Aperiodic crystals: from modulated phases to quasicrystals

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14 Polymorphism in molecular crystals J. Bernstein 15 Crystallography of modular materials G. Ferraris, E. Makovicky, S. Merlino 16 Diffuse x-ray scattering and models of disorder T.R. Welberry 17 Crystallography of the polymethylene chain: an inquiry into the structure of waxes D.L. Dorset

18 Crystalline molecular complexes and compounds: structure and principles F. H. Herbstein 19 Molecular aggregation: structure analysis and molecular simulation of crystals and liquids A. Gavezzotti

20 Aperiodic crystals: from modulated phases to quasicrystals T. Janssen, G. Chapuis, M. de Boissieu 21 Incommensurate Crystallography S. van Smaalen lUCr Texts on Crystallography

1

The solid state

A. Guinier, R. Julien

4 X-ray charge densities and chemical bonding P. Coppens 5 The basics of crystallography and diffraction, second edition C. Hammond 6 Crystal structure analysis: principles and practice W. Clegg, editor 7 Fundamentals of crystallography, second edition C. Giacovazzo, editor 8 Crystal structure refinement: a crystallographer's guide to SHELXL P. Miiller, editor 9 Theory and Techniques of Crystal Structure Determination U. Shmueli

Aperiodic Crystals From Modulated Phases to Quasicrystals TEDJANSSEN

Institute of Theoretical Physics, University

ofNijmegen

GERVAIS CHAPUIS Ecole Polytechnique Federale de Lausanne MARCDEBOISSIEU CNRS and Institut National Polytechnique de Grenoble

OXPORD UNIVERSITY PRESS

OXTORD

UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dares Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2007 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by the author using LaTeX Printed in Great Britain on acid-free paper by Biddies Ltd., www.biddles.co.uk ISBN 978-0-19-856777-6 (Hbk) 1 3 5 7 9 1 08 6 4 2

PREFACE In the eighteenth and nineteenth centuries the idea developed that, on the atomic level, crystals are constructed by regularly spaced unit cells. The mathematical theory for this idea was created by Bravais, Schoenflies, and Fedorov. After the discovery of X-rays in 1912 one really could prove this underlying periodic structure. Since then the prevailing idea has been that the ground state of matter at zero temperature is lattice periodic with a lattice constant of the order of a nanometre. Although there was actually no theoretical proof for this idea, it was generally accepted. In that view disordered systems, like glasses, form at most metastable configurations. In the 1960s, several materials became known where besides the reflections at the points of a reciprocal lattice, there are also sharp peaks which do not fit into this scheme. The first to describe this situation was Pirn de Wolff from Delft, who realized that the satellite peaks observed in anhydrous 7-Na2CC>3 are not defects, but belong really to the structure, which is in this case an aperiodic modulated crystal phase. Although at very low temperature, the ground state is again periodic, the intermediate state between the periodic high-temperature and low-temperature phases is thermodynamically stable and aperiodic, but not disordered! Fater several such materials were found. In fact, a substantial part of the minerals of the earth's crust turn out to be aperiodic. However, the term 'aperiodic' is not the proper characterization. As can be seen from the sharpness of the diffraction spots, these materials are just as well ordered as the usual crystals. One must agree that these materials should be called crystals as well. They have sharp diffraction peaks and flat facets. Nevertheless, these materials were not generally considered to be interesting. This changed with the discovery in 1982 of quasicrystals by Shechtman and collaborators. These materials were intriguing because they show sharp diffraction spots and a rotation symmetry which is not crystallographic, in the sense that the symmetry is not compatible with lattice periodicity. This shows that the new materials are well ordered, but aperiodic. In fact, they are quasiperiodic in the mathematical sense, and therefore they are called quasicrystals. After some years it turned out that the non-crystallographic symmetry is not an essential property. In that sense quasicrystals are not really different from modulated phases. They are all quasiperiodic crystals. Since the discovery approximately forty years have passed, and aperiodic crystals have been generally accepted as an interesting class of materials. This has brought about a change in well-accepted ideas about the structure of matter. Much work has been done on the determination of their structure and of their physical properties. Because of the aperiodic!ty the usual techniques, very often based on the presence of a Brillouin zone, cannot be used. New techniques had to be developed. The new materials have properties that are quite different from V

