This is a textbook for the senior undergraduate or graduate student beginning a serious study of X-ray crystallography It will be of interest both to those intending to become professional crystallographers and to those physicists, chemists, biologists, geologists, metallurgists and others who will use it as a tool in their research All major aspects of crystallography are covered - the geometry of crystals and their symmetry, theoretical and practical aspects of diffracting X-rays by crystals and how the data may be analysed to find the symmetry of the crystal and its structure Recent advances are fully covered, including the synchrotron as a source of X-rays, methods of solving structures from powder data and the full range of techniques for solving structures from single-crystal data A suite of computer programs is provided for carrying out many operations of data-processing and solving crystal structures - including by direct methods While these are limited to two dimensions they fully illustrate the characteristics of three-dimensional work These programs are required for many of the problems given at the end of each chapter but may also be used to create new problems by which students can test themselves or each other An introduction to X-ray crystallography An introduction to X-ray crystallography SECOND EDITION M.M WOOLFSON Emeritus Professor of Theoretical Physics University of York CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1970, 1997 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 1970 Second edition 1997 Typeset in Times 10/12 pt A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data Woolfson, M M An introduction to X-ray crystallography / M.M Woolfson - 2nd ed p cm Includes bibliographical references and index ISBN 521 41271 (hardcover) - ISBN 521 42359 (pbk.) X-ray crystallography I Title QD945.W58 1997 548'.83-dc20 96-5700 CIP ISBN 521 41271 hardback ISBN 521 42359 paperback Transferred to digital printing 2003 Contents Preface to the First Edition Preface to the Second Edition LI 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 The geometry of the crystalline state Page x xii The general features of crystals The external symmetry of crystals The seven crystal systems The thirty-two crystal classes The unit cell Miller indices Space lattices Symmetry elements Space groups Space group and crystal class Problems to Chapter 1 12 15 16 20 23 30 31 The scattering of X-rays 32 A general description of the scattering process Scattering from a pair of points Scattering from a general distribution of point scatterers Thomson scattering Compton scattering The scattering of X-rays by atoms Problems to Chapter 32 34 36 37 42 43 48 Diffraction from a crystal 50 Diffraction from a one-dimensional array of atoms Diffraction from a two-dimensional array of atoms Diffraction from a three-dimensional array of atoms The reciprocal lattice Diffraction from a crystal - the structure factor Bragg's law The structure factor in terms of indices of reflection Problems to Chapter 50 56 57 59 64 67 72 74 The Fourier transform 76 The Fourier series Numerical application of Fourier series 76 79 viii Contents 4.3 4.4 4.5 4.6 4.7 4.8 Fourier series in two and three dimensions The Fourier transform Diffraction and the Fourier transform Convolution Diffraction by a periodic distribution The electron-density equation Problems to Chapter 83 85 92 94 99 99 106 Experimental collection of diffraction data 108 The conditions for diffraction to occur The powder camera The oscillation camera The Weissenberg camera The precession camera The photographic measurement of intensities Diffractometers X-ray sources Image-plate systems The modern Laue method Problems to Chapter 108 112 118 125 130 135 140 143 150 151 154 The factors affecting X-ray intensities 156 6.1 6.2 6.3 6.4 6.5 6.6 Diffraction from a rotating crystal Absorption of X-rays Primary extinction Secondary extinction The temperature factor Anomalous scattering Problems to Chapter 156 162 169 173 175 179 188 The determination of space groups 190 Tests for the lack of a centre of symmetry The optical properties of crystals The symmetry of X-ray photographs Information from systematic absences Intensity statistics Detection of mirror planes and diad axes Problems to Chapter 190 196 