Point groups, space groups, crystals, molecules:
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World Scientific
1999
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| Beschreibung: | XXXV, 707 S. graph. Darst. |
| ISBN: | 9810237324 |
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| 100 | 1 | |a Mirman, Ronald |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Point groups, space groups, crystals, molecules |c R. Mirman |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1999 | |
| 300 | |a XXXV, 707 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Geometria elementar |2 larpcal | |
| 650 | 7 | |a Geometria |2 larpcal | |
| 650 | 7 | |a Grupos cristalográficos |2 larpcal | |
| 650 | 4 | |a Crystallography, Mathematical | |
| 650 | 4 | |a Group theory | |
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Datensatz im Suchindex
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| adam_text | Table of Contents
Preface v
I Transformations With a Point Fixed: Point Groups 1
1.1 SYMMETRIES OF SPACE AND OF OBJECTS 1
1.2 WHY STRUCTURES OF CRYSTALS AND MOLECULES ARE
LIMITED 3
I.2.a What are the essential properties of these objects? 3
I.2.b There are only a few categories of crystals 4
1.3 LATTICES 5
1.3.a Lattices and crystals 6
I.3.b Unit cells 7
1.4 THE ROTATION GROUP AND ITS FINITE SUBGROUPS . . 8
I.4.a The finite subgroups of the two dimensional rota¬
tion group 8
1.4.b The finite subgroups for three dimensions .... 8
I.4.c Regular polyhedra 10
I.4.c.i Determination of the regular polyhedra . 10
I.4.c.ii Other ways of determining these polyhedra 13
I.4.c.iii Dual polyhedra 15
I.4.c.iv Extensions and generalizations 16
1.4.d How the rotation group determines its finite sub¬
groups 17
I.4.d.i Orbits and poles and why there are differ¬
ent types 17
I.4.d.ii The number of orbits and the finite sub¬
groups 19
I.4.d.iii Implications and extensions 22
1.5 HOW TRANSLATIONS LIMIT ROTATION GROUPS OF CRYS¬
TALS 23
1.5.a Translations limit the trace of the rotation matrix 24
I.5.b Discreteness is essential 25
xiv
I.5.c Complex numbers, rotations and translations ... 26
I.5.d Why are rotational symmetries of lattices limited? 27
1.6 POINT GROUPS 27
I.6.a The symmetry operations on crystals and molecules 28
I.6.a.i Naming the point symmetry transforma¬
tions 28
I.6.a.ii The point symmetry transformations ... 29
I.6.a.iii Why only some transformations are con¬
sidered 33
I.6.b Naming and describing the point groups 33
I.6.b.i Point groups with only rotations 35
I.6.b.ii Restrictions on orders of point groups from
their characters 36
I.6.C Point groups with reflections 39
I.6.c.i Cyclic groups with reflections 39
I.6.c.ii Dihedral groups with reflections 40
I.6.c.iii Adding reflections to the groups of regu¬
lar polyhedra 41
I.6.d There are thus 32 point groups 42
I.6.e Why are these all the point groups? 44
I.6.f The molecular point groups 46
I.6.g Objects invariant under the point groups 47
1.7 STRUCTURE OF POINT GROUPS 50
I.7.a Point groups as semi direct products 50
I.7.a.i Dihedral groups 52
I.7.a.ii Cubic groups 52
I.7.a.iii The structure of improper point groups.. 53
I.7.b Classes of the point groups 55
1.8 DOUBLE GROUPS 57
I.8.a Definition of double groups 58
I.8.b Construction of double groups 58
I.8.c Double group classes 60
I.8.d The single group is a factor group 62
1.9 SIMPLICITY AND SYMMETRY IN A COMPLEX ENVIRON¬
MENT 64
n Crystal Structures and Bravais Lattices 66
H.1 CRYSTALS AND LATTICES 66
n.2 LATTICES AND CRYSTAL SYSTEMS 67
H.2.a Definition of a lattice 68
n.2.b Unit cells are parallelepipeds 68
n.2.c What is a Bravais lattice? 69
H.2.d The holohedry groups and their subgroups .... 69
XV
11.3 LATTICES IN TWO DIMENSIONS 71
n.3.a The two dimensional lattices 71
H.3.b Why we consider all these lattices, and all distinct 74
n.3.c Why are points added to unit cells? 75
H.3.d Where can points be put simultaneously in two
dimensions? 76
H.3.e The symmetry groups of two dimensional lattices 78
n.3.f Finding the two dimensional Bravais lattices ana¬
lytically 79
n.3.g What makes two lattices distinct? 80