vi

PREFACE

those of the usual lattice periodic crystals. This has led to new developments in chemistry, physics, crystallography, and even mathematics. The field has now become mature, although there are still many open questions. A number of books on modulated structures, and, in particular, on quasicrystals have appeared in the meantime. However, the common property of quasiperiodicity means that many of the techniques can be used on the whole family of quasiperiodic materials. Therefore, it seemed a good idea to us, to consider quasiperiodic crystals from a unified point of view, with applications in modulated phases and quasicrystals. Actually, it is very difficult to make a real distinction between these. It is easy to find materials where it is difficult to tell whether they belong to one class or to the other. The book is intended for materials scientists, physicists, and crystallographers. We do not assume previous knowledge of aperiodic crystals. We only assume a basic knowledge of solid state physics and crystallography. We have tried to avoid a heavy mathematical formulation, but on the other hand we want to be sufficiently precise. Proofs are not always given. Instead, there are many references to the literature, and in a number of cases we give a more rigorous discussion of some point in a paragraph one can skip if one wants. These parts are indicated by ft at the beginning and \ at the end. Sections intended to give some more fundamental background, and which could be skipped, are also indicated by the sign ft. One of us (T.J.) wants to thank the Tohoku University in Sendai, in particular Professor An-Pang Tsai, for the hospitality during a visit, when part of this book was written. We thank many people for interesting discussions, and want to mention Alia Arakcheeva, Michael Baake, Esther Belin-Ferre, S.I. Ben-Abraham, Roland Currat, Jean-Marie Dubois, Michal Dusek, Sonia Francoual, Franz Gahler, Denis Gratias, Yoshihiro Ishibashi, Aloysio Janner, Ronan McGrath, Marek Mihalkovic, Lukas Palatinus, Manuel Perez-Mato, Vaclav Petficek, Penelope Schobinger-Papamantellos, Pat Thiel, Andreas Schonleber, Hans-Rainer Trebin, Sander van Smaalen, and Akiji Yamamoto.

GLOSSARY

Atomic scattering factor Atomic surface (window or acceptance domain for quasicrystals) Basis of 3D direct lattice Basis of lattice in superspace Basis of reciprocal lattice Basis 3D reciprocal lattice Charge coupled device Decomposition modulation vector (Eq. 2.9) Decomposition reciprocal lattices composite (Eqs.2.13-14) Density Density of states Dimension superspace Dimension physical space Dimension internal space Dynamical structure factor Fourier module Fourier transform of /(r) (= ^ / /(k) exp(ik.r)) Frac: value of x minus the largest integer smaller than x Golden mean ((A/S - l)/2 fa 0.618 Internal (perpendicular) space Inverse golden mean ((%/5 + l)/2 fa 1.618 Irreducible representation /x Lattice in 3D Lattice in superspace Metric tensor Modulation function Modulation vector Normal coordinate mode characterized by k,z/ Number of unit cells in a sample Participation ratio Parts of the integral matrix T(R) Photomultiplier tube Photo-stimulated luminescence Physical (parallel) space Projections Vs —> VE Projection Vs —> V/ VII

/j(H) f2 a^ a si =(aj,aij) a* (aj.a*=27T(V,-) a*,b*,c* CCD Uij Z^ p(r) DOS n D d S*(q, w) M* /(k) Frac(x) $ V/ or Ej_ or Eperp T DM A £ g u x j( ) q Qki/ N P YE, F/, FM PMT PSL VE or E\\ or Epar K = KE TT/

viii

GLOSSARY Pseudo-Brillouin Zone PBZ Reciprocal lattice vector 3D K Reciprocal lattice in three dimensions, and in superspace, resp. A*, S* Reciprocal superspace vector ks (k,k/) Rigid motion {R|t} Scattering vectors q = kf — ki Scattering intensity /(H)=|F(H)| 2 Structure factor (=± £* exp(-27riH.ri)) F(H) Superspace (or hyperspace) Vs or Esup Vector from Fourier module H=J^. hja* Idem for modulated phase H = K+q e Vibration of atom j in the mode q,i/ qf,j Vibrational density of states VDOS Wave vectors (phonons etc.) k