208 210 215 227 229 The determination of crystal structures 231 Trial-and-error methods The Patterson function The heavy-atom method Isomorphous replacement The application of anomalous scattering Inequality relationships Sign relationships 231 233 249 255 267 274 282 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 7.1 7.2 7.3 7.4 7.5 7.6 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Contents 8.8 8.9 9.1 9.2 9.3 9.4 9.5 General phase relationships A general survey of methods Problems to Chapter 290 297 298 Accuracy and refinement processes 301 The determination of unit-cell parameters The scaling of observed data Fourier refinement Least-squares refinement The parameter-shift method Problems to Chapter 301 307 309 317 320 322 Physical constants and tables Appendices Program listings I STRUCFAC II FOUR1 III SIMP1 IV FOUR2 V FTOUE VI HEAVY VII ISOFILE VIII ISOCOEFF IX ANOFILE X PSCOEFF XI MINDIR XII CALOBS Solutions to Problems References Bibliography Index 325 327 328 333 335 336 339 346 349 350 352 353 354 366 367 395 397 399 Preface to the First Edition In 1912 von Laue proposed that X-rays could be diffracted by crystals and shortly afterwards the experiment which confirmed this brilliant prediction was carried out At that time the full consequences of this discovery could not have been fully appreciated From the solution of simple crystal structures, described in terms of two or three parameters, there has been steady progress to the point where now several complex biological structures have been solved and the solution of the structures of some crystalline viruses is a distinct possibility X-ray crystallography is sometimes regarded as a science in its own right and, indeed, there are many professional crystallographers who devote all their efforts to the development and practice of the subject On the other hand, to many other scientists it is only a tool and, as such, it is a meeting point of many disciplines - mathematics, physics, chemistry, biology, medicine, geology, metallurgy, fibre technology and several others However, for the crystallographer, the conventional boundaries between scientific subjects often seem rather nebulous In writing this book the aim has been to provide an elementary text which will serve either the undergraduate student or the postgraduate student beginning seriously to study the subject for the first time There has been no attempt to compete in depth with specialized textbooks, some of which are listed in the Bibliography Indeed, it has also been found desirable to restrict the breadth of treatment, and closely associated topics which fall outside the scope of the title - for example diffraction from semi- and non-crystalline materials, electron- and neutron diffraction - have been excluded For those who wish to go no further it is hoped that the book gives a rounded, broad treatment, complete in itself, which explains the principles involved and adequately describes the present state of the subject For those who wish to go further it should be regarded as a foundation for further study It has now become clear that there is wide acceptance of the SI system of units and by-and-large they are used in this book However the angstrom unit has been retained as a unit of length for X-ray wavelengths and unit-cell dimensions etc., since a great deal of the basic literature uses this unit A brief explanation of the SI system and some important constants and equations are included in the section Physical constants and tables on pp 325-326 I am deeply indebted to Dr M Bown and Dr S G Fleet of the Department of Mineralogy, University of Cambridge and to my colleague, Dr P Main, for reading the manuscript and for their helpful criticism which included suggestions for many improvements of treatment Solutions to Problems 388 Fig VIII-6 The anomalous-difference Patterson map FACTOR CONVERTING TO DENSITY/UNIT AREA IS 0.0065 OUTPUT WITH Y HORIZONTAL AND ORIGIN TOP LEFT o 100 72 23 -1 -16 -15 -3 -12 kh -10 -16 -27 -16 -9 -15 -36 -45 -30 -10 -2 -24 -32 -15 -7 -19 -2 -18 -30 -28 -11 12 19 16 -7 -29 -24 23 10 16 11 15 12 * • « ; 37.•>o -5 -11 \?o 18 14 -24 -33 -32 K ;18 10 11 12 13 14 -1 - 91 q 10 19 -8 -12 •Si 10 18 16 -2 -l -3 -10 14 16 -16 -3 -6 -10 -6 -8 -14 -4 / -12 -8 16 10 -8 -3 15 -4 - -3 -21 - 1 t ^ 37 20,1 -4 -25 -18 ^ V3 -36 -30 -12 -25 -4 20 & * -4 0.