11.4 THE SEVEN THREE DIMENSIONAL CRYSTAL SYSTEMS . . 84
H.4.a Labeling axes and faces 85
H.4.b Classification of lattices by their sides and angles 86
II.4.C Description of the seven systems 86
H.4.d Why the systems have the names they do 91
H.4.e The restrictions on projections of lattices from
the point groups 92
n.4.f What determines these systems? 92
H.4.g Where can axes be placed? 96
n.4.h What symmetry elements must a lattice contain? . 97
n.5 PRIMITIVE AND NON PRIMITIVE BRAVAIS LATTICES ... 98
H.5.a The conditions on added points in three dimensions 98
n.5.a.i Conditions given by two dimensional lattices 98
n.5.a.ii The points allowed by the two dimensional
sublattices 99
n.5.a.iii Adding points simultaneously 99
n.5.b Thus there are limits on lattices 102
n.5.c Finding the Bravais lattices analytically 102
n.6 DESCRIPTIONS OF THE FOURTEEN LATTICES 103
H.6.a Triclinic, monoclinic, orthorhombic and tetrago¬
nal systems 103
n.6.b The cubic system 105
n.6.c The trigonal and hexagonal systems 107
II.6.d The rhombohedral unit cell 108
II.7 THE SEVEN CRYSTAL SYSTEMS AND THEIR SYMMETRY
GROUPS Ill
H.7.a Systems, groups and lattices 112
H.7.b Derivation of the Bravais lattices from their sym¬
metry 113
n.7.b.i The monoclinic lattices 113
II.7.b.ii Tetragonal lattices 114
n.7.b.iii The orthorhombic lattices 116
H.7.b.iv The trigonal and hexagonal lattices .... 118
xvi
II.7.b.v The cubic system 119
n.7.c Seven lattice symmetry groups and fourteen
lattices 120
n.7.d Decreasing the symmetry 120
II.7.e Dilatations relating lattices 121
n.7.f Crystals and lattices 121
II.8 THE SYMMETRIES OF CUBIC AND HEXAGONAL
LATTICES 126
II.8.a The rotational symmetries of the cube 126
II.8.b The symmetry group as a subgroup of symmetric
groups 127
n.8.c The representation matrices of the symmetry
group 128
n.8.d Adjunction of the inversion 129
II.8.e Generalizations of the cube 129
II.8.f The symmetries of the hexagonal lattice 130
in Space Groups 132
111.1 GROUPS WITH DISCRETE TRANSLATIONS 132
111.2 SPACE GROUPS: DEFINITIONS AND NOTATIONS 133
III.2.a Denoting space group operations 134
III.2.b The affine group 134
ni.2.c Types of space groups 135
III.2.d Translations are invariant 135
III.2.e Semi direct products 136
111.3 SYMMORPHIC SPACE GROUPS 137
III.3.a A symmorphic group in two dimensions 138
III.3.b Glide planes 140
111.4 NONSYMMORPHIC SPACE GROUPS 141
III.4.a Screw axes 142
III.4.a.i The space group with a screw and the point
group 144
III.4.a.ii Limitations on crystallographic screws . . 145
III.4.a.iii Pairs of enantiomorphic screws 146
III.4.b The nonprimitive glide reflection 146
III.4.C Why are these the only non primitive operations? 148
in.4.c.i These are the — only — nonprimitive op¬
erations 149
III.4.c.ii Nonprimitive operations are determined
by primitive ones 151
III.4.d The factor group of a space group gives a point
group 152
III.4.d.i Cosets and coset representatives 153
xvii
in.4.d.ii The difference in factor groups of sym
morphic and nonsymmorphic groups ... 154
m.4.e How nonprimitive elements are restricted 154
m.4.f Example of a two dimensional nonsymmorphic
group 156
m.4.f.i How the crystal differs from the lattice . . 157
m.4.f.ii The space group operations 157
in.4.f.iii Shifting the origin cannot make the group
symmorphic 158
111.5 THE DIAMOND STRUCTURE 159
m.5.a The point group of the diamond space group . . . 162
HI.5.b Where are the nonprimitive elements placed? ... 164
m.5.b.i Placing glides 164
in.5.b.ii Placing screws 164
m.5.c Why is the space group of diamond so rich? .... 165
m.5.d Spinel 166
111.6 ENHOMOGENEOUS ROTATION GROUPS AND NONSYM¬
MORPHIC GROUPS 167
m.6.a Are nonsymmorphic space groups semi direct prod¬
ucts? 168
m.6.b Why are there nonsymmorphic groups? 168
III. 7 GEOMETRIC AND ARITHMETIC EQUIVALENCE OF CRYS¬
TAL CLASSES 169
111.8 THE SPACE GROUPS IN ONE AND TWO DIMENSIONS ... 172
HI.8.a The symmetry groups of linear objects (frieze
groups) 173
m.8.b Space groups with two translations 175
m.8.b.i Description of the two dimensional space
groups 176
in.8.b.ii Notation and illustrations for the wallpa¬
