the basics of crystallography and diffraction(third edirion), Study notes of Materials Physics

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I N T E R N A T I O N A L U N I O N O F C RY S TA L L O G R A P H Y T E X T S

O N C RY S TA L L O G R A P H Y

I U C r B O O K S E R I E S C O M M I T T E E

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

The Basics of

Crystallography

and

Diffraction

Third Edition

Christopher Hammond

Institute for Materials Research University of Leeds

INTERNATIONAL UNION OF CRYSTALLOGRAPHY

3 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 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 © Christopher Hammond 2009 First edition (1997) Second edition (2001) Third edition (2009) The moral rights of the author have been asserted Database right Oxford University Press (maker) 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 catalogue in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in the UK on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire

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vi Preface to the First Edition (1997)

nothing more than an extension of Bragg’s law. Finally, the important X-ray and electron diffraction techniques from polycrystalline materials are covered in Chapter 10. The Appendices cover material that, for ease of reference, is not covered in the text. Appendix 1 gives a list of items which are useful in making up crystal models and provides the names and addresses of suppliers. A rapidly increasing number of crystallography programs are becoming available for use in personal computers and in Appendix 2 I have listed those which involve, to a greater or lesser degree, some ‘self learning’ element. If it is the case that the computer program will replace the book, then one might expect that books on crystallography would be the first to go! That day, however, has yet to arrive. Appendix 3 gives brief biographical details of crystallographers and scientists whose names are asterisked in the text. Appendix 4 lists some useful geometrical relationships. Throughout the book the mathematical level has been maintained at a very simple level and with few minor exceptions all the equations have been derived from first principles. In my view, students learn nothing from, and are invariably dismayed and perplexed by, phrases such as ‘it can be shown that’—without any indication or guidance of how it can be shown. Appendix 5 sets out all the mathematics which are needed. Finally, it is my belief that students appreciate a subject far more if it is presented to them not simply as a given body of knowledge but as one which has been gained by the exertions and insight of men and women perhaps not much older than themselves. This therefore shows that scientific discovery is an activity in which they, now or in the future, can participate. Hence the justification for the historical references, which, to return to my first point, also help to show that science progresses, not by being made more complicated, but by individuals piecing together facts and ideas, and seeing relationships where vagueness and uncertainty existed before.

Preface to the Second Edition (2001)

In this edition the content has been considerably revised and expanded not only to provide a more complete and integrated coverage of the topics in the first edition but also to introduce the reader to topics of more general scientific interest which (it seems to me) flow naturally from an understanding of the basic ideas of crystallography and diffraction. Chapter 1 is extended to show how some more complex crystal structures can be understood in terms of different faulting sequences of close-packed layers and also covers the various structures of carbon, including the fullerenes, the symmetry of which finds expression in natural and man-made forms and the geometry of polyhedra. In Chapter 2 the figures have been thoroughly revised in collaboration with Dr K. M. Crennell including additional ‘familiar’ examples of patterns and designs to provide a clearer understanding of two-dimensional (and hence three-dimensional) symmetry. I also include, at a very basic level, the subject of non-periodic patterns and tilings which also serves as a useful introduction to quasiperiodic crystals in Chapter 4. Chapter 3 includes a brief discussion on space-filling (Voronoi) polyhedra and in Chapter 4 the section on space groups has been considerably expanded to provide the reader with a much better starting-point for an understanding of the Space Group representation in Vol. A of the International Tables for Crystallography.

Preface to Third Edition (2009) vii

Chapters 5 and 6 have been revised with the objective of making the subject-matter more readily understood and appreciated. In Chapter 7 I briefly discuss the human eye as an optical instrument to show, in a simple way, how beautifully related are its structure and its function. The material in Chapters 9 and 10 of the first edition has been considerably expanded and re-arranged into the present Chapters 9, 10 and 11. The topics of X-ray and neu- tron diffraction from ordered crystals, preferred orientation (texture or fabric) and its measurement are now included in view of their importance in materials and earth sciences. The stereographic projection and its uses is introduced at the very end of the book (Chapter 12)—quite the opposite of the usual arrangement in books on crystallography. But I consider that this is the right place: for here the usefulness and advantages of the stereographic projection are immediately apparent whereas at the beginning it may appear to be merely a geometrical exercise. Finally, following the work of Prof. Amand Lucas, I include in Chapter 10 a sim- ulation by light diffraction of the structure of DNA. There are, it seems to me, two landmarks in X-ray diffraction: Laue’s 1912 photograph of zinc blende and Franklin’s 1952 photograph of DNA and in view of which I have placed these ‘by way of symmetry’ at the beginning of this book.

