Aperiodic Crystals: From Modulated Phases to Quasicrystals (International Union of Crystallography Monographs on Crystallography)

Contents......Page 10
Preface......Page 6
Glossary......Page 8
1.1.1 History......Page 16
1.1.2 Description......Page 17
1.1.3 The role of space group symmetry in the structure determination......Page 21
1.1.4 Symmetry and physical properties......Page 22
1.1.5 Examples......Page 23
1.1.6 Conclusion......Page 24
1.2.1 History......Page 25
1.2.2 Classes and examples......Page 29
1.2.3 Modulated phases......Page 30
1.2.4 Incommensurate composites......Page 36
1.2.5 Quasicrystals......Page 38
1.2.6 Morphology......Page 43
1.3 Summary......Page 44
2.1.1 Periodicity and aperiodicity......Page 46
2.2.1 Modulated phases......Page 49
2.2.2 Composites......Page 55
2.2.3 Quasicrystals......Page 57
2.2.4 # Electromagnetic crystals in space-time......Page 59
2.3.1 Embedding......Page 61
2.3.2 Modulated phases......Page 72
2.3.3 Incommensurate composites......Page 76
2.3.4 Quasicrystals......Page 82
2.3.5 Is the classification into three types unique?......Page 90
2.4.1 Point group symmetry of diffraction patterns......Page 91
2.4.2 Superspace groups......Page 94
2.4.3 Examples......Page 96
2.4.4 Approximants......Page 98
2.4.5 Superspace groups for commensurate phases......Page 99
2.4.6 Consequences of superspace group symmetry......Page 101
2.5 Scaling symmetries......Page 103
2.6 Alternative descriptions......Page 106
2.7 Summary......Page 109
3.1 Model sets......Page 111
3.2 Introduction to tilings......Page 113
3.3.1 Substitutions with an alphabet......Page 116
3.3.2 Embedding of substitutional chains......Page 117
3.3.3 Tilings by substitution......Page 119
3.4.1 Construction of aperiodic tilings......Page 121
3.4.2 Embedding of tilings......Page 125
3.4.3 # Symmetry of tilings......Page 133
3.5 Approximants......Page 138
3.6 Coverings......Page 141
3.7 Random tilings......Page 142
3.8 Summary......Page 144
4.1.1 Diffraction from periodic and aperiodic crystals......Page 145
4.1.2 Indexing the diffraction pattern......Page 156
4.1.3 Mathematical questions......Page 158
4.2 Diffraction techniques......Page 161
4.2.1 X-ray area detectors......Page 162
4.2.3 Measurement techniques......Page 163
4.3.2 The structure factor of incommensurate structures......Page 166
4.3.3 Possible expressions of the modulation functions......Page 167
4.3.4 Additional expressions of modulation functions......Page 169
4.3.5 Practical aspects of structure determination and refinement......Page 171
4.3.6 Ab initio methods......Page 177
4.3.7 Relation between harmonics and satellite orders......Page 180
4.3.8 Composite structures......Page 181
4.3.9 Commensurately modulated structures......Page 183
4.4.2 The modulated phases of Na[sub(2)]CO[sub(3)]......Page 185
4.4.3 The composite structure of La[sub(2)]Co[sub(1.7)]......Page 191
4.4.4 Aperiodicity in the structures of elements......Page 194
4.4.5 p-chlorobenzamide......Page 198
4.4.6 Modular structures......Page 201
4.4.7 Conclusion......Page 204
4.5.1 A simple one-dimensional quasiperiodic model......Page 205
4.5.2 Structure determination of a one-dimensional quasicrystal......Page 209
4.5.3 Structure determination of icosahedral and decagonal phases......Page 218
4.6.1 Introduction......Page 219
4.6.2 Structure of the i-AlPdMn phase......Page 220
4.6.3 Atomic structure of the CdYb icosahedral phase......Page 256
4.6.4 Structure of the AINiCo decagonal phase......Page 266
4.7 Diffraction by an imperfect crystal......Page 275
4.7.1 Diffuse scattering when an average lattice exists......Page 276
4.7.2 Diffuse scattering when there is no average lattice......Page 280
5.1 Introduction......Page 283
5.2 The Landau theory of phase transitions......Page 284
5.3.1 Substrate models......Page 290
5.3.2 Spin models......Page 293
5.3.3 Models with continuous degrees of freedom......Page 296
5.5.1 Charge-density and spin-density systems......Page 302
5.5.2 Hume-Rothery compounds......Page 303
5.6.2 Phase transitions in a hexagonal model......Page 305
5.6.3 Simulation of organic incommensurate crystal structures......Page 314
5.7 Summary......Page 318
6.1 Introduction......Page 320
6.2 Tensorial properties......Page 321
6.3.1 Hydrodynamic theory of fluids and periodic crystals......Page 327
6.3.2 Hydrodynamic theory of aperiodic crystals and phason modes......Page 328
6.4.1 Introduction......Page 330
6.4.2 Simple models......Page 334
6.4.3 Eigenvectors and spectrum......Page 360
6.4.4 Damping......Page 361
6.5.1 Scattering......Page 362
6.5.2 Modulated phases and composites......Page 364
6.5.3 Quasicrystals......Page 365
6.6.2 Phason modes in modulated crystals......Page 371
6.6.3 Diffuse scattering and phason modes in icosahedral quasicrystals......Page 375
6.6.4 Phason modes in icosahedral quasicrystals......Page 379
6.6.5 Phason modes in other quasicrystals......Page 391
6.7 Non-linear excitations......Page 395
6.8.2 Simple models......Page 399
6.8.3 Electrical conductivity......Page 404
6.9 Summary......Page 405
7.1.1 The puzzling habit of the mineral calaverite......Page 407
7.1.2 The morphology of the TMA Zn phases......Page 411
7.1.3 The morphology of icosahedral and decagonal quasicrystals......Page 412
7.2.1 Introduction......Page 415
7.2.2 Structure of surfaces of aperiodic crystals......Page 418
7.2.3 Generalization of the morphological laws......Page 421
7.2.4 Physical properties of quasicrystalline surfaces......Page 423
7.3 Magnetic quasiperiodic systems......Page 425
A.1 Crystallographic operations in n dimensions......Page 429
A.2 Lattices......Page 431
A.3 Crystal classes......Page 432
A.4 Space groups......Page 434
A.5 Classification......Page 436
A.6 Space groups for aperiodic crystals......Page 440
A.7.1 Superspace groups for incommensurate phases......Page 442
A.8 Extinction rules......Page 444
A.9.1 Introduction......Page 445
A.9.2 Tables for irreducible representation of point groups: point groups of 5-, 8-, 10-, 12-fold and icosahedral symmetry......Page 446
A.9.3 Examples of superspace groups for modulated phases......Page 450
A.9.4 Superspace groups for quasiperiodic structures with five-, eight-, ten-, twelve-fold, or icosahedral symmetry......Page 451
B.1 Magnetic systems and time-reversal symmetry......Page 453
B.3 Magnetic space groups......Page 454
B.4 The magnetic groups for quasiperiodic crystals......Page 455
References......Page 458
C......Page 478
I......Page 479
R......Page 480
W......Page 481