per groups 178
111.9 THE THREE DIMENSIONAL SPACE GROUPS 180
III.9.a The symmorphic space groups 180
III.9.a.i Ambiguities of setting give more space
groups 181
III.9.a.ii Why are these the symmorphic space
groups? 181
III.9.b Enantiomorphic space groups 182
m.9.c Deriving the space groups by enumeration .... 182
IV Representations of Translation Groups 185
IV. 1 REPRESENTATIONS AND THE ROLE THEY PLAY 185
IV.2 REPRESENTATIONS OF TRANSLATIONS 186
xviii
IV.2.a The reciprocal space 186
IV.2.b Lattice points and vectors 188
IV.2.C Brillouin zones 189
IV.2.c.i Brillouin zones are of the lattice 190
IV.2.c.ii Symmetry of points in the Brillouin zone . 190
IV.2.c.iii Boundaries of Brillouin zones 191
IV.2.c.iv Examples of two dimensional Brillouin
zones 192
IV.2.d The representation basis states of the translations 193
IV.2.e Bloch s theorem 193
IV.2.f The translation irreducible representations .... 194
IV.3 THE WIGNER SEITZ CELLS 196
IV.3.a The centered rectangular lattice — what determines
its cell? 197
IV.3.b The space filling parallelohedra 199
IV.3.C Dual figures 201
IV.3.d Inscription in, and circumscription about, spheres 201
IV.3.e Illustrations of parallelohedra 202
IV.3.f Parallelohedra give lattices 203
IV.3.g Determining parallelohedra from their properties 204
IV.3.g.i The two dimensional projections of the
cells 204
IV.3.g.ii The cells are centrosymmetric 205
IV.3.g.iii The faces form bands 205
IV.3.g.iv Sharing by adjacent cells 207
IV.3.g.v Counting nearest neighbors 207
IV.3.h The number of sides of parallelohedra 210
IV.4 LATTICES CAN HAVE SEVERAL WIGNER SEITZ CELLS . . 212
IV.4.a Emphasizing the importance of understanding what
is geometry, what is convention 212
IV.4.b The centered tetragonal lattice 213
IV.4.C Implications of multiple Wigner Seitz cells 214
IV.5 WIGNER SEITZ CELLS FOR THE LATTICES 215
IV.5.a The cubic system 216
IV.5.a.i The simple cube 216
IV.5.a.ii The face centered cubic lattice 216
IV. 5.a.iii The body centered cubic lattice 218
IV.5.b The tetragonal system 221
IV.5.b.i The primitive tetragonal lattice 221
IV.5.b.ii The tetragonal Wigner Seitz cell is not a
stretched cubic Wigner Seitz cell 222
IV.5.b.iii The body centered tetragonal lattice ... 223
IV.5.b.iv The all face centered lattice 224
xix
IV.5.b.v Why the tetragonal has fewer lattices than
the cubic 224
IV.5.b.vi The differing Wigner Seitz cells of the
lattice 226
IV.5.b.vii How cells change 227
rv.5.b.vili Surfaces appear when edges disappear. 229
IV.5.b.ix Cells as limits of others 230
IV.5.C The orthorhombic system 230
IV.5.c.i The simple orthorhombic lattice 231
IV.5.c.ii The two face centered orthorhombic
lattice 231
IV.5.c.iii The all face centered orthorhombic lattice 232
IV.5.c.iv The body centered orthorhombic lattice . 232
IV.5.d The other systems 234
IV.5.e Why are there five categories of Wigner Seitz
cells? 235
IV.5.f Obtaining three dimensional cells from their cross
sections 237
IV. 5 .g Lattices as spaces and representations of their trans¬
lation groups 238
IV.6 BRILLOUIN ZONES FOR THE LATTICES 239
IV.6.a The cube and its relatives 240
IV.6.a.i The Brillouin zone of the simple cube . . . 240
IV.6.a.ii The Brillouin zone of the body centered
cubic lattice 241
IV.6.a.iii The Brillouin zone of the face centered cu¬
bic lattice 242
IV.6.b The Brillouin zones of the tetragonal lattices ... 243
IV.6.C The other systems 245
V Representations: Point Groups and Projective 247
V.I FORMULATING REPRESENTATIONS 247
V.2 REPRESENTATIONS OF THE CRYSTALLOGRAPHIC POINT
GROUPS 247
V.2.a The characters of point groups 248
V.2.b Representations and characters of cylic groups . . 248
V.2.c Dihedral group representations and characters . . 249
V.2.d Computation of characters 251
V.2.e The table of characters 254
V. 2 .f Representations of the tetrahedral and octahedral
groups 255
V.2.g Adjunction of reflections and the inversion .... 256
V.3 REPRESENTATIONS OF DOUBLE GROUPS 256
XX
V.3.a The representations of the double cyclic groups . 257
V.3.b Representations of D2* 258
V.3.c Representations of other double groups 259
