as (a+2b, -2a-b, c; 0 0 0). i) Decompose with AMPLIMODES the 80K structure into symmetry-adapted modes with respect to the parent phase. Identify the primary dominant distortion. Try to understand and characterize this dominant mode. ii) How many positional atomic parameters are necessary to refine this structure and how many are required to describe the dominant primary distortion? iii) Apply AMPLIMODES to the 1.7K structure. Compare the results with those of the structure at 80K. First, compare the total amplitudes for each irrep distortion mode, and check if the result is plausible. Second compare the polarization vectors of each distortion mode, by inspection, and then by calculating their dot products. You should

detect that there is some inconsistency in one of the models for one of the secondary modes (the atomic displacements of this secondary mode are more or less opposite in the two models, relative to those of the primary mode). So probably one of the structures is probably wrong (corresponding to a false minimum in the refinement). (files required: icsd_parent_BaMnO3.cif, icsd_BaMnO3_80K.cif, icsd_BaMnO3_1.7K.cificsd_BaMnO3_1.7K.cif)

Tutorial exercises on the use of MAXMAGN and other tools of the Bilbao Crystallographic Server for the analysis of magnetic structures.

J.M. Perez-Mato1, S. V. Gallego1, L. Elcoro1, E. Tasci2, G. de la Flor1, M.I. Aroyo1

1 Depto. Física de la Materia Condensada, Fac. de Ciencia y Tecnología, Universidad del Pais Vasco, UPV/EHU, Apdo. 644, Bilbao, Spain.

2 Physics Department, Middle East Technical University, Ankara, Turkey.

The program MAXMAGN in the Bilbao Crystallographic Server (http://www.cryst.ehu.es) is a computer tool, freely available in the web, which facilitates the systematic use and application of magnetic symmetry in the analysis and determination of commensurate magnetic structures.

It is a known fact that most of the reported magnetic structures are “1-k” magnetic phases, i.e. their magnetic orderings have a single propagation vector. Furthermore, most of them have spin configurations that possess one of the possible maximal magnetic symmetries compatible with its propagation vector. This second property is not well known and its importance and utility has not been yet exploited. The program MAXMAGN makes use of this property in a systematic way by calculating all possible maximal magnetic symmetries and deriving magnetic structural model consistent with them. The assumption of a maximal magnetic symmetry restricts the possible spin arrangements, and limits the number of free parameters or degrees of freedom to be determined experimentally. In simple cases, these restrictions are equivalent to those obtained applying the representation method, but in general they can be more restrictive.

The propagation vector of a 1-k magnetic structure can in principle be identified from diffraction experiments. Once this modulation vector is known, the few possible magnetic space groups (i.e. Shubnikov groups) consistent with this vector fulfilling that they have a maximal possible symmetry can be systematically determined. From their knowledge, all possible alternative spin models of maximal symmetry consistent with the observed propagation vector can be derived. This is the main purpose of MAXMAGN. The alternative models provided by the program can then be contrasted with and fitted to the experimental data.

Using as input only the knowledge of the space group of the paramagnetic phase (we shall call it in the following parent space group) and the propagation vector k, MAXMAGN first provides all possible magnetic space groups of maximal symmetry consistent with this propagation vector. In the following we shall call these groups “k-maximal magnetic groups”. If a paramagnetic structure is also introduced, the program determines the spin arrangements allowed for each of these possible k-maximal symmetries, and defines their refinable parameters. The output is organized in such a way that the program can be systematically applied to identify and analyse all possible alternative spin models. A CIF-like file can then be obtained for each of the alternative magnetic structures of k-maximal symmetry, which can then be refined in programs like JANA2006 [1] or FULLPROF [2], or they can be introduced in the program ISODISTORT [3] for mode analysis, or transformed with the structure editor STRCONVERT of the Bilbao Crystallographic Server. These CIF-like files can also be used for 3D visualization with VESTA[4] or Jmol [5]. A direct link to the tool MVISUALIZE, also in the Bilbao

http://www.cryst.ehu.es/

crystallographic server also allows an inmediate visualization of each of the alternative models.

The magnetic structure models provided by the program are given by default in a setting as similar as possible to the one of the parent paramagnetic phase (the so-called parent like setting), but they can also be obtained in a standard setting of the magnetic group considered, or alternatively in a setting defined by the user.

If none of the models with k-maximal symmetry are satisfactory, the program can descend to lower symmetries, adding in this controlled way additional degrees of freedom. The program also allows to derive, for a given spin model, all physically equivalent spin arrangements to which the structure can in principle be switched, as they correspond to twin-related or, in general, domain-related spin configurations.

The program MAXMAGN provides an alternative approach to the traditional representation method for the parameterization of magnetic structures, which in most cases is more intuitive and direct. The direct use of magnetic symmetry arguments allows to establish in many cases (when the active irrep is more than one-dimensional) additional constraints fulfilled by the magnetic phase.

In the following, by means of several exercises, we will go step by step, through the different capabilities of the program, using at some points some additional tools of the Bilbao Crystallographic Server. At the end of each example, a comparison with the representation method, i.e an alternative analysis using irreps (irreducible representations) of the parent space group, will also be presented.

Example 1. CrCl2

The paramagnetic structure of CrCl2 can be summarized as (Howard et al., PRB 72, 214114):

Pnnm (#58)

Lattice parameters: 6.8257 6.2139 3.4947 Asymmetric unit: Cl 0.35860 0.28930 0.00000 Cr 0.00000 0.00000 0.00000

Magnetic atom: Cr

The magnetic phase of this compound is known to have a propagation vector: k=(0, 1/2, 1/2). We can use MAXMAGN to explore the possible magnetic orderings of k-maximal symmetry compatible with the observed propagation vector. To have k-maximal symmetry means that the magnetic space group associated with the spin configuration, besides being a subgroup of the grey magnetic group Pnnm1' compatible with the observed propagation vector, fulfills that there exists no other subgroup of Pnnm1' (also compatible with the propagation vector) containing it as a subgroup.

a) Open the main page of MAXMAGN, introduce the number of the space group of the paramagnetic phase and the propagation vector, and submit. Four possible k-maximal magnetic space groups are then listed by the program (Figure 1). Their labels as standard Shubnikov groups (BNS setting) are shown in the first column. Although these labels are equal by pairs, they correspond to different subgroups of Pnnm1', and the second column shows the basis transformations necessary to describe each subgroup in the standard setting (click on the heading "Transformation matrix" to see its definition). This transformation is different for the listed subgroups with the same group-type label. The transformation is given by a 3x3 matrix P and a column vector p, such that the unit cell basis corresponding to the standard setting as, bs, cs is given by:

, , , ,s s s p p pa b c a b c P

where ap, bp, cp are the unit cell basis of the Pnnm space group, while the origin for the description of the subgroup in the standard setting must also be shifted by p, with this vector being given by its components in the parent basis.

It is important to stress that the list only includes one subgroup per conjugate class of subgroups. The other subgroups of each class are physically equivalent and correspond to domain-related configurations.

Figure 1: List of distinct k-maximal magnetic space groups for a parent space group Pnnm and a propagation vector (0, 1/2, 1/2), as given by MAXMAGN.

b) Click on the label of the first group for a direct link to MGENPOS with the listing of the Shubnikov group Pa21/c (#14.80) in BNS standard setting (see Figure 2). This is a type

IV magnetic space group, with eight distinct representative operations, namely four non-primed operations forming the group P21/c (#14), plus all these operations multiplied by the "antitranslation" {1'|1/2, 0, 0}. This MGENPOS page has additional links to the OG description of the group, the different possible Wyckoff positions, etc. But this information refers to the standard setting, which requires the use of an oblique cell, and therefore it is far from the description of the structure in the parent Pnnm phase. In general, a description of the magnetic subgroup in a setting as similar as possible to the one used for the parent phase is in practice more useful.