\b -32 -33 -24 i-r /o -20 -14 -12 y '11 -4 -34 '-39 -17 -5 -14 -14 -2 -4 19 23 \ -4.-14 -8 -6 -10 -6 19 21 -2 inn - 12 -11 -26 -26 -15 -1 11 17 12 -2 \ 25 ) 10 -8 -20 -20.-12 -4 19 20 f 16 -10 -15 12 2 ' \ f/\ -3 \ » -16 -1 n -11 20 -13 -33 -1 11 7 -8 -23 -12 16 23 - R -12 V \ -6 - -36 -18 -9 -17 - -8 19 - it 37 30 -8 -1 16 33 2 -4 -1 -5 -13 - -2 -3 -12 -11 -9 - - -4 - -9 11 26 - 1 -2 - -18 -24 -in 18 17 28 13 16 - -10 -?R -22 -9 15 33 - -22 -38 -28 -8 -10 -23 - -13 14 17 - -12 - -9 .-33 % 11 -30 -27 -9 29 1R -7 -10 -28 -23 13 -7 -18 32 -6 -11 28 -8 ^Z -8 -4 -4 -6 12 -7 -16 -11 - -8 -18 -11 -9 -13 -3 -9 - -11 - -13 -11 -18 11 31 -15 -8 -9 \ -6 -5 n -7 26 32 -3 -10 - 1 27 5?V 28 30 -3 -5 :/, -10 \ -5 ^22 -16 -6 -11 -5 -10 (7? 49\ * 23 11 -1 - - - - - - - -14 -2 -12 -3 10 N -2 13 - t - -5 - ^54 ; - "~23 - - - - -17 -27 -10 20 24 10 13 12 ^ -7 -6 -3 -4 - - - "13 14 -17 - 13 -3 10 11 c, \ 10 - 13 —a* -3 10 -10 12 "28 11 17 -11 ,-20 -15 12 16 « MINDIR 1.666 832.9 268.0 TOP LEFT ™L -9 11 26 13 - 10 - -9 IB _-, - Solutions to Problems 391 Table VIII-1 sign 10 12 10 11 12 13 0 0 0 10 11 12 13 10 11 12 13 / sign 2 2 2 2 2 3 3 3 3 3 3 + — — — + — + u + — — _ — + — + + + + + + u / 10 11 12 10 sign / 11 4 4 4 4 4 4 4 5 5 5 5 5 sign + + u = unobserved Table VII-2 THE SEED USED IS 3000 Z SOLN CFOM A 296 1.315 670.3 544 0.880 617.7 358 1.026 563.7 491 1.263 779.4 286 1.144 573.6 11 650 1.259 889.6 13 454 1.201 722.1 15 316 1.035 538.6 17 416 1.206 698.0 19 284 1.057 527.2 21 0 1000 000 23 821 080 916.2 25 355 079 588.6 27 659 1.245 887.8 29 464 0.950 598.1 31 393 077 614.2 33 268 666 832.9 35 349 1.317 707.8 37 651 1.176 846.7 39 409 048 610.2 41 443 204 715.1 43 528 1.263 805.9 45 1000.0 1000 1.000 47 475 1.045 655.3 49 381 1.271 706.7 SOLN 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 A 646.9 748,8 608.7 693.2 759.8 758.9 754.8 730.9 640.0 800.3 543.3 479.7 1000.0 565.3 770.3 698.5 810.1 659.9 784.5 711.3 800.8 634.9 664.3 524.3 668.3 Z 370.3 348.6 428.2 450.5 543.2 601.1 440.1 367.7 513.5 597.2 319.1 394.2 1000.0 411.5 608.2 557.7 513.1 411.6 571.9 356.1 505.2 421.3 258.9 481.8 549.7 CFOM 1.171 1.396 1.020 1.152 1.155 1.075 1.284 1.336 0.964 1.160 1.041 0.817 1.000 0.959 1.087 1.017 1.292 1.140 1.163 1.314 1.285 1.079 1.355 0.785 0.970 392 Solutions to Problems Chapter 9.1 The calculation using equations (9.2) and (9.4) gives the results presented in table IX-1 Table IX-1 h k e sin a(A) 7 6 64° 27' 66° 35' 70° 35' 77° 45' 64° 55' 73° 54' 0.8140 0.8421 0.8895 0.9560 0.8203 0.9231 6.013 6.012 6.008 6.005 6.020 6.016 A plot of the determined values of a against sin2 shows considerable scatter, in particular the last two points give values which are far too high A best straight line for the other points gives a = 6.002 ± 0.001 A Try the calculation again with a/b = 0.747 9.2 In the (hOl) layer of the reciprocal lattice we have s2 = 4S11^ ° = h2a*2 + l2c*2 + 2hla*c*cosP* A From the (300), (400) and (500) reflections one finds an average value of a* = 0.23091 A " and the next four reflections similarly give c* = 0.16496A" The final reflections give cos/?* from O5k (4 sin 8/A2) - h2a*2 - l2c*2 and the mean value of cos /?* is 0.499 97 which gives /?* = 60° 0' For a monoclinic lattice P = 180° - j8* = 120° 0' We now find a ^ ^ s i n j S ) " = 5.001 A and 9.3 The data is plotted on a reciprocal-lattice net and is divided into regions of sin 6/1 enabling table IX-2 to be constructed Plotting ln«/>/Z) indicates that the innermost point does not fit the general trend A straight line as close as possible to the other points gives K = 0.30 and B = 2.6 A" 9.4 (i) 0.0874eA" , (ii) 0.0060A Solutions to Problems 393 Table IX-2 sin0 A /sin 0\ \ *1 \ * 28/ c 12/O2 I ln2 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.025 0.065 0.125 0.205 0.305 481 235 129 85 72 483 265 143 81 50 964 500 272 166 122 99.1 80.6 59.3 14.7 -2.28 /sin 0\ * 0.158 0.255 0.353 0.452 0.552 -1.83 -1.52 -2.46 -2.84 7.0 9.5 CALOBS gives a reliability index of 0.401 for the original estimated coordinates and the map shown in fig IX-1 Revised coordinates estimated from the map are: Na (0.943,0.255), Na (0.475,0.605), O (0.014,0.690), N (0.196,0.390), C (0.114,0.200), C (0.391,0.155), C (0.333,0.390) Fig IX-1 The map obtained as the first stage in a Fourier refinement procedure FACTOR CONVERTING TO DENSITY/UNIT AREA I S TOP OUTPUT WITH Y HORIZONTAL AND ORIGIN 0.1016 LEFT 10 11 -8 H- - 1 - - -2 -12 -8 2 21./ \ 23.s 12 29 18 1 16 49 16 11 13 19 20 ' * - 49 11 12 18> s 30 ' \ 15 - -3 - - \ 19/ 15 - - -6 15 17 16 -1 -1 -2 -3 10 27 15 15 -8 -8 -6 -5 12 12* -7 -2 10 -5 -1 -1 -6 -1 -1 -10.