Crystallographic and metacrystallographic groups:
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| Language: | English |
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Amsterdam u.a.
North-Holland
1986
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 628 S. |
| ISBN: | 0444869557 |
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| 245 | 1 | 0 | |a Crystallographic and metacrystallographic groups |
| 264 | 1 | |a Amsterdam u.a. |b North-Holland |c 1986 | |
| 300 | |a XIX, 628 S. | ||
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| adam_text | Contents
Preface v
Contents ix
Part I. Concerning the mathematical language used in this book 1
Chapter 1. Sets, mappings, classifications and algebraic structures 3
1.1. Sets, mappings, permutations 3
1.2. Equivalence relation on a set; classifications 6
1.3. Algebraic structures 8
Chapter 2. A selection of definitions and theorems concerning groups 10
2.1. Definition of a group and isomorphism of groups 10
2.2. Some general theorems concerning the structure of groups 15
2.2.1. Subgroups of a group 15
2.2.2. Subsets of a group 16
2.2.3. Subgroup generated by a subset of a group 17
2.2.4. Cosets of a subgroup, normal subgroups, quotient groups 19
2.2.5. Homomorphism of groups 22
2.2.6. Normalizer and centralizer of a subgroup, conjugate subgroups,
classes of conjugate elements 25
2.2.7. Direct and semidirect product groups 27
2.2.8. Subgroups of index 2 29
2.3. Groups of permutations of a finite or infinite set 30
2.3.1. Symmetric group on a set 30
2.3.2. Symmetric group of degree n 33
2.3.3. Natural matrix representation of the symmetric group of
degree n 35
2.4. Groups of permutations of a set as invariance groups of its subsets 37
2.4.1. Invariance groups, generating groups and the symmetry group
of a subset 37
2.4.2. Effect of permutations of a set X and of a set Y on functions from
X into 7 41
2.4.3. Action of a group on a set and, in particular, on a function space;
invariance groups of a function, and its symmetry group 44
ix
x Contents
2.5. Automorphisms of a group; Cayley s Theorem 44
2.5.1. Permutations of a group 44
2.5.2. Automorphisms of a group 45
2.5.3. Automorphisms of cyclic groups and of dihedral groups 47
2.5.4. Cayley s Theorem 49
2.5.5. Holomorph of a group 50
2.5.6. Homomorphisms with the same kernel 50
2.5.7. Field of complex numbers, field of real numbers, and their
automorphisms 51
2.6. Direct product and semidirect products of two groups; group
extensions 52
2.6.1. Various kinds of products of groups 52
2.6.2. Direct product of groups 53
2.6.3. Semidirect products of two groups 53
2.6.4. Wreath products 56
2.6.5. Extensions of a group by another group 59
2.7. Subgroups of the direct product of two groups 59
2.7.1. Subdirect products 59
2.7.2. Subgroups of the direct product of two groups of which one is of
order 2 60
2.7.3. fc subgroups of the direct product of two groups 62
2.7.4. A general method of finding all subdirect products of
two groups 65
2.8. Permutation representations of groups 68
2.8.1. Permutation representations of a group which are generated by
its subgroups 68
2.8.2. Equivalence classes of permutation representations 71
2.8.3. The Main Theorem on Permutation Representations 73
2.8.4. Permutation representation of G as an /i subgroup of GxdP 75
2.9. Matrix groups , 76
2.9.1. Matrix group GL(n,C) 76
2.9.2. Matrix group GL(n,U) 78
2.9.3. Matrix groups O(n,U) and GL(n,Z) 79
2.9.4. Q , Z and O equivalence of matrix groups 81
2.10. Tables 82
Part II. Affine and Euclidean groups, the Newton group 85
Chapter 3. Leading to space time 87
3.1. Outline of Chapter 3 87
3.2. Complex and real vector spaces 88
3.3. Affine point spaces 92
3.4. Euclidean vector spaces and Euclidean point spaces 95
3.4.1. Euclidean vector spaces 95
Contents xi
3.4.2. Euclidean point spaces 98
3.4.3. Change from an arbitrary coordinate system to a Cartesian
coordinate system 99
3.5. Space time, £s(3) X £t(l) 101
3.6. Table 102
Chapter 4. Linear and affine groups, orthogonal and Euclidean groups, gross
classification of groups of isometries 103