Figure 2: List of the 8 representative operations of the magnetic space group Pa21/c (#14.80), as obtained when clicking on the group label in the list shown in Figure 1 (direct link to MGENPOS). Together with the translation lattice these operations span the full group in the setting that we consider as BNS standard.

b) Come back to the main output list shown in Figure 1 and click now on the column headed with "General positions" for the first magnetic subgroup (for the moment we leave unused the optional buttons of the previous two columns). We get then a table which on its left lists the operations of this magnetic space group in its standard setting, and on the right the same corresponding operations but in a setting that we call "parent-like" (see Figure 3). This non-standard setting keeps the origin and also the unit cell orientation of the parent/paramagnetic phase, but multiplying the cell parameters to produce a supercell consistent with the periodicity mantained by the propagation vector.

At the heading of the list one can find the transformation from the parent-like to the standard setting, similarly as we had in the previous table. Note that the listed transformation is however different as it transforms the used "parent-like" basis (and not the parent one) to the standard setting.

Figure 3: List of the representative operations (incomplete) of the magnetic space group Pa21/c (#14.80), as obtained when clicking on the column "General positions" of the initial list of MAXMAGN shown in Figure 1. The operations on the right are described in the so-called parent-like setting indicated on the heading of the table, which keeps the crystallographic directions of the parent phase, and introduces a supercell consistent with the propagation vector.

As k=(0,1/2,1/2) in this example, the parent like setting is (a,2b,2c;0,0,0), and a centering {1|0, 1/2, 1/2} is obliged as the lattice translation b+c is mantained by a spin wave with wave vector (0,1/2,1/2). You can localize this operation in the list. One can also see that this magnetic group maintains the inversion center at the origin, as it keeps the inversion operation {-1|0, 0, 0}, while the monoclinic axis that is maintained is along the x direction in the Pnnm setting.

c) Come back to the main ouput list shown in Figure 1 and click on the fourth column (headed with "General positions") for the second group in the list , which has the same label Pa21/c (#14.80). One can see that this second subgroup maintains the inversion operation {-1|0, 1/2, 0} (i.e. {-1|0, 1, 0} in the parent Pnnm setting), but the inversion at the origin {-1|0, 0, 0} is not present. This means that an ordering according to this second subgroup breaks the inversion center at the origin, while it keeps the inversion center that is located at the point (0,1/2,0) in the Pnnm1' grey group. Note that this alternative subgroup of Pnnm1', despite being the same type of magnetic space group

and therefore having the same label as group type as the first one in the list, is NOT equivalent and it is therefore listed as a distinct possible k-maximal magnetic symmetry.

d) Come back to the main output list shown in Figure 1, and click on the fourth column (headed with "General positions") to get the information on the third subgroup of type Ca2/m (#12.64). One can see that in this case the standard setting of the magnetic space group only differs from the parent-like one, by a permutation of the axes. The monoclinic axis of this subgroup is along the z direction of the Pnnm setting. A comparison with the fourth group of the same type shows that, as in the previous pair of subgroups of the same type, they maintain different alternative inversion centers from the parent phase.

e) The symmetry hierarchy can be further explored using the program k-SUBGROUPSMAG. By introducing as parent space group Pnnm1', the propagation vector and, as end subgroup Ps-1, i.e. one restricted to the presence of the space inversion operation and the antitranslation forced by the propagation vector, one obtains a hierarchical graph of the subgroups consistent with the propagation vector (0,1/2,1/2), as shown in Figure 4.

Figure 4: Graph (obtained with k-SUBGROUPSMAG) of all possible magnetic symmetries for a magnetic ordering with a paramagnetic structure with space group Pnnm and a propagation vector (0 1/2 1/2), assuming that at least space inversion is also maintained. Only the BNS label of the corresponding group type is indicated. The k-maximal magnetic groups are highlighted with elliptical frames. Only one subgroup per conjugate class is shown. The two Ps-1 space groups differ in the location of the inversion center. Only one of the two allows a non-zero magnetic moment at the origin, namely the one with the inversion center at the origin (where the Cr magnetic atom lies).

In this example, non-polarized magnetic neutron diffraction is not subject to any specific systematic absence for any of the two possible subgroups. We leave therefore unchecked the fifth column in the first output page headed with "systematic absences", which is a link to the tool MAGNEXT, also available in the Bilbao server, for the calculation of the systematic absences for non-polarized magnetic diffraction. We will see the usefulness of this option in other examples.

f) Come back now to the first input page of MAXMAGN with the main menu, click on the option "structure data of the paramagnetic phase will be included", keeping the space group and propagation vector, and submit. In the next pages introduce the structural data of CrCl2 listed above (either uploading a cif file or by hand), indicating that the Cr atom is magnetic. The structure data of the paramagnetic phase must be described

in the standard setting of the space group of the paramagnetic phase (to work in a non-standard setting, click on the option "Non-conventional setting"; thus, the space group of the paramagnetic phase will be obtained from the symmetry operations present at the cif file). The first output page lists now the same four k-maximal subgroups, but two of them are highlighted with a darker background (see Figure 5). These latter are those that allow a non-zero magnetic moment for at least some of the atoms at the Cr site. The Table includes now an additional column on the right under the heading "Magnetic structure".

Figure 5: List of distinct k-maximal magnetic space groups for a parent space group Pnnm and a propagation vector (0, 1/2, 1/2), as given by MAXMAGN, after having introduced the paramagnetic structure of CrCl2 in the first input steps. The groups with darker background are the only ones allowing a non-zero average magnetic moment for at least some of the Cr atoms.

g) Click in the "Magnetic structure" column for the second possible group with label Ca2/m (#12.64). The program lists the asymmetric unit of a magnetic structure satisfying this symmetry (see Figure 6). The first column tabulates the atomic positions, where one can see that both Cr and Cl split into two symmetry-independent sites. The second column indicates the complete orbit for each independent atom for each independent site (Wyckoff orbit), including the magnetic moment relations. Multiplicity and symmetry restrictions on the magnetic moment of each site are shown in the following columns, while on the final column, for magnetic atoms, a menu allows to give specific values to the allowed moment components along the crystallographic axes (units assumed: bohr magnetons). One can see that the two independent Cr sites must necessarily have their moment along different directions. Thus, this symmetry does not allow a collinear ordering.

Figure 6: Atomic positions and magnetic moments of the asymmetric unit for the subgroup of type Ca2/m (#12.64) listed N. 3 in Figure 5, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the components of the magnetic moment for the representative atom listed in the second column, while the last column on the right allows to introduce specific values for the symmetry free moment components.

h) Come back to the previous output page and explore the other alternative maximal symmetry Pa21/c (#14.80). In this case, both Cr and Cl site do not split (see Figure 7). The direction of the magnetic moment of the Cr atom is not restricted by symmetry, but its Wyckoff orbit shows that a collinear ordering can occur if the moments are restricted on the plane yz or along x (component mx has a different set of sign changes through the orbit than the components my and mz, which change sign from one position to another in the orbit in the same form). In accordance with the models reported for this structure, let us assume that the moments lie on the xy plane. According to the listing in Figure 7, this implies a non-collinear model if both components mx and my are non-zero. Introduce some non-zero values for the Mx and My components of Cr, say 3 and 1. Using the appropriate button obtain and save a mCIF file (a CIF-like file) of the resulting magnetic structure. The page showing the mCIF file to be saved has also a button that is a direct link to the tool MVISUALIZE in the Bilbao crystallographic server and allows an inmediate 3D visualization of the chosen magnetic model.

Figure 7: Atomic positions and magnetic moments of the asymmetric unit for the subgroup of type Pa21/c (#14.80) listed N. 1 in Figure 5, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the components of the magnetic moment for the representative atom listed in the second colum (if any), while the last column on the right allows to introduce specific values for the symmetry-free moment components.

i) Open the mCIF file as a text file and observe the listing of the symmetry operations. The centering and "anti-centering translations" are listed separately (see Figure 8). There are four centering translations and antitranslations including the identity, plus four rotational or roto-inversion operations. The 16 operations that were listed in a previous output page are the result of multiplying these 4 "rotational" or "roto-inversion" operations by the 4 possible translational or "anti-translational" operations.

Figure 8: Part of the mCIF file for the Pa21/c (#14.80) model of CrCl2 that contains the information on the magnetic space group in the used parent-like setting (a,2b,2c;0,0,0).

j) At the end of the mCIF file observe the listed atomic positions and moments of the asymmetric unit. To be noted that the file only includes the magnetic moment components (along the crystallographic axes) of a single Cr atom (see Figure 9). The values of the rest of Cr atoms in the unit cell are unambiguously determined by the application of the operations of the defined magnetic space group (see Figure 8). It is important to stress that not only the magnetic moments but also the positions of all atoms in the unit cell are obtained by the application of the magnetic symmetry operations to the asymmetric unit listed in the mCIF file.

