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Group-subgroup relations

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<strong>Group</strong>-SubgroupRelations ofSpace <strong>Group</strong>sI. SubgroupsII. Wyckoff-position splittingsIII. Supergroups of space groupsIV. Normalizers of space groups


Conjugate <strong>subgroup</strong>sConjugate <strong>subgroup</strong>sLet H1


MAXIMAL SUBGROUPSOFSPACE GROUPS


Subgroups of Space groupsCoset decomposition G:TG(I,0) (W2,w2) ... (Wm,wm) ... (Wi,wi)(I,t1) (W2,w2+t1) ... (Wm,wm+t1) ... (Wi,wi+t1)(I,t2) (W2,w2+t2) ... (Wm,wm+t2) ... (Wi,wi+t2)... ... ... ... ... ...(I,tj) (W2,w2+tj) ... (Wm,wm+tj) ... (Wi,wi+tj)... ... ... ... ... ...Factor group G/TGisomorphic to the point group PG of GPoint group PG = {I, W1, W2, ...,Wi}


Subgroups of space groupsTranslationengleche <strong>subgroup</strong>s H


Example: P12/m1Translationengleche<strong>subgroup</strong>s H


Example: P12/m1Translationengleche<strong>subgroup</strong>s H


EXERCISESProblem 2.20Construct the diagram of thet-<strong>subgroup</strong>s of P4mm using the ‘analogy’with the <strong>subgroup</strong> diagram of 4mm


International Tables for Crystallography, Vol. A1Example: P4mmeds. H. Wondratschek, U. MuellerMaximal <strong>subgroup</strong>s of space groups


ITA1 maximal <strong>subgroup</strong> dataMaximal <strong>subgroup</strong>s of P4mm (No. 99)Remarks{braces forconjugate<strong>subgroup</strong>s(P, p): OH = OG + p(aH,bH,cH)= (aG,bG,cG) P


DATA ITA1:Maximal SubgroupsTransformation matrix:(P,p)group G{e, g2, g3, ..., gi,...,gn-1, gn}<strong>subgroup</strong> H


Klassengleiche <strong>subgroup</strong>s H


Klassengleiche <strong>subgroup</strong>s H1, primeq>1, primeSubgroups of space groupsTH< TG{ P H=PGH=TeTe(I,0)(I,t1)(I,t2)(I,tj)(I,ta)... ...ta(1,0,0)Te ta(I,t1+ta)(I,t2+ta)(I,tj+ta)... ...INFINITE number of maximalisomorphic <strong>subgroup</strong>s


International Tables for Crystallography, Vol. A1Example: P-1eds. H. Wondratschek, U. MuellerSeries of maximal isomorphic<strong>subgroup</strong>s


Subgroups of space groupsKlassengleiche <strong>subgroup</strong>s H


International Tables for Crystallography, Vol. A1Example: P4mmeds. H. Wondratschek, U. MuellerMaximal <strong>subgroup</strong>s of space groups


Bilbao Crystallographic ServerProblem:MAXIMAL SUBGROUPS OFSPACE GROUPSMAXSUBspace group87Static databasesDefault settings ofthe space groups


Bilbao Crystallographic ServerDATA-BASE:MAXIMAL SUBGROUPSMAXSUB


DATABASE:MAXIMAL SUBGROUPSMAXSUB(P,p)group G{e, g2, g3, ..., gi,...,gn-1, gn}<strong>subgroup</strong> H


Bilbao Crystallographic ServerProblem:MAXIMAL ISOMORPHICSUBGROUPSSERIESspace group87Data generated online(max. index 131)Static databases


Bilbao Crystallographic ServerStaticDatabases


Problem 2.21The retrieval tool MAXSUB gives an access to thedatabase on maximal <strong>subgroup</strong>s of space groups aslisted in ITA1. Consider the maximal <strong>subgroup</strong>s of thegroup P4mm, (No.99) and compare them with themaximal <strong>subgroup</strong>s of P4mm derived in Problem 2.17(ITA Exercises). Comment on the differences, if any.


