Aroyo_GroupSubgr.pdf

7/27/2019 Aroyo_GroupSubgr.pdf

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INTERNATIONAL SCHOOLON FUNDAMENTAL

CRYSTALLOGRAPHY


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Mois I. Aroyo

Universidad del Pais Vasco, Bilbao, Spain

Group-Subgroup

Relations


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Group-Subgroup

Relations ofSpace Groups

I. Subgroups

II. Wyckoff-position splittings

III. Supergroups of space groups

IV. Normalizers of space groups


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Subgroups: Some basic results (summary)

Subgroup H < G

1. H={e,h1,h2,...,hk} G

2. H satisfies the group axioms of G

Proper subgroups H < G, andtrivial subgroup: {e}, G

Index of the subgroup H in G: [i]=|G|/|H|(order of G)/(order of H)

Maximal subgroup H of G

NO subgroup Z exists such that:H < Z < G


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Coset decomposition G:H

Group-subgroup pair H < G

left cosetdecomposition

right cosetdecomposition

G=H+g2H+...+gmH, giH,m=index of H in G

G=H+Hg2+...+Hgm, giHm=index of H in G

Normalsubgroups Hgj= gjH, for all gj=1, ..., [i]


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Conjugate subgroups

Conjugate subgroups Let H1


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MAXIMAL SUBGROUPS

OFSPACE GROUPS


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Subgroups of Space groups

Coset 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/TG

isomorphic to the point group PG of G

Point group PG = {I, W1, W2, ...,Wi}


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(I,0) (2,0) ( ,0) (m,0)

(I,t1) (2,t1) ( , t1) (m, t1)

(I,t2) (2,t2) ( , t2) (m,t2)

(I,tj) (2,tj) ( , tj) (m, tj)

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

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

1

1

1

1

TG TG 2 TG TG m1

Coset decomposition G:TGExample: P12/m1

Factor group G/TGPG PG = {1, 2, 1, m}

n1/2

n2/2

n3/2

n1

n2

n3

-1

-1

-1

inversion centres (1,t):1 at


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Translationengleche subgroups H


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a

c

=TGTG2P121

P =TGTG11

Example: P12/m1 Translationenglechesubgroups H


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index

[1]

[2]

[4]

Subgroup diagram of point

group 2/mTranslationengleiche subgroups of

space group P2/m

Example: P12/m1 Translationenglechesubgroups H


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EXERCISES

Problem 2.20

Construct the diagram of the

t-subgroups ofP4mm using the analogy

with the subgroup diagram of 4mm


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Maximal subgroups of space groups

International Tables for Crystallography, Vol. A1

eds. H. Wondratschek, U. MuellerExample: P4mm


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ITA1 maximal subgroup data

Remarks

(P,p): OH= OG +p(aH,bH,cH)= (aG,bG,cG) P

{ braces forconjugate

subgroups

Maximal subgroups ofP4mm (No. 99)


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subgroup H


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Klassengleiche subgroups H


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Klassengleiche subgroups H1, prime

P1(a,qb,c): q>1, prime... etc. INFINITE number of maximalisomorphic subgroups


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Series of maximal isomorphicsubgroups


eds. H. Wondratschek, U. MuellerExample: P-1


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Klassengleiche subgroups H


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Maximal subgroups of space groups


eds. H. Wondratschek, U. MuellerExample: P4mm


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Default settings of

the space groups

Bilbao Crystallographic Server

87

space group

Static databases

MAXIMAL SUBGROUPS OFSPACE GROUPS

Problem:MAXSUB


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MAXIMAL SUBGROUPSDATA-

BASE:

MAXSUB



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MAXIMAL SUBGROUPSDATABASE: MAXSUB

group G {e, g2, g3, ..., gi,...,gn-1, gn}

subgroup H


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Data generated online(max. index 131) Static databases


space group

87

MAXIMAL ISOMORPHICSUBGROUPS

Problem:SERIES

i C i S


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StaticDatabases



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Problem 2.21

The retrieval tool MAXSUB gives an access to the

database on maximal subgroups of space groups aslisted in ITA1. Consider the maximal subgroups of thegroup P4mm, (No.99) and compare them with themaximal subgroups ofP4mm derived in Problem 2.17

(ITA Exercises). Comment on the differences, if any.


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GENERAL SUBGROUPS

OFSPACE GROUPS


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Chains of maximal subgroups

G

Z

1Z

1

Z2 Z

2

H

Z1

Group-subgroup pair

G > H : G, H, [i], (P, p)

Pairs: group - maximal subgroup

Zk > Zk+1, (P, p)k

(P, p) =n

k=1(P, p)k

General subgroups H


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Theorem Hermann, 1929:

Subgroups of space groups

General subgroups H


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Group-subgroup relations

Applications

Possible low-symmetry structures

Domain structure

Prediction of new structures

Symmetry modes

AIM

G> H, [i]

chains of maximal subgroups

Hk[i] H

classification of Hk

[i]

H

Applications

Aim

G

H

[i]

Group-Subgroup Relations


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Crystallographiccomputing programs

THE GROUP-SUBGROUPS SUITE



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SUBGROUPS OF SPACEGROUPS

SUBGROUPGRAPHProblem:

subgroup index

99

4

[i]=[iP].[iL]


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Study the group--subgroup relations 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 the

group-subgroup pair.