CONTENTS Preface

v

Glossary 1

Introduction

1

1.1

1 1 2

1.2

1.3 2

vii

Periodic crystals 1.1.1 History 1.1.2 Description 1.1.3 The role of space group symmetry in the structure determination 1.1.4 Symmetry and physical properties 1.1.5 Examples 1.1.6 Conclusion Quasiperiodic crystals 1.2.1 History 1.2.2 Classes and examples 1.2.3 Modulated phases 1.2.4 Incommensurate composites 1.2.5 Quasicrystals 1.2.6 Morphology Summary

6 7 8 9 10 10 14 15 21 23 28 29

Description and symmetry of aperiodic crystals

31

2.1

31 31 34 34 40 42 44 46 46 57 61 67 75 76 76 79 81 83

2.2

2.3

2.4

Aperiodic and quasiperiodic lunctions 2.1.1 Periodicity and aperiodicity Quasiperiodic structures 2.2.1 Modulated phases 2.2.2 Composites 2.2.3 Quasicrystals 2.2.4 ft Electromagnetic crystals in space-time Description in superspace 2.3.1 Embedding 2.3.2 Modulated phases 2.3.3 Incommensurate composites 2.3.4 Quasicrystals 2.3.5 Is the classification into three types unique? Symmetry 2.4.1 Point group symmetry ol diffraction patterns 2.4.2 Superspace groups 2.4.3 Examples 2.4.4 Approximants IX

CONTENTS

x

2.5 2.6 2.7

2.4.5 Superspace groups for commensurate phases 2.4.6 Consequences of superspace group symmetry Scaling symmetries Alternative descriptions Summary

84 86 88 91 94

3

Mathematical models 3.1 Model sets 3.2 Introduction to tilings 3.3 Substitutional chains 3.3.1 Substitutions with an alphabet 3.3.2 Embedding of substitutional chains 3.3.3 Tilings by substitution 3.4 Aperiodic tilings 3.4.1 Construction of aperiodic tilings 3.4.2 Embedding of tilings 3.4.3 ft Symmetry of tilings 3.5 Approximants 3.6 Coverings 3.7 Random tilings 3.8 Summary

96 96 98 101 101 102 104 106 106 110 118 123 126 127 129

4

Structure 4.1 Diffraction 4.1.1 Diffraction from periodic and aperiodic crystals 4.1.2 Indexing the diffraction pattern 4.1.3 Mathematical questions 4.2 Diffraction techniques 4.2.1 X-ray area detectors 4.2.2 Neutron area detectors 4.2.3 Measurement techniques 4.3 Determination of modulated phases and composites 4.3.1 Introduction 4.3.2 The structure factor of incommensurate structures 4.3.3 Possible expressions of the modulation functions 4.3.4 Additional expressions of modulation functions 4.3.5 Practical aspects of structure determination and refinement 4.3.6 Ab initio methods 4.3.7 Relation between harmonics and satellite orders 4.3.8 Composite structures 4.3.9 Commensurately modulated structures 4.4 Typical examples of modulated phases and composites 4.4.1 Introduction

130 130 130 141 143 146 147 148 148 151 151 151 152 154 156 162 165 166 168 170 170

CONTENTS

4.5

4.6

4.7

5

6

4.4.2 The modulated phases of Na 2 CO 3 4.4.3 The composite structure of La2Coi.7 4.4.4 Aperiodicity in the structures of elements 4.4.5 p-chlorobenzamide 4.4.6 Modular structures 4.4.7 Conclusion Structure determination of quasicrystals 4.5.1 A simple one-dimensional quasiperiodic model 4.5.2 Structure determination of a one-dimensional quasicrystal 4.5.3 Structure determination of icosahedral and decagonal phases Examples of quasicrystal structures 4.6.1 Introduction 4.6.2 Structure of the i-AlPdMn phase 4.6.3 Atomic structure of the CdYb icosahedral phase 4.6.4 Structure of the AINiCo decagonal phase Diffraction by an imperfect crystal 4.7.1 Diffuse scattering when an average lattice exists 4.7.2 Diffuse scattering when there is no average lattice