Preface to Third Edition (2009)

I have considerably expanded Chapters 1 and 4 to include descriptions of a much greater range of inorganic and organic crystal structures and their point and space group sym- metries. Moreover, I now include in Chapter 2 layer group symmetry—a topic rarely found in textbooks but essential to an understanding of such familiar things as the patterns formed in woven fabrics and also as providing a link between two- and three-dimensional symmetry. Chapters 9 and 10 covering X-ray diffraction techniques have been (partially) updated and include further examples but I have retained descriptions of older techniques where I think that they contribute to an understanding of the geometry of diffraction and reciprocal space. Chapter 11 has been extended to cover Kikuchi and EBSD patterns and image formation in electron microscopy. A new chapter (Chapter 13) introduces the basic ideas of Fourier analysis in X-ray crystallography and image formation and hence is a development (requiring a little more mathematics) of the elementary treatment of those topics given in Chapters 7 and 9. The Appendices have been revised to include polyhedra in crystallography in order to complement the new material in Chapter 1 and the biographical notes in Appendix 3 have been much extended. It may be noticed that many of the books listed in ‘Further Reading’ are very old. However, in many respects, crystallography is a ‘timeless’ subject and such books to a large extent remain a valuable source of information. Finally, I have attempted to make the Index sufficiently detailed and comprehen- sive that a reader will readily find those pages which contain the information she or he requires.

