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J. Bernstein, Israel G. R. Desiraju, India J. R. Helliwell, UK T. Mak, China P. Müller, USA P. Paufler, Germany H. Schenk, The Netherlands P. Spadon, Italy D. Viterbo (Chairman), Italy
IUCr Monographs on Crystallography 1 Accurate molecular structures A. Domenicano, I. Hargittai, editors 2 P.P. Ewald and his dynamical theory of X-ray diffraction D.W.J. Cruickshank, H.J. Juretschke, N. Kato, editors 3 Electron diffraction techniques, Vol. 1 J.M. Cowley, editor 4 Electron diffraction techniques, Vol. 2 J.M. Cowley, editor 5 The Rietveld method R.A. Young, editor 6 Introduction to crystallographic statistics U. Shmueli, G.H. Weiss 7 Crystallographic instrumentation L.A. Aslanov, G.V. Fetisov, J.A.K. Howard 8 Direct phasing in crystallography C. Giacovazzo 9 The weak hydrogen bond G.R. Desiraju, T. Steiner 10 Defect and microstructure analysis by diffraction R.L. Snyder, J. Fiala and H.J. Bunge 11 Dynamical theory of X-ray diffraction A. Authier 12 The chemical bond in inorganic chemistry I.D. Brown 13 Structure determination from powder diffraction data W.I.F. David, K. Shankland, L.B. McCusker, Ch. Baerlocher, editors 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 22 Structural crystallography of inorganic oxysalts S.V. Krivovichev 23 The nature of the hydrogen bond: outline of a comprehensive hydrogen bond theory G. Gilli, P. Gilli 24 Macromolecular crystallization and crystal perfection N.E. Chayen, J.R. Helliwell, E.H. Snell
IUCr Texts on Crystallography 1 The solid state A. Guinier, R. Julien 4 X-ray charge densities and chemical bonding P. Coppens 7 Fundamentals of crystallography, second edition C. Giacovazzo, editor 8 Crystal structure refinement: a crystallographer’s guide to SHELXL P. Müller, editor 9 Theories and techniques of crystal structure determination U. Shmueli 10 Advanced structural inorganic chemistry Wai-Kee Li, Gong-Du Zhou, Thomas Mak 11 Diffuse scattering and defect structure simulations: a cook book using the program DISCUS R. B. Neder, T. Proffen 12 The basics of crystallography and diffraction, third edition C. Hammond 13 Crystal structure analysis: principles and practice, second edition W. Clegg, editor Crystal Structure Analysis Principles and Practice
Second Edition
Alexander J. Blake School of Chemistry, University of Nottingham William Clegg Department of Chemistry, University of Newcastle upon Tyne Jacqueline M. Cole Cavendish Laboratory, University of Cambridge John S.O. Evans Department of Chemistry, University of Durham Peter Main Department of Physics, University of York Simon Parsons Department of Chemistry, University of Edinburgh David J. Watkin Chemical Crystallography Laboratory, University of Oxford
Edited by William Clegg
1 3 Great Clarendon Street, Oxford ox26dp 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 Dar es 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 © Alexander J. Blake, William Clegg, Jacqueline M. Cole, John S.O. Evans, Peter Main, Simon Parsons, and David J. Watkin, 2009 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First edition first published 2001, reprinted 2006 Second edition first published 2009 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 Crystal structure analysis : principles and practice / William Clegg ...[et al.]. — 2nd ed. p. cm. — (International Union of Crystallography book series; 13) ISBN 978–0–19–921946–9 (hardback) — ISBN 978–0–19–921947–6 (pbk.) 1. X-ray crystallography. 2. Crystals—Structure. I. Clegg, William, 1949– QD945.C79 2009 548 .81—dc22 2009011644 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wilts
ISBN: 978–0–19–921946–9 ISBN: 978–0–19–921947–6 13579108642 Preface
The material in this book is derived from an intensive course in X-ray structure analysis organized on behalf of the Chemical Crystallogra- phy Group of the British Crystallographic Association and held every two years since 1987. As with a crystal structure derived from X-ray diffraction data, the course contents have been gradually refined over the years and they reached a stage in 1999 (the seventh course) where we considered they could be published, and hence made available to a far wider audience than can be accommodated on the course itself. The result was the first edition of this book, published in 2001. The authors were the principal lecturers on the course in 1999 and they revised and expanded the material, while converting the lecture notes into