V.4 PROJECTIVE REPRESENTATIONS 259
V.4.a Definition of projective representation 260
V.4.b Factor systems 261
V.4.b.i The multiplicator 262
V.4.b.ii The Abelian group formed by the factor
classes 262
V.4.b.iii All factors give projective representations 263
V.4.b.iv The factors are roots of unity 265
V.4.b.v Properties of factor systems and projec¬
tive representations 266
V.4.c Some groups with projective representations . . . 267
V.4.c.i Projective representation of cyclic groups
and products 267
V.4.c.ii Dihedral groups 269
V.5 CENTRAL EXTENSIONS 270
V.5.a Nonuniqueness of central extensions 271
V.5.a.i Symmetric group S3 as a central extension 271
V.5.a.ii Why central extensions are not unique . . 272
V.5.a.iii Central extension of the four group .... 272
V.5.b Central extensions and factor systems 274
V.5.b.i The multiplicator and central extensions . 276
V.5.b.ii The meaning of projective representation 276
V.5.b.iii The covering group 277
V.5.b.iv Why only one factor system is considered. 277
V.5.c Forming a central extension using a specific factor
system 277
VI Induced Representations 280
VI. 1 INDUCING AND SUBDUCING TO FIND
REPRESENTATIONS 280
VI.2 SUBDUCED REPRESENTATIONS 281
VI.2.a Conjugate representations 281
VI.2.b Orbit of a representation 283
VI.2.C Little groups 284
VI.2.d The multiplicity of an orbit is representation in¬
dependent 286
VI.2.e There is but one orbit in the decomposition .... 287
VI.2.f The set of subduced representations of a normal
subgroup — Clifford s theorem 288
VI.3 INDUCED REPRESENTATIONS 288
xxi
VI.3.a Representation finding using induction 289
VI.3.b Definition of induced representation 291
VI.3.C The matrices give a representation of the group . 292
VI.4 EXAMPLES OF INDUCED REPRESENTATIONS 293
VI.4.a Representations of S3 as examples of induction . 294
VI.4.b Induced representations of D4 298
VI.4.C The representations of tetrahedral group T (A4) .301
VI.4.d Induced representations of octahedral group O . . 303
VI.4.e Inducing from a nonnormal subgroup 305
VI.4.f Representations of S4 from S3 306
VI. 5 PROPERTIES OF INDUCED REPRESENTATIONS 308
VI.5.a Basis functions of induced representations .... 308
VI.5.b Conjugate representations, little groups, and
orbits 312
VI.5.C Conjugate subgroups 313
VI.5.d The characters of the induced representation . . . 314
VI. 5.e The Frobenius reciprocity theorem 317
VI.6 IRREDUCIBIUTY AND COMPLETENESS FOR ARBITRARY
SUBGROUPS 319
VI.7 INDUCING FROM A NORMAL SUBGROUP 321
VI.7.a Proof of irreducibility of the induced
representation 322
VI. 7.b Irreducibility of representations induced from nor¬
mal subgroup 323
VI.7.C Allowable representations of the little group . . . 324
VI.7.d All representations are given by the allowable rep¬
resentations 325
VI.7.e Obtaining only nonequivalent representations . . 327
VI. 7.f Finding the induced representation using the little
group 327
VI.7.g The matrices of the induced representations ... 328
VI.7.h Induced representations of direct product groups 329
VI.7.i Semi direct product groups 330
VI.7.i.i Cases for which induced representations
can be found as Kronecker products. ... 331
VI.7.i.ii The representations that are of the form
of Kronecker products 332
VI.7.i.iii Subgroups with indices 2 335
VII Representations of Space Groups 337
VII. 1 FINDING THE REPRESENTATIONS 337
VII.2 CONCEPTS FOR THE REPRESENTATIONS 338
xxii
vn.2.a The type of space group representations studied
here 339
VH.2.b Induced representations in the terminology of
space groups 340
VH.2.C Orbits 341
VII.2.d The star of a vector 342
VH.2.e Classification of positions of Brillouin zones . . 342
VH.2.f The cosets formed from the translations 343
VII.2.g Little groups and reciprocal lattice vectors .... 343
VH.2.h Little co groups and little groups 344
Vn.3 REPRESENTATIONS OF LITTLE GROUPS AND LITTLE CO
GROUPS 347
VH.3.a Small representations, and why they help .... 347
VII.3.a.i Translation representations are scalars in
little groups 348
VH.3.a.iiRepresentations can be obtained from the
little co group 349
