ISBN: 3-540-67464-0
TITLE: From Quasicrystals to More Complex Systems
AUTHOR: Axel, F.; Denoyer, F.; Gazeau, J.P. (Eds.)
TOC:

COURSE 1
Dynamics and Transport Properties of Aperiodic Crystals
by T. Janssen
1. Structure and symmetry 1
2. Phonons 5
3. Domain wall motion 11
4. Electrons 13
5. Tensorial properties 16
6. Surface effects 17
7. Transport properties 19
8. Concluding remarks 20
COURSE 2
Diffraction Experiments on Quasicrystals and Related Phases
by F. Dnoyer
1. Introduction 23
2. Decagonal quasicrystals 24
2.1 Generalities 24
2.2 X-ray structure determination of decagonal quasicrystals 24
3. Decagonal symmetry and twinning 30
3.1 Tenfold twinning of Al_13Fe_4 and Al_13Fe_4-type structure 30
3.2 Twinning in the structure of decagonal phases and fine structure of diffraction peaks 32
3.3 Twinning and main characteristic features of diffraction patterns 32
3.4 Description of microstructures in terms of phason-strain quasicrystals 35
4. Quasicrystal transformations 41
5. Icosahedral short range order in glasses 42
6. Conclusion 45
COURSE 3
Electronic Properties of Quasicrystals. A Comparison with Approximant Phases and Disordered Systems
given by C. Berger
written by C. Berger and T. Grenet
1. Introduction 49
1.1 Quasiperiodic order 50
1.2 Quasicrystals, crystals and amorphous phases 52
1.3 Samples of high structural quality in ternary alloys 53
1.4 Unexpected physical properties 54
2. Conductivity and density of states in quasicrystals 54
2.1 Low electrical conductivity values in i-phases 54
2.2 Low electronic density of states in quasicrystals 57
3. Comparison with other metallic alloys 59
3.1 Scale of conductivity in metallic alloys 59
3.2 Effect of diffraction 60
3.3 Quantum interference effects in disordered systems 61
3.4 Disordered insulator and Anderson localization 65
4. Periodicity as an approach to quasiperiodicity 65
4.1 Approximant phases 66
4.2 Experimental electrical conductivity in approximant phases 66
4.3 Calculated electronic properties in approximants 68
4.4 Low dimensional perfect quasiperiodic models 69
5. Quasicrystals as ordered structures of high symmetry 72
5.1 Pseudo Brillouin zone 72
5.2 Role of local atomic clusters 74
6. Towards a metal-insulator transition in quasicrystals: Comparison with disordered systems 75
6.1 Some experimental evidence for the approach to a metal-insulator transition in quasicrystals 76
6.2 Crossing of the metal-insulator transition in i-AlPdRe 77
7. Conclusion 79
COURSE 4
Exact Electron States in 1D (Quasi-) Periodic Arrays of Delta-Potentials
given by P. Kramer
written by P. Kramer and T. Kramer
1. Introduction and scope 85
2. Finite periodic strings at negative energy 89
2.1 Preview: An energy gauge for crystals 89
2.2 Bloch and bound states in a single band 90
2.3 The string S^n 95
2.4 Rational Bloch labels 96
2.5 Bound states and clusters of the string S^n 96
2.6 Participation number 98
2.7 Supercell interpretation 99
2.8 Large n limit 99
3. Finite quasiperiodic strings at negative energy 100
3.1 Preview: Energy gauge in Fibonacci strings 100
3.2 Substitutional systems and their invariants 101
3.3 Recursive calculation of the transfer matrix 102
4. Periodic strings at positive energy 106
4.1 The S-matrix 107
4.2 The S-matrix for the periodic string S^n 108
5. Quasiperiodic strings at positive energy 110
5.1 The Fibonacci-atlas 110
6 Conclusion 112
COURSE 5
Random Tiling Models for Quasicrystals
by E. Cockayne
1. Introduction 115
1.1 Basic definitions 116
1.2 Generation of quasicrystalline tilings 117
1.3 Randomization of tilings 120
1.4 A zoo of tiling models 121
2. Mathematics of random tilings 123
2.1 Entropy density and phason elastic constants 123
2.2 Long-wavelength behavior and stability 126
2.3 Diffraction 127
3. Random tiling results 128