a book format. Because of its origin, the book represented a snapshot of the intensive course, which has continued to evolve, especially as the subject of chemical crystallography has undergone significant changes, mainly due to the widespread availability of area detector technology, the exponential increase in computing power and improvements in soft- ware, and greater use of synchrotron radiation and powder diffraction. Nevertheless, the underlying principles remain valid, and the particular application of those principles can be adapted to new developments for some time to come. By the time of the eleventh course in 2007, its contents and the team of principal lecturers had changed markedly, and we were asked to con- sider a second edition of the book reflecting these developments. This has been encouraged and assisted by the use of a consistent template for the 2007 course notes, and these have been used as the basis for this new edition. Nevertheless, any readers who participated in the 2007 course will detect a number of changes, particularly in the inclusion of some material not covered in the lecture notes, some updating, and differences of style made necessary by a non-interactive format. Since this book, like its first edition, owes its origins to the course, we acknowledge here our large debt to those who have dedicated much effort to the organization of the course since its inception; without them this book would never have existed, even as an idea. The first five courses were held at the University of Aston, where the local organizers Phil Lowe and Carl Schwalbe set a gold standard of course administration and smooth operation, establishing many of the enduring characteristics valued by participants ever since. Following the move to the Univer- sity of Durham, Vanessa Hoy and then Claire Wilson developed these firm foundations to even further heights of excellence, presenting a
v vi Preface
challenge to Andres Goeta, who took over for the 2009 Course. Through- out the course’s history Judith Howard has provided overall guidance and expertise, particularly in fund raising, and has spared the course lecturers much concern with the practicalities of maintaining and pro- moting the course. Several organizations, including the EPSRC, IUCr, BCA and commercial sponsors, have been long-standing and generous supporters of the course. The first course in 1987 was the brainchild of David Watkin, who worked extremely hard to launch it and establish it as the enduring success that it has become. His role as course director was taken over in the mid-1990s by Bob Gould, who passed on the baton to Sandy Blake after 1999; from 2011 the director will be Simon Parsons. The template for the lecture notes on which this book is based was developed by Horst Puschmann, and Amber Thompson has looked after assembling and producing the notes for the last few courses. Many colleagues have made contributions to the course over the years, in lectures and in the crucial group tutorial sessions: a book format can never reflect the intensive interaction and lively atmosphere. These and the social aspects of the course are probably at least as important in the memories of participants as the formal lecture presentations. One aspect of the tutorial group sessions of the course has been retained in modified form in the book. Most chapters include exercises, for which answers are provided in an appendix. Readers are encouraged to tackle the exercises at leisure and not consult the answers until they are satis- fied with their own efforts. In the spirit of the tutorials, these exercises may also prove beneficial as a basis for group discussion. Over the twenty years of the course and its eleven occasions, we have seen former students return as group tutors, and tutors move on into lecturer roles. Course participants, from many countries, are established practising crystallographers in academic and industrial posts around the world. Courses elsewhere have been developed, modelled on our experience. We dedicate this book to the hundreds of students who have been the course’s primary beneficiaries and whose hard work and commitment, intellectually and socially, have contributed much to its success. Bill Clegg, Newcastle University November 2008 Editor, on behalf of the authors Acknowledgements
We are grateful to authors and publishers for their permission to reproduce some of the Figures that appear in this book, as follows.