Problem 2.22

What is [iL] for P41212 > P21, [i]=4 ?


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Problem 2.23

Explain the difference between the contractedand completegraphs of the t-subgroups ofP4mm (No. 99) obtained bythe program SUBGROUPGRAPH. Compare the complete

graph with the results of Problem 2.4 and 2.17 of ITAExercises.

Explain why the t-subgroup 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 subgroup entries differ.


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PROBLEM: Domain-structure analysis

G H[i]

number of domain statestwins and antiphase domains

symmetry groups of the domainstates; multiplicity and degeneracy

twinning operation

Ph t iti d i t t


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Phase transitions domain structures

Homogeneous(parent) phase

Deformed(daughter) phase

Domain

G H

symmetryreduction

Domain structureA 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 crystalsis not limited; they differ in theirlocations in space, in their orientations, in their shapes and in their

space groups but all belong to the same space-group type of H.

Domainstates

The domains belong to a finite (small) number ofdomain 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-subgrouprelations of the space groups G and H.


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Hermann, 1929:

For each pair G>H, index [i], there

exists a uniquely defined intermediatesubgroup M,GM H, such that:

Mis a t-subgroup ofG

His a k-subgroup ofM

G

M

H

iP

iL

SUBGROUPS CALCULATIONS: HERMANN

Domain-structure analysis

iP=PG/PH

iL=ZH,p/ZG,p=VH,p/VG,p

[i]=[iP].[iL]with twinsantiphase


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Problem:

P421m

Cmm2

Pba2

ik = 2

it = 2

i = 4

G

H

Fm3m

Pm3m Pm3m Pm3m Pm3m

i = ik = 4

G=M

H

P6222

P3221

i = it = 2

H =M

G

twin domainsantiphase

domains

twin and

antiphase

domains

Quartz Cu3Au Gd2(MoO4)3

t-subgroup k-subgroup

HERMANNCLASSIFICATIONOF DOMAINS

Index [i] for a group subgroup pair G>HEXAMPLE


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Index [i] for a group-subgroup pair G>H

Lead vanadate Pb3(VO4)2

R-3m

C2/m

P21/c

iP=PG/PH

iL=ZH,p/ZG,p

EXAMPLE

High-symmetry phase R-3m

Low-symmetry phase P21/c

[i]=[iP].[iL]INDEX:

[iP]=3

[iL]=2

ZG,p=1

ZH,p=?

|PG|=12

|PH|=?


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Ferroelastic Domains in P21/c phase

number of ferroelastic domains: iP = 12:4=3

number of different subgroups P21/c: 3

Maximal-subgroup graph

number of domains= index [i] = [iP].[iL]=6

SUBGROUPGRAPH

Pb3(VO4)2:

EXERCISES


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EXERCISES

How many and what kind of domain states?

High symmetry phase: P2/mLow symmetry phase: P1, small unit-cell deformation

(A)

High symmetry phase: P2/mLow symmetry phase: P1, duplication of the unit cell

(B)


High symmetry phase: P4mmLow symmetry phase: P2, index 8

(C)


High symmetry phase: P42bcLow symmetry phase: P21, index 8

(D)

Hint: Determine the index [i]=[iP].[iL]



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Problem 2.24

At 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 INPUT data should be introduced?

What program can be used?

Hint: The program INDEX could be useful



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iL=ZH,p/ZG,p=(fG/fH)ZH,c/ZG,c


Problem: INDEXINDEX [i] for G>H

index [iL]

221

99

iL=VH,p/VG,p= (fG/fH) VH,c/VG,c{


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BaTiO3: Ferroelectric Domains in P4mm phase

number of ferroelectric domains: 6

number of different subgroups P4mm: 3


index [i] = iP= 48 : 8=6

Domain structure analysis: Twinning


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Coset decomposition of G:H

left: G>H, G=H+(V2,v2)H + ...+ (Vn,vn)H

right: G>H, G=H+H(W2,w2) + ...+ H(Wn,wn)

Domain-structure analysis: Twinningoperation

99

221


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BaTiO3: Ferroelectric Domains in P4mm phase

Twinning operations

Coset decomposition:

(1,0)

coset representatives: qi

(3,0) (3-1

,0)polarization: Pi= qiP

(1,0)

Pm3m : P4zmm, index 6

(3-1

,0)(3,0)