VII.3.a.iii The induced representation matrices . . 350
VII.3.a.iv The point group part of the matrices . . 351
VH.3.b The central extensions of the little co group ... 351
VII.3.C Symmorphic space groups 352
VII.4 INDUCING REPRESENTATIONS OF NONSYMMORPHIC
SPACE GROUPS 353
VH.4.aRepresentations, allowable and not 353
VH.4.bInducing representations of symmorphic space
groups 354
Vn.4.cRepresentations of nonsymmorphic groups .... 355
VII.4.c.i These matrices form a representation . . . 357
VII.4.c.ii The representations are irreducible .... 357
VII.4.c.iii All inequivalent representations are
obtained 359
Vn.4.dThe allowable representations of the little group . 359
VII.4.d.i Obtaining the allowable representations
from representations of the little co group 359
Vn.4.d.iiThe cases of internal vectors 361
VII.4.d.iii Special positions on the boundary .... 361
VII.5 WHAT THE PROCEDURE IS, AND WHAT IT MEANS ... 364
VII.5.a The procedure in essence 364
VII.5.b What operators are diagonal? 365
VII.5.C The meaning of space group representations . . 367
VH.5.d What is the dimension of the space group repre¬
sentation? 369
xxiii
VH.5.e Representations can contain more than one mo¬
mentum magnitude value 370
VII.6 THE SQUARE AS AN EXAMPLE 372
VH.6.a The reciprocal lattice vectors of the square ... 372
VH.6.b General vectors and vectors giving symmetry . . 372
VII.6.C The stars of the points of the square . 373
VH.6.d The square and the rectangle 376
VH.6.e Little groups of general, and of special, vectors . 377
VII.6.f Representations of the space group of the
square 377
VII.6.f.i The little groups and the representations 377
VII.6.f.ii What determines the little groups? 379
VH.6.g The square with nonsymmorphic glides 379
VH.6.h A nonsymmorphic group of the rectangle .... 380
VH.6.i What determines the space group and its repre¬
sentations? 383
VII.6.i.i The representation basis functions 383
VII.6.i.ii How the glide affects basis functions ... 384
VII.7THE CUBIC AND DIAMOND STRUCTURES 384
VII.7.aThe representations of the factors for Fm3m ... 389
VH.7.bThe representations of the factors for Fd3m . . . 389
VH.7.b.i Representations at Point T 390
VH.7.b.nPoints with nonsymmorphic little groups . 391
Vffl Spin and Time Reversal 393
Vm.l MORE COMPLICATED CRYSTALS 393
Vm.2 TIME REVERSAL 394
VET. 2.a Antilinear and antiunitary operators 395
Vm.2.b The general form of an antilinear operator .... 396
VHI.2.C The general form of the time reversal operator . 398
Vm.2.d Kramer s Theorem 402
VIII.3COMPLEX CONJUGATE REPRESENTATIONS 403
VllU.a When are conjugate representations equivalent? 403
VIII.3.b Operators mixing representations and their con¬
jugates 404
VIIL3.C Classification of groups under complex conjuga¬
tion 405
Vin.3.c.i Potentially real, and pseudo real, repre¬
sentations 405
Vin.3.c.ii Ambivalent groups and their character
sets 406
Vm.3.c.iii Why physically are groups so classified? 407
VW.3.civ Reality and the rotation group 409
xxiv
WI.3.d Equivalent sets can be distinguishable 410
VIII.3.d.i Mathematical equivalence, and physical
equivalence 411
Vin.3.d.ii Complexity of half integral angular mo¬
mentum statefunctions 411
VIII.4 COLOR GROUPS 411
VIII.4.a The conditions on color groups 413
VIII.4.b Point groups with time reversal 414
Vin.4.b.i Cyclic color groups 415
Vin.4.b.ii Dihedral color groups 416
Vin.4.b.iii Gray groups 418
VIII.4.C Magnetic groups 419
VHI.4.d An example of a magnetic group 421
VIII.4.e Construction of a magnetic group 421
VIII.4.e.i The square has other magnetic groups . 422
VIII.4.e.ii Why the square has these magnetic
groups 422
VIII.4.f The orthorhombic magnetic crystal 424
VIII.4.g What determines which groups are different? . . 426
VIII.5 MAGNETIC BRAVAIS LATTICES 426
VIII.5.a Properties of the magnetic Bravais lattices .... 428
VIII.5.b Two dimensional magnetic Bravais lattices .... 429
VHI.5.C Three dimensional magnetic Bravais lattices . . . 429
VIII.6 MAGNETIC SPACE GROUPS 431
VIII.6.a Types of colored space groups 432
VIII.6.b A space group obtained from the primitive cubic
lattice 433