Figures 1.1, 1.8, 2.1, 2.2, 22.1, 22.2, and 22.4 from W. Clegg: Crystal Structure Determination. Oxford University Press, Oxford, 1998. Figures 1.2 and 4.1 from C. Giaccovazzo, H. L. Monaco, D. Viterbo, F. Scordari, G. Gilli, G. Zanotti and M. Catti: Fundamentals of Crystal- lography. Oxford University Press, Oxford, 1992. Figures 1.3 and 1.5 from G. Harburn, C. A. Taylor and T. R. Welberry: Atlas of Optical Transforms. G. Bell, London, 1975. Figure 1.9 from J. P. Glusker and K. N. Trueblood: Crystal Structure Analysis – A Primer, Second Edition. Oxford University Press, Oxford, 1985. Figures 2.3 and 2.4 from Traidcraft plc, Gateshead, UK. Figures 5.6 and 5.7 from Rigaku Corporation, Sevenoaks, Kent, UK. Figure 6.1 from Stoe & Cie GmbH, Darmstadt, Germany. Figures 8.5–8.8 from W. Clegg, J. Chem. Educ. 81, 908; copyright 2004. American Chemical Society. Figure 10.1, reprinted by permission from G. N. Ramachandran and R. Srinavasan, Nature, 190, 161; copyright 1961. Macmillan Magazines Ltd. Figure 14.3 from R. B. Neder and T. Proffen: Diffuse Scattering and Defect Structure Simulations: A Cook Book Using the Program DISCUS. Oxford University Press, Oxford, 2008. Figure 14.14 from International Tables for Crystallography, Volume A. Kluwer Academic Press, Dordrecht, The Netherlands. Copyright 1983, International Union of Crystallography. Figure 17.2 from Panalytical Ltd, Cambridge, UK. Figure 17.4 from R. Haberkorn, personal communication to J. S. O. Evans, 1999. Figure 18.1 from S. Parsons, Acta Crystallogr. D59, 1995. Copyright International Union of Crystallography, 2003. Figure 22.3 from Diamond Light Source, Didcot, Oxfordshire, UK.
vii This page intentionally left blank Contents
1 Introduction to diffraction 1 1.1 Introduction 1 1.2 X-ray scattering from electrons 1 1.3 X-ray scattering from atoms 1 1.4 X-ray scattering from a unit cell 2 1.5 The effects of the crystal lattice 2 1.6 X-ray scattering from the crystal 3 1.7 The structure-factor equation 4 1.8 The electron-density equation 5 1.9 A mathematical relationship 6 1.10 Bragg’s law 6 1.11 Resolution 7 1.12 The phase problem 8
2 Introduction to symmetry and diffraction 9 2.1 The relationship between a crystal structure and its diffraction pattern 9 2.2 Translation symmetry in crystalline solids 10 2.3 Symmetry of individual molecules, with relevance to crystalline solids 12 2.4 Symmetry in the solid state 16 2.5 Diffraction and symmetry 18 2.6 Further points 20 Exercises 24
3 Crystal growth and evaluation 27 3.1 Introduction 27 3.2 Protect your crystals 27 3.3 Crystal growth 28 3.4 Survey of methods 28 3.4.1 Solution methods 28 3.4.2 Sublimation 33 3.4.3 Fluid-phase growth 33 3.4.4 Solid-state synthesis 34 3.4.5 General comments 34
ix x Contents
3.5 Evaluation 35 3.5.1 Microscopy 35 3.5.2 X-ray photography 36 3.5.3 Diffractometry 36 3.6 Crystal mounting 36 3.6.1 Standard procedures 36 3.6.2 Air-sensitive crystals 38 3.6.3 Crystal alignment 39
4 Space-group determination 41 4.1 Introduction 41 4.2 Prior knowledge and information other than from diffraction 42 4.3 Metric symmetry and Laue symmetry 43 4.4 Unit cell contents 43 4.5 Systematic absences 44 4.6 The statistical distribution of intensities 47 4.7 Other points 48 4.8 A brief conducted tour of some entries in International Tables for Crystallography, Volume A 50 Exercises 52