- 13 -1 10 11 V -2 -5 -6 -7 -4 -5 13 13 16 -1 )v 17 -7 18 15 19 15 17 20 10 13 21 22 \ \ \ 22 < 14 N if/ % isb 38 > • • -6 -2 15 -2 24 14 14 10 -8 -8 -6 -5 12 16 -1 -1 -2 -3 10 12 23 11 10 8 -1 -6 8 \6 i 14 13 -1 -1 -5 -7 -4 -6 -2 -5 -1 20 11 12 17 22 17 15 8 -2 -7 -4 - 10 1 32- ^ o ' > \1 -2 -1 24 -2 -5 16 25 -1 12 26 27 28 - 15 4 10 -3 -10 -1 ,.48-^7^,9 ^ A \\10 LtK)r-94>\38 /S7\ ^V 40 ' 12 ' q 11 30 49 35 16 / " _15< 1 11 13 16 ,2f 21/ 29 3 10 12 / 16 ,30 46' 13 -10 - -1 -5 -1 -1 12 -7 22< *27~ 31 j 15 22 v 10 ^0^-1^ 10 ^ ^ 20{ 19 -1 1 21*- 12 18 M -t / 29 19r-15 15' - - - f -2 5 / I - 1 - -10 - 21 A - 40-* 7 15-r - 23 / L -2 5>, 1? / 2C 10 53 /°' 11 3*2 ( ^ 24N -1 92 15 -8- 18 7? 15 10 -1 -1 12 - -1 \ -4 11 ?*=• -7 20 -2 r^ - ior 15 23 14 12 12 .11 - 5 s 15 16 9 -2 ^ -1 11^ 21 \ \ 15 10 10 17 -1 49 ^ ^ ^ 22 24* l) 10 ^ \ " -2 17 13 20 12 19 13 1 -"o" -6 18 10 8 -2 17 16 16 i i ? ^21_._ 14" 11 15 12 x \ 14 13 •} s •s 16 ^ \ 12 -12 : 7" -8 -\ -8 q A, 394 Solutions to Problems Structure factors calculated with these new estimates gave a reliability index of 0.360 It should be noted that the origin is free to move along the y direction in this space group and that as the refinement proceeds there may be a gradual drift of y coordinates towards higher or lower values However, this does not prevent the refinement from taking place References Abrahams, S.C & Robertson, J.M (1948), Acta Cryst 1, 252 Alcock, N.W & Sheldrick, G.M (1967), Acta Cryst 23, 35 Archer, E M (1948), Ada Cryst 1, 64 Baker, T W., George, J.D., Bellamy, B A & Causer, R (1966), Nature 210, 720 Beevers, C.A & Lipson, H.S (1934), Phil Mag 17, 825-6 Bertaut, E.F (1955), Ada Cryst 8, 544 Bhuiya, A.K & Stanley, E (1963), Ada Cryst 16, 981 Blow, D M & Crick, F H C (1959), Ada Cryst 12, 794 Bokhoven, C, Schoone, J.C & Bijvoet, J.M (1951), Ada Cryst 4, 275 Bond, W.L (1960), Ada Cryst 13, 814 Bragg, W L (1913), Proc Camb Phil Soc 17, 43 Bragg, W.L, James, R W & Bosanquet, C.H (1921), Phil Mag 41, 309 Brown, G M & Bortner, M H (1954), Ada Cryst 7, 139 Buerger, M J (1950a), Ada Cryst 3, 465 Buerger, M J (1950b), Proc Nat Acad Sci 36, 376 Clews, C.J.B & Cochran, W (1948), Ada Cryst 1, Cochran, W (1951), Ada Cryst 4, 376 Cochran, W (1952), Ada Cryst 5, 65 Cochran, W (1953), Ada Cryst 6, 260 Cochran, W (1955), Ada Cryst 8, 473 Cochran, W & Penfold, B R (1952), Ada Cryst 5, 644 Cochran, W & Woolfson, M M (1955), Ada Cryst 8, Cruickshank, D.W.J, Pilling, D.E, Bujosa, A, Lovell, F.M & Truter, M.R (1961), Computing Methods and the Phase Problem in X-ray Crystal Analysis Oxford: Pergamon de Jong, W.F & Bouman, J (1938), Z Krist 98, 456 Dunitz, J.D (1949), Ada Cryst 2, Dutta, S.N & Woolfson, M.M (1961), Ada Cryst 14, 178 Farquhar, M.C.M & Lipson, H (1946), Proc Phys Soc Lond 58, 200 Fiocco, G & Thompson, E (1963), Phys Rev Letters 10, 89 Foster, F & Hargreaves, A (1963a), Acta Cryst 16, 1124 Foster, F & Hargreaves, A (1963b), Acta Cryst 16, 1133 Fowweather, F & Hargreaves, A (1950), Acta Cryst 3, 81 Friedrich, W, Knipping, P & von Laue, M (1912) Interferenz-Ersceinungen bei Rontgenstrahlen Sitsungsberichte der Kgl Bayerischen Akademie der Wissenschaften zu Miinchen, pp 303-22 Groth, P (1910), Chemische Kristallographie Leipzig: Engelmann Harker, D (1936), J Chem Phys 4, 381 Harker, D & Kasper, J S (1948), Acta Cryst 1, 70 Honl, H (1933), Ann Phys (Leipzig) 18, 625-57 Howells, E.R, Phillips, D.C & Rogers, D (1950), Acta Cryst 3, 210 Hughes, E.W (1941), J Am Chem Soc 63, 1737 395 396 References International Tables for X-ray Crystallography, Vols I, II, III Birmingham: The Kynoch Press International Tablesfor Crystallography, Vol A, Space-Group Symmetry Dordrecht: Reidel International Tablesfor Crystallography, Vol B, Reciprocal Space Dordrecht: Reidel International Tables for Crystallography, Vol C, Mathematical, Physical and Chemical Tables Dordrecht: Reidel Karle, I L & Karle, J (1964a), Acta Cryst 17, 1356 Karle, I L & Karle, J (1964b), Acta