Table Of ContentBEYOND
QUASICRYSTALS
Les Houches, March 7-18, 1994
Editors
Fran~oise Axel, Universire Paris VII-Denis Diderot
Denis Gratias, CECN-CNRS, Vitry
Springer-Verlag Berlin Heidelberg GmbH
ISBN 978-3-540-59251-8 ISBN 978-3-662-03130-8 (eBook)
DOI 10.1007/978-3-662-03130-8
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© Springer-Verlag Berlin Heidelberg 1995
Originally published by Springer-Verlag Berlin Heidelberg New York in 1995
The school was supported by:
Universite Joseph Fourier de Grenoble
Centre National de la Recherche Scientifique (CNRS-fonnation pennanente)
Direction des Recherches et Etudes Techniques (DGA-DRET)
xn
Commission of the European Communities -DG
Minisrere des Affaires Etrangeres
Obituary
Professor Gerhard Fritsch, who lectured in our School "Beyond
Quasicrystals" on transport properties in quasiperiodic media, died
suddenly and unexpectedly on April 14, 1994.
Gerhard Fritsch was born on July 3, 1940 in Marktredwitz in Nothern
Bavaria where he also spent his school time. Afterwards he studied
physics at the Technical University (TU) of Munich and got there his
PhD in 1969 with a work on "Measurements on the thermic and electric
conductivity of sodium directly at the melting point".
From 1969 to 1975 he worked as Scientific Research Assistant at the Physics
Department of the TU Munich. From 1975 to 1977 he was Research
Associate at the University of lllinois (USA). In 1977 Gerhard Fritsch
was appointed as a Senior Research Scientist at the Physics Department
of the TU Munich and became in 1979 full Professor for Experimental
Physics at the University of the Armed Forces Munich. In 1984 he spent
a year at the University of California at Los Angeles (USA).
Besides the stimulating work in his Institute Professor Fritsch was
committed to the development of the knowledge in modem physics for
teachers. He served as co-editor of the German journal "Physik in
unserer Zeit" for more than two decades.
In research, he combined interests for fundamental topics as the transport
properties of amorphous and quasi crystalline alloys or the determination
of the diffuse X-ray scattering of sodium with interests for more applied
studies as measurements of the properties of materials by means of micro
beam X-rays.
To his wife, who was with him in Les Houches and to their young daughter,
we present the expression of our heartfelt sympathy.
Foreword
This book is the collection of most of the written versions of the
Courses given at the Winter School "Beyond Quasicrystals" in Les Houches
(March 7-18, 1994).
The School gathered lecturers and participants from all over the
world and was prepared in the spirit of a general effort to promote
theoretical and experimental interdisciplinary communication between
mathematicians, theoretical and experimental physicists on the topic of the
nature of geometric order in solids beyond standard periodicity and
quasi periodicity.
The overall structure of the book reflects the wish of the editors to
pose this fundamental question of geometric order in solids from both the
experimental and theoretical point of view.
The first part is devoted more specifically to quasicrystals. These
materials were the common starting point of most of the audience and
present a first concrete example of a non-trivial geometric order. We chose
to focus on a few fundamental aspects of quasicrystals related to hidden
symmetries in solids which are not easily found in standard textbooks on the
topic, not to reach an exhaustive survey which is already available
elsewhere.
Y. Meyer redevelops here a few of the ideas presented in his 1972
book "Number theory and harmonic analysis". It is a unique occasion for
most physicists of the quasicrystal community to discover Y. Meyer's very
early synthetic definition of a "quasicrystal". Basics of geometry, structural
and dynamical properties of quasicrystals are given by T. Janssen, whereas
the more experimental crystallographic aspects are developed by W. Steurer
and P. Mikulic. "Special" symmetries in quasicrystals form the basis of the
lectures by J. Patera, P. Kramer and A. Janner, respectively on
"Pentacrystals", on non commutative geometry, the other on multimetrical
analysis. All three extend the field of crystallography by introducing more
general concepts than simple isometries.
The fundamental question of understanding the propagation of non
trivial order by local interaction "matching rules" in quasi crystals is
discussed by A. Katz in connection with the problem of atomic diffusion by
P. Kalugin. Defects in aperiodic materials from quasicrystals to amorphous
are presented and compared by N. Rivier.
The second part deals with deterministic aperiodic order "beyond"
quasip eriodicity. Deterministic sequences are a major example of quasicrystal
generalization. Different properties are presented here from basics in the
VI
Courses by M. Mendes France and J.-P. Allouche, to automatic walks by M.
Dekking with possible relationships to diffusion. Their Fourier transforms
are discussed in great details by M. Queffelec based on the general notions
of measures and substitution dynamical systems. V. Berthe presents and
discusses their entropies, Z. Y. Wen the subwords appearing in the
Fibonacci substitution and its generalizations. Algebraic properties appear
in the Course on trace maps by J. Peyriere.
Realizations in physics of such deterministic aperiodic sequences are
at present of two kinds
- SchrOdinger type equations having such a deterministic sequence for a
potential. This very important topic is presented in the Courses by A. Siito
and H. Kunz.