4.1. Outline of Chapter 4 103
4.2. Linear transformations of a vector space (real or complex) 104
4.2.1. Linear group 3) of a vector space V(n), and the general linear
complex matrix group GL(n,C) 104
4.2.2. Referring a group of linear transformations of V(n) to different
bases of V(n) 106
4.2.3. Active and passive interpretation of matrix equations 107
4.3. Orthogonal transformations of a Euclidean vector space 108
4.4. Affine transformations of an affine point space A{n) 110
4.4.1. Affine group stf and the general inhomogeneous linear group
GIL(n) 110
4.4.2. V(n) .T{n) isomorphism 114
4.5. Isometries of a Euclidean point space 115
4.6. Similarity transformations of a Euclidean point space 117
4.7. Referring a group of affine transformations of A(n) and, in particular,
a group of isometries to different coordinate systems in A(n) 117
4.8. Equivalence classes of coordinate systems 119
4.9. Orientation preserving subgroups of the affine group 120
4.10. Equivalence classes of groups of isometries 121
4.10.1. General case 121
4.10.2. Equivalence classes of groups of rotations 123
4.11. On some uses of Jacobi matrices 125
4.12. Some properties common to all subgroups of the affine group 127
4.13. Tables 129
Chapter 5. Matrix representations of groups 131
5.1. Carrier spaces of matrix representations of a group 131
5.2. Some general properties of matrix representations of groups 132
5.2.1. Reducibility and decomposability of matrix groups 132
5.2.2. Reducibility, decomposability and equivalence of matrix
representations of groups 135
5.2.3. On matrix representations of finite Abelian groups, finite cyclic
groups, and finite transitive groups of permutations 136
5.3. Properties of carrier spaces of reducible and decomposable matrix
representations of groups 136
xii Contents
5.3.1. General case 136
5.3.2. The case of the reducible representations that contain the
identity representation 138
5.4. A criterion for the irreducibility of a matrix group 141
5.5. Direct product of two matrix representations of a group 142
5.6. The fundamental equations of the theory of matrix representations of
finite groups 146
5.7. Table 147
Chapter 6. Euclidean groups ?(3), S(2), S{ ) and the Newton group
fs(3) X «f,(l) 148
6.1. Isometries of space, and the Euclidean group ?(3) 148
6.1.1. Mostly on terminology and notation 148
6.1.2. Group % of all (proper and improper) rotations about a point,
and its subgroup £%+ of proper rotations 149
6.1.3. Isometries other than rotations and translations: screw
displacements and glide reflections 151
6.1.4. Some generalities concerning groups of isometries 153
6.2. Finite and some infinite groups of rotations of space 154
6.2.1. Finite subgroups of ® = @+ X J 154
6.2.2. Finite and infinite axial groups of rotations 159
6.2.3. Symmetry groups of a line lS and of a plane 2S in space 161
6.3. Euclidean groups ^(2) of a plane and ^(1) of a line 162
6.3.1. Homomorphism of 2S onto S(2) 162
6.3.2. Finite groups of rotations of a plane 163
6.3.3. The Euclidean group £{Y), and the time group Sl 164
6.4. Structure preserving group of space time: the Newton group jV 165
6.4.1. The Newton group Jf, and its subgroups ~sl, st, W and jV+ 165
6.4.2. A gross classification of the subgroups of the Newton group J/~ 168
6.4.3. Subgroups of the space time rotation group %st 169
6.5. Tables 175
Part III. Discrete point sets in space and discrete groups of isometries;
in particular, crystallographic groups 185
Chapter 7. Leading to crystallographic groups 187
7.1. Introducing the idea of a group theoretical classification of discrete point
sets in space 187
7.2. Generating groups of point sets in space, discrete point sets, and discrete
groups of isometries 188
7.3. Invariance groups and the symmetry group of a point set; global and
local symmetry 191
7.4. Gross classification of all discrete groups of isometries and of all discrete
point sets; crystallographic groups 195
Contents xiii
7.5. Some events in the history of crystallographic groups 197
7.6. Table 200
Chapter 8. Lattice groups and point lattices 201
