Crystals and Crystal Structures Crystals and Crystal Structures Richard J. D. Tilley Emeritus Professor, University of Cardiff Copyright # 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com 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, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Crystallography. I. Title. QD905.2.T56 2006 548–dc22 2006016562 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13 978-0-470-01820-0 (HB) ISBN-10 0-470-01820-8 (HB) ISBN-13 978-0-470-01821-7 (PB) ISBN-10 0-470-01821-6 (PB) Typeset in 10.5/12.5 pt Times by Thomson Digital Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wilts This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. For Elizabeth, Gareth and Richard Contents Preface xi 1 Crystals and crystal structures 1 1.1 Crystal families and crystal systems 1 1.2 Morphology and crystal classes 3 1.3 The determination of crystal structures 5 1.4 The description of crystal structures 5 1.5 The cubic close-packed (A1) structure of copper 7 1.6 The body-centred cubic (A2) structure of tungsten 8 1.7 The hexagonal (A3) structure of magnesium 8 1.8 The halite structure 9 1.9 The rutile structure 9 1.10 The fluorite structure 10 1.11 The structure of urea 11 1.12 The density of a crystal 12 Answers to introductory questions 13 Problems and exercises 14 2 Lattices, planes and directions 17 2.1 Two-dimensional lattices 17 2.2 Unit cells 20 2.3 The reciprocal lattice in two dimensions 20 2.4 Three-dimensional lattices 23 2.5 Alternative unit cells 25 2.6 The reciprocal lattice in three dimensions 26 2.7 Lattice planes and Miller indices 29 2.8 Hexagonal lattices and Miller-Bravais indices 31 2.9 Miller indices and planes in crystals 32 2.10 Directions 33 2.11 Lattice geometry 35 Answers to introductory questions 36 Problems and exercises 38 viii CONTENTS 3 Two-dimensional patterns and tiling 43 3.1 The symmetry of an isolated shape: point symmetry 43 3.2 Rotation symmetry of a plane lattice 45 3.3 The symmetry of the plane lattices 46 3.4 The ten plane crystallographic point symmetry groups 48 3.5 The symmetry of patterns: the 17 plane groups 50 3.6 Two-dimensional ‘crystal structures’ 55 3.7 General and special positions 57 3.8 Tesselations 59 Answers to introductory questions 61 Problems and exercises 62 4 Symmetry in three dimensions 67 4.1 The symmetry of an object: point symmetry 67 4.2 Axes of inversion: rotoinversion 69 4.3 Axes of inversion: rotoreflection 72 4.4 The Hermann-Mauguin symbols for point groups 73 4.5 The symmetry of the Bravais lattices 74 4.6 The crystallographic point groups 76 4.7 Point groups and physical properties 79 4.8 Dielectric properties 80 4.9 Refractive index 83 4.10 Optical activity 84 4.11 Chiral molecules 85 4.12 Second harmonic generation 86 4.13 Magnetic point groups and colour symmetry 87 Answers to introductory questions 88 Problems and exercises 89 5 Building crystal structures from lattices and space groups 93 5.1 Symmetry of three-dimensional patterns: space groups 93 5.2 The crystallographic space groups 94 5.3 Space group symmetry symbols 96 5.4 The graphical representation of the space groups 98 5.5 Building a structure from a space group 101 5.6 The structure of diopside, CaMgSi2O6 103 5.7 The structure of alanine, C3H7NO2 105 Answers to introductory questions 109 Problems and exercises 109 6 Diffraction and crystal structures 113 6.1 The position of diffracted beams: Bragg’s law 114 6.2 The geometry of the diffraction pattern 115 CONTENTS ix 6.3 Particle size 119 6.4 The intensities of diffracted beams 121 6.5 The atomic scattering factor 122 6.6 The structure factor 123 6.7 Structure factors and intensities 125 6.8 Numerical evaluation of structure factors 126 6.9 Symmetry and reflection intensities 128 6.10 The temperature factor 130 6.11 Powder X-ray diffraction 132 6.12 Electron microscopy and structure images 135 6.13 Structure determination using X-ray diffraction 140 6.14 Neutron diffraction 142 6.15 Protein crystallography 142 6.16 Solving the phase problem 144 6.17 Photonic crystals 149 Answers to introductory questions 151 Problems and exercises 152 7 The depiction of crystal structures 155 7.1 The size of atoms 155 7.2 Sphere packing 156 7.3 Metallic radii 159 7.4 Ionic radii 160 7.5 Covalent radii 162 7.6 Van der Waals radii 163 7.7 Ionic structures and structure building rules 163 7.8 The bond valence model 164 7.9 Structures in terms of non-metal (anion) packing 166 7.10 Structures in terms of metal (cation) packing 168 7.11 Cation-centred polyhedral representations of crystals 169 7.12 Anion-centred polyhedral representations of crystals 171 7.13 Structures as nets 174 7.14 The depiction of organic structures 175 7.15 The representation of protein structures 177 Answers to introductory questions 181 Problems and exercises 182 8 Defects, modulated structures and quasicrystals 187 8.1 Defects and occupancy factors 187 8.2 Defects and unit cell parameters 189 8.3 Defects and density 190 8.4 Modular structures 191 x CONTENTS 8.5 Polytypes 194 8.6 Crystallographic shear phases 196 8.7 Planar intergrowths and polysomes 200 8.8 Incommensurately modulated structures 202 8.9 Quasicrystals 208 Answers to introductory questions 210 Problems and exercises 211 Appendices 215 Appendix 1 Vector addition and subtraction 215 Appendix 2 Data for some inorganic crystal structures 217 Appendix 3 Schoenflies symbols 221 Appendix 4 The 230 space groups 223 Appendix 5 Complex numbers 227 Appendix 6 Complex amplitudes 229 Answers to problems and exercises 231 Bibliography 239 Formula index 245 Subject index 249 Preface Crystallography – the study of crystals, their struc- as engineering. Two further examples will suffice tures and properties – has antecedents predating to demonstrate this. The scientific breakthroughs modern science. Despite this long history, the in the first half of the 20th century, leading to subject remains dynamic and is changing rapidly. applications as diverse as nuclear power and semi- Protein crystallography is but one example. An conductor technology, were built, to a considerable understanding of many biological processes at a extent, upon detailed understanding of metallic molecular level, already leading to a fundamental and inorganic crystal structures. In the latter change in the treatment of many chronic illnesses, half of the 20th century, molecular biology, now has been built upon protein crystal structure stud- embedded in crystallography, has lead to a deep ies of awesome complexity. Other areas of rapid rooted change in medicine, as a result of crystal growth stand out. The development of computer structure studies, starting with the structures of software has revolutionised the way in which crys- biologically important molecules such as insulin, tal structures can be depicted. The presentation of via the epoch making determination of the crystal such structures as nets, for example, leads to images structure of DNA to recent studies of enormous that compare in beauty with the best of abstract art. protein structures. Computers themselves, the microprocessors that With all of these aspects in mind, this book do the work, and the displays that present the is designed as an introductory text for students information, as well as the host of mobile devices and others that need to understand crystals and that are essentially computers in one form or crystallography without necessarily becoming another, rely almost entirely upon a fundamental crystallographers.