xi 170 176 179 183 186 189 190 190 194 203 204 204 205 241 251 260 261 265

Origin and stability 5.1 Introduction 5.2 The Landau theory of phase transitions 5.3 Semi-microscopic models 5.3.1 Substrate models 5.3.2 Spin models 5.3.3 Models with continuous degrees of freedom 5.4 Composites 5.5 Electronic instabilities 5.5.1 Charge-density and spin-density systems 5.5.2 Hume-Rothery compounds 5.6 Numerical modelling of aperiodic crystals 5.6.1 Introduction 5.6.2 Phase transitions in a hexagonal model 5.6.3 Simulation of organic incommensurate crystal structures 5.7 Summary

268 268 269 275 275 278 281 287 287 287 288 290 290 290

Physical properties 6.1 Introduction 6.2 Tensorial properties 6.3 Hydrodynamics of aperiodic crystals

305 305 306 312

299 303

xii

CONTENTS

6.4

6.5

6.6

6.7 6.8

6.9 7

6.3.1 Hydrodynamic theory of fluids and periodic crystals 6.3.2 Hydrodynamic theory of aperiodic crystals and phason modes Phonons and phasons: theory 6.4.1 Introduction 6.4.2 Simple models 6.4.3 Eigenvectors and spectrum 6.4.4 Damping 6.4.5 Calculation of phonons Phonons: Experimental findings 6.5.1 Scattering 6.5.2 Modulated phases and composites 6.5.3 Quasicrystals Phasons: Experiment 6.6.1 Introduction 6.6.2 Phason modes in modulated crystals 6.6.3 Diffuse scattering and phason modes in icosahedral quasicrystals 6.6.4 Phason modes in icosahedral quasicrystals 6.6.5 Phason modes in other quasicrystals Non-linear excitations Electrons 6.8.1 Introduction 6.8.2 Simple models 6.8.3 Electrical conductivity 6.8.4 Realistic potentials Summary

Other topics 7.1 The morphology of aperiodic crystals 7.1.1 The puzzling habit of the mineral calaverite 7.1.2 The morphology of the TMA Zn phases 7.1.3 The morphology of icosahedral and decagonal quasicrystals 7.2 Surfaces 7.2.1 Introduction 7.2.2 Structure of surfaces of aperiodic crystals 7.2.3 Generalization of the morphological laws 7.2.4 Physical properties of quasicrystalline surfaces 7.3 Magnetic quasiperiodic systems

A Higher-dimensional space groups A.I Crystallographic operations in n dimensions A.2 Lattices

312 313 315 315 319 345 346 347 347 347 349 350 356 356 356 360 364 376 380 384 384 384 389 390 390 392 392 392 396 397 400 400 403 406 408 410 414 414 416

CONTENTS A.3 A.4 A.5 A.6 A.7

B

xiii

Crystal classes Space groups Classification Space groups for aperiodic crystals Notation A.7.1 Superspace groups for incommensurate phases A.7.2 General space groups A.8 Extinction rules A.9 Tables A.9.1 Introduction A.9.2 Tables for irreducible representation of point groups: point groups of 5-, 8-, 10-, 12-fold and icosahedral symmetry A.9.3 Examples of superspace groups for modulated phases A.9.4 Superspace groups for quasiperiodic structures with five-, eight-, ten-, twelve-fold, or icosahedral symmetry

436

Magnetic symmetry of quasiperiodic systems

438

B.I B.2 B.3 B.4

438 439 439 440

Magnetic systems and time-reversal symmetry Magnetic point groups Magnetic space groups The magnetic groups for quasiperiodic crystals