8.1. Some affine properties of lattice groups and point lattices 201
8.2. Some metric properties of lattice groups and point lattices 205
8.2.1. Discrete groups of translations are lattice groups 205
8.2.2. Positive definite symmetric quadratic forms associated with a
point lattice, and corresponding positive definite symmetric
matrices 207
8.2.3. Unit cells of a point lattice 210
8.2.4. Reciprocal point lattices 211
8.2.5. Reduced bases of a point lattice, reduced unit cells and reduced
metric forms 212
8.3. Bravais classification of point lattices and of lattice groups 214
8.3.1. Definition of Bravais classes of point lattices and of
lattice groups 214
8.3.2. Holohedry of a point lattice; holohedral groups of rotations 215
8.3.3. Bravais groups and the symmetry group of a point lattice 217
8.3.4. Criteria for point lattices to belong to the same Bravais class 220
8.3.5. Example illustrating the meaning of Hoi A, BrA and FA 222
8.4. Some properties of finite integral matrix groups and, in particular, of
Bravais groups 224
8.5. Determining Bravais classes of point lattices 228
8.5.1. Outline of a method 228
8.5.2. Bravais classes of point lattices in a plane (n = 2) 229
8.5.3. Point lattices in space (n = 3) 235
8.6. Bravais systems of point lattices, of lattice groups, of Bravais groups and
of Bravais flocks 239
8.6.1. Definition of Bravais systems 239
8.6.2. The case of « = 3, 2 and 1 240
8.6.3. Example of a group which belongs to a holohedral class but is not
holohedral 242
8.7. Tables of Bravais classes and Bravais systems of point lattices in space
and in a plane 243
8.8. Tables 245
Chapter 9. Space groups 250
9.1. General properties of space groups 250
9.1.1. Lattice and point group of a space group; rotation groups
associated with a space group 250
9.1.2. Properties common to all space groups 252
9.1.3. Symmorphic and non symmorphic space groups 255
9.1.4. Space groups referred to their natural coordinate systems 257
xiv Contents
9.1.5. Subgroups of a space group which are themselves space groups;
their gross classification 260
9.1.6. Subgroups of a space group which are groups of rotations
(crystallographic groups of rotations) 261
9.2. Introductory remarks concerning classification of space groups 262
9.2.1. Bieberbach s Theorem 262
9.2.2. Starting point for classifications of space groups 263
9.3. Various classifications of space groups 264
9.3.1. Geometric classes and arithmetic classes of space groups; Bravais
flocks of arithmetic classes of space groups 264
9.3.2. Bravais classes and subclasses of space groups; exceptional space
groups 269
9.4. Symmorphic space groups 270
9.5. Non symmorphic space groups 274
9.6. Various classifications of space groups (9.3 continued): systems of space
groups 276
9.7. Symbols used for affine classes of space groups of a line (n = 1), of a
plane (n = 2) and of space (« = 3) 279
9.8. Automorphisms and normalizers of space groups 282
9.9. Tables 284
Chapter 10. Classification of crystals by means of space groups 290
10.1. Structure of a space group F as it manifests itself in the structure of
crystals generated by F 290
10.2. Wyckoff s classification of simple crystals 293
10.2.1. /( classes of simple point sets generated by a discrete group K
of isometries 293
10.2.2. Wyckoff classes of simple crystals; examples 295
10.3. Symmetry groups of crystals 298
10.3.1. Generalities concerning the symmetry groups of simple and
composite crystals 298
10.3.2. Examples of determining the symmetry group of a
simple crystal 300
10.3.3. Point lattices which have a common symmetry group 302
10.4. Tables 304
Chapter 11. Decomposable discrete groups of isometries 306
11.1. Some general properties of decomposable discrete groups of isometries 306
11.2. Net groups 2K(3) in space and line groups J/C(2) in a plane 307
11.2.1. Properties common to the groups 2K(3) and 1K(2) 307
11.2.2. Derivation of the line groups 1K(2) in a plane 311
11.2.3. Net groups 2/C(3) in space 312
11.2.4. Groups 2/C(3) and T#C(2) as subgroups of space groups F(3)
and F(2) 313
Contents xv