3.1 Monte Carlo simulation 128
3.2 Combinatorics 129
3.3 Transfer matrix method 130
3.4 Bethe Ansatz method 132
4. Atomic models for quasicrystals 132
4.1 HREM/diffraction-based 133
4.2 Models based on realistic interatomic forces 136
4.3 Other models 138
5. Quasicrystal phase transformations 139
5.1 Phason unlocking 139
5.2 Quasicrystal <-> (micro)crystal 139
6. Conclusions 140
COURSE 6
Model Sets: A Survey
by R.V. Moody
1. Introduction 145
2. Model sets 147
3. Geometric side 149
4. Arithmetic side 150
4.1 The icosian model sets 150
4.2 p-adic model sets 152
5. Analytic side 155
6. Dynamical systems side 157
7. Diffraction 160
7.1 Comments 163
COURSE 7
Acceptance Windows Compatible with a Quasicrystal Fragment
given by J. Patera
written by Z. Maskov, J. Patera and E. Pelantov
1. Introduction 167
2. Notation and auxiliary facts 171
3. Local invariance and the forward growth 174
4. The maximal acceptance window 178
5. Example: Analysis of two-dimensional quasicrystal data 180
6. Comments and remarks 189
COURSE 8
Counting Systems with Irrational Basis for Quasicrystals
given by J.P. Gazeau
written by J.P. Gazeau and R. Krejcar
1. Introduction 195
2. The set of tau-integers 197
3. Tau-integer labelling of the Fibonacci chain 199
4. Tau-integer labelling of diffraction pattern 202
5. Tau-integer labelling of two-dimensional structures 205
6. Arithmetics and algebra of the beta-integers 211
COURSE 9
Acoustic-Like Excitations in Strongly Disordered Media
given by E. Courtens
written by E. Courtens and R. Vacher
1. Introduction 219
2. The case of mass-fractal media 221
2.1 The structure of mass fractals 222
2.2 The concept of mutually self-similar series of MSSS 226
2.3 The dynamics of mass fractals 227
3. The case of glasses 234
3.1 What is already established 236
3.2 Spectroscopy of acoustic excitations
in the terahertz regime  three remarks 240
3.3 Some studies near the end of acoustic branches in glasses 243
4. Conclusions 249
COURSE 10
Intermittent Dynamics and Ageing in Glassy Systems
by J.-Ph. Bouchaud
1. Introduction 261
2. A simple model: Traps and intermittent dynamics 263
3. Relation with mode-coupling descriptions 265
4. Self-induced quenched disorder and open questions 267
COURSE 11
A Short Introduction to Ergodic Theory and Its Applications
by F.M. Dekking
1. Dynamical systems 273
1.1 Examples 273
1.2 Recurrence 277
1.3 Ergodic theorem 278
1.4 Unique ergodicity 280
1.5 Expected recurrence time 281
2. Spectral properties of dynamical systems 282
2.1 The spectrum of a dynamical system 282
2.2 Mixing 283
3. Entropy of dynamical systems 284
3.1 Isomorphism 286
3.2 Entropy and Hausdorff dimension 287
4. Epilogue 288
COURSE 12
Fractality and the Kinetics of Chaos
by G.M. Zaslavsky
1. Introduction 291
2. Mapping the dynamics 295
3. Topological zoo (singular zones) 297
4. Self-similar hierarchy of islands 300
5. Quasi-traps 301
6. Boundary layer as a quasi-trap 303
7. Fractal and multifractal space-time of kinetics 304
8. Dimension spectrum of the multifractal space-time 307
9. Fractional kinetics 309
10. Conclusions 312
COURSE 13
Long-Tailed Distributions in Physics
given by M.F. Shlesinger
written by M.F. Shlesinger, J. Klafter and G. Zumofen
1. Introduction 315
2. Fractal time 319
3. Slow relaxations 321
4. Fractal space processes 321
5. Nonlinear dynamics 324
COURSE 14
Distribution of Galaxies: Scaling vs. Fractality
by R. Balian
1. Introduction 329
2. Algebra of point distributions 330
2.1 Densities and correlations 330
2.2 Counts in cells 332
3. Scale invariance 333
3.1 Scaling of correlations 333
3.2 The void probability 333
3.3 Scaling of counts in cells 334
4. Fractality 336
4.1 Correlation dimension 336
4.2 Hausdorff dimension for occupied cells 337
4.3 Renyi index 338
4.4 Multifractal dimension 340
5. Conclusion 343
END