Contents

X-ray photograph of zinc blende (Friedrich, Knipping and von Laue, 1912) xiv

4 Crystal symmetry: point groups, space groups,

xii Contents

• 1 Crystals and crystal structures X-ray photograph of deoxyribonucleic acid (Franklin and Gosling, 1952) xv
  • 1.1 The nature of the crystalline state
  • 1.2 Constructing crystals from close-packed hexagonal layers of atoms
  • 1.3 Unit cells of the hcp and ccp structures
  • 1.4 Constructing crystals from square layers of atoms
  • 1.5 Constructing body-centred cubic crystals
  • 1.6 Interstitial structures
  • 1.7 Some simple ionic and covalent structures
  • 1.8 Representing crystals in projection: crystal plans
  • 1.9 Stacking faults and twins
  • 1.10 The crystal chemistry of inorganic compounds
    • 1.10.1 Bonding in inorganic crystals
    • 1.10.2 Representing crystals in terms of coordination polyhedra
  • 1.11 Introduction to some more complex crystal structures - structures 1.11.1 Perovskite (CaTiO 3 ), barium titanate (BaTiO 3 ) and related - alumina 1.11.2 Tetrahedral and octahedral structures—silicon carbide and
    • 1.11.3 The oxides and oxy-hydroxides of iron
    • 1.11.4 Silicate structures
    • 1.11.5 The structures of silica, ice and water
    • 1.11.6 The structures of carbon
  • Exercises
• 2 Two-dimensional patterns, lattices and symmetry
  • 2.1 Approaches to the study of crystal structures
  • 2.2 Two-dimensional patterns and lattices
  • 2.3 Two-dimensional symmetry elements
  • 2.4 The five plane lattices
  • 2.5 The seventeen plane groups
  • 2.6 One-dimensional symmetry: border or frieze patterns
  • 2.7 Symmetry in art and design: counterchange patterns
  • 2.8 Layer (two-sided) symmetry and examples in woven textiles
  • 2.9 Non-periodic patterns and tilings
  • Exercises
• 3 Bravais lattices and crystal systems x Contents
  • 3.1 Introduction
  • 3.2 The fourteen space (Bravais) lattices
  • 3.3 The symmetry of the fourteen Bravais lattices: crystal systems
    • space-filling polyhedra 3.4 The coordination or environments of Bravais lattice points:
  • Exercises
  • crystals symmetry-related properties and quasiperiodic
  • 4.1 Symmetry and crystal habit
  • 4.2 The thirty-two crystal classes
  • 4.3 Centres and inversion axes of symmetry
  • 4.4 Crystal symmetry and properties
  • 4.5 Translational symmetry elements
  • 4.6 Space groups
  • 4.7 Bravais lattices, space groups and crystal structures
  • 4.8 The crystal structures and space groups of organic compounds
    • 4.8.1 The close packing of organic molecules
    • 4.8.2 Long-chain polymer molecules
  • 4.9 Quasiperiodic crystals or crystalloids
  • Exercises
  • Miller indices and zone axis symbols 5 Describing lattice planes and directions in crystals:
  • 5.1 Introduction
  • 5.2 Indexing lattice directions—zone axis symbols
  • 5.3 Indexing lattice planes—Miller indices
  • 5.4 Miller indices and zone axis symbols in cubic crystals
  • 5.5 Lattice plane spacings, Miller indices and Laue indices
  • 5.6 Zones, zone axes and the zone law, the addition rule
    • 5.6.1 The Weiss zone law or zone equation
    • 5.6.2 Zone axis at the intersection of two planes
    • 5.6.3 Plane parallel to two directions
    • 5.6.4 The addition rule
    • Weber symbols and Miller-Bravais indices 5.7 Indexing in the trigonal and hexagonal systems:
  • 5.8 Transforming Miller indices and zone axis symbols
    • lattices 5.9 Transformation matrices for trigonal crystals with rhombohedral
  • 5.10 A simple method for inverting a 3 × 3 matrix
  • Exercises
• 6 The reciprocal lattice Contents xi
  • 6.1 Introduction
  • 6.2 Reciprocal lattice vectors
  • 6.3 Reciprocal lattice unit cells
  • 6.4 Reciprocal lattice cells for cubic crystals
    • lattice vectors 6.5 Proofs of some geometrical relationships using reciprocal
    • 6.5.1 Relationships between a , b , c and a ∗ , b ∗ , c ∗
    • 6.5.2 The addition rule
    • 6.5.3 The Weiss zone law or zone equation
    • 6.5.4 d -spacing of lattice planes ( hkl )
    • 6.5.5 Angle ρ between plane normals (h 1 k 1 l 1 ) and (h 2 k 2 l 2 )
    • 6.5.6 Definition of a ∗ , b ∗ , c ∗ in terms of a , b , c
    • 6.5.7 Zone axis at intersection of planes (h 1 k 1 l 1 ) and (h 2 k 2 l 2 )
    • 6.5.8 A plane containing two directions [u 1 v 1 w 1 ] and [u 2 v 2 w 2 ]
  • 6.6 Lattice planes and reciprocal lattice planes
  • 6.7 Summary
  • Exercises
• 7 The diffraction of light
  • 7.1 Introduction
  • 7.2 Simple observations of the diffraction of light
  • 7.3 The nature of light: coherence, scattering and interference
    • and nets 7.4 Analysis of the geometry of diffraction patterns from gratings
    • microscope and the eye 7.5 The resolving power of optical instruments: the telescope, camera,
  • Exercises
  • W. H. and W. L. Bragg and P. P. Ewald 8 X-ray diffraction: the contributions of Max von Laue,
  • 8.1 Introduction
  • 8.2 Laue’s analysis of X-ray diffraction: the three Laue equations
  • 8.3 Bragg’s analysis of X-ray diffraction: Bragg’s law
  • 8.4 Ewald’s synthesis: the reflecting sphere construction
  • Exercises
• 9 The diffraction of X-rays
  • 9.1 Introduction
    • equation and its applications 9.2 The intensities of X-ray diffracted beams: the structure factor
    • nodes 9.3 The broadening of diffracted beams: reciprocal lattice points and
    • 9.3.1 The Scherrer equation: reciprocal lattice points and nodes
    • 9.3.2 Integrated intensity and its importance
      • coherence length 9.3.3 Crystal size and perfection: mosaic structure and
  • 9.4 Fixed θ, varying λ X-ray techniques: the Laue method
    • precession methods 9.5 Fixed λ, varying θ X-ray techniques: oscillation, rotation and
    • 9.5.1 The oscillation method
    • 9.5.2 The rotation method
    • 9.5.3 The precession method
  • 9.6 X-ray diffraction from single crystal thin films and multilayers
  • 9.7 X-ray (and neutron) diffraction from ordered crystals
  • 9.8 Practical considerations: X-ray sources and recording techniques
    • 9.8.1 The generation of X-rays in X-ray tubes
    • 9.8.2 Synchrotron X-ray generation
    • 9.8.3 X-ray recording techniques
  • Exercises
• 10 X-ray diffraction of polycrystalline materials
  • 10.1 Introduction
    • diffraction techniques 10.2 The geometrical basis of polycrystalline (powder) X-ray
    • polycrystalline materials 10.3 Some applications of X-ray diffraction techniques in
    • 10.3.1 Accurate lattice parameter measurements
    • 10.3.2 Identification of unknown phases
    • 10.3.3 Measurement of crystal (grain) size
    • 10.3.4 Measurement of internal elastic strains
  • 10.4 Preferred orientation (texture, fabric) and its measurement
    • 10.4.1 Fibre textures
    • 10.4.2 Sheet textures
  • 10.5 X-ray diffraction of DNA: simulation by light diffraction
  • 10.6 The Rietveld method for structure refinement
  • Exercises
• 11 Electron diffraction and its applications
  • 11.1 Introduction
  • 11.2 The Ewald reflecting sphere construction for electron diffraction
  • 11.3 The analysis of electron diffraction patterns
  • 11.4 Applications of electron diffraction - 11.4.1 Determining orientation relationships between crystals Contents xiii - 11.4.2 Identification of polycrystalline materials - 11.4.3 Identification of quasiperiodic crystals
    • 11.5 Kikuchi and electron backscattered diffraction (EBSD) patterns
      • 11.5.1 Kikuchi patterns in the TEM - in the SEM 11.5.2 Electron backscattered diffraction (EBSD) patterns
    • 11.6 Image formation and resolution in the TEM
    • Exercises
  • 12 The stereographic projection and its uses
    • 12.1 Introduction
    • 12.2 Construction of the stereographic projection of a cubic crystal
    • 12.3 Manipulation of the stereographic projection: use of the Wulff net
    • 12.4 Stereographic projections of non-cubic crystals
    • 12.5 Applications of the stereographic projection
      • 12.5.1 Representation of point group symmetry
      • 12.5.2 Representation of orientation relationships
      • 12.5.3 Representation of preferred orientation (texture or fabric)
    • Exercises
  • 13 Fourier analysis in diffraction and image formation
    • 13.1 Introduction—Fourier series and Fourier transforms
    • 13.2 Fourier analysis in crystallography
    • 13.3 Analysis of the Fraunhofer diffraction pattern from a grating
    • 13.4 Abbe theory of image formation - crystallography Appendix 1 Computer programs, models and model-building in
• Appendix 2 Polyhedra in crystallography - in the text Appendix 3 Biographical notes on crystallographers and scientists mentioned
• Appendix 4 Some useful crystallographic relationships - their use in crystallography Appendix 5 A simple introduction to vectors and complex numbers and - diffraction in electron diffraction patterns Appendix 6 Systematic absences (extinctions) in X-ray diffraction and double
• Answers to Exercises
  • Further Reading
  • Index