Figure 9: Part of the mCIF file for the Pa21/c (#14.80) model of CrCl2 that contains the information on the atomic positions and magnetic moments of the asymmetric unit.

The obtained mCIF file can be read by the refinement programs JANA2006 or FULLPROF, and if diffraction data were available could be used as a starting symmetry-constrained model to be refined. One must stress in this case that care should be taken that the diffraction data is indexed consistently with the parent-like unit cell that is being employed to describe the structure. The mCIF file obtained from MAXMAGN can also be

read by VESTA or Jmol for visualization purposes, and it can also be employed with ISODISTORT to analyse the model in terms of irrep modes. The tool MVISUALIZE in the Bilbao Crystallographic Server can also read these files, and provides an inmediate 3D visualization with Jmol without the need of any program installation.

k) Load the mCIF file in the VESTA program to visualize the model. Something similar to the representation shown in Figure 10(a) can be obtained.

a b

Figure 10: (a) Magnetic ordering in CrCl2 according to the magnetic space group Pa21/c (#14.80) using the parent-like setting (a,2b,2c;0,0,0), and having arbitrarily restricted the spins to the plane xy. (b) Magnetic ordering equivalent to the one in (a) corresponding to a twin-related configuration. Its symmetry is given by a magnetic subgroup conjugate to the one associated with the structure in (a). The two arrangements are related for instance by the lost operation {2z'|0, 0, 0} that is present in the paramagnetic phase.

l) Go back to the main output page and click on "alternatives (domain-related)" for the same group. The program prompts then a page with the list of distinct subgroups belonging to the same conjugation class (see Figure 11). One can see that there is an alternative subgroup which is conjugate to the one chosen by the program. This conjugate subgroup will correspond to a physically equivalent configuration that is domain-related with the first one (the one chosen by the program). By clicking "choose" on this second magnetic space group we change the choice of subgroup of type Pa21/c (#14.80) to this second one. We obtain then, following the same procedure as above, a magnetic configuration different from the one in Figure 4(a), but equivalent. It corresponds to its transformation by one of the lost operations belonging to the parent grey group. The result to be obtained is shown in Figure 4(b). One can see there that the individual local couplings of FM or AFM type between neighbouring specific sites should not be considered as an absolute part of the magnetic structural model, as they depend on which of the two equivalent descriptions is chosen.

Figure 11: Listing of the two subgroups of type Pa21/c (#14.80) of the group Pnnm1' that form a conjugation class, and could describe a magnetic ordering with propagation vector (0, 1/2, 1/2) of maximal symmetry in CrCl2. The first subgroup is the one chosen by default by MAXMAGN, while the second one corresponds to a domain-related physically equivalent configuration. The user can change the choice of the representative subgroup, and therefore the chosen configuration, by clicking on the button of the last column. The 3x3 matrix and column vector listed indicates the transformation of each subgroup to the standard setting of the group type, and defines unambiguously each subgroup.

m) Come back to the main output page of the magnetic structure under the group Pa21/c (#14.80) described in the parent-like setting (Figure 7), after having switched back to the default choice of the subgroup, and click on the button "use an alternative setting". An mCIF with a description using a smaller unit cell that avoids the centering {1|0, 1/2, 1/2} can be obtained with this option (see Figure 12). The cost is having an oblique unit cell with respect to the Pnnm setting. This description is specially appropiate for ab-initio calculations, as it uses a minimal unit cell. To do that introduce as desired basis transformation from the parent: (a,b+c,-b+c;0,0,0). The input matrix follows the same rule as in other cases, i.e. each matrix column represents a transformed basis vector. The output provided for this user-defined setting is more limited than for the default choice, but the corresponding mCIF file can be equally produced. Introduce arbitrary values for the moment components, similar to those in the previous setting. Taking into account the basis transformation, the moment chosen previously would be (3, 0 .5, 0.5) in this new basis. Download the corresponding mCIF file, and visualize it with VESTA (see Figure 13) or MVISUALIZE.

Figure 12: Menu to introduce an alternative user-chosen setting (unit cell) to describe the magnetic structure, once one of the possible k-maximal magnetic symmetries has been chosen. The unit cell introduced must be consistent with the periodicity of the magnetic ordering, and this is cross-checked by the program.

Figure 13: Magnetic structure as in Figure 10(a), using a primitive unit cell basis related with the parent unit cell by (a,b+c,-b+c;0,0,0).

n) Open the tool STRCONVERT at the Bilbao crystallographic server and introduce the mCIF file. This tool allows various editing and save operations of the information of the mCIF. Here we apply the button "Transform the structure to P1 setting" to obtain a listing of all the atoms in the unit cell with their magnetic moments (see Figure 14). This can be specially useful for an ab-initio calculation using a code that works without symmetry and requires the information on all atoms in a primitive unit cell. The orbit of magnetic atoms in the primitive cell used is reduced to four atoms. The first two Cr atoms, Cr1_1 and Cr1_2, are those in the Pnnm unit cell at the origin and at (1/2, 1/2, 1/2) transformed to the new basis, while the two additional ones are those related by a translation c of the parent unit cell. The latter have therefore opposite moments as obliged for a magnetic ordering with a propagation vector (0, 1/2, 1/2). Therefore the specific features that define this model with k-maximal magnetic symmetry reduce to the

moment relation (mx, -my, -mz) of Cr1_2 with respect to the moment (mx, my, mz) of Cr1_1, the remaining relations being obliged by the value of the propagation vector.

Figure 14: Main menu page (partial) of STRCONVERT after introducing the mCIF file of the magnetic model of k-maximal symmetry using a setting (a, b+c,-b+c;0,0,0) with respect to the parent Pnnm, and after having clicked "Transform the structure to P1 setting". The program lists positions and magnetic moments of all atoms in the unit cell.

p) Obtain the equivalent domain-related configuration described in the same primitive magnetic unit cell following the same steps as above. Use then STRCONVERT again to list all atoms in the unit cell:

Figure 15: Main menu page (partial) of STRCONVERT after introducing the mCIF file of the alternative domain-related magnetic model of k-maximal symmetry using a setting (a, b+c,-b+c;0,0,0) with respect to the parent Pnnm, and after having clicked "Transform the structure to P1 setting". The program lists positions and magnetic moments of all atoms in the unit cell, which can be compared with those in Figure 14.

In Figure 15, the atom listed as Cr1_4 is the one listed as Cr1_2 in the previous model. The relation of the moment of this atom with the one of Cr1_1 is now (-mx, my, mz), i.e. just the opposite spin coupling, although it describes a fully equivalent spin configuration. This means that if the spin configuration were a collinear arrangement restricted along x or on the plane yz, individual local FM or AFM couplings between neighbouring atoms have no absolute meaning as description of the spin configuration.

Epilogue to this first example: The case of CrCl2 was considered by Izyumov et al. in their book "Neutron Diffraction of Magnetic Materials" [6] as a paradigmatic example, which according to these authors demonstrated the insufficiency of the magnetic space groups to describe the spin correlations occuring in many magnetic structures. This reference assumed that the magnetic structure of CrCl2 is a collinear arrangement on the plane xy, with the moments of the Cr atoms at the origin and at (1/2, 1/2, 1/2) being equal and opposite. From the considerations above, it is clear that the model considered in this reference is incompatible with any of the two possible k-maximal symmetries, and its symmetry is limited to its common subgroup Ps-1 (see Figure 4), where the two Cr sites would be symmetry independent. Therefore the opposite values of the moments in the two Cr sites, which were assumed in that model, would be an important constraint satisfied by this magnetic phase that was not explained by its magnetic symmetry. A revision of the original reports on the magnetic structure of CrCl2 shows however that the collinear model on the xy plane assumed by Izyumov et al. has not very solid ground. The experimental data presented in these old publications are extremely poor and their fit to more symmetrical configurations did not seem to have been checked. Furthermore, some recent unpublished ab-initio calculations and new refinements using the scarce poor data available in the literature indicate that the monoclinic arrangement of k-maximal symmetry discussed above is probably a more appropiate model for the magnetic phase of this compound.