11.3. Three dimensional line groups rK(3) 315
11.3.1. Properties common to all line groups f/C(3) 315
11.3.2. Derivation of the line groups J/C(3) 318
11.3.3. Hermann Mauguin notation for line groups */f(3) 325
11.3.4. Crystallographic line groups 7C(3) 326
11.4. Point sets generated by the line groups 1K(3) 327
11.4.1. General case 327
11.4.2. One sided and two sided line strips and the groups which
generate them 330
11.5. Helical groups H(3) in space 332
11.6. Tables 337
Part IV. Metacrystallographic groups; in particular, colour groups, magnetic
groups, spin groups 347
Chapter 12. Leading to metacrystallographic groups 349
12.1. Introducing the idea of a group theoretical classification of functions
denned on a crystal by means of their invariance groups 349
12.2. Metacrystallographic groups and their gross classification 350
12.3. Metacrystallographic groups of kinds (i) and (ii) 353
12.3.1. Metacrystallographic groups of kinds (i) and (ii) in the sense
of action (A2) 353
12.3.2. Metacrystallographic groups of kinds (i) and (ii) in the sense
of an action different from (A2) 354
12.4. Metacrystallographic groups of kind (iii) 355
12.5. A sketch of the early history of metacrystallographic groups 356
12.5.1. The Heesch Shubnikov groups 356
12.5.2. Magnetic space groups; magnetic symmetry 359
12.5.3. Metacrystallographic groups other than the Heesch Shubnikov
groups 360
12.6. On terminology 360
12.7. On two group theoretical classifications of functions denned on crystals 361
Chapter 13. Colour groups and colour functions (coloured point sets) 363
13.1. Definition of d colour groups 363
13.2. Properties common to all colour groups 365
13.2.1. Colour groups and permutation representations 365
13.2.2. On finding the colour group family of a discrete group K of
isometries from the matrix representations of K 368
13.2.3. Equivalence classes of colour groups 369
13.3. Crystallographic colour groups of rotations 370
13.4. Crystallographic infinite colour groups 373
13.4.1. Colour lattice groups and colour space groups 373
13.4.2. Two colour groups (dichromatic groups, black and white
groups) 374
xvi Contents
13.5. On classifications of colour functions (that is, of coloured point sets) 375
13.5.1. General characteristics of colour functions 375
13.5.2. Equivalence classes of d colour functions 377
13.6. Classification of mass valued functions defined on crystals 380
13.7. Colour W groups 383
13.8. Multiple colour groups 384
13.9. Table 386
Chapter 14. Action of the Newton group on functions denned on space time;
in particular, its action on electromagnetic quantities 387
14.1. Action of the Newton group on functions having values in the carrier
space of one of its representations 387
14.2. Scalar and vector functions denned on space time 389
14.2.1. Action of the Newton group on scalar functions and on vector
functions 389
14.2.2. Symmetry groups of scalar functions and of vector functions
which are constant in space and time 392
14.3. Action of the Newton group on electromagnetic quantities; macroscopic
point of view 394
14.3.1. Covariance of the Maxwell equations and its consequences 394
14.3.2. Invariance groups of the constitutive relations of a medium 396
14.4. Action of the Newton group on electromagnetic quantities; atomic
point of view 398
14.5. Tables 399
Chapter 15. Invariance groups of magnetic and electric dipole arrangements
in crystals; in particular, magnetic groups 400
15.1. Invariance groups of magnetic dipole arrangements (that is, of spin
arrangements): magnetic groups 400
15.2. Properties common to all magnetic groups 402
15.3. Relation of magnetic groups to two colour groups 406
15.4. Crystallographic magnetic groups of rotations (that is, crystallographic
magnetic point groups) 406
15.5. Non crystallographic magnetic groups of rotations 408
15.6. Invariance groups of electric dipole arrangements 408
15.7. Tables 410
Chapter 16. Magnetic lattice groups 414
16.1. Subgroups of index 2 of an H dimensional lattice group 414
16.2. Bravais classification of magnetic lattice groups and magnetic point