GENERAL SUBGROUPSOFSPACE GROUPS


General <strong>subgroup</strong>s H H : G, H, [i], (P, p)Z 1Z 1Z 1Pairs: group - maximal <strong>subgroup</strong>Z k > Z k+1 , (P, p) kZ 2 Z 2(P, p) = nk=1 (P, p) kH


Subgroups of space groupsGeneral <strong>subgroup</strong>s H


<strong>Group</strong>-Subgroup <strong>Group</strong>-<strong>subgroup</strong> <strong>relations</strong> RelationsGApplicationsApplications Possible low-symmetry structures Domain structure[i] Prediction of new structures Symmetry modesHAimAIMG > H, [i]chains of maximal <strong>subgroup</strong>s[i]H k ∼ H[i]classification of H k ∼ H


Crystallographiccomputing programsTHE GROUP-SUBGROUPS SUITE


Bilbao Crystallographic ServerProblem:SUBGROUPS OF SPACEGROUPSSUBGROUPGRAPH994<strong>subgroup</strong> index[i]=[iP].[iL]


Problem 2.22Study the group--<strong>subgroup</strong> <strong>relations</strong> between thegroups G=P41212, No.92, and H=P21, No.4 using theprogram SUBGROUPGRAPH. Consider the cases withspecified index e.g. [i]=4, and not specified index of thegroup-<strong>subgroup</strong> pair.What is [iL] for P41212 > P21, [i]=4 ?


Problem 2.23Explain the difference between the contracted and completegraphs of the t-<strong>subgroup</strong>s of P4mm (No. 99) obtained bythe program SUBGROUPGRAPH. Compare the completegraph with the results of Problem 2.4 and 2.17 of ITAExercises.Explain why the t-<strong>subgroup</strong> graphs of all 8 space groupsfrom No. 99 P4mm to No. 106 P42bc have the same`topology' (i.e. the same type of `family tree'), only thecorresponding <strong>subgroup</strong> entries differ.


PROBLEM:Domain-structure analysisG[i]Hnumber of domain statestwins and antiphase domainstwinning operationsymmetry groups of the domainstates; multiplicity and degeneracy


Phase transitions domain structuresHomogeneous(parent) phaseGsymmetryreductionHDeformed(daughter) phaseDomain structureDomainDomainstatesA connected homogeneous part of a domain structure or of atwinned crystal is called a domain. Each domain is a single crystal.The number of such crystals is not limited; they differ in theirlocations in space, in their orientations, in their shapes and in theirspace groups but all belong to the same space-group type of H.The domains belong to a finite (small) number of domain states.Two domains belong to the same domain state if their crystalpatterns are identical, i.e. if they occupy different regions of spacethat are part of the same crystal pattern.The number of domain states which are observed after a phasetransition is limited and determined by the group-<strong>subgroup</strong><strong>relations</strong> of the space groups G and H.


Domain-structure analysisSUBGROUPS CALCULATIONS: HERMANNHermann, 1929:GFor each pair G>H, index [i], therei Pexists a uniquely defined intermediateM<strong>subgroup</strong> M, G M H, such that:M is a t-<strong>subgroup</strong> of Gi LH is a k-<strong>subgroup</strong> of MHwith[i]=[iP].[iL]iP=PG/PHtwinsiL=ZH,p/ZG,p=VH,p/VG,pantiphase


Problem:CLASSIFICATIONOF DOMAINSHERMANNQuartz Cu 3 Au Gd 2 (MoO 4 ) 3GP6 2 22G = MF m¯3mGP¯42 1 mi t = 2t-<strong>subgroup</strong>k-<strong>subgroup</strong>C mm2H = MHi k = 2P3 2 21Pm¯3m Pm¯3m Pm¯3m Pm¯3mHPba2i = i t = 2i = i k = 4i = 4twin domainsantiphasedomainstwin andantiphasedomains


EXAMPLEIndex [i] for a group-<strong>subgroup</strong> pair G>HLead vanadate Pb3(VO4)2INDEX:[i]=[iP].[iL]R-3mHigh-symmetry phase R-3mZG,p=1|PG|=12iP=PG/PH[iP]=3C2/miL=ZH,p/ZG,pP2 1 /c[iL]=2Low-symmetry phase P21/c|PH|=?ZH,p=?