Comparison with the representation method: The little group of the propagation vector (0, 1/2, 1/2) is the full group Pnnm. As can be seen using REPRES (in the Bilbao crystallographic server) there are two 2-dim irreps of Pnnm for this wave vector: T1+ and T1- (T is the standard label for the vector (0, 1/2, 1/2) in the Brillouin zone). Therefore, for the grey group Pnnm1' there are two analogous irreps: mT1+ and mT1- , which can be relevant for a magnetic ordering (the symbol m is used to distinguish irreps odd for time reversal from even ones). The magnetic representation for this wave vector and limited to the magnetic moments of the Cr atoms is however restricted to the irrep mT1+, contained three times (see for instance the results using Basireps of the FullProf suite [2]): Mrepr = 3 mT1+

This means that any arbitrary spin configuration with a propagation vector (0, 1/2, 1/2) transforms according to the irrep mT1+, and can therefore be described in terms of 6 basis functions associated with this irrep. In other words, the assumption of the magnetic ordering complying with a single irrep does not introduce any constraint, and the representation method is of no use in this example. This should be compared with the constraints introduced by the assumption of one of the two possible k-maximal magnetic groups discussed above, where the number of free parameters describing the spin configuration is limited to three in both cases.

We can use ISOCIF and ISODISTORT [3] to perform the mode decomposition of the monoclinic model of k-maximal symmetry Pa21/c (#14.80) represented in Figure 10(a). ISOCIF should be used to

transform the mCIF produced above in step h) into a standard setting. This mCIF file in standard setting can be then uploaded in ISODISTORT (method 4) as the distorted structure to be decomposed into symmetry modes with respect to the Pnnm structure, which should have been uploaded previously as parent structure. The mode decomposition provided by ISODISTORT shows that the spin configuration corresponds as expected to mT1+, but restricted to a special direction in the representation space corresponding to the k-maximal symmetry discussed above, which is listed here as a so-called isotropy subgroup of the irrep mT1+. The number of listed basis modes complying with this specific symmetry is only three, as expected from the description done above using directly the magnetic space group.

Summarizing, any arbitrary spin configuration in CrCl2 can be associated with the irrep mT1+, and therefore in this case the representation method in its traditional form is of no use. A spin ordering restricted to have one of the k-maximal magnetic symmetries discussed above corresponds to the choice of a special direction of higher symmetry within the mT1+ representation (i.e. a specific linear combination of the three pairs of mT1+ basis functions), so that the number of free parameters in the possible combination of basis modes is restricted from 6 to 3. To assign a k-maximal magnetic group is therefore NOT equivalent to the assignment of an irrep, and introduces additional constraints.

Example 2. Orthorhombic HoMnO3 (see MAGNDATA #1.20)

The paramagnetic structure of HoMnO3 can be summarized as (Muñoz, A. et al., Inorg. Chem. (2001) 40 1020 - 1028):

Space group: Pnma (#62) Lattice parameters: 5.83536 7.36060 5.25722 Asymmetric unit: Ho1 - 0.08390 0.25000 0.98250 Mn1 - 0.00000 0.00000 0.50000 O1 - 0.46220 0.25000 0.11130 O2 - 0.32810 0.05340 0.70130 Magnetic atoms: Ho1, Mn1 The magnetic phase of this compound is known to have a propagation vector k=(1/2, 0, 0) and its antiferromagnetic magnetic order induces a switchable electric polarization, being therefore a multiferroic in the broad sense that is presently employed. We can use MAXMAGN to explore the possible magnetic orderings of k-maximal symmetry with this propagation vector, to demonstrate that the parent space group and the propagation vector is sufficient information to predict that this system, if fully magnetically ordered, has a great probability of being multiferroic. a) Introduce in MAXMAGN the propagation vector and the structural data of the parent structure of HoMnO3, either using the data above or with the corresponding CIF file, indicating the magnetic character of Mn and Ho. A list of four possible k-maximal magnetic space groups are obtained (see Figure 16)

Figure 16: List of distinct k-maximal magnetic space groups for a parent space group Pnma and a propagation vector (1/2, 0, 0), as given by MAXMAGN, after having introduced the paramagnetic structure of HoMnO3 in the first input steps. The groups with darker background are those allowing a non-zero average magnetic moment for at least some of the Mn or Ho atoms. In this case, the four alternative symmetries are possible.

It should be noticed that this list is comprehensive in the sense that it includes a representative of all the classes of magnetic subgroups equivalent by conjugation with respect the parent space group. The list therefore encompasses all possible non domain-equivalent magnetic symmetries, which are consistent with the observed magnetic propagation vector, and have no supergroup (magnetic group) above them that also fulfils this condition. The determination of this list only requires the knowledge of the parent space group and the propagation vector k. The space groups are determined by mathematically searching among the subgroups of the grey group Pnma1’ of the parent phase all maximal subgroups which have as Bravais magnetic lattice the one defined by the vector k, i.e. the sub-lattice of the parent lattice formed by the parent lattice translations L that satisfy exp(i2k.L)= 1, but having also as "antitranslations" (i.e. translations combined with time reversal) parent lattice translations satisfying exp(i2k.L)= -1. The condition of the subgroups being maximal is considered in an extended form, disregarding intermediate subgroups of type II (grey groups), as by definition they contain the time reversal operation and therefore they cannot describe the symmetry of a magnetic phase. b) Explore the listed four possible models of maximal symmetry by clicking on the last column headed with "magnetic structure". Check first that the models with monoclinic centrosymmetric symmetry Pc21/c(#14.82) and Pa21/m(#11.55) require that a half of the Mn atoms remain disordered with null magnetic moment. This means that a fully ordered magnetic arrangement of the Mn of maximal symmetry under the observed propagation can only be achieved under the non-centrosymmetric symmetries Pana21 (#33.149) or Pbmn21 (#31.129). The point symmetry in both cases is the grey polar point

group mm21'. As shown by the listed transformation matrices, in both cases the polar axis is along the c axis of the Pnma setting. A multiferroic character of the magnetic phase should therefore be expected if all magnetic Mn atoms order and the phase symmetry is maximal. The polar direction is also predicted to be along the c axis.

Figure 17: Atomic positions and magnetic moments (partial) of the asymmetric unit of HoMnO3 for the subgroup of Pnma1' of type Pc21/c (#14.82), listed N. 3 in Figure 16, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the components of the magnetic moment for the representative magnetic atom listed in the second column (if any), while the last column on the right allows to introduce specific values for the symmetry-free moment components. Both the Ho and Mn split into two independent sites. The bar in the fifth column for some of the magnetic sites indicates that the magnetic moment at this site is symmetry-forced to be zero.

c) Using the program k-SUBGROUPSMAG construct the graph of all magnetic subgroups (conjugate classes) consistent with the observed propagation vector, and check that apart from the monoclinic k-maximal subgroups there is only a third possible centrosymmetric symmetry, namely a subgroup of type Ps-1, which is a common subgroup of the two centrosymmetric monoclinic groups of maximal symmetry (see Figure 18). Check the operations belonging to this subgroup using the options of k-SUBGROUPSMAG.