0

0

V

0

0

-V

V

0

0

-V

0

0

0

V

0

0

-V

0

F l t i D i i A 2 B TiO


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Ferroelectric Domains in Amm2 BaTiO3

number of ferroelectric domains: 12

number of different subgroups Amm2 : 6


index [i] = iP= 48 : 4=12

SUBGROUPGRAPH

F l t i D i i A 2 B TiO


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Ferroelectric Domains in Amm2 BaTiO3

Number of domains = 48/4=12

12 eq. directions for the order parameter:

Order of m-3m = 48

Amm2: Q(0,1/2,1/2)(m-3m, mm2)

high symmetry Pm-3morder parameter:

irrep T1u (vector representation)

Order of mm2 = 4

P

P P

P

(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)

(1/2,1/2,0)(-1/2,1/2,0)(-1/2,-1/2,0)

(1/2,-1/2,0)

(0,1/2,1/2)(0,-1/2,1/2)(0,-1/2,-1/2)(0,1/2,-1/2)


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SrTiO3 has the cubic perovskite structure, space groupPm-3m. Upon cooling below 105K, the coordinationoctahedra are mutually rotated and the space group is

reduced 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 the

low-temperature form of SrTiO3 using the computertools of the Bilbao Crystallographic server.

Problem 2.25


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GENERATION

OFSPACE GROUPS

Generation of space groups


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Generation of space groups

Set of generators of a group is a set of group

elements such that each element of the group can beobtained as an ordered product of the generators

g1 - identityg2, g3, g4 - primitive translations

Composition series: P1 Z2 Z3 ... Gindex 2 or 3

Crystallographic groups are solvable groups

W=(gh) * (gh-1) * ... * (g2) * g1kh kh-1 k2

g5, g6 - centring translationsg7, g8,... , - generate the rest of elements


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Generation of sub-cubic point groups

Generation of orthorhombic and


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tetragonal groups

1

2

4

422

222


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Generation of sub-hexagonal point groups

Generation of trigonal and hexagonal


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g ggroups

1

3

6

622

32

EXERCISES


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Problem 2.26 (A)

Generate the space group C2mm using theselected generators

Compare the results of your calculation with the

coordinate triplets listed under General positionof the ITA data of C2mm

General Layout: Right-hand page


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y g p g

EXERCISES


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Problem 2.26 (B)

Generate the space group P4mm using theselected generators.


coordinate triplets listed under General positionof the ITA data ofP4mm

Construct the composition series for the space groupP4mm in analogy with the composition series of4mm

1 2 4 4mm

[2] [2] [2]

2z 4z mx

Hint:

Space group P4mm


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Space group P4mm

EXERCISES


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Problem 2.27 (additional)

Generate the space group P42/mbcusing theselected generators.


coordinate triplets listed under General positionof the ITA data ofP42/mbc

Construct the composition series for the space

group P42/mbcin analogy with the compositionseries of4/mmm

1 2 4 422

[2] [2] [2]

2z 4z 2y

Hint:

[2]

1

4/mmm


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RELATIONS BETWEENWYCKOFF POSITIONS

Symmetry reduction

Relations between Wyckoff


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Symmetry reduction

G= Pmm2 > H = Pm, [i] = 2

S0, G= Pmm2 S1, H = Pm2h m.. (12, y , z) 2c 1 (x , y , z)

2f .m. (x, 12, z) 1b2 m (x2,

1

2, z2)1b1 m (x1,12, z1)

Splitting of Wyckoff positionsSYMMETRY REDUCTION

2h m.. (1/2,y,z) 2c1 (x,y,z)

2f.m. (x,1/2,z) 1b m (x2,1/2,z2)1b m (x1,1/2,z1)

ypositions

Consider the group


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EXAMPLE -subgroup pair P4mm>Pmm2

Determine the splitting schemes for WPs 1a,1b, 2c, 4d, 4e

[i]=2,a=a, b=b, c=c

group P4mm subgroup Pmm2

Group-subgroup pair


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EXAMPLE P4mm>Pmm2, [i]=2

a=a, b=b, c=c

P4mm Pmm2

2c 2mm. 1/2 0 z01/2 z

1/2 0 z 1c mm2

0 1/2 z 1b mm2

Data on Relations between WyckoffP i i i I i l T bl f


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Positions in International Tables forCrystallography, Vol. A1

Example

Splitting of WyckoffProblem:


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Splitting of Wyckoff positions

Applications

Phase transitions

Derivative structures

Symmetry modes

Objetivo

G> H, (P,p), WG

splitting of WG in suborbits

relation between the suborbits and WHi

Splitting of Wyckoffpositions

Applications

AIM

Problem:WYCKSPLIT



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groupsubgroup

Transformation

matrix (P,p)

Two-level input:

Choice of theWyckoff positions



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Two-level output:

Relations betweencoordinate triplets


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Problem 2.28

Consider the group-subgroup pair P4mm (No.99) > Cm(No.8) of index [i]=4 and the relation between the bases

a=a-b, b=a+b, c=c. Study the splittings of the Wyckoffpositions for the group-subgroup pair by the programWYCKSPLIT.