5 Background theory for data collection 53 5.1 Introduction 53 5.2 A step-wise theoretical journey through an experiment 53 5.3 The geometry of X-ray diffraction 55 5.3.1 Real-space considerations: Bragg’s law 55 5.3.2 Reciprocal-space considerations: the Ewald sphere 56 5.4 Determining the unit cell: the indexing process 58 5.4.1 Indexing: a conceptual view 58 5.4.2 Indexing procedure 60 5.5 Relating diffractometer angles to unit cell parameters: determination of the orientation matrix 62 5.6 Data-collection procedures and strategies 64 5.6.1 Criteria for selecting which data to collect 64 5.6.2 How best to measure data: the need for reflection scans 65 5.7 Extracting data intensities: data integration and reduction 67 5.7.1 Background subtraction 67 5.7.2 Data integration 68 5.7.3 Crystal and geometric corrections to data 68 Exercises 72
6 Practical aspects of data collection 73 6.1 Introduction 73 Contents xi
6.2 Collecting data with area-detector diffractometers 73 6.3 Experimental conditions 75 6.3.1 Radiation 75 6.3.2 Temperature 76 6.3.3 Pressure 77 6.3.4 Other conditions 77 6.4 Types of area detector 77 6.4.1 Multiwire proportional chamber (MWPC) 77 6.4.2 Phosphor coupled to a TV camera 78 6.4.3 Image plate (IP) 78 6.4.4 Charge-coupled device (CCD) 78 6.5 Some characteristics of CCD area-detector systems 80 6.5.1 Spatial distortion 81 6.5.2 Non-uniform intensity response 81 6.5.3 Bad pixels 81 6.5.4 Dark current 81 6.6 Crystal screening 82 6.6.1 Unit cell and orientation matrix determination 84 6.6.2 If indexing fails 86 6.6.3 Re-harvest the reflections 86 6.6.4 Still having problems? 87 6.6.5 After indexing 87 6.6.6 Check for known cells 87 6.6.7 Unit cell volume 88 6.7 Data collection 88 6.7.1 Intensity level 88 6.7.2 Mosaic spread 89 6.7.3 Crystal symmetry 89 6.7.4 Other considerations 90 Exercises 91
7 Practical aspects of data processing 93 7.1 Data reduction and correction 93 7.2 Integration input and output 93 7.3 Corrections 94 7.4 Output 95 7.5 A typical experiment? 95 7.6 Examples of more problematic cases 96 7.7 Twinning and area-detector data 98 7.8 Some other special cases (in brief) 99 Exercises 101
8 Fourier syntheses 103 8.1 Introduction 103 8.2 Forward and reverse Fourier transforms 104 8.3 Some mathematical and computing considerations 107 8.4 Uses of different kinds of Fourier syntheses 108 xii Contents
8.4.1 Patterson syntheses 109 8.4.2 E-maps 109 8.4.3 Full electron-density maps, using (8.2) or (8.3) as they stand 109 8.4.4 Difference syntheses 110 8.4.5 2Fo − Fc syntheses 111 8.4.6 Other uses of difference syntheses 112 8.5 Weights in Fourier syntheses 112 8.6 Illustration in one dimension 113 8.6.1 Fc synthesis 114 8.6.2 Fo synthesis, as used in developing a partial structure solution 114 8.6.3 Fo − Fc synthesis 114 8.6.4 Full Fo synthesis 114 Exercises 115
9 Patterson syntheses for structure determination 117 9.1 Introduction 117 9.2 What the Patterson synthesis means 118 9.3 Finding heavy atoms from a Patterson map 121 9.3.1 One heavy atom in the asymmetric unit of P1 121 9.3.2 One heavy atom in the asymmetric unit of P21/c 122 9.3.3 One heavy atom in the asymmetric unit of P212121 124 9.3.4 One heavy atom in the asymmetric unit of Pbca 124 9.3.5 One heavy atom in the asymmetric unit of P21 125 9.3.6 Two heavy atoms in the asymmetric unit of P1 and other space groups 125 9.4 Patterson syntheses giving more than one possible solution, and other problems 126 9.5 Patterson search methods 128 9.5.1 Rotation search 129 9.5.2 Translation search 129 Exercises 131
10 Direct methods of crystal-structure determination 133 10.1 Amplitudes and phases 133 10.2 The physical basis of direct methods 134 10.3 Constraints on the electron density 135 10.3.1 Discrete atoms 135 10.3.2 Non-negative electron density 136 10.3.3 Random atomic distribution 137 10.3.4 Maximum value of ∫ ρ3(x)dV 139 Contents xiii