417 419 421 425 427 427 429 429 430 430 431 435

References

443

Index

463

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1

INTRODUCTION 1.1

1.1.1

Periodic crystals

History

People have always been intrigued by the beautiful shapes of crystals. The symmetrical objects with their flat, shiny surfaces and beautiful optical properties have been used for decoration since ancient times. Scientific interest in them also started very early. Johannes Kepler was fascinated by the beauty of snow crystals. In 1611 he wrote his On the six-cornered snowflake in which he tried to explain the six-fold symmetry of snow crystals on the basis of a closed-packing of spheres. Rene Descartes studied snowflakes in Amsterdam and wrote his observations of their morphology in February 1634. Robert Hooke showed detailed drawings of the many different shapes of snowflakes in his book Micrographia (1665). So, two important ingredients of crystallography, symmetry and morphology, were studied in detail in the seventeenth century. During the French Revolution, Adrien-Marie Legendre wrote his Elements de geometric (1794), in which he gave a precise definition of symmetrical polyhedra. Shortly afterwards a new step forward was formulated in the Traite de mineralogie by Rene Just Haiiy. He described the construction of a regularly shaped crystal out of smaller symmetrical units (cubes, octahedra, and tetrahedra). He omitted icosahedra, 'because in crystals no five-fold symmetries occur'. From his composition procedure follows the law of rational indices, which states that the planes of crystal surfaces intersect properly chosen axes in points with integer coordinates. The macroscopic symmetry is based on the symmetry of the microscopic periodic arrangement of the unit building blocks. Lattices and their symmetries were studied for the first time by Moritz Ludwig Frankenheim in 1835. Previously, he and Johann Hessel had derived the 32 crystal classes. In 1856 Frankenheimer showed the fourteen lattice classes in three dimensions. Auguste Bravais then derived the Bravais classes on the basis of purely geometrical reasoning. The next step was the derivation of the full Euclidean symmetry of periodic patterns in three dimensions, the space groups. The mineralogist Evgraph Fedorov and the mathematician Arthur Schoenflies gave, largely independently, the full classification of the 230 space groups in 1890. An interesting account of the history of symmetry in crystallography can be found in (Burckhardt, 1988). The macroscopic consequences of the microscopic symmetry could be checked on the morphology, in particular the law of rational indices. The proof that the microscopic structure of an ideal crystal has space group symmetry had to wait till the discovery of X-rays by Max von Laue in 1912. This started a

1

2

INTRODUCTION

new era of crystallography. Structure determination was then possible, based on the work of the Braggs, and it was highly successful. This led to the idea that the ground state of matter was an arrangement of atoms and molecules with lattice periodicity, which means with space group symmetry. In this view, which prevailed for decades, each ideal crystal could be described by one of the 230 space groups. At non-zero temperature there are always defects, and metastable glassy matter may exist, but in principle the ground state is lattice periodic. This view changed after 1960 when states of matter were discovered which reasonably should be classified as crystals, but which lacked the lattice periodicity. Before starting in that direction, the main topic of this book, we shall briefly review the role symmetry plays in conventional, lattice periodic crystals. 1.1.2 Description Two ingredients of the description of (lattice periodic) crystals are space group symmetry, and morphology. Macroscopically, the point group symmetry can be seen in the morphology. A crystal is invariant under certain rotations, reflections, and combinations of these. The group of all these operations is the point group. At the atomic level, accessible via neutron and X-ray diffraction and via highresolution electron microscopy, one may distinguish the lattice periodicity. Then proper and improper rotations (as the combinations of a rotation and a reflection are called) leave the framework of the translations invariant, but, to keep the positions of the atoms invariant, they sometimes have to be complemented by a translation. These combinations leave the distances the same. A space group is a group of rigid, distance-preserving transformations, also called Euclidean transformations. Its elements are pairs {-R|t} of an orthogonal transformation R and a translation t acting on a position vector r as {fl|t}r = Rr + 1. After the choice of an origin and a basis, the transformation is given by matrices:

Such transformations are physically important, because they leave the physics the same because distances between the particles do not change. A group of Euclidean transformations is a space group if its translation group, all the translations in the group, is generated by three independent basis vectors. Each element of the translation group can be written as

The set of all orthogonal transformations R in the space group elements forms the point group, which is isomorphic to the quotient of the space group and its translation group: K = G/A, where G is the space group, A its translation

PERIODIC CRYSTALS

3

group, and K its point group. The translation parts of space group elements may belong to the translation group or not. If a in g = {R\a} belongs to the translation group, it is called a primitive translation. Otherwise, it is a non-primitive translation. In three dimensions space group elements with non-primitive translations are transformations with either a screw axis or a glide plane. Notice that the translation part a depends on the origin. Therefore, a non-primitive translation may disappear as such, when the origin is shifted. If there is an origin such that all translation parts of a space group become primitive translations, then the group is called a symmorphic space group. Otherwise it is a non-symmorphic space group. An example of a non-symmorphic group in two dimensions is given in Fig. 1.1.