X-ray photograph of deoxyribonucleic acid

The photograph of the ‘B’ form of DNA taken by Rosalind Franklin and Raymond Gosling in May 1952 and published, together with the two papers by J. D. Watson and F. H. C. Crick and M. H. F. Wilkins, A. R. Stokes and H. R. Wilson, in the 25 April issue of Nature, 1953, under the heading ‘Molecular Structure of Nucleic Acids’. The specimen is a fibre (axis vertical) containing millions of DNA strands roughly aligned parallel to the fibre axis and separated by the high water content of the fibre; this is the form adopted by the DNA in living cells. The X-ray beam is normal to the fibre and the diffraction pattern is characterised by four lozenges or diamond-shapes outlined by fuzzy diffraction haloes and separated by two rows or arms of spots radiating out- wards from the centre. These two arms are characteristic of helical structures and the angle between them is a measure of the ratio between the width of the molecule and the repeat-distance of the helix. But notice also the sequence of spots along each arm; there is a void where the fourth spot should be and this ‘missing fourth spot’ not only indicates that there are two helices intertwined but also the separation of the helices along the chain. Finally, notice that there are faint diffraction spots in the two side lozenges, but not in those above and below, an observation which shows that the sugar-phosphate ‘backbones’ are on the outside, and the bases on the inside, of the molecule. This photograph provided the crucial experimental evidence for the correctness of Watson and Crick’s structural model of DNA—a model not just of a crystal structure but one which shows its inbuilt power of replication and which thus unlocked the door to an understanding of the mechanism of transmission of the gene and of the evolution of life itself.

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2 Crystals and crystal structures

A

A

B

B

H I K L

C

C

D

D

E

E

161 F^ G

1 32

Fig. 2

a b (^) c

d

Fig. 1.1. ‘Scheme VII’(from Hooke’s Micrographia, 1665), showing crystals in a piece of broken flint (Upper—Fig. 1), crystals from urine (Lower—Fig. 2) and hypothetical sketches of crystal structures A–L arising from the packing together of ‘bullets’.

1.1 The nature of the crystalline state 3

observations of the angles between the faces of quartz crystals, but was developed much more fully as a general law by Rome de L’Isle∗^ in a treatise entitled Cristallographie in

1783. He measured the angles between the faces of carefully-made crystal models and proposed that each mineral species therefore had an underlying ‘characteristic primitive form’. The notion that the packing of the underlying building blocks determines both the shapes of crystals and the angular relationships between the faces was extended by René Just Haüy.∗^ In 1784 Haüy showed how the different forms (or habits) of dog-tooth spar (calcite) could be precisely described by the packing together of little rhombs which he called ‘molécules intégrantes’ (Fig. 1.2). Thus the connection between an internal order and an external symmetry was established. What was not realized at the time was that an internal order could exist even though there may appear to be no external evidence for it.

E

f

d

h

s

s 

Fig. 1.2. Haüy’s representation of dog-tooth spar built up from rhombohedral ‘molécules integrantes’ (from Essai d’une théorie sur la structure des cristaux, 1784).

∗ (^) Denotes biographical notes available in Appendix 3.