10.3.5 Equal atoms 139 10.3.6 Maximum entropy 140 10.3.7 Equal molecules and ρ(x) = const. 140 10.3.8 Structure invariants 140 10.3.9 Structure determination 141 10.3.10 Calculation of E values 142 10.3.11 Setting up phase relationships 142 10.3.12 Finding reflections for phase determination 142 10.3.13 Assignment of starting phases 144 10.3.14 Phase determination and refinement 144 10.3.15 Figures of merit 144 10.3.16 Interpretation of maps 145 10.3.17 Completion of the structure 146 Exercises 147
11 An introduction to maximum entropy 149 11.1 Entropy 149 11.2 Maximum entropy 150 11.2.1 Calculations with incomplete data 150 11.2.2 Forming images 152 11.2.3 Entropy and probability 152 11.3 Electron-density maps 153
12 Least-squares fitting of parameters 155 12.1 Weighted mean 155 12.2 Linear regression 156 12.2.1 Variances and covariances 158 12.2.2 Restraints 158 12.2.3 Constraints 160 12.3 Non-linear least squares 162 12.4 Ill-conditioning 164 12.5 Computing time 165 Exercises 167
13 Refinement of crystal structures 169 13.1 Equations 169 13.1.1 Bragg’s law 170 13.1.2 Structure factors from the continuous electron density 170 13.1.3 Electron density from the structure amplitude and phase 170 13.1.4 Structure factor from a parameterized model 172 13.2 Reasons for performing refinement 172 13.2.1 To improve phasing so that computed electron density maps more closely represent the actual electron density 172 13.2.2 To try to verify that the structure is ‘correct’ 173 xiv Contents
13.2.3 To obtain the ‘best’ values for the parameters in the model 175 13.3 Data quality and limitations 175 13.3.1 Resolution 175 13.3.2 Completeness 176 13.3.3 Leverage 176 13.3.4 Weak reflections and systematic absences 176 13.3.5 Standard uncertainties 177 13.3.6 Systematic trends 177 13.4 Refinement fundamentals 177 13.4.1 w, the weight 178 13.4.2 Y1, the observations 178 13.4.3 Y2, the calculations 179 13.4.4 Issues 180 13.5 Refinement strategies 180 13.6 Under- and over-parameterization 182 13.6.1 Under-parameterization 182 13.6.2 Over-parameterization 183 13.7 Pseudo-symmetry, wrong space groups and Z > 1 structures 183 13.8 Conclusion 184 Exercises 186
14 Analysis of extended inorganic structures 189 14.1 Introduction 189 14.2 Disorder 190 14.2.1 Site-occupancy disorder 191 14.2.2 Positional disorder 192 14.2.3 Limits of Bragg diffraction 193 14.3 Phase transitions 194 14.4 Structure validation 195 14.5 Case history 1 – BiMg2VO6 196 14.6 Case history2–Mo2P4O15 199 Exercises 203
15 The derivation of results 205 15.1 Introduction 205 15.2 Geometry calculations 205 15.2.1 Fractional and Cartesian co-ordinates 205 15.2.2 Bond distance and angle calculations 207 15.2.3 Dot products 208 15.2.4 Transforming co-ordinates 208 15.2.5 Standard uncertainties 209 15.2.6 Assessing significant differences 211 15.3 Least-squares planes and dihedral angles 211 15.3.1 Conformation of rings and other molecular features 213 15.4 Hydrogen atoms and hydrogen bonding 213 Contents xv
15.5 Displacement parameters 214 15.5.1 βs, Bs and Us 215 15.5.2 ‘The equivalent isotropic displacement parameter’ 215 15.5.3 Symmetry and anisotropic displacement parameters 216 15.5.4 Models of thermal motion and geometrical corrections: rigid-body motion 217 15.5.5 Atomic displacement parameters and temperature 218 Exercises 219
16 Random and systematic errors 221 16.1 Random and systematic errors 221 16.2 Random errors and distributions 222 16.2.1 Measurement errors 222 16.2.2 Describing data 222 16.2.3 Theoretical distributions 225 16.2.4 Expectation values 227 16.2.5 The standard error on the mean 229 16.3 Taking averages 229 16.3.1 Testing for normality using a histogram 230 16.3.2 The χ 2 test for normality 231 χ 2 16.3.3 Averaging data when red 1 232 16.4 Weighting schemes 232 16.4.1 Weights used in least-squares refinement with single-crystal diffraction data 233 16.4.2 Robust-resistant weighting