Cryst 17, 835 Karle, J & Hauptmann, H (1956), Acta Cryst 9, 635 Klug, A (1958), Acta Cryst 11, 515 Lipson, H & Woolfson, M.M (1952), Acta Cryst 6, 439 Main, P & Woolfson, M.M (1963), Acta Cryst 16, 731 Milledge, H J (1962), Proc Roy Soc A, 267, 566 Mills, O.S (1958), Acta Cryst 11, 620 Nakanishi, H & Sarada, Y (1978), Acta Cryst B34, 332 Okaya, Y., Saito, Y & Pepinsky, R (1955), Phys Rev 98, 1857 Patterson, A.L (1934), Phys Rev 46, 372 Perales, A & Garcia-Blanco, S (1978), Acta Cryst B34, 238 Pitt, G J (1948), Acta Cryst 1, 168 Powder Diffraction File, (Ed W L Berry) Joint Committee on Powder Diffraction Standards (JCPDS), 1601 Park Lane, Swarthmore, PA 19081, U.S.A Pradhan, D., Ghosh, S & Nigam, G.D (1985), Structure and Statistics in Crystallography, (Ed A J C Wilson) Guilderland, New York: Adenine Press Reitveld, H.M (1967), Acta Cryst 22, 151 Renninger, M (1937), Z Krist 97, 107 Rogers, D & Wilson, A J.C (1953), Acta Cryst 6, 439 Rollett, J S (1965) Computing Methods in Crystallography Oxford: Pergamon Sayre, D (1952), Acta Cryst 5, 60 Sim, G A (1957), Acta Cryst 10, 536 Sim, G A (1959), Acta Cryst 12, 813 Sim, G A (I960), Acta Cryst 13, 511 Stanley, E (1964), Acta Cryst 17, 1028 Thomas, J.T., Robertson, J.H & Cox, E.G (1958), Acta Cryst 11, 599 Weiss, O., Cochran, W & Cole, W.F (1948), Acta Cryst 1, 83 Wilson, A.J.C (1942), Nature, 150, 152 Wilson, A.J.C (1949), Acta Cryst 2, 318 Wilson, A.J.C (1950), Acta Cryst 3, 397 Woolfson, M M (1954), Acta Cryst 7, 61 Woolfson, M M (1956), Acta Cryst 9, 804 Woolfson, M.M (1958), Acta Cryst 11, 393 Woolfson, M.M & Fan, H.-F (1995), Physical and Non-Physical Methods of Solving Crystal Structures Cambridge: Cambridge University Press Wooster, W A (1938), Crystal Physics Cambridge: Cambridge University Press Wrinch, D M (1939), Phil Mag 27, 98 Yao, J.-X (1981), Acta Cryst A37, 642 Zachariasen, W H (1952), Acta Cryst 5, 68 Zachariasen, W.H (1967), Acta Cryst 23, 558 Bibliography The books given in the list below, many of them specialist texts, between them cover a much wider range of material and go to much greater depth than does this introductory textbook They represent a selection of a very large number of books in the field A comprehensive list is given in the Crystallographic Book List edited by Helen D Megaw and published in 1965 by the International Union of Crystallography Commission on Crystallographic Teaching An update, produced by Dr J H Robertson, appears in the Journal of Applied Crystallography (1982) 15, 640-76 Arndt, U.W & Willis, B.T.M Single Crystal Diffractometry Cambridge: University Press, 1966 AzarorT, L.V., Kaplow, R., Kato, N., Weiss, R.J., Wilson, A.J.C & Young, R.A X-ray Diffraction New York: McGraw-Hill, 1974 Bacon, G.E Neutron Diffraction Oxford: University Press, 1962 Brown, P J & Forsyth, J.B The Crystal Structure of Solids London: Edward Arnold, 1973 Buerger, M.J X-ray Crystallography New York: Wiley, 1958 Cracknell, A.P Crystals and their Structures Oxford: Pergamon, 1969 Glusker, J.P & Trueblood, K.N Crystal Structure Analysis: A Primer Oxford: University Press, 1985 Guinier, A (translated by Lorrain, P & Lorrain, D.S.) X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies San Francisco: Freeman, 1963 Henry, N.F.M., Lipson, H & Wooster, W.A The Interpretation of X-ray Diffraction Photographs London: Macmillan, 1960 James, R.W X-ray Crystallography London: Methuen, 1948 James, R.W The Optical Principles of the Diffraction of X-rays (The Crystalline State, Vol 2) London: Bell, 1965 Ladd, M.F.C & Palmer, R.A Structure Determination by X-ray Crystallography New York: Plenum, 1978 Lipson, H & Cochran, W The Determination of Crystal Structures (The Crystalline State, Vol 3) London: Bell, 1966 Lonsdale, K Crystals and X-rays London: Bell, 1948 McKie, D & McKie, C Essentials of Crystallography Oxford: Blackwell, 1986 Pepinsky, R., Robertson, J.M & Speakman, J.C (Eds.) Computing Methods and the Phase Problem in X-ray Crystal Analysis: Glasgow Oxford: Pergamon, 1961 Phillips, F.C An Introduction to Crystallography London: Longmans, 1971 Ramachandran, G.N & Srinivasan, R Fourier Methods in Crystallography New York: Wiley, 1970 Ramaseshan, S & Abrahams, S.C (Eds.) Anomalous Scattering Copenhagen: Munksgaard, 1975 Rollet, J.S Computing Methods in Crystallography