Figure 18: Graph (obtained with k-SUBGROUPSMAG) of all possible magnetic symmetries for a magnetic ordering with a paramagnetic structure with space group Pnma and a propagation vector (1/2 0 0). Only the BNS label of the corresponding group type is indicated. The k-maximal magnetic groups are highlighted with elliptical frames. Only one subgroup per conjugate class is shown. Except for the two monoclinic k-maximal subgroups and its common subgroup Ps-1, all other possible symmetries are polar and allow an induced electric polarization. The three non-polar subgroups are not possible for a full magnetic ordering of HoMnO3, as they require that some Mn sites remain with zero moment.

d) Click on the option "go to a subgroup" for the subgroup of Pnma1' of type Pc21/c (#14.82), listed N. 3 in Figure 16, and in the following menu choose as generators of the chosen subgroup the inversion operation {-1|0, 0, 0} and the anti-translation {1'|1/2, 0, 0}. Submit this subgroup and check in the output that by this means the subgroup Ps-1 has been chosen. Click on the option "magnetic structure" of the next output page, and observe in the next output that a magnetic structure subject to this minimal centrosymmetric symmetry still requires that some Mn atoms remain disordered with moment zero. You can therefore predict that whatever is the magnetic ordering of HoMnO3 with propagation vector (1/2,0,0), if it involves all Mn atoms it must necessarily break the centrosymmetry of the structure, and the system is bound to have (if the system is an insulator) a magnetically induced electric polarization with ferroelectric properties. e) Construct a magnetic structure complying with the maximal symmetry Pana21 (#33.149) following the same procedure as in the example 1. Notice that in this case the difference between the standard setting and the parent-like is only an origin shift. The experimental diffraction data indicates that the Mn spins are essentially collinear along the x direction. Therefore introduce a non-zero value for the Mx component of the Mn independent atom, although as shown in the listing both polar symmetries allow an arbitrary direction for the Mn moment, that would have non-collinear character. Keep disordered the moment of the Ho atoms maintaining at zero their symmetry-allowed x and z components. Notice that this symmetry break splits into two the Ho site and the two oxygen sites of the parent structure. This means that many additional structural degrees of freedom are in principle triggered by the magnetic ordering and they can be taken into account in a controlled and systematic way, if the magnetostructural coupling is sufficiently strong to be detectable, using the magnetic space group for defining the constraints on the atomic positions. The atomic positions listed for the split atomic sites of the asymmetric unit satisfy among them the relations coming from the Pnma

symmetry, but their separate listing within the new asymmetric unit would allows their independent refinement. A symmetry-consistent crosscheck of their possible deviation from the Pnma relations due magnetostructural couplings is therefore possible.

Figure 19: Atomic positions and magnetic moments (partial) of the asymmetric unit of HoMnO3 for the subgroup of Pnma1' of type Pana21 (#33.149), listed N. 1 in Figure 16, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the moment components of the representative magnetic atom listed in the second column (if any), while the last column on the right allows to introduce specific values for its symmetry-free moment components. The Ho split into two independent sites, while the Mn remains a single independent site. The bar in the fifth column for some of the magnetic sites indicates that the magnetic moment at this site is symmetry-forced to be zero.

f) Produce an mCIF file of the Pana21 (#33.149) model and visualize it with VESTA or Jmol. If only the magnetic atoms are visualized, it will be something similar to Figure 20(a).

a b

Figure 20: (a) Possible k-maximal magnetic ordering for HoMnO3 according to the magnetic space group Pana21(#33.149) using the parent-like setting (2a,b,c;0,0,0), and having restricted the spins along x. (b) Magnetic ordering equivalent to the one in (a) corresponding to a twin-related configuration. Its symmetry is given by a magnetic subgroup conjugate to the one associated with the structure in (a). The two arrangements are related for instance by the lost inversion operation {-1|0, 0, 0} present in the paramagnetic phase. This operation also switches the polarity and therefore the two magnetic configurations have opposite magnetically induced polarizations along the c direction.

g) Come back to the main output list of k-maximal magnetic groups and click for "alternatives (domain related)" of the listed group Pana21(#33.149) to change to the second conjugate subgroup of this type. Follow then the same procedure as before and obtain the mCIF file of the corresponding magnetic arrangement. It is shown in Figure 20(b). This arrangement is twin related with the previous one. The lost inversion operation for instance transforms one into the other. The two configurations are physically equivalent and correspond to domains having opposite magnetically induced electric polarizations along c.

h) Come back to the main output list of k-maximal magnetic groups (Figure 16) and follow the same procedure for the second possible polar group Pbmn21 (#31.129). Notice in the output that in this phase if there is some magnetic ordering of the Ho atoms it can only happen along the b axis.

Figure 21: Atomic positions and magnetic moments (partial) of the asymmetric unit of HoMnO3 for the subgroup of Pnma1' of type Pbmn21 (#31.129), listed N. 2 in Figure 16, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the moment components of the representative magnetic atom listed in the second column (if any), while the last column on the right allows to introduce specific values for its symmetry-free moment components. The Ho splits into two independent sites, while the Mn remains a single independent site. The bar in the fifth column for some of the magnetic sites indicates that the magnetic moment at this site is symmetry-forced to be zero.

i) Restrict again the model to have the moments along the x axis and obtain the corresponding mCIF files for the two twin related configurations. They are depicted in Figure 22. This is actually the model that has been reported for HoMnO3 (see MAGNDATA 1.20). Note that the symmetry allows canting of the spins along the y and z directions, which would break the perfect collinearity, while keeping the symmetry.

a b

Figure 22: (a) Possible k-maximal magnetic ordering for HoMnO3 according to the magnetic

space group Pbmn21 (#31.129), using the parent-like setting (2a,b,c;0,0,0), and having restricted the spins along x. (b) Magnetic ordering equivalent to the one in (a) corresponding to a twin-related configuration. Its symmetry is given by a magnetic subgroup conjugate to the one associated with the structure in (a). The two arrangements are related for instance by the lost inversion operation {-1|0, 0, 0} present in the paramagnetic phase. The inversion also switches the polarity and therefore the two magnetic configurations will have opposite values for the magnetically induced polarization along the c direction. This is the magnetic arrangement that has been reported for this compound (Muñoz, A. et al., Inorg. Chem. (2001) 40 1020 - 1028. See entry 1.20 of MAGNDATA).

We have seen above that the atomic positions become split because of the symmetry break, and the symmetry relations that they have to fullfill rigorously in the magnetic phase are described by the same symmetry operations that are valid for the magnetic moments, which are listed in the mCIF file. The presence or not of time reversal in these symmetry operations is irrelevant for the atomic positions, which are then subject to the constrains of an effective space group obtained by disregarding the presence of time reversal in the operations. This effective space group is the one used for the labelling of the magnetic space group in the OG description. Thus, in our case, the group Pbmn21 (#31.129) in BNS notation is the group P2bm'n21' (#31.7.218) and in this case the effective space group for the atomic positions (and eletron density) is of the same type as the one used for the BNS notation, namely the space group Pmn21(#31).

j) Come back to the main list of k maximal groups (Figure 16) and click on the column "general positions" for the group Pbmn21 (#31.129) (previously switch back the group to its default choice) in order to see the symmetry operations of the group in the parent-like setting, and derive the effective space group for the atomic positions in this setting :

Figure 23: General positions or representative symmetry operations of the subgroup of Pnma1' of type Pbmn21 (#31.129), listed N. 2 in Figure 16, in the parent-like setting (2a,b,c;0,0,0), as obtained when clicking in the column "general positions".

One can see that the primitive unit cell of the effective lattice for the non-magnetic degrees of freedom is half the size of the magnetic unit cell, since for the non-magnetic degrees of freedom the operation {1'|1/2 0 0} is fully equivalent to a centering translation {1|1/2 0 0}. The effective space group is then given by the operations: {1| 0 0 0 }, { 2001| 3/4 0 1/2} , {m010| 0 1/2 0 }, { m100| 3/4 1/2 1/2 }, plus the centering translation {1| 1/2 0 0}. Therefore, the effective primitive unit cell for the atomic positions remains the parent unit cell (a,b,c;0,0,0). If we use this parent basis for the atomic positions instead of the parent-like one, the operations of the effective space group constraining the atomic positions is then given by the operations:

{1| 0 0 0 }, { 2001| 1/2 0 1/2 } , { m010| 0 1/2 0 }, { m100| 1/2 1/2 1/2 } or

x,y,z -x+1/2,-y,z+1/2 x,-y+1/2,z -x+1/2,y+1/2,z+1/2

This is indeed a space group of type Pmn21(#31) in a non-standard setting. The standard setting can be reached with the transformation (-b, a, c; 1/4, 1/4, 0) (you can check it using IDENTIFY GROUP in the Bilbao Crystallographic Server).