Fig. 1.1. A two-dimensional periodic array. The rectangle is the unit cell. It contains two dumbbell-shaped molecules. The array is left invariant by the translations along the edges of the unit cell, and by a rotation of TT around a corner. Moreover, there are two glide operations: mirrors along a line followed by a translation along the line. The operations are {Ijmiai + m^a^}, {2|0}, {m^Kai +a2)/2}, and {mj,|(ai + a.^)/"1}. The symbol for this space group is p2gg. Its point group is 2mm, which has 4 elements. Indicated are a unit cell and 4 glide lines. The lattice of a space group is invariant under the point group. This means that on a lattice basis the point group transformations are represented by integer matrices M defined by

4

INTRODUCTION

On the other hand, on an orthogonal basis the matrices corresponding to R are orthogonal matrices, which can be written on a properly chosen basis as

Under a basis transformation the trace of a matrix does not change. Therefore, the traces of M(R) and M'(R) are the same. Now, M(R) is an integer matrix, and consequently l+2cos() is an integer, which is only possible if = TT, 27T/3,7T/2,7T/3 or 0. If R is a rotation, it is a two-, three-, four-, or six-fold rotation, or the identity. This strongly restricts the number of possible rotations in a point group of a lattice. The condition that l+2cos() is an integer is called 'the crystallographic condition'. If a system is left invariant by a lattice of translations, there is a finite region in space such that every point in space can be reached by a lattice translation from a uniquely determined point in this region. The structure is a repetition of a basic unit, called the unit cell. A crystal is described by the contents of its unit cell. The positions of the atoms in the unit cell are TJ (j = 1 , . . . s). Each position has a site symmetry, the group that leaves the point fixed modulo lattice vectors. This site symmetry characterizes the different positions, the Wyckoff positions. Once a unit cell is chosen, every point in space corresponds to a uniquely determined point in the unit cell, from which it can be reached by lattice translations. However, the unit cell is itself not uniquely determined. A convenient choice is the parallelepiped spanned by the basis vectors. With respect to this basis, the coordinates of the unit cell range from 0 to 1. However, another choice is the same unit cell shifted, such that the coordinates run from — ^ to +^. A choice that takes the symmetry better into account is the Wigner-Seitz cell defined as the set of points closer to a chosen lattice point than to any other lattice point. Such cells all have the same shape and size, and they have the full point group symmetry of the lattice. The notion of space groups may also be used for other dimensions. For two dimensions, these are called plane groups. The crystallographic condition is the same and the only non-primitive translations occur in transformations with a glide mirror. These transformations consist of a mirror operation together with a translation along the mirror line. For higher dimensions, the notion of space groups may also be extended, as we shall do further on in this book. Then the groups are called space groups, in general. These are briefly discussed in Appendix A. The facets of a crystal are parallel to planes spanned by lattice translations, called net planes. Net planes in a crystal may be characterized by their normals.

PERIODIC CRYSTALS

5

It is convenient to choose these in a particular way, as reciprocal lattice vectors. These are introduced as follows. The density of a crystal may be decomposed into a Fourier series, a sum of plane waves with wave vectors K. Because the crystal is invariant under translations a of the lattice, this must hold also for the constituting plane waves. Therefore, exp(27r«K.r) and exp(27r«K.(r + a)) must be the same for all a from the lattice. The wave vectors K for which all products K.a are integers, or in other words for which K.a = 0 (mod 1), form a lattice, called the reciprocal lattice. For a basis a.j (j = 1, 2, 3) of the translation lattice, the basis of the reciprocal lattice is given by the reciprocal basis a* satisfying the relations which in three-dimensional space is equivalent to