lattices 419
16.2.1. Magnetic point lattices, their Bravais groups and their symmetry
groups 419
Contents xvii
16.2.2. Bravais classes of magnetic point lattices, of magnetic lattice
groups and of spin lattices 423
16.2.3. Determination of the Bravais classes of magnetic point lattices;
in particular, the case of dimensions n = 2 and n = 3 426
16.3. Tables 431
Chapter 17. Magnetic space groups 435
17.1. Some general remarks concerning the derivation of magnetic
space groups 435
17.1.1. Outline of the method; terminology and notation 435
17.1.2. Relation of magnetic space groups to net groups and to
two colour space groups 436
17.1.3. Two kinds of magnetic space groups 437
17.1.4. Point group and lattice of a magnetic space group 438
17.2. T subgroups of space groups, and magnetic space groups of the
kind MT 440
17.3. /^ subgroups of space groups, and magnetic space groups of the
kind MR 443
17.3.1. Some general properties of /^ subgroups of space groups 443
17.3.2. ^ subgroups of symmorphic space groups 449
17.3.3. .R subgroups of non symmorphic space groups 453
17.4. Tables of M dimensional space groups and magnetic space groups for
« = 3, 2 and 1 463
17.5. Tables 465
Chapter 18. Classification of magnetic crystals by means of magnetic groups 475
18.1. Introduction: magnetic crystals and magnetic symmetry 475
18.2. Spin arrangements invariant under a space group or a magnetic
space group 476
18.2.1. Simple spin arrangements 476
18.2.2. Ferromagnetic, antiferromagnetic and ferrimagnetic simple spin
arrangements; ferromagnetic space groups 482
18.2.3. Composite spin arrangements 487
18.3. Spin arrangements invariant under a group belonging to the magnetic
family of a decomposable group of isometries; helical spin
arrangements 488
18.4. Symmetry group and classification label of a magnetically
ordered crystal 489
18.4.1. Introductory remarks 489
18.4.2. Example: magnetic structure of DyCrOj 490
18.5. Digression on ferroelectric space groups 494
18.6. Table 495
xviii Contents
Chapter 19. Macroscopic symmetry of crystals; especially of
magnetic crystals 496
19.1. Crystalline medium and its symmetry 496
19.1.1. Transition from the atomic description of a crystal to the
macroscopic description 496
19.1.2. Neumann s Principle and Curie s Principle 497
19.2. Example: magnetoelectric crystalline medium and magnetoelectric
point groups 499
19.3. Table 504
Chapter 20. Spin groups 505
20.1. Definition of spin groups (that is, vector groups) 505
20.2. Gross classification of spin groups 506
20.2.1. Spin only, space only and non trivial spin groups 506
20.2.2. Structure of those spin groups which are symmetry groups of
vector valued functions 509
20.3. Spin groups as invariance groups of spin arrangements 510
20.3.1. Non trivial spin groups that correspond to magnetic groups 510
20.3.2. Classification of spin arrangements by means of spin groups 511
20.4. Equivalence classes of non trivial spin groups 515
20.5. Relation between non trivial spin groups and colour groups 516
20.6. Crystallographic spin groups 516
20.6.1. Spin groups of rotations (spin point groups) 516
20.6.2. Infinite spin groups 518
20.7. Table 519
Chapter 21. Digression to quantum mechanics 520
21.1. Introduction, and outline of Chapter 21 520
21.2. Unitary vector space and quantum mechanical description of an atomic
system 521
21.2.1. Unitary vector space 521
21.2.2. The unitary vector space associated with an atomic system 523
21.3. Semilinear transformations of a complex vector space, and
corepresentations of groups 523
21.3.1. The case of an arbitrary n dimensional complex vector space 523
21.3.2. The case of a unitary vector space: unitary and antiunitary
transformations 528
21.3.3. Example: a corepresentation of the time group 530
21.4. Invariance groups of an atomic system 530
21.4.1. Definition of an invariance group 530
21.4.2. Time inversion in quantum mechanics 532
21.4.3. Corepresentations of magnetic groups 533
Contents xix
Part V. Beyond traditional crystallography 535
Chapter 22. Modulated crystals and superspace groups 537
22.1. Introduction 537