VIII.6.c Implications and extensions 433
VIII.7 REPRESENTATIONS OF GROUPS WITH ANTBLINEAR OP¬
ERATORS 434
VIII.7.a Corepresentations 435
vm.7.b Multiplication rules for corepresentations .... 436
VHI.7.C Representation Spaces 437
VIII.7.d Constructing the corepresentations 437
VIII.7.e Transformations of the corepresentations .... 440
vni.7.f Reducibility of corepresentations 441
VIII. 7.g The types of corepresentations 441
vm.7.h The case of inequivalent subgroup representa¬
tions, A and A+ 442
VIII.7.i Equivalent subgroup representations 443
Vin.7.i.i Transforming to reduced form 444
Vm.7.i.ii The case with the sign positive 446
Vm.7.i.iii The case with the minus sign 447
XXV
VHI.7.J A simple corepresentation 448
VIII.7.k Which representations are which? 449
vni.7.1 The Herring test 451
VIII.7.m Are corepresentations representations? 453
VIII.8 SPIN AND COREPRESENTATIONS 454
VIII.8.a Integral spin 454
VIII.8.b Half integer spin 454
VIII.8.C Classifying the corepresentations 455
VIII.9 APPLICATION OF COREPRESENTATIONS TO MAGNETIC
SPACE GROUPS 456
VHI.9.a The translation operators 456
VHI.9.b Corepresentations of the magnetic space groups 457
Vin.9.b.i The procedure for unitary groups .... 457
Vin.9.b.ii How the procedure is changed for mag¬
netic space groups 458
VHI.9.C Projective corepresentations of magnetic groups 459
VHI.9.d Time reversal invariance and gray space groups . 462
Vin.9.d.i Unitary magnetic little groups 462
VIII.9.d.ii Magnetic little groups with antiunitary
operators 463
Vin.9.d.iii Degeneracies depend on the magnetic
corepresentation 463
VHI.9.e The physical meaning of the representations . . 463
Vm.lO REPRESENTATIONS OF DOUBLE SPACE GROUPS ... 464
IX Tensors, Groups and Crystals 467
K.1 MACROSCOPIC PHYSICAL PROPERTIES OF CRYSTALS . . 467
K.2 TENSORS FOR THE ROTATION GROUPS 468
IX.2.a The number of representations symbolized by a
tensor 469
DC.2.b Pseudo tensors 473
IX.2.b.i Tensors formed from products of vectors . 473
K.2.b.ii Restrictions on crystal properties 474
K.2.c Tensors relating vectors 474
K.2.d Required symmetry in indices 475
IX.3 TENSORS AND SYMMETRY 476
IX.3.a Foundations of the tensor analysis of crystal prop¬
erties 477
IX.3.a.i Neumann s principle 477
IX.3.a.ii Is Neumann s principle obvious? 478
IX.3.a.iii Why tensor components are point group
scalars 479
K.3.b Equilibrium and non equilibrium properties .... 480
xxvi
K.3.C Magnetic tensors and the effect of time reversal . 481
D£.3.d The number of independent components 481
DC.3.e The meaning of the number of independent com¬
ponents . . 484
K.3.f Requirements imposed by symmetry elements on
tensors 484
D£.3.g How group theory provides information about ten¬
sors 487
IX.4 RANK 1 TENSORS ELECTRIC AND MAGNETIC DIPOLE
MOMENTS 487
DCS SECOND RANK TENSORS 489
DC5.a Thermal conductivity 489
K.5.a.i The meaning of the symmetric and anti¬
symmetric parts 490
K.5.a.ii The physical quantities given by the sym¬
metric parts 491
DC5.b Thermal expansion 492
K.5.C Stress and strain 494
DC5.d Second rank tensors for group C3V 495
DC5.e Nonzero components for different crystal
systems 496
K.6 THIRD RANK TENSORS 497
DC6.a Piezoelectricity 497
K.6.b The Hall effect 498
DC.6.C Optical activity 499
K.6.d Groups that can, and cannot, have these tensor
properties 500
K.7 FOURTH RANK TENSORS 501
K.7.a Relating stress and strain 501
DC7.b Photoelasticity 503
DC8 THE EFFECT OF ffiREDUCffilUTY ON THE PHYSICS OF
TENSORS 505
K.9 THE USEFULNESS OF TENSORS IN ANALYZING
CRYSTALS 511
X Groups, Vibrations, Normal Modes 512
X. 1 WHAT GROUPS TELL US ABOUT MOLECULES AND CRYS¬
TALS 512
X.2 VEBRATIONAL STATES AND SYMMETRY 513
X.2.a Normal modes 514
X.2.b The simple harmonic oscillator 516
X.2.C Group theory of the one dimensional simple har¬
monic oscillator 517
xxvii
X.2.d The linear triatomic molecule 518
X.3 WHY, AND HOW, GROUP THEORY IS RELEVANT TO VI¬
BRATIONS 520
X.3.a Symmetry and normal coordinates 522
X.3.a.i Kinetic energy, potential energy and nor¬
mal coordinates 523
X. 3 .a.ii Proof of the existence of normal
coordinates 525
X.3.b Finding normal coordinates 526