The vectors K of the reciprocal lattice, as defined here, satisfy K.a = 0 (mod 1) (mod for every lattice vector a, because 1), since hi and HJ are integers. The Wigner-Seitz cell of the reciprocal lattice is called the Brillouin zone. An example in two dimensions is given in Fig. 1.2. In three dimensions the vectors of the reciprocal lattice are often written as K = ha* + kb* + Ic*. The basis vectors of the lattice also form a basis for the space. Therefore, any point in space r has 3 coordinates with respect to this basis:

In the unit cell the values of £j are between 0 and 1. They are called lattice coordinates. The notation of a point in terms of these is r = [£1,^2, £3]-

Fig. 1.2. The relation between the basis a,b and the reciprocal basis a*,b*. A net plane is then given by K.a = constant integer c for given K and all a in the plane. Each net plane is characterized by the smallest perpendicular vector K from the reciprocal lattice. The distance between two adjacent parallel net planes is given by the inverse of the length of K. Then the morphologically most important net planes are those with the smallest |K|.

6 1.1.3

INTRODUCTION The role of space group symmetry in the structure determination

The space group symmetry is very helpful in the structure determination. Suppose the crystal has a density function /o(r) and space group G. If {-R|t} is an element of G, the following relation holds.

The Fourier transform of this relation gives

For a vector k satisfying Rk = k this implies that either k.t = 0 (mod 1) or /o(k) = 0. This is a very important relation, because the Fourier transform of the density, or the potential, plays an important role in scattering experiments. In such an experiment, a neutron or X-ray beam with wave vector k» is scattered by the sample into beams with wave vector k/. The scattering vector is k = k/-kj. Quantum mechanics teaches us that the scattering is determined by the matrix element of the scattering potential between incoming and outgoing states. In this case we have

The relation of the Fourier transforms leads to extinction rules /o(k) = 0 for k = Kk in the case that (in a non-symmorphic space group with a screw axis or a glide plane) k.t ^ 0 (mod 1), which is a great help in the structure determination. The main steps in the structure determination are as follows. The latter is based on the measured intensities of the Bragg peaks:

where the sum is over all particles in the unit cell. The sum is called the structure factor. The positions of the Bragg peaks are K = ha* + kb* + Ic*. A basis for the direct lattice is then formed by three vectors a,b,c. Together with the composition this gives the number of structure units in the unit cell. The symmetry of the diffraction pattern contains at least the point group of the space group. Assuming a point group, all possible space groups then can be found in the International Tables for Crystallography, and these space groups are distinguished by the extinction rules. The difference between space groups with the same point group is the difference in the presence of screw axes and glide planes. These are associated with well-defined extinction rules. The next step is the choice of a model. For that purpose, the positions TJ of the particles in the unit cell are expressed in free parameters, compatible with the space group. Because the space group links positions of atoms, the number of parameters is, generally, much less than three times the number of particles

PERIODIC CRYSTALS

7

in the unit cell. The parameters are varied to yield a minimum in the difference between calculated and observed intensities, which gives the final answer. That is a very short account of crystal structure determination. It is just a reminder. For details we refer to introductory texts on crystallography. There are a number of problems. A very difficult one is the phase problem. The intensities are measured experimentally, but these are connected to the absolute value of the structure factor. This means that the structure is not yet uniquely determined. Several methods have been developed (such as direct methods, the method of charge-flipping, and the method of maximum entropy) to help to solve this problem. Another problem is the fact that different models may give comparable differences between calculated and measured intensities, because these differences are never zero. The structure determination has become a powerful technique, applicable to very large unit cells, with many atoms. Its principal basis is the knowledge of the space groups. 1.1.4 Symmetry and physical properties Another important role of the space group symmetry is the possibility it gives to characterize electrons and lattice vibrations. From the theory of applications of group theory in physics it is known that elementary excitations and electron states may be described using representations of the symmetry group. A matrix representation of a group G is a mapping from the group to a group of nonsingular d-dimensional matrices D(G) such that _D(