- Multilayers systems which are now being studied mostly by diffractionnists
Finally, an introductory Course with basics for multifractal analysis
is given by J. Peyriere.
Almost all the authors have made at our request a gigantic effort,
for which we are happy to heartily thank them, to render the contents of
their Course accessible to non specialists: in the present instance, this
School was directed to experimental physicists with the hope to develop the
basis for joint investigations on long-range order in solids. We also are
particularly grateful to the following colleagues who generously provided
help in shaping this book:
E. Cockayne, R. Collela, M. Dekking, F. Delyon, F. Denoyer, F. Gahler,
A. Janner, T. Janssen, P. Kramer, M. Mendes France, M. Queffelec, S. Van
Smaalen.
This very intense moment of scientific communication between
mathematicians, physicists and material scientists, and this book would
never have existed without the constant encouragement of Michele Leduc
(Directeur du Centre de Physique des Houches), her very competent and
helping staff in Les Houches Mesdames G. Chioso et B. Rousset, and
Madame Grosseaux's efficiency and patience (Les Editions de
Physique/Springer-Verlag); we would like them to accept the expression of
our sincere gratitude.
But particular acknowledgements, particular thanks, are due to
Madame Fran~oise Kakou. She took care of the entire administrative
organization of the School, of the preparation of this book and its annexes
with her usual perfect skill, competence, efficiency and kindness, without
which we would not have succeeded.
Readers, be merciful: if you find errors of any kind, please write to
us in view of future printings ...
Fran~oise Axel Denis Gratias
CONTENTS
Quasicrystals
COURSE 1
Quasicrystals, diophantine approximation and algebraic numbers
by Yves Meyer
1. Introduction.................................................................................... 3
2. Almost-periodic functions. Poisson summation formula and algebraic numbers.. .... 5
3. Model sets and quasicrystals. ................................................................ 9
4. Quasicrystals and diophantine approximation. .. .. . . . . . .. .. . . . . .. . .. . .. . . . . .. . .. . . . .. . .. .. . 10
5. Poisson summation formula and quasicrystals ............. ..................... .......... 14
6. Conclusion ...... ...... ......... ...... .......................................................... 15
COURSE 2
The pentacrystals
by J. Patera
1. Introduction... . . . .. . .. . .. . . .. . . .. . .. . . .. .. . .. . .. .. . . .. .. . . .. .. . . . . . .. . .. . .. . . .. . .. .. . . .. . .. .. .. 17
2. Preliminaries.......... ......................................................................... 18
3. The pentacrystal map. ......................................................................... 20
4. DefInition of quasicrystals. . . . .. . .. . . .. .. . .. .. .. . . . .. . .. . .. .. .. .. . .. . . .. . .. . . .. . .. . . . .. .. . .. . . 22
5. Phasons......................................................................................... 23
6. Quasiaddition .................................................................................. 24
7. Examples....................................................................................... 25
COURSE 3
Elements of a multimetrical crystallography
by A. Janner
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ... . . . . .. . . . . . ... . . . . . .. . . . .. . . . . . . . . . . . . .. . . . . . .. .. . . . . . 33
1. Introduction.................................................................................... 33
2. Close-packed structures.. . . . . . .. . . . .. ... . .. . . .. .. .. .. . .. .. . . . . . .. .. . .. . . .. . .. .. .. . .. . .. . . . . .. 35
2.1 The 2-dimensional case.................................................................. 35
2.2 The 3-dimensional case.................................................................. 35
3. Multimetrical symmetry of the 20 hexagonal lattice...... .. ....... ........ ..... ...... ..... 37
4. Binary integral quadratic forms and quadratic fIelds. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5. IndefInite ternary integral quadratic forms.................................................. 41
6. Quadratic forms oflattices of3D close-packed structures.. .. .. .. .. ... .. .. .. . .. .. . . .. .. .. 44
6.1 Reduced metric tensors.................................................................. 44
6.2 IndefInite binary quadratic forms....................................................... 45
6.3 IndefInite ternary quadratic forms ...................................................... 46
7. Multimetrical point group of the hexagonal close-packed lattice. . . ... . . . . .. . . . . . . . . . . . . . 47
8. Multimetrical space groups of crystal structures...... ......... .... ............... ..... .... 49
VIII
8.1 Hexagonal close-packed structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2 The Wurtzite structure. .................................................................. .... 51
9. Concluding remarks........................................................................... 52
COURSE 4
Non-commutative models for quasicrystals
by P. Kramer and J. Garcia-Escudero
1. Why non-commutative models for quasicrystals ? .................................... .... 55
2. Free groups and their automorphisms ................. . . . . . . . . . . . . . . . . . . .. . . .. .. . .. . . . . . . . . 56
3. Non-commutative crystallography..................... ... .................................. 57
4. Structure and geometry of the group Aut(F2) .. • • .. • • • • • .. • • .. • .. • .. • .. • • .. • .. • .. .. • .. • • • • • 58
5. Free groups and automorphisms for n > 2 ................................................ 66
6. Non-commutative models and symmetries for 2D quasiperiodic patterns.............. 66
7. Automata for the triangle and Penrose patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8. Survey of other results.. .. . .. . . . . .. . .. . .. . .. . .. . . .. . . . . .. . . .. . . .. . .. . .. . .. . .. . . .. .. . .. . . . . . . . . 72
COURSE 5
From quasiperiodic to more complex systems
by T. Janssen
1. Structures........ .................................................................. ... .......... 75
1.1 Introduction ............................................................................... 75
1.2 Classes of quasiperiodic structures. .. . .. . . . . . . .. . . .. . .. . .. . .. . . .. . .. . . . . . .. . . . . . . .. .. . . 78
1.3 Embedding of quasiperiodic systems.................................................. 81