schemes and outliers 234 16.4.3 Assessing weighting schemes 235 16.5 Analysis of the agreement between observed and calculated data 238 16.5.1 R factors 238 16.5.2 Significance testing 239 16.6 Estimated standard deviations and standard uncertainties of structural parameters 240 16.6.1 Correlation and covariance 240 16.6.2 Uncertainty propagation 242 16.7 Systematic errors 242 16.7.1 Systematic errors in the data 243 16.7.2 Data thresholds 244 16.7.3 Errors and limitations of the model 244 16.7.4 Assessment of a structure determination 247 Exercises 250
17 Powder diffraction 251 17.1 Introduction to powder diffraction 251 17.2 Powder versus single-crystal diffraction 252 xvi Contents
17.3 Experimental methods 254 17.4 Information contained in a powder pattern 258 17.4.1 Phase identification 258 17.4.2 Quantitative analysis 259 17.4.3 Peak-shape information 260 17.4.4 Intensity information 261 17.5 Rietveld refinement 261 17.6 Structure solution from powder diffraction data 264 17.7 Non-ambient studies 265 Exercises 268
18 Introduction to twinning 271 18.1 Introduction 271 18.2 A simple model for twinning 271 18.3 Twinning in crystals 272 18.4 Diffraction patterns from twinned crystals 274 18.5 Inversion, merohedral and pseudo-merohedral twins 276 18.6 Derivation of twin laws 279 18.7 Non-merohedral twinning 280 18.8 The derivation of non-merohedral twin laws 282 18.9 Common signs of twinning 283 18.10 Examples 285 Exercises 296
19 The presentation of results 299 19.1 Introduction 299 19.2 Graphics 300 19.3 Graphics programs 300 19.4 Underlying concepts 301 19.5 Drawing styles 302 19.6 Creating three-dimensional illusions 306 19.7 The use of colour 307 19.8 Textual information in drawings 307 19.9 Some hints for effective drawings 308 19.10 Tables of results 309 19.11 The content of tables 310 19.11.1 Selected results 310 19.11.2 Redundant information 311 19.11.3 Additional entries 311 19.12 The format of tables 312 19.13 Hints on presentation 312 19.13.1 In research journals 312 19.13.2 In theses and reports 313 19.13.3 On posters 313 19.13.4 As oral presentations 313 19.13.5 On the web 314 19.14 Archiving of results 315 Contents xvii
20 The crystallographic information file (CIF) 319 20.1 Introduction 319 20.2 Basics 319 20.3 Uses of CIF 321 20.4 Some properties of the CIF format 321 20.5 Some practicalities 323 20.5.1 Strings 323 20.5.2 Text 324 20.5.3 Checking the CIF 325
21 Crystallographic databases 327 21.1 What is a database? 327 21.2 What types of search are possible? 327 21.3 What information can you get out? 328 21.4 What can you use databases for? 328 21.5 What are the limitations? 328 21.6 Short descriptions of crystallographic databases 328
22 X-ray and neutron sources 333 22.1 Introduction 333 22.2 Laboratory X-ray sources 333 22.3 Synchrotron X-ray sources 335 22.4 Neutron sources 339
A Appendix A: Useful mathematics and formulae 343 A.1 Introduction 343 A.2 Trigonometry 343 A.3 Complex numbers 344 A.4 Waves and structure factors 345 A.5 Vectors 346 A.6 Determinants 348 A.7 Matrices 348 A.8 Matrices in symmetry 349 A.9 Matrix inversion 350 A.10 Convolution 351
B Appendix B: Questions and answers 353
Index 385 This page intentionally left blank Introduction to diffraction 1 Peter Main
1.1 Introduction
The subsequent chapters in this book will assume some basic knowledge of crystal-structure determination. As readers will be at very different levels, we wish to make sure you have available some of the fundamen- tals of the subject that will be developed in the book. It is not necessary to understand everything in this introduction before reading further, but we hope that it will provide helpful reference material for some of the chapters.