Comparison with the representation method: The little group of the propagation vector (1/2, 0, 0) is the full group Pnma. As can be seen using REPRES (in the Bilbao crystallographic server) there are two 2-dim irreps of Pnma for this wave vector: X1 and X2 (X is the standard label for the vector (0, 1/2, 1/2) in the Brillouin one). Therefore, for the grey group Pnma1' there are two analogous irreps: mX1 and mX2 , which can be relevant for a magnetic ordering (the symbol m is used to distinguish irreps odd for time reversal from even ones). The magnetic representation for this wave vector and limited to the spins of the Mn atoms decomposes into these irreps in the following form (see for instance the results using Basireps of the FullProf suite [2]): Mrepr = 3 mX1 + 3 mX2

Therefore, the assumption of the magnetic ordering complying with one of the two single irreps reduces the numbers of degrees of freedom of the spin configuration from 12 to 6, i.e. there are 6 independent spin basis functions to be considered, and therefore 6 refinable parameters when describing a spin configuration of the Mn atoms complying with one of the two irreps. This should be compared with the constraints introduced by the assumption of one of the four possible k-maximal magnetic groups discussed above, where the number of free refinable parameters describing the spin configuration is limited to three in any of the four cases. Using ISODISTORT as in the previous example one can check that the k-maximal subgroups Pbmn21 and Pa21/m correspond to spin configurations according to the irrep mX1, but restricted within a special "direction" in the representation space. This means that the mX1 basis functions must be combined in a specific form such that the number of free parameters reduces from 6 to 3. Pbmn21 and Pa21/m are so-called isotropy subgroups (or epikernels) of the irrep mX1. Similarly, Pana21 and Pc21/c are isotropy subgroups (epikernels) of the irrep mX2 and the restriction of the configuration to one of these alternative symmetries reduces the number of effective basis functions from 6 to 3.

Summarizing, the spin ordering in HoMnO3 complies with one of the k-maximal magnetic symmetries discussed above, namely Pbmn21 (#31.129), and it corresponds to the choice of a special direction of higher symmetry within the space of mX1 distortions, so that the number of free symmetry-adapted basis modes is restricted from 6 to 3. To assign this k-maximal magnetic group is NOT equivalent to the assignment of the irrep mX1. It introduces additional constraints.

Example 3: KFe3(OH)6(SO4)2 (MAGNDATA #1.25)

The paramagnetic structure of this compound can be found in the Inorganic Crystal Structure Database (ICSD, #34344):

Space group R-3m (#166)

Lattice parameters: 7.30400 7.30400 34.53600 90.00 90.00 120.00 Asymmetric unit: Fe 1 - 0.5 0.0 0.5 K 1 - 0.00000 0.00000 0.00000 O 1 - 0.00000 0.00000 0.39340 O 2 - 0.21800 -0.21800 -0.05430 O 3 - 0.12470 -0.12470 0.13510

S 1 - 0.00000 0.00000 0.30840 Magnetic atom: Fe The magnetic phase of this compound is known to have a propagation vector k=(0, 0, 3/2). The Fe atoms in this compound lie on layers forming 2D Kagome lattices, and their spin arrangement was studied by Inami, T. et al., J. of Magn. and Magn. Mat. (1998) 177, 752 (see MAGNDATA #1.25). In this reference no symmetry arguments were applied and the model proposed was found, according to the authors, by checking 24 different alternative possible configurations. The exercise below with MAXMAGN shows however that there are only two configurations of highest symmetry which should be first considered as being the most probable, and in fact, the model proposed by this experimental study is one them. a) Introduce the data in MAXMAGN using the cif file or the listing above, and flag Fe as magnetic. Four possible k-maximal magnetic symmetries are listed as possible, from which two of them allow a non-zero magnetic moment of the Fe site (see Figure 24). Check that there are no twin-related configurations described by alternative conjugate groups, by clicking on "alternative (domain-related)".

Figure 24: List of distinct k-maximal magnetic space groups for a parent space group R-3m and a propagation vector (0, 0, 3/2), as given by MAXMAGN, after having introduced the paramagnetic structure of KFe3(OH)6(SO4)2 in the first input steps. The groups with darker background are those allowing a non-zero average magnetic moment for at least some of the Fe atoms. Only two of the symmetries are therefore possible for the magnetic Fe ordering.

b) Click on the fifth column to see the systematic absences of any of the two allowed groups (they are the same):

Figure 25: Systematic absences for the k-maximal subgroup of type RI-3c, listed N. 2 in Figure 24, as given by MAXMAGN, after having clicked on the column headed with "systematic absences".

To understand the listing of systematic absences obtained (see Figure 25) one has to take into account that the used (h,k,l) indices correspond to the parent-like basis (a, b, 2c). Firstly, the absences for any reflection (h,k,l) are given. On one hand, the centering translation {1| 1/3 2/3 1/3 } yields the extinction for h+2k+l different from a multiple of 3. Secondly, the antitranslation {1'|0,0,1/2} forces the absence of magnetic diffraction for l even.

If one transforms the (h,k,l) indexation into a description in terms of the propagation vector k=3/2c*, the meaning of these absences becomes rather simple:

*

* * * * *

02 32

cH ha kb lc mk ha kb l m

The l index is therefore 2lo+3m, with lo being the index of a neighboring reflection associated with non-magnetic diffraction index in the parent basis. Therefore, the systematic absence for l even just means that second order magnetic reflections with m=2 are forbidden, and therefore magnetic and non-magnetic diffraction reflections do not overlap. This is a general property of all magnetic space groups of type IV, i.e. those that have antitranslations. The other systematic absence is a signature of the R centering.

On the other hand, the specific absence of all magnetic reflections of type (0,0,l) is much more specific. It is related with the existence of the three-fold symmetry in the magnetic group. The breaking of this extinction would imply that the symmetry of the ordering is necessarily lower, having broken the three-fold rotation symmetry.

c) Come back to the main list of k-maximal groups (Figure 24) and using the button in the last column for the group RI-3m (#166.102) obtain a magnetic structure complying with this symmetry. The listing obtained (see Figure 26) shows that the moment of the representative Fe site in the asymmetric unit is forced to lie along the (1,0,0) direction. Note that the parent-like setting used is (a,b,2c;0,0,0), and if it were fitted to a diffraction pattern, the diffraction data should indexed consistently with this magnetic unit cell.

Figure 26: Atomic positions and magnetic moments (partial, only Fe site shown) of the asymmetric unit of KFe3(OH)6(SO4)2 for the subgroup of type RI-3m (#166.102), listed N. 4 in Figure 24, as obtained when clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the moment components of the representative magnetic atom, while the last column on the right allows to introduce specific values for its symmetry-free moment components.

d) Assign an arbitrary initial value to the component Mx of Fe, and save the resulting mCIF file. This file could be introduced for a refinement of the model in the refinement programs JANA2006 or FULLPROF, if diffraction data were available, of it could be subject to an irrep mode analysis with ISODISTORT. We only use it here for the visualization of the model with VESTA or Jmol (see Figure 27).

a

b

Figure 27: Possible magnetic ordering with k-maximal symmetry for KFe3(OH)6(SO4)2 according to the magnetic space group RI-3m (#166.102) using the parent-like setting (a,b,2c;0,0,0). The Fe magnetic moment is only scale free. (a) Arrangement of Fe spins within a unit cell. (b) view of one of the layers on the xy plane showing the Kagome lattice of Fe atoms.

e) Come back to the main list of k-maximal groups and using the button in the last column for the group RI-3c (#167.108) obtain a magnetic structure complying with this symmetry. The listing obtained (see Figure 28) shows that the moment of the representative Fe site in the asymmetric unit is forced to have its component on the plane xy along the (1,2,0) direction, but a component along the z axis is allowed.

Figure 28: Atomic positions and magnetic moments (partial, only Fe site shown) of the asymmetric unit of KFe3(OH)6(SO4)2 for the subgroup of type RI-3c (#167.108), listed N. 2 in Figure 24, as obtained when

clicking in the column "Magnetic structure". The table indicates the Wyckoff orbit of positions and moments corresponding to all atoms that are symmetry related with the one listed as representative in the asymmetric unit (all described in the parent-like setting). The number of symmetry related atoms within the used unit cell (multiplicity) is given in the fourth column. The fifth column indicates the symmetry restrictions on the value of the moment components of the representative magnetic atom, while the last column on the right allows to introduce specific values for its symmetry-free moment components.

f) Assuming that the plane xy is the easy plane for the magnetic ordering, assign an arbitrary initial value to the Mx of Fe, keeping zero the Mz component, and save the resulting mCIF file. This mCIF file could then be introduced for a refinement of the model in the programs JANA2006 or FULLPROF, if diffraction data were available, or it could subject to an irrep mode analysis with ISODISTORT. We only use it here for the visualization of the model with VESTA or Jmol (see Figure 29). This is in fact the model proposed by Inami, T. et al., J. of Magn. and Magn. Mat. (1998) 177, 752 (see MAGNDATA #1.25) after checking 24 different configurations.

a

b

Figure 29: Possible magnetic ordering with k-maximal magnetic symmetry for KFe3(OH)6(SO4)2 according to the magnetic space group RI-3c (#167.108), described in the parent-like setting (a,b,2c;0,0,0). A symmetry allowed z-component of the spins is not included. (a) Arrangement of Fe spins within a magnetic unit cell. (b) view of one of the layers on the xy plane showing the Kagome lattice of Fe atoms.