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SUPERGROUPS OF

SPACE GROUPS

Supergroups of space groups


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Definition: The group G is a supergroup of H ifH is a subgroup of G, GH

If H is a proper subgroup of G, HH

Types of minimalsupergroups:

If H is a maximal subgroup of G, HHtranslationengleiche (t-type)klassengleiche (k-type)

isomorphicnon-isomorphic

minimal non-isomorphic k- and t-

supergroups types

ITA1 data:

The Supergroup Problem


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The Supergroup Problem

Given a group-subgrouppair G>H of index [i]

Determine: all Gk>Hof index [i], GiG

H

G G2 G3 Gn...

all Gk>H contain H as subgroupH

G

[i] [i]

Gk=H+Hg2+...+Hgik

Example: Supergroup problem


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a p e: Supe g oup p ob e

Group-subgroup pair

P422>P222

P422

[2]

Supergroups P422 of

the group P222

P222

P4z22

P222

P4x22 P4y22

[2]

P422= 222+(222)(4,0)

P4z22= 222+(222)(4z,0)

P4x22= 222+(222)(4x,0)P4y22= 222+(222)(4y,0)

Are there more

supergroups P422 of P222?

Example: Supergroups P422 of P222


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p p g p

Group Supergroup Relations


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Group-Supergroup Relations

Applications

Possible high-symmetry structures

Prediction of phase transitions

Prototype structures

AIM

G> H, [i]

to obtain the Gk[i] G

Applications

AIM

G

H

[i]

Group-Supergroup Relations



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eds. H. Wondratschek, U. Mueller

Minimal Supergroup Data

Incomplete data

P222

P4z22 P4x22

P4y22[2]Space-group type only

No transformation

matrix

P4z22(...)

...

...



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supergroup

space group

index

OutputSupergroups

optionnormalizers

SUPERGROUPS OF SPACEGROUPS

Problem: SUPERGROUPSMINSUP


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Problem 2.30

Consider 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 subgroup?


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NORMALIZERS OF

SPACE GROUPS


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Normalizers of space groups


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Normalizers N(G) : g-1{G}g = {G}

p g p

Euclidean

Affine{



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Example: Pmmn

the symmetry of

symmetry


p g p

Euclidean

Affine{



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Example: Pmmn

the symmetry of

symmetry


p g p

Euclidean

Affine{



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Example: Pmmn

the symmetry of

symmetry


p g p

Euclidean

Affine{



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Example: Pmmn

the symmetry of

symmetry


p g p

Euclidean

Affine{



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p g p

Normalizers N(G) : Euclidean

Affine

Space group: Pmmn (a,b,c)

Euclidean normalizer:Pmmm (1/2a,1/2b,1/2c)

{the symmetryof symmetry

g-1{G}g = {G}

NormalizersNormalizers for specialized metrics


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Space group:Pmmn (a,b,c), a=b

Euclidean normalizer forspecialized metrics:P4/mmm (1/2a,1/2b,1/2c)

Applications: Equivalent point configurations

Wyckoff sets

Equivalent structure descriptions

International Tables for Crystallography, Vol. A, Chapter 15


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Example: Pmmn


E. Koch and W. Fischer


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Normalizersof space groupsProblem: NORMALIZER

Example NORMALIZER: Space group Pnnm (59)


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Example NORMALIZER: Space group Pnnm (59)


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Symmetry-equivalent

Normalizers

W k ff S


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Symmetry equivalentWyckoff positions

Wyckoff Sets

a b

Symmetry-equivalentWyckoff positions

Wyckoff Sets


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Wyckoff positions

Table 14.2.3.2

(selection)

International Tables forCrystallography, Vol. A

Fischer and Koch, Chapter 14.

BilbaoCrystallographic

Server

Problem 2 31


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Problem 2.31

Using 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 ofP222. Compare your

results with the data used in Problem 2.25 of the ITAExercises.

Determine the assignment of Wyckoff positions into

Wyckoff sets with respect to the different Euclideannormalizers ofP222 (for general and specialized metrics)and comment on the differences, if any.


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Problem 2.29

Study the splittings of the Wyckoff positions for the group-subgroup pair P42/mnm (No.136) > Cmmm (No.65) of index

2 by the program WYCKSPLIT.

Both transformation matrices a+b, -a+b, c and a+b, -a+b,c; 0, 0, 1/2 specify the samesubgroupCmmm (No.65) of

P42/mnm (No.136). Compare the splitting schemes of theWyckoff positions for the two transformation matrices andexplain the differences, if any.

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