X.3.b.i Valence bond lengths and interbond
angles 527
X.3.b.ii Coordinates that are not independent. . . 527
X.4 CHARACTERS AND COUNTING 528
X.5 EXAMPLES OF APPLICATION OF SYMMETRY TO VIBRA¬
TIONS 529
X.5.a Group theory, symmetry and the vibrations of
water 530
X.5.a.i The vibrational representations 531
X.5.a.ii The symmetry coordinates 533
X.5.a.iii Internal coordinates for water 536
X.5.b Ammonia 537
X.5.b.i The vibrational representations 538
X.5.b.ii Displacements in the normal modes .... 539
X.5.b.iii The two dimensional E modes 541
X.5.b.iv Internal coordinates for ammonia 542
X.5.b.v Why is group theory relevant here? .... 543
X.5.c How vibrational spectra depend on the molecule . 544
X.5.d Breaking of symmetry 549
X.6 MULTIPLE EXCITATIONS 550
X.7 TRANSITIONS AND SELECTION RULES 552
X.7.a Transitions due to electric dipole moments .... 554
X.7.b Polarizability and the Raman effect 556
X.7.b.i Raman scattering is of second order. . . . 556
X.7.b.ii The angular momentum selection rules. . 557
X.7.b.iii Exclusion rule 558
X.7.b.iv Raman scattering for crystals 558
X.8 HOW REASONABLE ARE THE APPROXIMATIONS? 559
X.9 HOW DIFFERENT GROUPS GIVE DIFFERENT VIBRATIONAL
SPECTRA 561
xxviii
XI Bands, Bonding, and Phase Transitions 564
M.1 WHY IS SYMMETRY RELEVANT? 564
XLl.a Different types of objects can be studied
separately 565
XLl.b Degeneracy, necessary and accidental 566
XI.1.C What can we know about objects in crystals? ... 566
XLl.d Symmetry varies; how is it useful? 567
XI.l.e Selection rules, perhaps not exact, but still pro¬
ductive 569
XI.2 ELECTRON STATES IN CRYSTALS 569
XI.2.a Labels for states are necessary 570
XI.2.b Bands, and why they are 570
XI.2.C What group theory tells about electron bands . . . 571
XI.2.c.i Point groups differ at different points in
reciprocal space 572
XI.2.c.ii How these bands illustrate the meaning of
space group representations 573
XI.2.d Symmetry reduction 574
XI.2.e The equation governing the objects 576
XI.2.f Translational symmetry and bands 577
XI.2.g Energy eigenvalues 578
XI.2.h Energies and statefunctions for cubic lattices ... 579
XI.2.h.i The simple cubic lattice 579
XI.2.h.ii The body centered cubic lattice 584
XI.2.h.iii The face centered cubic lattice 585
XI.2.i Energy bands for nonsymmorphic groups 586
XI.2.U Sticking together of bands 586
XI.2.i.ii Irreducible representations on the zone
faces 587
XI.2.i.iii The little group need not be a subgroup of
the point group 589
XI.2.J The close packed hexagonal structure 590
XI.2.j.i Nonsymmorphic elements of the crystal . 590
XI.2.j.ii Special positions of the Brillouin zone . . . 592
XI.2.j.iii Representations, and the effect of nonsym¬
morphic elements 594
XI.2.k What do we learn from this group theoretical anal¬
ysis? 596
XI.3 THE EFFECT OF TIME REVERSAL 596
XI.3.a The energy surfaces must have inversion
symmetry 597
XI.3.b Degeneracy due to time reversal 597
XI. 3.c Degeneracy at general points 598
xxix
XI.4 LATTICE VIBRATIONS 598
XI.4.a The dynamical matrix 600
XI.4.b Transitions in the simple cubic lattice 600
XI. 5 ATOMS IN CRYSTALS AND ENERGY LEVEL SPLITTING . . 601
XI.5.a Level splitting in crystals 601
XI.5.a.i Some different possibilities 602
XI.5.a.ii Steps in the decomposition 602
XI.5.b The octahedral group as an example 603
XI.5.b.i The intermediate case 603
XI.5.b.ii The case of a weak crystal field 605
XI.5.b.iii Splitting of the octahedral levels 605
XI.6 SPIN ORBIT COUPLING 606
XI.6.a Spin orbit coupling and removal of degeneracy . . 606
XI.6.b Cubic symmetry 607
XI. 7 MOLECULAR ORBITALS 607
XI.7.a Relating molecular orbitals and atomic orbitals . . 608
XI.7.b The types of orbitals we consider 609
XI.7.C Benzene 610
XI.7.d Bonding and antibonding states 611
XI.7.d.i Tetrahedral carbon 611
XI.7.d.ii Trigonal carbon 612
XI.8 CHANGE OF PHASE 613
XI.8.a First and second order phase transitions 613
XI.8.b Limitations on the analysis 615
XI.8.C Specifying the crystal thermodynamics 616
XI.8.d Equilibrium 616
XI.8.e How symmetry change is found 617
XI.8.f The order parameter 618
XI.8.f.i Magnetic ordering 618
