Crystallography of Quasicrystals (eBook)

Born in Vienna, Austria on December 24, 1950, Walter Steurer studied chemistry at the University of Vienna. After finishing his doctoral dissertation under the supervision of Prof. Hans Nowotny at the Institute of Physical Chemistry, he moved, in 1980, to the University of Munich, Germany, where he worked as a research assistant and lecturer. In 1987 he concluded his habilitation thesis in the field of crystallography and mineralogy. In 1990 he receive the Victor-Moritz-Goldschmid Award of the Deutsche Mineralogische Gesellschaft (DMG) for his contributions to the higher-dimensional structure analysis. His research topics were uncommon crystals and their phase transitions, and included incommensurably modulated structures, quasicrystals and polytypes. For a short period, from 1992 until 1993, he was professor of crystallography at the University of Hanover, Germany. After rejecting calls to the Universities of Hamburg and Munich, Germany, he has been full professor of crystallography at the Laboratory of Crystallography of the ETH and University of Zurich since the fall of 1993. In 2008, Walter Steurer received the Jean Marie Dubois Award for Excellence in Quasicrystal Research. At present, his research topics comprise structural studies of aperiodic crystals and their phase transformations, the modeling of order/disorder phenomena, and higher-dimensional crystallography. Sofia Deloudi, was born in Athens, Greece on March 27, 1980. She studied interdisciplinary natural sciences at the ETH Zurich and concluded her doctoral dissertation under the supervision of Prof. Walter Steurer at the Laboratory of Crystallography of the ETH. At present, she is a junior scientist with her focus on modeling of quasiperiodic structures and higher-dimensional crystallography.

Preface 6
Contents 8
Part I Concepts 15
1 Tilings and Coverings 21
1.1 1D Substitutional Sequences 23
1.1.1 Fibonacci Sequence (FS) 24
1.1.2 Octonacci Sequence 27
1.1.3 Squared Fibonacci Sequence 28
1.1.4 Thue--Morse Sequence 29
1.1.5 1D Random Sequences 30
1.2 2D Tilings 30
1.2.1 Archimedean Tilings 32
1.2.2 Square Fibonacci Tiling 33
1.2.3 Penrose Tiling (PT) 35
1.2.4 Heptagonal (Tetrakaidecagonal) Tiling 45
1.2.5 Octagonal Tiling 50
1.2.6 Dodecagonal Tiling 52
1.2.7 2D Random Tilings 56
1.3 3D Tilings 57
1.3.1 3D Penrose Tiling (Ammann Tiling) 57
1.3.2 3D Random Tilings 58
References 59
2 Polyhedra and Packings 62
2.1 Convex Uniform Polyhedra 63
2.2 Packings of Uniform Polyhedra with Cubic Symmetry 67
2.3 Packings and Coverings of Polyhedra with Icosahedral Symmetry 69
3 Higher-Dimensional Approach 73
3.1 nD Direct and Reciprocal Space Embedding 75
3.2 Rational Approximants 80
3.3 Periodic Average Structure (PAS) 82
3.4 Structure Factor 84
3.4.1 General Formulae 84
3.4.2 Calculation of the Geometrical Form Factor 85
3.5 1D Quasiperiodic Structures 90
3.5.1 Reciprocal Space 90
3.5.2 Symmetry 92
3.5.3 Example: Fibonacci Structure 93
3.6 2D Quasiperiodic Structures 104
3.6.1 Pentagonal Structures 106
3.6.2 Heptagonal Structures 113
3.6.3 Octagonal Structures 120
3.6.4 Decagonal Structures 133
3.6.5 Dodecagonal Structures 159
3.6.6 Tetrakaidecagonal Structures 167
3.7 3D Quasiperiodic Structures with Icosahedral Symmetry 182
3.7.1 Reciprocal Space 183
3.7.2 Symmetry 186
3.7.3 Example: Ammann Tiling (AT) 189
References 198
Part II Methods 201
4 Experimental Techniques 204
4.1 Electron Microscopy 207
4.2 Diffraction Methods 208
4.3 Spectroscopy 212
References 213
5 Structure Analysis 215
5.1 Data Collection Strategy 217
5.2 Multiple Diffraction (Umweganregung) 218
5.3 Patterson Methods 220
5.4 Statistical Direct Methods 224
5.5 Charge Flipping Method (CF) 225
5.6 Low-Density Elimination 226
5.7 Maximum Entropy Method 228
5.8 Structure Refinement 232
5.9 Crystallographic Data for Publication 235
References 236
6 Diffuse Scattering and Disorder 240
6.1 Phasonic Diffuse Scattering (PDS) on the Example of the Penrose Rhomb Tiling 244
6.2 Diffuse Scattering as a Function of Temperatureon the Example of d-Al--Co--Ni 245
References 250
Part III Structures 252
7 Structures with 1D Quasiperiodicity 255
References 256
8 Structures with 2D Quasiperiodicity 257
8.1 Heptagonal Phases 258
8.1.1 Approximants: Borides, Borocarbides, and Carbides 260
8.1.2 Approximants: -Gallium 262
8.2 Octagonal Phases 262
8.3 Decagonal Phases 264
8.3.1 Two-Layer and Four-Layer Periodicity 264
8.3.2 Six-Layer Periodicity 281
8.3.3 Eight-Layer Periodicity 283
8.3.4 Surface Structures of Decagonal Phases 285
8.4 Dodecagonal Phases 287
References 291
9 Structures with 3D Quasiperiodicity 298
9.1 Mackay-Cluster Based Icosahedral Phases (Type A) 301
9.2 Bergman-Cluster Based Icosahedral Phases (Type B) 302
9.3 Tsai-Cluster-Based Icosahedral Phases (Type C) 307
9.4 Example: Icosahedral Al--Cu--Fe 312
9.5 Surface Structures of Icosahedral Phases 317
References 320
10 Phase Formation and Stability 327
10.1 Formation of Quasicrystals 328
10.2 Stabilization of Quasicrystals 330
10.3 Clusters 334
10.4 Phase Transformations of Quasicrystals 339
10.4.1 Quasicrystal Quasicrystal Transition 340
10.4.2 Quasicrystal Crystal Transformation 343
10.4.3 Microscopic Models 351
References 355
11 Generalized Quasiperiodic Structures 364
11.1 Soft Quasicrystals 365
11.2 Photonic and Phononic Quasicrystals 367
11.2.1 Interactions with Classical Waves 368
11.2.2 Examples: 1D, 2D and 3D Phononic Quasicrystals 371
References 375
Glossary 377
Index 381