Pb3(VO4)2: Ferroelastic Domains in P21/c phaseSUBGROUPGRAPHMaximal<strong>subgroup</strong>graphnumber of domains= index [i] = [iP].[iL]=6number of ferroelastic domains: iP = 12:4=3number of different <strong>subgroup</strong>s P21/c: 3


EXERCISES(A)High symmetry phase: P2/mLow symmetry phase: P1, small unit-cell deformationHow many and what kind of domain states?Hint: Determine the index [i]=[iP].[iL](B)High symmetry phase: P2/mLow symmetry phase: P1, duplication of the unit cellHow many and what kind of domain states?(C)(D)High symmetry phase: P4mmLow symmetry phase: P2, index 8How many and what kind of domain states?High symmetry phase: P42bcLow symmetry phase: P21, index 8How many and what kind of domain states?


Problem 2.24At high temperatures, BiTiO3 has the cubic perovskitestructure, space group Pm-3m (No. 221). Upon cooling, itdistorts to three slightly deformed structures, all threebeing ferroelectric, with space groups P4mm (No. 99),Amm2 and R3m. Can we expect twinned crystals of thelow symmetry forms? If so, how many kinds of domains?What program can be used?What INPUT data should be introduced?Hint: The program INDEX could be useful


Bilbao Crystallographic ServerProblem:INDEX [i] for G>HINDEX22199index [iL]{iL=ZH,p/ZG,p=(fG/fH)ZH,c/ZG,ci L=VH,p/VG,p= (fG/fH) VH,c/VG,c


BaTiO 3 : Ferroelectric Domains in P4mm phaseMaximal<strong>subgroup</strong>graphindex [i] = iP= 48 : 8=6number of ferroelectric domains: 6number of different <strong>subgroup</strong>s P4mm: 3


Domain-structure analysis: TwinningoperationCoset decomposition of G:Hleft:right:G>H, G=H+(V2,v2)H + ...+ (Vn,vn)HG>H, G=H+H(W2,w2) + ...+ H(Wn,wn)22199


BaTiO 3 : Ferroelectric Domains in P4mm phaseTwinning operationsCoset decomposition:Pm3m : P4zmm, index 6coset representatives: qi(1,0)(1,0)(3,0) (3,0) (3 -1 ,0)(3 -1 ,0)polarization: Pi= qiP00V-V000000V-VV-V0000


Ferroelectric Domains in Amm2 BaTiO 3SUBGROUPGRAPHMaximal<strong>subgroup</strong>graphindex [i] = iP= 48 : 4=12number of ferroelectric domains: 12number of different <strong>subgroup</strong>s Amm2 : 6


Ferroelectric Domains in Amm2 BaTiO 3(m-3m, mm2)high symmetry Pm-3morder parameter:irrep T 1u (vector representation)Amm2: Q(0,1/√2,1/√2)PPOrder of m-3m = 48Order of mm2 = 4Number of domains = 48/4=1212 eq. directions for the order parameter:(0,1/√2,1/√2)(0,-1/√2,1/√2)(0,-1/√2,-1/√2)(0,1/√2,-1/√2)(1/√2,0,1/√2)(-1/√2,0,1/√2)(1/√2,0,-1/√2)(1/√2,0,-1/√2)(0,1/√2,1/√2)(0,-1/√2,1/√2)(0,-1/√2,-1/√2)(0,1/√2,-1/√2)(1/√2,1/√2,0)(-1/√2,1/√2,0)(-1/√2,-1/√2,0)(1/√2,-1/√2,0)PP


Problem 2.25SrTiO3 has the cubic perovskite structure, space groupPm-3m. Upon cooling below 105K, the coordinationoctahedra are mutually rotated and the space group isreduced to I4/mcm; c is doubled and the conventionalunit cell is increased by a factor of four.Determine the number and the type of domains of thelow-temperature form of SrTiO3 using the computertools of the Bilbao Crystallographic server.