Oxford: Pergamon, 1965 Schenk, H., Wilson, A.J.C & Parthasarathy, S (Eds.) Direct Methods, Macromolecular Crystallography and Crystallographic Statistics Singapore: World Scientific, 1987 Stout, G.H & Jensen, L.H X-ray Structure Determination New York: Macmillan, 1968 397 398 Bibliography Taylor, C.A & Lipson, H Optical Transformations London: Bell, 1964 Wilson, A.J.C X-ray Optics London: Methuen, 1962 Wilson, A.J.C (Ed.) Structure & Statistics in Crystallography Guilderland, New York: Adenine, 1985 Woolfson, M.M Direct Methods in Crystallography Oxford: University Press, 1961 Woolfson, M.M & Fan Hai-fu Physical and Non-Physical Methods of Solving Crystal Structures Cambridge: University Press, 1995 Index Abrahams, S C 233 absorption edges 166 absorption of X-rays 162 et seq calculations of 164 for sphere and cylinder 165 acentric intensity distribution 219 Alcock, N.W 307 alum 2-amino-4,6-dichloropyrimidine 255 2-amino-4-methyl-6chloropyrimidine 255 ANOFILE 269,352 anomalous scattering 179 et seq and Friedel's law 185 and solution of crystal structures 267 et seq anthracene 240 arbitrary signs, rules for selection 279 Archer, E M 246 L-arginine dihydrate 293 Arndt, U.W 143 atomic absorption coefficient 165 atomic scattering factor 46 atomic vibrations 175 axial ratios, determination of 13 Baker, T.W 305 beam trap 113 Beever-Lipson strips 81 bending magnets Bertaut, E F 287 Bhuiya, A K 320 biaxial crystals 205 et seq Bijvoet, J M 264 Blow, D M 266 body-centred lattice 20 Bokhoven, C 264 Bond, W.L 304 Bortner, M.H 231 Bosanquet, C H 174 Bouman, J 133 Bragg, W.L 174 Bragg's law 67 et seq., 136 Bravais lattices \1 et seq Brown, G.M 231 Buerger, M.J 130, 135,246,249 calcite 8, 200 CALOBS 323, 366 Cauchy inequality 274 cell dimensions from layer lines 119 central-limit theorem 215, 286 centre of symmetry 21, 23, 67 190 detection of intensity statistics 215 et seq centric intensity distribution 219 characteristic radiation 144 characteristic temperature 179 cleavage 1, Clews, C.J.B 255 Cochran, W 255, 283, 285, 287, 290, 291, 309 Cochran distribution 291 coherence of Thomson scattering 39 combined figure of merit 294 Compton scattering 42 et seq relation to Thomson scattering 43 from atoms 46 cones of diffraction 54, 56, 58 continuous white radiation 144 convolution 94 et seq coordinate errors from errors in data 331 et seq from series termination 317 counter Geiger-Muller 140 scintillation 141 standard errors of measurement 142 Cox, E.G 288 Crick, F.H.C 266 Cruickshank, D W J 153, 318 crystal classes 9, 30 faces, formation of 13 faces, normals to habit optics 196 et seq symmetry systems crystal monochromator 149 cube, symmetry elements of cubic system 10 optics of 197, 205 cuprous chloride azo-methane complex 298 dead-time of Geiger-Muller counter 140 of scintillation counter 141 Debye theory for temperature factor 179 Debye-Waller factor 178 deconvolution of powder lines 116 delta function 45, 90 densitometers 139 density of film 138 density-exposure relationships for film 138 dextro-rotation 193 diad axis detection from intensities 227 diad screw axis from systematic absences 211 dicyclopentadienyldi-iron tetracarbonyl 220, 223, 300 difference Patterson function 285 diffraction conditions for 58, 108 et seq from a crystal 58, 99 from linear array 50 et seq from three-dimensional array 57 et seq from two-dimensional array 56 et seq relationship to Fourier transform 92 ripples from series termination 317 diffractometers 140 et seq 4,6-dimethyl-2-hydroxypyrimidine 240 Dirac (5-function 45, 90 disordered structures 233 divergence of X-ray beam 137 double refraction 200 Dunitz, J D 233 Dutta, S.N 279 electromagnetic radiation 32, 38 electron density contours 103 equation 99 et seq grid spacing for 105 in projection 103 electron, radius of 39 electrons in atoms 43 et seq enantiomorph and phase relationships 297 enantiomorphic relationships in unit cell 25, 194 energy in diffracted beam 156 et seq energy states of electrons in atoms 46 equi-inclination method for Weissenbeg camera 130 errors in electron-density maps 312 et seq 399 400 Ewald sphere (sphere of reflection) 111 extraordinary rays and waves 201 face-centred lattices 20 detection by systematic absences 210 facets 1, Fan, H-f, 266 Farquhar, M.C.M 303,322 figures of merit 293 film packs 308 