1.4 Superspace groups.................................................... ................... 84
1.5 Action of symmetry groups in 3-dimensional space.............. . . . . . . . . . . . . . . . .. . . 86
1.6 Scale symmetries......................................................................... 89
1.7 Hierarchy of structures .................................................................. 91
1.8 Physical origin of quasiperiodicity . . .. . . .. . . .. . .. . .. . . .. . . . . .. . . . . .. . . .. . . . . . . . . . . . . .. . 93
2. Diffraction............. . . .. . . . .. . . . . . .. . .. . . . . .. . . .. . . .. . .. . .. . . . . . . . . . . . . .. . .. . . . . .. . .. . . . . . .. . 94
2.1 Structure factor. . .. . . . . .. .. . . . . . .. . .. . .. . .. . .. . . .. . .. . . . . . . . . . . . .. . . .. . . . . .. . .. . .. . . . . . . . . 94
2.2 Structure factor of quasiperiodic structures .............................. . . . . . . . . . .. . . 97
2.3 Influence of symmetry ................................................... ................ 99
2.4 Thermal vibrations ....................................................................... 100
2.5 Disorder................................................................................... 101
3. Phonons........................................................................................ 104
3.1 Phonons in IC phases.................................................................... 104
3.2 Spectra..................................................................................... 110
3.3 Phonons in quasicrystals ................................................................ 111
3.4 Neutron scattering from quasiperiodic structures..................................... 116
4. Substitutional chains.......................................................................... 122
4.1 Introduction .................................................................. ............. 122
4.2 Atomic surfaces........................................................................... 124
4.3 Fractal atomic surfaces................................................................... 127
5. Electrons........................................................................................ 132
5.1 Models..................................................................................... 132
5.2 Spectra..................................................................................... 135
5.3 Wave functions........................................................................... 136
IX
COURSE 6
Matching rules and quasiperiodicity: the octagonal tHings
by A. Katz
1. Introduction........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2. Quasiperiodic tHings..................................... ..................................... 142
2.1 Quasiperiodicity ......................... ................................................. 142
2.2 The atonric surfaces ............... ......... ... ......... ... ... ... ... ...... ......... ... .... 143
2.3 The cut algorithm... ... ...... ......... ... ................ ............ ... ........ ... ....... 144
2.4 Canonical or "Penrose like" tilings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.4.1 Definition............................................................................ 144
2.4.2 The oblique tiling................................................................... 145
2.4.3 Octagonal tHings. ........................... ... ............ ... ..... .......... ....... 146
3. The composition-decomposition method................................................... 148
3.1 Self-sinrilarity............................................................................. 148
3.2 Inflation and quasiperiodicity............................ ............................... 150
4. The method of forbidden planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1 Position of the problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.2 Non-transversality conditions. ...... ... ................................................. 153
4.3 The forbidden planes..... ...................................................... ... ....... 154
5. Decoration of the tiles....... ........ .................. ............ ...... ... ................... 156
5.1 A simple case. .......... ........ ..................... ......... ... ... ... ............... .... 156
5.2 The Ammann decoration of vertices......... ........................... ... ... ....... ... 159
6. The main theorem............................ ....................................... ... ....... 162
6.1 Position and intersections of the forbidden planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2 Systems of data.................................................. ... ........ .............. 163
6.3 Propagation of order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4 Proof of the theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.4.1 The pushing procedure............... .................................... .......... 166
6.4.2 The cone of planes................................................................. 168
6.5 Quasiperiodic tilings and "special tilings" .................... ........................ . 170
7. Generalised Ammann tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.1 Definitions................................................................................. 174
7.2 Symmetry considerations.................. .......................................... .... 174
7.3 Setting the method........................................................................ 175
7.3.1 Systems of data in £1........................................................ ....... 177
7.4 Reduction to "bad prisms" .......................................................... .... 177
7.5 Proof of the theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.6 Order in generalised Ammann tilings of the [ust kind ............ . . . . . . . . . . . . . . . . .. . 181
7.7 Generalised Ammann tilings of the second kind: an example of weak: rules. . . . . . 183
8. Conclusion..................................................................................... 188
COURSE 7
A mechanism for diffusion in quasicrystals
by P. A. Kalugin............................................................................... 191
COURSE 8
Experimental aspects of the structure analysis of aperiodic
materials
by W. Steurer
1. Introduction................................................................................... . 203
2. What are aperiodic materials ? ............................................................... 204
3. Experimental probes for distinguishing between crystals and aperiodic structures.... 206