Xt.8.f.ii Alloys 619
XI.8.g Expansion of the density 620
XI.8.h Physically irreducible
representations 622
XI.8.i Symmetry restrictions on the expansion
coefficients 622
XI.8.J Implications of the requirement that $ be a mini¬
mum 624
XI.8.k Conditions from necessity of terms being zero . . 625
XI.8.k.i The third order term must be zero 625
XI.8.k.ii The fourth order term 626
XI.8.k.iii The requirement of spatial homogeneity . 626
XI.8.1 Halving the symmetry always allows a transition . 627
XL8.mActive and passive representations 627
XXX
XI.9 CLASSICAL VIEWS, QUANTUM VIEWS, REALITY 627
A Symbols and definitions 630
B The Point Groups 631
C Objects Invariant Under the Point Groups 637
D Two Dimensional Space Groups 650
E Point Group Character Tables 655
E.1 DENOTING THE REPRESENTATIONS 655
E.2 THE CHARACTER TABLES 656
References 674
Index 683
|
| any_adam_object | 1 |
| author | Mirman, Ronald |
| author_facet | Mirman, Ronald |
| author_role | aut |
| author_sort | Mirman, Ronald |
| author_variant | r m rm |
| building | Verbundindex |
| bvnumber | BV012731464 |
| callnumber-first | Q - Science |
| callnumber-label | QD455 |
| callnumber-raw | QD455.3.G75 |
| callnumber-search | QD455.3.G75 |
| callnumber-sort | QD 3455.3 G75 |
| callnumber-subject | QD - Chemistry |
| classification_rvk | UQ 1350 |
| ctrlnum | (OCoLC)41503080 (DE-599)BVBBV012731464 |
| dewey-full | 548/.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 548 - Crystallography |
| dewey-raw | 548/.7 |
| dewey-search | 548/.7 |
| dewey-sort | 3548 17 |
| dewey-tens | 540 - Chemistry and allied sciences |
| discipline | Chemie / Pharmazie Physik |
| format | Book |
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| id | DE-604.BV012731464 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:32:42Z |
| institution | BVB |
| isbn | 9810237324 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008656464 |
| oclc_num | 41503080 |
| open_access_boolean | |
| owner | DE-703 DE-11 |
| owner_facet | DE-703 DE-11 |
| physical | XXXV, 707 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Mirman, Ronald Verfasser aut Point groups, space groups, crystals, molecules R. Mirman Singapore [u.a.] World Scientific 1999 XXXV, 707 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometria elementar larpcal Geometria larpcal Grupos cristalográficos larpcal Crystallography, Mathematical Group theory Kristallographie (DE-588)4033217-2 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Kristallographie (DE-588)4033217-2 s Gruppentheorie (DE-588)4072157-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008656464&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mirman, Ronald Point groups, space groups, crystals, molecules Geometria elementar larpcal Geometria larpcal Grupos cristalográficos larpcal Crystallography, Mathematical Group theory Kristallographie (DE-588)4033217-2 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| subject_GND | (DE-588)4033217-2 (DE-588)4072157-7 |
| title | Point groups, space groups, crystals, molecules |
| title_auth | Point groups, space groups, crystals, molecules |
| title_exact_search | Point groups, space groups, crystals, molecules |
| title_full | Point groups, space groups, crystals, molecules R. Mirman |
| title_fullStr | Point groups, space groups, crystals, molecules R. Mirman |
| title_full_unstemmed | Point groups, space groups, crystals, molecules R. Mirman |
| title_short | Point groups, space groups, crystals, molecules |
| title_sort | point groups space groups crystals molecules |
| topic | Geometria elementar larpcal Geometria larpcal Grupos cristalográficos larpcal Crystallography, Mathematical Group theory Kristallographie (DE-588)4033217-2 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| topic_facet | Geometria elementar Geometria Grupos cristalográficos Crystallography, Mathematical Group theory Kristallographie Gruppentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008656464&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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