GENERATIONOFSPACE GROUPS


Generation of space groupsCrystallographic groups are solvable groupsComposition series:P1 Z2 Z3 ... Gindex 2 or 3Set of generators of a group is a set of groupelements such that each element of the group can beobtained as an ordered product of the generatorskh kh-1 k2W=(gh) * (gh-1) * ... * (g2) * g1g1 - identityg2, g3, g4 - primitive translationsg5, g6 - centring translationsg7, g8,... , - generate the rest of elements


Generation of sub-cubic point groups


Generation of orthorhombic andtetragonal groups122224422


Generation of sub-hexagonal point groups


Generation of trigonal and hexagonalgroups13326622


EXERCISESProblem 2.26 (A)Generate the space group C2mm using theselected generatorsCompare the results of your calculation with thecoordinate triplets listed under General positionof the ITA data of C2mm


General Layout: Right-hand page


EXERCISESProblem 2.26 (B)Generate the space group P4mm using theselected generators.Compare the results of your calculation with thecoordinate triplets listed under General positionof the ITA data of P4mmHint:Construct the composition series for the space groupP4mm in analogy with the composition series of 4mm2z 4z mx1 2 4 4mm[2] [2] [2]


Space group P4mm


EXERCISESProblem 2.27 (additional)Generate the space group P42/mbc using theselected generators.Compare the results of your calculation with thecoordinate triplets listed under General positionof the ITA data of P42/mbcHint:Construct the composition series for the spacegroup P42/mbc in analogy with the compositionseries of 4/mmm2z 4z 2y1 2 4 422[2] [2] [2]1[2]4/mmm


RELATIONS BETWEENWYCKOFF POSITIONS


Relations Symmetry between reduction WyckoffpositionsG = Pmm2 > H = Pm, [i] = 2S 0 , G = Pmm2 S 1 , H = Pm2h m.. ( 1 2, y, z) 2c (x, y, z)2h m.. (1/2,y,z) 2c 1 (x,y,z)2f .m. (x, → 12 , z) 1b 2 m (x 2 , 1 2 , z 2)1b 1 m (x 1 , 1 2 , z 1)2f .m. (x,1/2,z)1b m (x 2,1/2,z 2)1b m (x 1,1/2,z 1)Splitting of Wyckoff positionsSYMMETRY REDUCTION


Consider the groupEXAMPLE-<strong>subgroup</strong> pair P4mm>Pmm2[i]=2, a’=a, b’=b, c’=cDetermine the splitting schemes for WPs 1a,1b, 2c, 4d, 4egroup P4mm<strong>subgroup</strong> Pmm2


<strong>Group</strong>-<strong>subgroup</strong> pairEXAMPLEP4mm>Pmm2, [i]=2a’=a, b’=b, c’=cP4mmPmm22c 2mm. 1/2 0 z01/2 z1/2 0 z 1c mm20 1/2 z 1b mm2


Data on Relations between WyckoffPositions in International Tables forCrystallography, Vol. A1Example


Problem:Splitting of WyckoffpositionsWYCKSPLITSplitting of Wyckoff positions Phase transitionsApplicationsApplications Derivative structures Symmetry modesObjetivoAIMG > H, (P, p), W G splitting of W Gin suborbits relation between the suborbits and W H i


Bilbao Crystallographic Server<strong>subgroup</strong>groupTransformationmatrix (P,p)Two-level input:Choice of theWyckoff positions


Bilbao Crystallographic ServerTwo-level output:Relations betweencoordinate triplets


Problem 2.28Consider the group-<strong>subgroup</strong> pair P4mm (No.99) > Cm(No.8) of index [i]=4 and the relation between the basesa’=a-b, b’=a+b, c’=c. Study the splittings of the Wyckoffpositions for the group-<strong>subgroup</strong> pair by the programWYCKSPLIT.