Comparison with the representation method: The little group of the propagation vector (0, 0, 3/2) is the full group R-3m. As can be seen using REPRES (in the Bilbao crystallographic server) there are six irreps of R-3m for this wave vector. Therefore, for the grey group R-3m1' there are six analogous irreps which can be relevant for a magnetic ordering, i.e. which are odd for time reversal, namely the 1-dim irreps mT1+, mT2+, mT1- ,mT2- and the 2-dim irreps mT3+ and mT3- (the symbol m is used to indicate that they are odd for time reversal). The so-called magnetic representation for this wave vector and limited to the spins of the Fe atoms decomposes into irreps in the following form (see for instance the results using Basireps of the FullProf suite [2]):

Mrepr = 2 mT1- + mT2- + 3 mT3-

In contrast with the multidimensional irreps, which were relevant in the previous examples, for a 1-dim irrep there is a one to one correspondence between the irrep and a magnetic space group. Thus, in this example the assumption of a magnetic ordering complying with one of the two 1-dim irreps is equivalent to the assigning of a specific magnetic space group. The magnetic symmetry associated with a magnetic distortion according to either mT1- or mT2- is in fact given by the group RI-3c (#167.108) or RI-3m (#166.102), respectively, which are the k-maximal magnetic space groups

discussed above. This is a general property: magnetic space groups associated with magnetic orderings complying with a single 1-dim irrep are k-maximal in the sense used here. In the present case the magnetic ordering corresponds to the 1-dim irrep mT1-.

Summarizing, in contrast to the previous examples, to say that the magnetic phase of KFe3(OH)6(SO4)2 has symmetry RI-3c (#167.108) or that it complies with the irrep mT1- of R-3m are equivalent statements in what concerns the resulting constraints for the atomic magnetic moments. One should however consider that the magnetic symmetry assignment includes additional information, as it also comprehends the constraints on all degrees of freedom of the material, including non-magnetic ones, macroscopic properties (subject to the associated point group symmetry -3m1') and the knowledge of symmetry-related systematic absence rules in unpolarized neutron magnetic diffraction patterns (see program MAGNEXT in the Bilbao Crystallographic Server).

Example 4: Na2MnF5 (MAGNDATA #1.55) (Nuñez, P. et al., Solid State Commun. (1994) 92 , 601)

Paramagnetic structure

Space Group: P21/c (#14)

unit cell parameter: 7.7197 5.2402 10.8706 90.000 108.991 90.000 F1 - 0.04480 0.15100 0.16290 F2 - -0.10830 0.29780 -0.07580 F3 - 0.25140 0.16200 0.00310 F4 - 0.61790 0.21920 0.13470 F5 - 0.49180 0.24420 -0.12280 Mn1 - 0.00000 0.00000 0.00000 Mn2 - 0.50000 0.00000 0.00000 Na1 - 0.15350 -0.01070 0.36470 Na2 - 0.34410 0.44390 0.17996 (or a similar model from ICSD #61206) Magnetic atoms: Mn1, Mn2 Propagation vector (0, 1/2, 0) Following a similar procedure as in previous examples, and assuming that the moments are oriented along the x direction obtain the two possible magnetic structural models of k-maximal symmetry represented in its two twinned forms in Figure 30. The magnetic ordering reported by Nuñez et al. is in fact the one having the monoclinic symmetry Pbc, with the additional restriction of making equal the moment moduli of the two independent Mn sites, and the moments restricted along the x axis. This magnetic ordering breaks the centrosymmetry of the paramagnetic phase, the symmetry being polar.

Derive from the output of MAXMAGN that non-collinear AFM cantings along y and z are allowed in this phase.

Pbc (a,2b,c;0 1/4 0)

Pbc (a,2b,c;0 3/4 0)

Ps-1 (a,-c,2b;0 0 0)

Ps-1 (a,-c,2b;0 1/2 0)

Figure 30: Magnetic moment arrangements assumed along x for the Mn atoms of Na2MnF5 subject to each of the possible magnetic space groups of maximal symmetry. Symmetry independent Mn atoms are distinguished by arbitrary different moment values. Half of the Mn moments must remain disordered in the centrosymmetric arrangements. The two twin-related configurations for each group type are related by the lost binary rotation. Additional trivial twinned configurations are obtained by switching all magnetic moments (time reversal operation).

Comparison with the representation method: This is a similar case to that of example 1. The assumption of a single irrep active does not constraint the spin configuration as the magnetic representation of the two Mn atoms only includes a single irrep. The little group of the propagation vector (0, 1/2, 0) is the full group P21/c. As can be seen using REPRES (in the Bilbao crystallographic server) there is only one 2-dim irrep of P21/c for this wave vector: Z1 (Z is the standard label for the vector (0, 1/2, 0) in the Brillouin one). Therefore, for the grey group P21/c1' there is one analogous irrep: mZ1, which can be relevant for a magnetic ordering (the symbol m is used to distinguish irreps that are odd for time reversal from even ones). The magnetic representation for this wave vector and the magnetic moments of the two independent Mn atoms can only contain this irrep (see for instance the results using Basireps of the FullProf suite [2]): Mrepr = 6 mZ1

This means that any arbitrary spin configuration with a propagation vector (0, 1/2, 0) transforms according to the irrep mZ1, and can be described in terms of 12 basis functions associated with this irrep. In other words, the assumption of the magnetic ordering complying with a single irrep does

not introduce any constraint, and the representation method is of no use for this case. This should be compared with the constraints introduced by the assumption of one of the two possible k-maximal magnetic groups discussed above, where the number of free parameters describing the spin configuration is limited to six in both cases.

Summarizing, any arbitrary spin configuration in Na2MnF5 with the observed propagation vector can be associated with the irrep mZ1, and therefore in this case the representation method in its traditional form is of no use. A spin ordering restricted to have one of the k-maximal magnetic symmetries shown above corresponds (see example 1) to the choice of a special direction of higher symmetry within the mZ1 representation (i.e. a specific linear combination of the six pairs of mZ1 basis functions), so that the number of free parameters in the possible combination of basis modes is restricted from 12 to 6. Hence, to assign a k-maximal magnetic group is NOT equivalent to the assignment of an irrep, and introduces additional constraints.

REFERENCES [1] jana.fzu.cz/ [2] www.ill.eu/sites/fullprof/ [3] iso.byu.edu [4] jp-minerals.org/vesta/ [5] jmol.sourceforge.net/ [6] Izyumov, Yu. A., Naish, V. E. & Ozerov, R. P. (1991). Neutron Diffraction of Magnetic Materials. Dordrecht: Kluwer Academic Publishers.

http://jana.fzu.cz/

http://www.ill.eu/sites/fullprof/

http://iso.byu.edu/

http://jp-minerals.org/vesta/

http://jmol.sourceforge.net/

Hands-on Exercises on the tools of the Magnetic Section of the Bilbao Crystallographic Server (www.cryst.ehu.es)

M1. Upload the cif file of LaMnO3 in STRCONVERT. Change to magnetic option. Transform to P1 to produce the whole set of atomic positions within the unit cell. Introduce magnetic moments along x of the four listed Mn atoms according to the sign relations: 1,-‐1,-‐1,1. Use findsym to find the MSG, and the description using this MSG. Visualize the magnetic structure with Jmol. Introduce a non-‐zero component my at the single symmetry-‐independent Mn atom, and transform again to P1 to observe that the resulting my values for 4 Mn atoms within the unit cell have the same sign (expected weak FM along y). Create an mcif file of the structure with STRCONVERT, open it with a text editor and localize the different data items: unit cell, atomic positions, symmetry operations. Download the mcif file in VESTA and visualize with this tool. (file required: 1.LaMnO3_parent.cif).