films density-exposure relationship of 138 double coated 135 fog level 138 filters for X-rays 168 Fiocco, G 41 fluorene 231 Foster, F 226 FOUR1 81,333 FOUR2 104, 336 Fourier refinement 309 et seq Fourier series determination of coefficients 78 form of 76, 79 in two and three dimensions 83 et seq numerical examples 79 et seq periodic nature of 82 four-circle diffractometer 141 Fourier transform calculation of 88 et seq convolution 94 et seq definition of 87 of crystal 156,157 of Dirac ^-function 91 of Gaussian function 90 of point lattice 92 of product of functions 97 relationship with diffraction 92 et seq Fowweather, F 229 frequency doubling in crystals 195 et seq Friedel's law 73, 115, 186, 210, 235, 267 Friedrich 68 FTOUE 225,339 Garcia-Blanco, S 74 Gaussian function 90 Geiger-Muller counter 140 Ghosh, S 226 glide planes 212 detection by systematic absences 212 L-glutamine 292 goniometer head 119 goniometers, optical grid spacing for electron-density maps 105 Groth 303 Index Hargreaves, A 226,229 Harker, D 246, 274 Harker-Kasper inequalities 274 et seq Hartree self-consistent field method 46 Hauptman, H 291 Hauy HEAVY 254,346 heavy-atom method 249 et seq for non-centrosymmetrical structures 252 Helliwell, J R 153 hexad axis hexagonal system 10, 197, 205 Honl 185 Howells, E.R 225 Hughes, E.W 318 Huygens' construction 197 hypercentric intensity distribution 226 /-centred lattice 20 image-plate systems 150 imperfect crystals 164 incoherent scattering 42, 46, 48 indices of reflection 65, 70, 73 inequality relationships 274 et seq., 287 integrated reflection 156 integrating method for intensity measurement 139 intensity of diffracted beam 67 of radiation 32 photographic measurement of 137 statistics 215 et seq wedges 135 intensifying screen 135 International Tables for Crystallography 23 International Tables for X-ray Crystallography 254 et seq inverse-square law 33 inversion axes in crystals in unit cell 23 ISOCOEFF 264,350 ISOFILE 259, 349 isomorphous replacement 255 et seq and phase ambiguity 261, 264 and anomalous scattering 274 James, R.W 174 Karle, I L 290, 292 Karle,J 290,291,292 Kasper, J S 274 Klug, A 287 Knipping 68 kx unit 303 laevo-rotation laser 41 lattice Bravais 17 space 16 et seq Laue equations 58, 70, 119 group 210 method 151 et seq photograh 208 layer lines from oscillation camera 119 layer-line screens 126 least-squares refinement 311 et seq weighting schemes for 318 limiting sphere 112 linear absorption coefficient 163 calculation for compounds 167 linearly dependent structure factors 280 Lipson, H 226, 303, 322 Lorentz factor 161 Lp factor 161 Main, P 305 mass absorption coefficient 167 matrix notation 319 metaboric acid 287 microscope, use for crystal optics 201 et seq Miller indices 15 et seq., 70, 117 Milledge, H.J 233 Mills, O S 220, 300 MINDIR 293,354 minimum function 249 mirror planes in a crystal 12 in a unit cell 21, 22 detection in cell from intensities 227 moment tests for symmetry elements 226 monoclinic system 9, 11, 205, 207 mosaic crystal 163, 173, 174 multiple isomorphous replacement (MIR) 265 N(z) test 225 et seq Nigam, G D 226 non-centrosymmetrical structure, detection by intensity statistics 219 et seq normalized structure factor 224 octahedron 1, Okaya, Y 272 optic axis 198 et seq optical activity 193 et seq optical goniometers 6, 31 ordinary rays and waves 201 origin-fixing signs 279, 280 orthorhombic system 11, 205, 207 oscillation camera 118 et seq effect of mis-setting 121 indexing of photographs 122 setting of crystals 121 193 Pc(u) function Ps(u) function 272 272 401 Index para-chlor-idoxy benzene 246 parameter-shift method 320 et seq para-mtroanilinG 233 Patterson, A L 235 Patterson function 233 et seq and space-group dependent vectors 240 dimensions of 236 for centrosymmetric function 238 peaks between heavy atoms 238 peaks between parallel groups 240 peak shapes 237 sharpening of 249 symmetry of 246 Patterson-Harker section 246 Penfold, B R 290 Pepinsky, R 272 Perales, A 74 phase problem 103 phase of structure factor 66, 102 phase relationships 290 et seq phase shift, scattering 33 phase-vector diagram 36, 37, 53, 66 phase velocity 205 phenylammonium salt of fosfomycin 74, 229 Phillips, D.C 244 225 photographic measurement of intensities 135 er seq piezoelectric effect 190 et seq detection