SUPERGROUPS OFSPACE GROUPS


Supergroups of space groupsDefinition:The group G is a supergroup of H ifH is a <strong>subgroup</strong> of G, G≥HIf H is a proper <strong>subgroup</strong> of G, HHIf H is a maximal <strong>subgroup</strong> of G, HHTypes of minimalsupergroups:translationengleiche (t-type)klassengleiche (k-type)non-isomorphicisomorphicITA1 data:minimal non-isomorphic k- and t-supergroups types


The Supergroup ProblemGiven a group-<strong>subgroup</strong>pair G>H of index [i]Determine: all Gk>Hof index [i], Gi≃GG...G G2 G3 Gn[i][i]HHall Gk>H contain H as <strong>subgroup</strong>Gk=H+Hg2+...+Hgik


Example: Supergroup problem<strong>Group</strong>-<strong>subgroup</strong> pairP422>P222Supergroups P422 ofthe group P222P422P4z22P4x22 P4y22[2][2]P222P222P4z22= 222 +(222)(4z,0)P4x22= 222 +(222)(4x,0)P4y22= 222 +(222)(4y,0)P422= 222 +(222)(4,0)Are there moresupergroups P422 of P222?


Example: Supergroups P422 of P222


<strong>Group</strong>-Supergroup Relations<strong>Group</strong>-Supergroup RelationsGApplicationsApplications Possible high-symmetry structures Prediction of phase transitions[i] Prototype structuresHAIMAIMG > H, [i]to obtain the G k[i]∼ G


International Tables for Crystallography, Vol. A1eds. H. Wondratschek, U. MuellerMinimal Supergroup DataIncomplete dataSpace-group type onlyNo transformationmatrixP4z22[2]P222P4z22(...)P4x22...P4y22...


Bilbao Crystallographic ServerProblem:SUPERGROUPS OF SPACEGROUPSSUPERGROUPSMINSUPsupergroupspace groupindexOutputSupergroupsoptionnormalizers


Problem 2.30Consider the group--supergroup pair H < G with H = P222,No. 16, and the supergroup G= P422, No. 89, of index [i]=2.Using the program MINSUP determine all supergroups P422of P222 of index [i]=2.How does the result depend on the normalizer of thesupergroup and/or that of the <strong>subgroup</strong>?


NORMALIZERS OFSPACE GROUPS


Normalizers of space groups


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffine


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffineExample: Pmmnthe symmetry ofsymmetry


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffineExample: Pmmnthe symmetry ofsymmetry


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffineExample: Pmmnthe symmetry ofsymmetry


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffineExample: Pmmnthe symmetry ofsymmetry


Normalizers of space groupsNormalizers N(G) :g -1 {G}g = {G}{EuclideanAffinethe symmetryof symmetrySpace group: Pmmn (a,b,c)Euclidean normalizer:Pmmm (1/2a,1/2b,1/2c)


Normalizers for specialized metricsNormalizersSpace group:Pmmn (a,b,c),a=bEuclidean normalizer forspecialized metrics:P4/mmm (1/2a,1/2b,1/2c)Applications:Equivalent point configurationsWyckoff setsEquivalent structure descriptions


International Tables for Crystallography, Vol. A, Chapter 15Normalizers of space groupsE. Koch and W. FischerExample: Pmmn


Problem:Normalizersof space groupsNORMALIZER


Example NORMALIZER: Space group Pnnm (59)


Example NORMALIZER: Space group Pnnm (59)


NormalizersSymmetry-equivalentWyckoff positionsWyckoff Setsab


Symmetry-equivalentWyckoff positionsWyckoff SetsInternational Tables forCrystallography, Vol. AFischer and Koch, Chapter 14.Table 14.2.3.2(selection)BilbaoCrystallographicServer


Problem 2.31Using the computer tool NORMALIZER determine theEuclidean normalizer of the group P222 (general metric)and the Euclidean normalizers of enhanced symmetry forthe cases of specialized metric of P222. Compare yourresults with the data used in Problem 2.25 of the ITAExercises.Determine the assignment of Wyckoff positions intoWyckoff sets with respect to the different Euclideannormalizers of P222 (for general and specialized metrics)and comment on the differences, if any.


Problem 2.29Study the splittings of the Wyckoff positions for the group<strong>subgroup</strong>pair P42/mnm (No.136) > Cmmm (No.65) of index2 by the program WYCKSPLIT.Both transformation matrices a+b, -a+b, c and a+b, -a+b,c; 0, 0, 1/2 specify the same <strong>subgroup</strong> Cmmm (No.65) ofP42/mnm (No.136). Compare the splitting schemes of theWyckoff positions for the two transformation matrices andexplain the differences, if any.

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