M2. Using k-‐SUBGROUPSMAG obtain the k-‐maximal subgroups for the parent space group Pnma for a propagation vector k=(1/2,0,0).

M3. Upload the mcif file of HoMnO3 in STRCONVERT. Introduce a non-‐zero spin component of Ho along x or along z, and try to obtain all the atomic positions and moments within the unit cell by transforming to trivial symmetry P1. Check that the program in both cases (for Ho moment along x or z) gives and error/warning: this means that if the Ho atoms are magnetically ordered in this phase, their magnetic moments can only be directed along b. Come back to the description using the MSG using the back button of the browser and introduce now a non-‐zero spin component for the Ho atoms along b and transform again to P1 to observe the resulting values for the symmetry-‐related Ho atoms within the unit cell. Come back to the descritption using the MSG, and visualize the magnetic structure with Jmol and/or VESTA. Using MVISUALIZE observe the differences between the parent unit cell, the standard unit cell and the unit cell actually used in the description. (file required: 2.HoMnO3.mcif).

M4. Upload the mcif file of HoMnO3 in STRCONVERT. Among the listed symmetry operations identify the anticentering operation {1’|1/2,0,0}. Identify also in the list the operations {2z|3/4,0,1/2} and {2z’|1/4,0,1/2}. Copy/paste the list of symmetry operations and introduce them in the program “IDENTIFY MAGNETIC GROUP” and check the MSG of the structure. (file required: 2.HoMnO3.mcif).

M5. Using MAGNEXT in its option B, obtain the systematic absences for the symmetry operations {2z| 0 0 0 } (-‐x,-‐y,z,+1); and for {2z| 0 0 ½ } (-‐x,-‐y,z+1/2,+1). Obtain the systematic absences for the corresponding primed operations. From the output of the program show, that although the operation {2'z| 0 0 0 } does not result in any general systematic absence, it may however give place to a systematic absence of all reflections of type (0,0,l) if the magnetic ordering is only along the z direction.

M6. Using MAGNEXT derive the systematic absences for the magnetic structure of LaMnO3 due to its MSG Pn'ma', and those additional ones due to the fact that the spins are aligned along a. Obtain by hand the systematic absence on reflections of type (0,0,l) due to the symmetry operation {2'z|1/2,0,1/2} present in Pn'ma', check the

result using the option B of MAGNEXT. Compare with the results given by the program for this type of reflections for the group Pn'ma'.

M7. Obtain with MAXMAGN the four possible alternative models of maximal symmetry for the magnetic structure of HoMnO3, which are compatible with its propagation vector k= (1/2,0,0) (upload as starting data the cif file of its parent Pnma structure). Obtain the symmetry constraints for the moments of the Ho atoms, in each case. Check that the two possible orthorhombic symmetries can be distinguished by the systematic absence of all reflections of type (h,0,l)+k, which will happen for one of the groups and not the other, if the spins are aligned along a. See tutorial of MAXMAGN, example 2, for a more detailed tutorial exercise. (file required: 3.HoMnO3_parent.cif).

M8. Using k-‐SUBGROUPSMAG explore all possible symmetries for the magnetic structure of HoMnO3, which are compatible with its propagation vector. Check that there are two different possible MSGs of the same type, namely of type Pa21. From the output of the program for the two groups, determine what makes them different.

M9. Using k-‐SUBGROUPSMAG obtain the k-‐maximal subgroups for the parent space group Pmmm for a propagation vector k=(1/2,0,0), and compare with those obtained for Pnma. Check that in the case of a parent Pmmm symmetry the inversion symmetry is not lost in any of the possible MSGs. This happens for a parent Pnma symmetry because the mirror planes and binary axes include non-‐trivial translations: a) Using the BNS magnetic unit cell (2ap, bp,cp) that results from the propagation vector, check that the gray group Pnma1’ described in this unit cell contains the "centering" {1|1/2,0,0} and the anticentering {1’|1/2,0,0}, and also the mirror plane operations {mz|1/4,0,1/2} and {mz|1/4,0,1/2}, and their "primed" ones. (use GENPOS if necessary) b) Demonstrate that the magnetic ordering necessarily destroys the operation {1|1/2,0,0} while keeping the anticentering {1’|1/2,0,0}. c) Using this previous property demonstrate that necessarily the operations {mz|1/4,0,1/2} , {mz|1/4,0,1/2}, and their primed ones cannot be conserved as they are incompatible with the antitranslation {1’|1/2,0,0} d) Demonstrate that the presence of the operation {1’|1/2,0,0} makes incompatible the space inversion operation with the presence of any binary rotations around c. e) From this knowledge and similar considerations for the binary operations along x enumerate the possible maximal magnetic points groups of the magnetic ordering if the parent space group is Pnma.

M10. Use MTENSOR to obtain some of the crystal tensor properties of the magnetic phase of HoMnO3 (electric polarization, magnetization, linear magnetoelectric tensor, cuadratic magnetoelectricity,...). The same for the magnetic phase of LaMnO3. (Upload the corresponding mcif files in STRCONVERT, copy the list of symmetry operations in the output of STRCONVERT and paste in the option B of MTENSOR, but deleting the translational parts, so that the point-‐group operations are left). (files required: 2.HoMnO3.mcif and 4.LaMnO3.mcif).

M11. From the knowledge of its parent space group and its propagation vector (P21/c and k=(0,1/2,0)), use k-‐SUBGROUPSMAG to explore all possible symmetries of the magnetic structure of Na2MnF5 and check that the system is probably a multiferroic of type II, with the magnetic ordering breaking the symmetry into a polar phase. Assuming that the Fe spins are aligned along a, obtain with MAXMAGN the

two possible alternative models of maximal symmetry. See tutorial of MAXMAGN, example 4. (file required: 5.Na2MnF5_parent.cif).

M12. Use ISODISTORT to obtain the possible active irreps the possible resulting magnetic symmetries and corresponding models for Na2MnF5 (parent space group P21/c and k=(0,1/2,0)). (file required: 5.Na2MnF5_parent.cif).

M13. Use ISODISTORT to obtain the possible active irreps the possible resulting magnetic symmetries and corresponding models for NiO (parent space group Fm-‐3m and k=(0,1/2,0)). (file required: 6.NiO_parent.cif).

M14. Using k-‐SUBGROUPSMAG and MAGMODELIZE obtain an mcif file of the alternative model for NiO with symmetry Cc2/m, which can result if the irrep mL3+ is the active one, and visualize it. (file required: 6.NiO_parent.cif).

M15. Show that the weak FM of the magnetic phase of NiF2 (MAGNDATA 0.36) is due to the non-‐magnetic atoms: from MAGNDATA submit the structure to STRCONVERT, remove the non magnetic atoms, and transform the structure to P1, in order to apply findsym and find the actual MSG if the non-‐magnetic atoms are ignored. Check that an antitranslation exists rel that relates the Ni atom at the origin with the one at (1/2,1/2,1/2), such that the MSG is of type IV, and therefore it forbids any FM.

LIST OF PARTICIPANTS

Asya Teneva-ManoylovaBorislava Georgieva

Changsong XuDesislava Vasileva

Dimitur VasilevErdinç Öz

Florian ThöleGülçin Aşkın

Hussien OsmanIuliana Pasuk

Ivan Aleksandrov KostovKristina Ivanova

Lyubomir MihaylovaLyubomira Plamenova Ermenkova

Maia Garcia VergnioryMariya Kalapsazova

Mehmet GünayMehmet Selim Aktuna

Mirela Tsokeva VrazhilovaNadezhda Kazakova

Nadire NayirNatalya FedorovaPavel GavryushkinRadoslav Raikov

Rosario Isabel Vilaplana CerdáRusi Ivanov Rusev

Svilen MitsievTina Radmilova Tasheva

Tsvetomila Ivanova LazarovaUmut Adem

Vyara Nikolaeva VelchevaYoyo Hinuma

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