of 191 Pitt, G.J 240 planes of relection 68 point group 11, 20, 31 polarization factor 161 potassium dihydrogen phosphate 195 powder camera 112 et seq Powder Diffraction File (PDF) 118 powder diffractometer 115 et seq Pradham, D 226 precession camera 130 et seq primary extinction 169 et seq primitive lattice 17 probability curves for MIR 266 progressive-wave, equation of 33 PSCOEFF 273, 353 pseudo-symmetry 233 pyroelectric effect 192 et seq detection of 193 quartz 2, 193 quenching of Geiger-Muller counter 140 racemic mixture 195 radiation damping 182 RANTAN 292 ray axes 205 ray-velocity surface 197, 205 Rayleigh scattering 32 reciprocal lattice 59 et seq and indices of reflection 60 weighted 73 reciprocity law for film 138 reflecting power of crystal 156 refractive index 196 reliability index (residual) 232 Renninger reflections 241 et seq residual (reliability index) 232 for randomly incorrect structures 311 for trial structures 311 rhombohedral lattice 17 rhombohedron Rietvelt, H.M 117 Rietveld refinement 117 Robertson, J.H 288 Robertson, J M 233 Rogers, D 225, 226 rotating anode X-ray source 145 rotation axes in crystals in unit cells 23 Roussin's red ethyl ester 288 Saito, Y 272 Sayre, D 283,285 Sayre's equation 284, 290 salicyclic acid 309 scaling of data 307 et seq scattering angle 33 scattering factor 46 scattering from centrosymmetric distribution 44 scattering from general distribution 36 scattering length 33 scattering of X-rays by atoms 43 et seq scattering phase shift 33, 34, 39, 170, 179 scattering power 40 scattering, total by electron 40 scattering vector 35, 44, 58, 108, 180 Schoone, J.C 264 scintillation counter 141 screw axes 23, 28 sealed hot-cathode X-ray tube 143 secondary extinction 173 et seq series termination errors 317 Sheldrick, G M 307 sign relationships 283 et seq breakdown of 289 probability of 285 et seq Sim,G.R 252 SIMP1 82, 335 Simpson's rule 82 Sim weighting 253 Slater's wave functions 46 space groups 23 et seq space lattices 16 et seq spacing of reflecting planes 69 et seq sphere of reflection 111, 158 standard errors of counter measurements 142 Stanley, E 320 statistical disorder 233 stereographic projection 32 et seq stereo-images 104 STRUCFAC 73, 328 structure factor 65, 72, 99 phase angle of 66 structure invariants 280 superposition methods 246 symbolic addition method 292 symmetry elements in crystal et seq in unit cell 20 et seq table of 24 symmetry information from photographs 208 et seq synchrotron radiation 145 et seq pulsed form 146 polarization 146 power spectrum 146 vertical spread of beam 146 synchrotron sources 145 et seq systematic absences 210 et seq tangent formula 291, 292 temperature factor 175 et seq determination from observed data 308 et seq tetrad axis 6, 13 tetraethyl diphosphine disulphide 279 tetragonal system 10, 197, 205 tetrahedral bonding of carbon 194 tetrahedron 1,2,3,4-tetraphenylcyclobutane 233 thallium hydrogen tartrate 303 Thomas, J.T 288 Thomas-Fermi method 46 Thompson, E 41 Thomson scattering 37 et seq., 181 experimental measurement of 41 relationship to Compton scattering 46 m-toluidene dihydrochloride 229, 230 transpose of matrix 320 triad axis trial-and-error methods 231 triclinic system 9, 10, 205, 207 trigonal system 10, 11, 197, 205 triple-product sign relationship 278, 285 undulator 147 uniaxial crystals 197, 205 identification of 200 unit cell 12 et seq symmetry of 20 et seq unit-cell parameters, determination of 301 unitary structure factor 222 et seq van der Waals 301 vibrations of atoms 175 volume of reciprocal-lattice cell of unit cell 62 von Laue 67 62 402 wave front 197 wave function 43 wave-particle duality 43 wave surface 197, 205 weighted reciprocal lattice weighting Fourier coefficients 252, 253 weighting scheme for least-squares refinement 318 Weiss, O 303 Weissenberg camera 125 et seq equi-inclination method 130 Index extension and contraction of spots 137 layer-line screens 126 Weissenberg photographs 126 et seq coordinates of reflections in 126 et seq wiggler 147 Willis, B.T.M 143 Wilson, A J.C 215,226,309 Wilson plot 309,322 Wilson statistics 251 et seq Woolfson, M M 226, 252, 266, 279, 285, 287, 305 Wrinch, D M 246 X-ray photographs, symmetry from 208 et seq X-ray sources 143 et seq xylose 103 Yao, J-X 292 Zachariasen, W H 287 172 283, 285,