22.2. (1 + l) dimensional superspace groups introduced 538
22.2.1. One dimensional modulated point lattices and their symmetry
groups 538
22.2.2. An example of a one dimensional modulated point lattice 540
22.2.3. One dimensional modulated point lattices embedded in a
two dimensional Euclidean point space, and (1 + l) dimensional
superspace groups 541
22.3. (3 + l) dimensional superspace groups introduced 544
22.3.1. Three dimensional modulated point lattices 544
22.3.2. Three dimensional modulated point lattices embedded in a
four dimensional Euclidean point space, and (3 + l) dimensional
superspace groups 545
22.4. An outline of the theory of (3 I l) dimensional superspace groups 548
22.4.1. Restricted Bravais group of a (3 + l) dimensional point lattice 548
22.4.2. Determining the restricted Bravais groups and the Bravais
superspace groups 552
22.4.3. Classifications of the (3 + l) dimensional superspace groups 555
22.5. Tables 558
Notes and second thoughts 559
Bibliography 575
Addendum to Bibliography 598
Notes added in proof 600
Some specific notations concerning groups 603
Author Index 604
Subject Index 608
N.B. The sections and subsections of the chapters are referred to in the text by
indicating the numbers of the chapter, section and subsection, in that order. For
example, 2.8.3 means Chapter 2, Section 8, Subsection 3; and 7.4 means Chapter 7,
Section 4.
|
| any_adam_object | 1 |
| author | Opechowski, W. |
| author_facet | Opechowski, W. |
| author_role | aut |
| author_sort | Opechowski, W. |
| author_variant | w o wo |
| building | Verbundindex |
| bvnumber | BV000582556 |
| classification_rvk | UQ 1350 |
| ctrlnum | (OCoLC)246873362 (DE-599)BVBBV000582556 |
| discipline | Physik |
| format | Book |
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| id | DE-604.BV000582556 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:16:04Z |
| institution | BVB |
| isbn | 0444869557 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000360277 |
| oclc_num | 246873362 |
| open_access_boolean | |
| owner | DE-12 DE-19 DE-BY-UBM DE-384 DE-703 DE-355 DE-BY-UBR DE-706 DE-83 DE-188 |
| owner_facet | DE-12 DE-19 DE-BY-UBM DE-384 DE-703 DE-355 DE-BY-UBR DE-706 DE-83 DE-188 |
| physical | XIX, 628 S. |
| publishDate | 1986 |
| publishDateSearch | 1986 |
| publishDateSort | 1986 |
| publisher | North-Holland |
| record_format | marc |
| spelling | Opechowski, W. Verfasser aut Crystallographic and metacrystallographic groups Amsterdam u.a. North-Holland 1986 XIX, 628 S. txt rdacontent n rdamedia nc rdacarrier Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Kristallographie (DE-588)4033217-2 gnd rswk-swf Raumgruppe (DE-588)4177070-5 gnd rswk-swf Kristallographie (DE-588)4033217-2 s Gruppentheorie (DE-588)4072157-7 s DE-604 Raumgruppe (DE-588)4177070-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000360277&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Opechowski, W. Crystallographic and metacrystallographic groups Gruppentheorie (DE-588)4072157-7 gnd Kristallographie (DE-588)4033217-2 gnd Raumgruppe (DE-588)4177070-5 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4033217-2 (DE-588)4177070-5 |
| title | Crystallographic and metacrystallographic groups |
| title_auth | Crystallographic and metacrystallographic groups |
| title_exact_search | Crystallographic and metacrystallographic groups |
| title_full | Crystallographic and metacrystallographic groups |
| title_fullStr | Crystallographic and metacrystallographic groups |
| title_full_unstemmed | Crystallographic and metacrystallographic groups |
| title_short | Crystallographic and metacrystallographic groups |
| title_sort | crystallographic and metacrystallographic groups |
| topic | Gruppentheorie (DE-588)4072157-7 gnd Kristallographie (DE-588)4033217-2 gnd Raumgruppe (DE-588)4177070-5 gnd |
| topic_facet | Gruppentheorie Kristallographie Raumgruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000360277&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT opechowskiw crystallographicandmetacrystallographicgroups |