1.2 X-ray scattering from electrons
The scattering of X-rays from electrons is called Thomson scattering. It occurs because the electron oscillates in the electric field of the incoming X-ray beam and an oscillating electric charge radiates electromagnetic waves. Thus, X-rays are radiated from the electron at the same frequency ◦ as the primary beam. However, most electrons radiate π radians (180 ) out of phase with the incoming beam, as shown by a mathematical model of the process. The motion of an electron is heavily damped when the X-ray frequency is close to the electron resonance frequency. This occurs near an absorption edge of the atom, changing the relative phase of the radiated X-rays to π/2 and giving rise to the phenomenon of anomalous (resonant) scattering. 8 Oxygen
1.3 X-ray scattering from atoms 6
There is a path difference between X-rays scattered from different parts of the same atom, resulting in destructive interference that depends f upon the scattering angle. This reduction in X-rays scattered from an atom with increasing angle is described by the atomic scattering fac- Carbon tor, illustrated in Fig. 1.1. The value of the scattering factor at zero scattering angle is equal to the number of electrons in the atom. The atomic scattering factors illustrated are for stationary atoms, but atoms 0 (sin u)/λ are normally subject to thermal vibration. This movement modifies the scattering factor and must always be taken into account. Fig. 1.1 Atomic scattering factors.
1 2 Introduction to diffraction
30 Sm If anomalous scattering takes place, the atomic scattering factor is altered to take this into account. This occurs when the X-ray frequency 20 f is close to the resonance frequency of an electron. Only some of the elec- trons in the atom are affected and they will scatter the X-rays roughly π 10 /2 out of phase with the incident beam. Electrons scattering exactly π/2 out of phase are represented mathematically by an imaginary com- 0 ponent of the scattering factor and they cease to contribute to the real part. The exact phase change is very sensitive to the X-ray frequency. –10 This is shown in Fig. 1.2 that displays the real and imaginary parts of Δf the contribution to the atomic scattering factor of the anomalously scat- tering electrons as a function of wavelength. The remaining electrons –20 in the atom are unaffected by this change in wavelength. Such informa- –30 tion on atomic scattering factors is obtained from quantum-mechanical calculations. 1.84 1.85 λ(Å) 1.4 X-ray scattering from a unit cell Fig. 1.2 Real (f ) and imaginary (f”) con- tributions to anomalous scattering for the X-rays scattered from each atom in the unit cell contribute to the overall example of a samarium atom. scattering pattern. Since each atom acts as a source of scattered X-rays, the waves will add constructively or destructively in varying amounts depending upon the direction of the diffracted beam and the atomic positions. This gives a complicated diffraction pattern whose amplitude and phase vary continuously, as can be seen in the two-dimensional optical analogue in Fig. 1.3.
1.5 The effects of the crystal lattice
The diffraction pattern of the crystal lattice is also a lattice, known as the reciprocal lattice. The name comes from the reciprocal relationship between the two lattices – large crystal lattice spacings result in small spacings in the reciprocal lattice and vice versa. The direct cell parame- ters are normally represented by a, b, c, α, β, γ and the reciprocal lattice ∗ ∗ ∗ ∗ ∗ ∗ ∗ parameters by a , b , c , α , β , γ . The direction of a is perpendicu- lar to the directions of b and c and its magnitude is reciprocal to the
Fig. 1.3 Holes in an opaque sheet and their optical diffraction pattern. 1.6 X-ray scattering from the crystal 3
∗ ∗ spacing of the lattice planes parallel to b and c; similarly for b and c . A two-dimensional example of the relationship between the direct and reciprocal lattices is shown in Fig. 1.4.
1.6 X-ray scattering from the crystal
A combination (convolution) of a single unit cell with the crystal lat- tice gives the complete crystal. The X-ray diffraction pattern is therefore given by the product of the scattering from the unit cell and the recip- rocal lattice, i.e. it is the scattering pattern of a single unit cell observed only at reciprocal lattice points. This can be seen in Fig. 1.5, which shows the unit cell of Fig. 1.3 repeated on a lattice and its corresponding diffrac- tion pattern. The underlying intensity is the same in both patterns. The positions of the reciprocal lattice points are given by the crystal lattice; the value of the diffraction pattern at a reciprocal lattice point is given by the atomic arrangement within the unit cell.
x10 0 1 1 1 2 2 3 3 h 4 5 y 6 7 k
Fig. 1.4 Direct lattice (left) and the corresponding reciprocal lattice (right).
Fig. 1.5 The unit cell of Fig. 1.3 repeated on a lattice and its diffraction pattern. 4 Introduction to diffraction
1.7 The structure-factor equation
There are many factors affecting the intensity of X-rays in the diffraction pattern. The one that depends only upon the crystal structure is called the structure factor. It can be expressed in terms of the contents of a single unit cell as: