introduction x ray

7/27/2019 Introduction X Ray

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This is a textbook for the senior undergraduate or graduate studentbeginning a serious study of X-ray crystallography. It will be of interestboth to those intending to become professional crystallographers and tothose physicists, chemists, biologists, geologists, metallurgists and otherswho will use it as a tool in their research. All major aspects of crystallographyare covered - the geometry of crystals and their symmetry, theoretical andpractical aspects of diffracting X-rays by crystals and how the da ta may beanalysed to find the symmetry of the crystal and its structure. Recentadvances are fully covered, including the synchrotron as a source of X-rays,methods of solving structures from powder data and the full range oftechniques for solving structures from single-crystal data. A suite ofcomputer programs is provided for carrying out many operations ofdata-processing and solving crystal structures - including by directmethods. While these are limited to two dimensions they fully illustrate thecharacteristics of three-dimensional work. These programs are required formany of the problems given at the end of each chapter but may also be usedto create new problems by which students can test themselves or each other.


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An introduction to X-ray crystallography


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An introduction toX-ray crystallographySECOND EDITIONM.M. WOOLFSONEmeritus Professor of Theoretical PhysicsUniversity of York

CAMBRIDGEUNIVERSITY PRESS


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PUBLI SHED BY THE PRESS SYNDI CATE OF THE UNI VERSI TY OF CAM BRI DGEThe Pitt Building, Trumpington Street, Cambridge CB2 1RP, United KingdomCAM BRI DGE UNI VERSI TY PRESSThe Edinburgh Building, Cambridge CB2 2RU, United Kingdom40 West 20th Street, New York, NY 10011-4211, USA10 Stamford Road, Oakleigh, Melbourne 3166, Australia Cambridge University Press 1970, 1997This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published 1970Second edition 1997Typeset in Times 10/12 ptA catalogue record for this book is available from the British LibraryLibrary of Congress cataloguing in publication dataWoolfson, M. M.An introduction to X-ray crystallography / M.M. Woolfson. - 2nd ed.p. cm .Includes bibliographical references and index.ISBN 0 521 41271 4 (hardcover). - ISBN 0 521 42359 7 (pbk.)1. X-ray crystallo graph y. I. Title.QD945.W58 1997548'.83-dc20 96-5700 CI PISBN 0 521 41271 4 hardbackISBN 0 521 42359 7 paperback

Transferred to digital printing 2003


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Contents

PagePreface to the First Edition xPreface to the Second Edition xii1 The geometry of the crystalline state 1

LI Th e general features of crystals 11.2 Th e externa l sym me try of crystals 11.3 Th e seven crysta l system s 71.4 Th e thirty- two crystal classes 91.5 The un it cell 121.6 M iller ind ices 151.7 Spa ce lattices 161.8 Sym metry elements 201.9 Space gro ups 231.10 Space gr ou p and crystal class 30Problems to Chap ter 1 31

2 The scattering of X-rays 322.1 A general descr iption of the scatter ing proc ess 322.2 Scattering from a pair of po ints 342.3 Scattering from a genera l distr ibu tion of po int scatterers 362.4 Tho mso n scattering 372.5 Co mp ton scattering 422.6 Th e scattering of X-rays by ato m s 43Problem s to Ch apter 2 48

3 Diffraction from a crystal 503.1 Diffraction from a one -dim ensio nal arra y of ato m s 503.2 Diffraction from a two -dim ensio nal arra y of atom s 563.3 Diffraction from a three -dim ensio nal arra y of ato m s 573.4 Th e reciproca l lattice 593.5 Diffraction from a crystal - the stru ctu re factor 643.6 Bra gg's law 673.7 Th e stru ctur e factor in term s of indices of reflection 72Problems to Chap ter 3 74

4 The Fourier transform 764.1 Th e Fou rier series 764.2 Nu me rical app lication of Fo ur ier series 79


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viii Contents4.3 Fo uri er series in two and three dim ension s 834.4 The Fou rier transform 854.5 Diffraction an d the Fo uri er transform 924.6 Co nvo lution 944.7 Diffraction by a period ic distrib utio n 994.8 The electron-density equation 99Prob lems to Ch apter 4 106

5 Experim ental collection of diffraction data 1085.1 Th e cond itio ns for diffraction to occu r 1085.2 Th e pow der cam era 1125.3 Th e oscillation cam era 1185.4 Th e W eissenberg cam era 1255.5 The precession cam era 1305.6 The pho togr aph ic measu remen t of intensities 1355.7 Diffrac tometers 1405.8 X-ray sourc es 1435.9 Ima ge-p late systems 1505.10 The mo dern Laue metho d 151Problem s to Ch apter 5 154

6 The factors affecting X-ray intensities 1566.1 Diffraction from a ro tat ing crystal 1566.2 Ab sorp tion of X-rays 1626.3 Prim ary extinction 1696.4 Secon dary extinction 1736.5 The tem per atu re factor 1756.6 An om alou s scattering 179Problem s to Ch apter 6 188

7 The determination of space group s 1907.1 Tests for the lack of a cen tre of sym me try 1907.2 The optical prop erties of crystals 1967.3 The symm etry of X-ray pho togr aph s 2087.4 Info rm ation from systema tic absences 2107.5 Inte nsity statistics 2157.6 Dete ction of m irro r planes and diad axes 227Problem s to Ch apter 7 229

8 The determination of crystal structures 2318.1 Trial-an d-erro r meth ods 2318.2 Th e Pa tter so n function 2338.3 The heavy-atom meth od 2498.4 Isom orp hou s replacement 2558.5 The application of anom alous scattering 2678.6 Ineq uality relation ships 2748.7 Sign relation ships 282


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Contents8.8 Gen eral phase relationships 2908.9 A general survey of m etho ds 297Prob lems to Ch apter 8 298

9 Accuracy and refinement processes 3019.1 Th e determ ination of unit-cell para me ters 3019.2 Th e scaling of observed da ta 3079.3 Fo uri er refinement 3099.4 Lea st-squ ares refinement 3179.5 The parameter-shift me thod 320Prob lems to Ch apter 9 322

Physical constants and tables 325Appendices 327Program listingsI STRU CFA C 328II FO UR 1 333III SIM P1 335IV FO UR 2 336V FT OU E 339VI HEA VY 346VII ISO FIL E 349VII I ISO CO EFF 350IX AN OF ILE 352X PSC OE FF 353XI MI ND IR 354XII CAL OBS 366Solutions to Problems 367References 395Bibliography 397Index 399


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Preface to the First Edition

In 1912 von Laue proposed that X-rays could be diffracted by crystals andshortly afterwards the experim ent which confirmed this brilliant predictionwas carried out. At that time the full consequences of this discovery couldnot have been fully appreciated. From the solution of simple crystalstructures, described in terms of two or three parameters, there has beensteady progress to the point where now several complex biologicalstructures have been solved and the solution of the structures of somecrystalline viruses is a distinct possibility.

X-ray crysta llogra phy is som etimes rega rded as a science in its own rightand, indeed, there are many professional crystallographers who devote alltheir efforts to the development and practice of the subject. On the otherhand, to many other scientists it is only a tool and, as such, it is a meetingpoint of many disciplines - mathematics, physics, chemistry, biology,medicine, geology, metallurgy, fibre technology and several othe rs. How ever,for the crystallographer, the conventional boundaries between scientificsubjects often seem rather nebulous.In writing this book the aim has been to provide an elementary textwhich will serve either the undergraduate student or the postgraduatestudent beginning seriously to study the subject for the first time. There hasbeen no attempt to compete in depth with specialized textbooks, some ofwhich are listed in the Biblio grap hy. Indee d, it has also been found desirableto restrict the breadth of treatment, and closely associated topics which falloutside the scope of the title - for example diffraction from semi- andnon-crystalline materials, electron- and neutron diffraction - have beenexcluded. For those who wish to go no further it is hoped that the bookgives a rounded, broad treatment, complete in itself, which explains theprinciples involved and adequately describes the present state of the

subject. For those who wish to go further it should be regarded as afoundation for further study.It has now become clear that there is wide acceptance of the SI systemof units and b y-and-larg e they are used in this boo k. How ever the angstro munit has been retained as a unit of length for X-ray wavelengths andunit-cell dimensions etc., since a great deal of the basic literature uses thisunit. A brief explanation of the SI system and some important constantsand equations are included in the section Physical constants and tables onpp. 325-326.I am deeply ind ebted to D r M . Bown an d D r S. G. Fleet of the

Department of Mineralogy, University of Cambridge and to my colleague,Dr P . M ain, for readin g the man usc ript and for their helpful criticism w hichincluded suggestions for many improvements of treatment.


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Preface to the First EditionMy thanks are also due to Professor C. A. Taylor of the University ofCardiff for providing the material for figs. 8.9 and 8.10 and also to Mr W.Spellman and M r B. Cooper of the University of York for help with some ofthe illustrations.

M.M.W.


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Preface to the Second Edition

Since the first edition of this book was published in 1970 there have beentremendous advances in X-ray crystallography. Much of this has been dueto technological developments - for example new and powerful synchrotronsources of X-rays, improved detectors and increase in the power ofcomputers by many orders of magnitude. Alongside these developments,and sometimes prompted by them, there have also been theoreticaladvances, in particular in methods of solution of crystal structures. In thissecond edition these new aspects of the subject have been included anddescribed at a level which is app ropriate to the natu re of the book, which isstill an introductory text.A new feature of this edition is that advantage has been taken of theready availability of powerful table-top com puters to illustrate the proceduresof X-ray crystallography w ith FORTRAN computer p rograms. These arelisted in the appendices and available on the W orld Wide Web*. While theyare restricted to two-dimensional applications they apply to all thetwo-dimensional space groups and fully illustrate the principles of the m orecomplicated three-dimensional programs that are available. The Problemsat the end of each chapter include some in which the reader can use theseprograms and go through simulations of structure solutions - simulationsin that the known structure is used to generate what is equivalent toobserved d ata. M ore realistic exercises can be produced if readers will workin pairs, one providing the other with a data file containing simulatedobserved data for a synthetic structure of his own invention, while the otherhas to find he solution. It can be great fun as well as being very educational!I am particularly grateful to Professor J. R. Helliwell for providingmaterial on the new Laue method and on image-plate methods.

M. M. WoolfsonYork 1996

*http: //www.cup.cam.ac.uk/onlinepubs/412714/412714top.html

xu


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1 The geometry of the crystalline state

1.1 The general features of crystalsMaterials in the crystalline state are commonplace and they play animportant part in everyday life. The household chemicals salt, sugar andwashing soda; the industrial materials, corundum and germanium; and theprecious stones, diamon ds and emeralds, are all examples of such materials.A superficial examination of crystals reveals many of their interestingcharacteristics. The most obvious feature is the presence of facets andwell-formed crystals are found to be completely bounded by flat surfaces -flat to a degree of precision capable of giving high-quality plane-mirrorimages. Planarity of this perfection is not common in nature . It may be seenin the surface of a still liquid but we could scarcely envisage that gravitationis instrumental in moulding flat crystal faces simultaneously in a variety ofdirections.It can easily be verified that the significance of planar surfaces is notconfined to the exterior morphology but is also inherent in the interiorstructure of a crystal. Crystals frequently cleave along preferred directionsand, even when a crystal is crudely fractured, it can be seen through amicroscope that the apparently rough, broken region is actually a m yriad ofsmall plane surfaces.Another feature which may be readily observed is that the crystals of agiven material tend to be alike - all needles or all plates for example - whichimplies that the chemical nature of the material plays an im portant role indetermining the crystal habit. This suggests strongly that the macroscopicform of a crystal depends on structural arrangements at the atomic ormolecular level and that the underlying factor controlling crystal formationis the way in which atom s and molecules can pack together. The flatnessofcrystal surfaces can then be attributed to the presence of regular layers ofatoms in the structure and cleavage would correspond to the breaking ofweaker links between particular layers of atoms.

1.2 The external symm etry of crystalsMany crystals are very regular in shape and clearly exhibit a great deal ofsymmetry. In fig. l.l(a) there is shown a well-formed crystal of alum whichhas the shape of a perfect octahedron; the quartz crystal illustrated in fig.l.l(ft) has a cross-section which is a regular hexagon. However with manyother crystals such symmetry is not evident and it might be thought thatcrystals with symmetry were an exception rather than a rule.Although the crystals of a particular chemical species usually appear to


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The geometry of the crystalline stateFig. 1.1.(a) Alum crystal.(b) Quartz crystal.

ia) (b)

have the same general habit a detailed examination reveals considerablevariation in size and shape. In particular one may find a selection of platycrystals looking somewhat like those shown in fig. 1.2(a). The shapes ofthese seem to be quite unrelated but, if they are rearranged as infig.1.2(b), arather striking relationship may be noted. A lthough the relative sizes of thesides of the crystal cross-sections are very different the normals to the sides(in the plane of the figure) form an identical set from crystal to crystal.Furthermore the set of normals is just that which would be obtained from aregular hexagonal cross-section although none of the crystals in fig. 1.2displays the characteristics of a regular polygon. While this illustration isessentially two-dimensional the same general observations can be made inthree dimensions. Although the crystals of a given species vary greatly in theshapes and sizes of corresponding faces, and may appear to lack symmetryaltogether, the set of normals to the faces will be identical from crystal tocrystal (although a crystal may occasionally lack a particular face completely)and will usually show symmetry that the crystals themselves lack. Forexample, fig. 1.3(a) shows the set of normals for an octahedron. Thesenormals are drawn radiating from a single point and are of equal length.This set may well have been derived from a solid such as that shown in fig.13(b) but the symmetry of the normals reveals that this solid has faceswhose relative orientations have the same relationship as those of theoctahedron.The presentation of a three-dimensional d istribution of normals as donein fig. 1.3 makes difficulties both for the illustrator and also for the viewer.The normals have a common origin and are of equal length so that theirtermini lie on the surface of a sphere. It is possible to represent a sphericaldistribution of points by a perspective projection on to a plane and thestereographic projection is the one most commonly used by the crystallog-rapher. The projection p rocedure can be followed infig.1 A(a). Points on thesurface of the sphere are projected on to a diam etral p lane with projectionpoint either 0 or O\ where 00' is the diameter normal to the projectionplane. Each point is projected from whichever of O or O' is on the oppositeside of the plane and in this way all the projected points are containedwithin the diametral circle. The projected points may be conventionallyrepresented as above or below the projection plane by full or open circles.Thus the points A, B, C and D project as A\ B\ C and D' and, when viewedalong 00', the projection plane appears as in fig. lA(b).


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1.2 The external symme try of crystalsFig. 1.2.(a) Set of apparentlyirregular plate-likecrystals.(b) Crystals rearrangedto show parallelism offaces and underlyinghexagonal symmetry.

(b)

Fig. 1.3.(a) Set of normals to thefaces of an octahedron.(b) Solid whose faceshave same set of normalsas does an octahedron.

(a) (b)


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The geometry of the crystalline stateFig. 1.4.(a) The stereographicprojection of pointsfrom the surface of asphere on to adiametral plane.(b) The finalstereographicprojection.

(b)

We now consider the symmetry elements which may be present incrystals - or are revealed as intrinsically present by the set of normals to thefaces.Centre of symmetry (for symbol see section below entitled 'Inversionaxes')A crystal has a centre of symmetry if, for a point within it, faces occur inparallel pairs of equal dimensions on opposite sides of the point andequidistant from it. A selection of centrosymmetric crystals is shown in fig.1.5(a). However even when the crystal itself does not have a centre ofsymmetry the intrinsic presence of a centre is shown when normals occur in


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1.2 The external symme try of crystalscollinear pairs. The way in which this shows up on a stereographic pro-jection is illustrated infig.1.5(b).

Fig. 1.5.(a) A selection ofcentrosymmetriccrystals.(b) The stereographicprojection of a pair ofcentrosymmetricallyrelated faces.

Mirror plane (written symbol m; graphical symbol )This is a plane in the crystal such that the halves on opposide sides of theplane are mirror images of each other. Some crystal forms possessingmirror planes are shown in fig. 1.6(a). Mirror planes show up clearly in astereographic projection when the projecting plane is either parallel to orperpendicular to the mirror plane. The stereographic projections for each ofthe cases is shown infig.1.6(b).

(b)

Fig. 1.6.(a) Crystals with mirrorplanes.(b) The stereographicprojections of a pair offaces related by a mirrorplane when the mirrorplane is (i) in the planeof projection; (ii)perpendicular to theplane of projection.


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The geometry of the crystalline stateRotation axes (written symbols 2, 3, 4, 6; graphical symbolsAn H-fold rotation axis is one for which rotation through 2n/n leaves theappearance of the crystal unchanged. The values of n which may occur(apart from the trivial case n = 1) are 2, 3,4 and 6 and ex amp les of twofold(diad), threefold (triad), fourfold (tetrad) and sixfold (hexad) axes areillustrated in fig. 1.7 together with the stereographic projections on planesperpendicular to the symmetry axes.

Fig. 1.7.(a) Perspective viewsand views down theaxis for crystalspossessing diad, triad,tetrad and hexad axes.(b) The correspondingstereographicprojections.

Inversion axes (written symbols 1, 2, 3, 4, 6; graphical symbolso, none, A , , $ )The inversion axes relate crystal planes by a combin ation of rotatio n andinversion t hro ug h a centre. Th e op eratio n of a 4 axis may b e followed in fig.1.8(


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13 The seven crystal systemsFig. 1.8.(a) A perspective viewof the operation of aninverse tetrad axis.(b) Stereographicprojections for T, 2, 3, 4and 6.

2 =

light from the mirror-like facets of the crystal to define their relativeorientations.1.3 The seven crystal system s

Even from a limited observation of crystals it would be reasonable tosurmise that the symm etry of the crystal as a who le is somehow connectedwith the sym me try of some smaller subu nit w ithin it. If a crystal is fracturedthen the small plane surfaces exposed by the break , no m atter in what partof the bod y of the crystal they originate, show th e same ang ular relationsh ipsto the faces of the w hole crystal and, indeed, are often p arallel to th e crystalfaces.Th e idea of a structural subun it was first advan ced in 1784 by Haiiy w ho


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The geometry of the crystalline state

was led to his conclusions by observing the cleavage of calcite. This has athreefold axis of symmetry and by successive cleavage Haiiy extracted fromcalcite crystals small rhomboids of calcite. He argued that the cleavageprocess, if repeated many times, would eventually lead to a small, in-divisible, rhombohedral structural unit and that the triad axis of the crystalas a whole derives from the triad axis of the subunit (seefig.1.10(fo) fordescription of rhombohedron).Haiiy's ideas lead to the general consideration of how crystals may bebuilt from small units in the form of parallelepipeds. It is found that,generally the character of the subunits may be inferred from the nature ofthe crystal symmetry. Infig.1.9 is a cube built up of small cubic subunits; itis true that in this case the subunit could be a rectangular parallelepipedwhich quite accidentally gave a crystal in the shape of a cube. However ifsome other crystal forms which can be built from cubes are examined, forexample the regular octahedron and also the tetrahedron infig.1.9, then itis found th at the special angles between faces are those corresponding to acubic subunit and to no other.It is instructive to look at the symmetry of the subunit and the symmetryof the whole crystal. The cube has a centre of symmetry, nine mirror planes,six diad axes, four triad axes and three tetrad axes. All these elements ofsymmetry are shown by the octahedron but the tetrahedron, having sixmirror planes, three inverse tetrad axes and four triad axes, shows lesssymmetry than the cube. Some materials do crystallize as regular tetrahedraand this crystal form implies a cubic subunit. Thus, in some cases, thecrystal as a whole may exhibit less symmetry than its subunit. The commoncharacteristic shown by all crystals having a cubic subunit is the set of four

Fig. 1.9.Various crystal shapeswhich can be built fromcubic subunits:(left) cube;(centre) octahedron;(right) te trahedron. 1ii


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1.4 The thirty-two crystal classestriad axes - and conversely all crystals having a set of four triad axes are cubic.Similar considerations lead to the conclusion that there are sevendistinct types of subunit and we associate these with seven crystal systems.The subunits are all parallelepipeds w hose shapes are com pletely defined bythe lengths of the three sides a, b, c (or the ratios of these lengths) and thevalues of the thr ee ang les a, /?, y (fig. 1.10(a)). The m ain cha racteristics of theseven crystal systems and their subunits are given in table 1.1.

1.4 Th e thirty-two crystal clas sesIn table 1.1 there is given the essential symmmetry for the seven crystalsystems but, for each system, different sy mm etry arra ng em en ts are possible.A crystal in the triclinic system, for example, may or may not have a centreof symmetry and this leads us to refer to the two crystal classes 1 and 1within the triclinic system. As has been p reviously no ted 1 is the sym bol fora centre of symmetry and the symbol 1, representing a onefold axis,correspo nds to n o sym metry at all. These two crystal classes may be show nconveniently in terms of stereograp hic projections as in fig. 1.1 \{a) and (b).Th e projections sh ow the set of planes ge nerated from a genera l crystal faceby the complete group of symmetry elements.The possible arrangem ents for the monoclinic system are now considered.

Fig. 1.10.(a) A generalparallelepiped subunit.(b) A rhombohedronshowing the triad axis.(c) The basic h exagonalsubunits which arepacked as shown togive hexagonalsymmetry.

(a)

(b)

(c)


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10 The geometry of the crystalline stateTable 1.1. The seven crystal systemsSystemTriclinicMonoclinicOrthorhombicTetragonalTrigonal

Hexagonal

Cubic

Subuni tNo special relationshipsa^b^c0 5* a = 7 = 90a^b^ca = p = y = 90a = b 7 ca = p = y = 90a = fr = c(X = 0 = y ^ 90

(see fig. 1.10(6))or as hexagonala = b ^ ca = P = 90, 7 = 120(see fig. 1.10(c))a = b = ca = p = y = 90

Essential symmetry of crystalN o n eDiad axis or mirror plane(inverse diad axis)Three orthogonal diad or inversediad axesOne tetrad or inverse tetradaxisOne triad or inverse triadaxis

One hexad or inverse hexadaxisFour triad axes

Fig. 1.11.Stereographicprojections representingthe crystal classes {a) 1an d (b) I .

(a) (b)

These, illustra ted in fig. 1.12, hav e (a) a diad axis, (b) a mir ror plane an d (c) adiad axis and mirror plane together. The orthorhombic and trigonalsystems give rise to the classes shown in fig. 1.13.Some interesting points may be observed from a study of these diagrams.Fo r example, the com binatio n of symbols 3m implies that the mirror planecontains the triad axis and the trigonal symmetry demands therefore that aset of three m irror planes exists. On the oth er ha nd , for the crystal class 3/m,the mirror plane is perpendicular to the triad axis; this class is identical tothe hexagonal class 6 and is usually referred to by the latter name.It may also be noted that, for the orthorhombic class mm, the symmetryassociated with the third axis need not be stated. This omission is per-missible due to the fact that the two o rthog ona l mir ror planes automaticallygenerate a diad axis along the line of their intersection and a name such as2mm therefore contains redundant information. An alternative name formm is 2m and again the identity of the third symm etry element may be inferred.

Fo r the seven systems togethe r there are thirty -tw o crystal classes and all


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1A The thirty-two crystal classes 11Fig. 1.12.Stereographic projectionsrepresenting the threecrystal classes in themonoclinic system (a) 2,(b) m and (c) 2/m.

Fig. 1.13.Stereographicprojections representingthe three crystal classesin the orthorhombicsystem and the sixclasses in the trigonalsystem.

32Trigonal Or thorhombic

crystals may be assigned to one or o ther of these classes. While the generalnature of the basic subunit determines the crystal system, for each systemthere can be different elements of symmetry associated with the crystal. If amaterial, satisfying some minimization-of-potential-energy criterion, crys-tallizes with some element of symmetry, it strongly implies that there issome corresponding symmetry within the subunit itself. The collection ofsymmetry elements which characterizes the crystal class, and which mustalso be considered to be associated with the basic subunit, is called a pointgroup. It will be seen later that the point group is a macroscopic


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manifestation of the symmetry with which atoms arrange themselves withinthe subunits.

1.5 The unit cellWe shall now turn our attention to the composition of the structuralsubunits of crystals. The parallelepiped-shaped volume which, when re-produced by close packing in three dimensions, gives the whole crystal iscalled the unit cell. It is well to note that the unit cell may no t be an entitywhich can be uniquely defined. In fig. 1.14 there is a two-dimensionalpattern which can be thought of as a portion of the arrangement of atomswithin a crystal. Several possible choices of shape and origin of unit cellare shown and they are all perfectly acceptable in that reproducing theunit cells in a close-packed two-dimensional array gives the correctatomic arrangement. However in this case there is one rectangular unitcell and this choice of unit cell conveys more readily the special rectangularrepeat features of the overall pattern and also shows the mirror plane ofsymmetry. Similar arguments apply in three dimensions in that manydifferent triclinic unit cells can be chosen to represent the structural

Fig. 1.14.A two-dimensionalpattern and somepossible choices of unitcell.


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7.5 The unit cell 13arrangement. One customarily chooses the unit cell which displays thehighest possible symmetry, for this indicates far m ore clearly the symmetryof the underlying structure.In 1.3 and 1.4 the ideas were advanced that the symmetry of the crystalwas linked with the symmetry of the unit cell and that the disposition ofcrystal faces depends on the shape of the unit cell. We shall now explore thisidea in a little more detail and it helps, in the first instance, to restrictattention to a two-dimensional model. A crystal made of square unit cells isshown infig.1.15. The crystal is apparently irregular in shape but, when theset of normals to the faces is examined we have no doub t th at the unit cellhas a tetrad axis of symmetry. The reason why a square unit cell with atetrad axis gives fourfold symmetry in the bulk crystal can also be seen. Ifthe formation of the faces AB and BC is favoured because of the lowpotential energy associated with the atomic arrangement at these boundariesthen CD, DE and the other faces related by tetrad symmetry are alsofavoured because they lead to the same condition at the crystal boundary.Fo r the two-dimensional crystal in fig. 1.16 the set of normals reveals amirror line of symmetry and from this we know that the unit cell isrectangular. It is required to determine the ratio of the sides of the rectanglefrom measurements of the angles between the faces. The mirror line can belocated (we take the normal to it as the b direction) and the angles made tothis line by the faces can be found. Infig.1.17 the face AB is formed by pointswhich a re separated by la in one direction and b in the other. The angle 0,which the normal AN makes with the b direction, is clearly given by

tan 6 = b/2a. (1.1)

Fig. 1.15.A two-dimensionalcrystal made up of unitcells with a tetrad axis ofsymmetry.


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14 The geometry of the crystalline stateFig. 1.16.A two-dimensionalcrystal built ofrectangular units.

//\ \ \ \

\ \N \

S.\ \ \

\\

a

\\ \ /

//

s.

/

\\ \

/

\ \

\\

Fig. 1.17.The relationship betweenthe crystal face AB andthe unit cell.

/

B

\ /

a/

If the neighbo uring poin ts of the face were separated by na and mb then onewould havembta n 0 =

or- = - t a n 0 .a m (1.2)

The angles 0 for the crystal in fig. 1.16 are 32 12 r, 43 24' and 51 33' so tha twe have* = 0 . 6 3 0 ^ = 0 . 9 4 6 ^ . = 1.260^-. (1.3)


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16 The geometry of the crystalline stateFig. 1.19.The lines A'B' and CD'which are parallel to thecrystal faces AB and CDhave intercepts on theunit-cell edges of theform a/h and b/k whereh and k are integers.

intercepts a/2 and b/4 is also parallel to AB. However, we are hereconcerned with the smallest possible integers and these are referred to as theMiller indices of the face.In three dimensions a plane may always be found, parallel to a crystalface, which makes intercepts a/h, b/k and c/l on the unit-cell edges. Thecrystal face in fig. 1.20 is based on the unit cell shown with OA = 3a,OB = 4b and OC = 2c . The plane A'B'C is parallel to ABC and hasintercepts OA\ OB' and OC given by a/4, b/3 and c/6 (note that thecondition for parallel planes OA/OA' = OB/OB' = OC /OC is satisfied).This face may be referred to by its Miller indices and ABC is the face (436).The Miller indices are related to a particular unit cell and are thereforenot uniquely defined for a given crystal face. Returning to our two-dimensionalexample, the un it cell infig.1.21 is an alternative to that shown in fig. 1.19.The face AB which was the (1,2) face for the cell infig.1.19 is the (1,1) facefor the cell infig.1.21. However, no matter which unit cell is chosen, one canfind a triplet of integers (generally small) to represent the Miller indices ofthe face.

1.7 Space latticesIn figs. 1.19 and 1.21 are shown alternative choices of unit cell for atwo-dimensional repeated pat tern . The two unit cells are quite different inappearance but when they are packed in two-dimensional arrays they eachproduce the same spatial distribution. If one point is chosen to represent theunit cell - the top left-hand corner, the centre or any other defined point -then the array of cells is represented by a lattice of points and theappearance of this lattice does not depend on the choice of unit cell. Oneproperty of this lattice is that if it is placed over the structural pattern theneach point is in an exactly similar environm ent. This is illustrated infig.1.22where the lattice corresponding to figs. 1.19 and 1.21 is placed over thetwo-dimensional pattern and it can be seen that, no matter how the lattice isdisplaced parallel to itself, each of the lattice points will have a similarenvironment.If we have any repeated pattern in space, such as the distribution ofatoms in a crystal, we can relate to it a space lattice of points which definescompletely the repetition characteristics w ithout reference to the details ofthe repeated motif. In three dimensions there are fourteen distinctive space


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1.7 Space lattices 17Fig. 1.20.The plane A'B'C isparallel to the crystalface AB C and makesintercepts on the celledges of the form a/h,b/k an d c/l where h, kand / are integers.

Fig. 1.21.An alternative unit cellto that shown in fig.1.19. The faces AB andCD now have differentMiller indices.

Fig. 1.22.The lattice (small darkcircles) represents thetranslational repeatnature of the patternshown. *

lattices known as Bravais lattices. The unit of each lattice is illustrated in fig.1.23; lines connect the points to clarify the relationships between them.Firstly there are seven simple lattices based on the unit-cell shapesappropriate to the seven crystal systems. Six of these are indicated by thesymbol P which means 'primitive', i.e. there is one point associated witheach unit cell of the structure; the primitive rhom bohedral lattice is usuallydenoted by R. But other space lattices can also occur. Consider the spacelattice corresponding to the two-dimensional pattern given infig.1.24. Thiscould be considered a primitive lattice corresponding to the unit cell showndashed in outline but such a choice would obscure the rectangular repeatrelationship in the pattern. It is appropriate in this case to take the unit cellas the full line rectangle and to say that the cell is centred so that pointsseparated by \a t\b are similar. Such a lattice is non-primitive. The threepossible types of non-primitive lattice are:


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18 The geometry of the crystalline stateFig. 1.23.The fourteen Bravaislattices. Theaccompanying diagramsshow the environment ofeach of the lattice points. MonoclinicP

MonoclinicC

OrthorhombicP OrthorhombicC

Orthorhombic/

OrthorhombicF

(1st Part)


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1.7 Space latticesFig. 1.23. (cont)

19

Fig. 1.23. (2nd Part)


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Fig. 1.24.Two-dimensionl patternshowing two choices ofunit cell - generaloblique (dashed outline)and centred rectangular(full outline).

C-face centring - in which there is a translation vector \a, \b in the Cfaces of the basic unit of the space lattice. A and B-face centringmay also occur;F-face cent rin g- equivalent to simultaneous A, B and C-face centring; and/-centring - where there is a translation vector \a , \b, \c giving a point atthe intersection of the body diagonals of the basic unit of the spacelattice.The seven non-primitive space lattices are displayed infig.1.23. Any spacelattice corresponds to one or other of the fourteen shown and no otherdistinct space lattices can occur. For each of the lattices, primitive andnon-primitive, the constituent points have similar environments. A fewminutes' study of the figures will confirm the truth of the last statement.

1.8 Symm etry elementsWe have noted tha t there are seven crystal systems each related to the typeof unit cell of the underlying structure. In addition there are thirty-twocrystal classes so that there are differing degrees of symmetry of crystals allbelonging to the same system. This is associated with elements of symmetrywithin the unit cell itself and we shall now consider the possibilities for thesesymmetry elements.The symmetry elements which were previously considered were thosewhich may be displayed by a crystal ( 1.2) and it was stated that there arethirty-two possible arrangem ents of symmetry elements or point groups. Acrystal is a single unrepeated object and an arrangement of symmetryelements all associated with one point can represent the relationships of acrystal face to all symmetry-related faces.The situation is different when we consider the symmetry within the unitcell, for the periodic repeat pattern of the atomic arrangement gives newpossibilities for symmetry elements. A list of symmetry elements which canbe associated with the atomic arrangement in a unit cell is now given.


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1.8 Symm etry elements 21

Fig. 1.25.(a) A centrosymmetricunit cell showing thecomplete family of eightdistinct centres ofsymmetry.(b) A unit cell showingmirror planes.(c) A unit cell showingglide planes.(d) A view down atetrad axis of symmetryshowing the othersymmetry axes whicharise.(e) The operation of a2X axis.(/) The operation of 3i

Centre of symmetry (1)This is a point in the unit cell such that if there is an atom at vector position rthere is an equivalent atom located a t r. The unit cell in fig. 1.25(a) hascentres of symmetry at its corners. Since all the corners are equivalentpoints the pairs of atoms A and A related by the centre of symmetry at O arerepeated at each of the other corners. This gives rise to other centres ofsymmetry which bisect the edges of the cell and lie also at the face and body

.

0ST.- &.. i: rLLJ*(M

id)

U'l

A\


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22 The geometry of the crystalline statecentres. While these extra points are also centres of symmetry they are notequivalent to those at the corners since they have different environments.

Mirror plane (m)In fig. 1.25(b) there is shown a unit cell with mirror planes across twoopposite (equivalent) faces. The plane passing through 0 generates thepoints A2 and B2 from Ax and Bv The repeat distance perpendicular to themirror plane gives equivalent points A\, B\, A2 and B'2 and it can be seenthat there arises another mirror plane displaced by \a from the one through O.Glide planes (a, b, c, n, d)The centre of symmetry and the mirror plane are symm etry elements whichare observed in the morphology of crystals. Now we are going to consider asymmetry element for which the periodic nature of the pattern plays afundamental role. The glide-plane symmetry element operates by acombination of mirror reflection and a translation.The description of this symm etry element is simplified by reference to thevectors a, b and c which define the edges of the unit cell. Fo r an a-glide planeperpendicular to the b direction (fig. 1.25(c)) the point A1 is first reflectedthrough the glide plane to Am and then displaced by â to the point A2. Itmust be emphasized that Am is merely a construction point and the netresult of the operation is to generate A2 from Av The repeat of the patterngives a point A2 displaced by b from A2 and we can see that A2 and A x arerelated by ano ther glide plane parallel to the one through O and displacedby ^b from it. One may similarly have an a-glide plane perpendicular to cand also b- and c-glide planes perpendicular to one of the other directions.An n-glide plane is one which, if perpendicular to c, gives a displacementcomponent â + |b.The diamond glide-plane d is the most complicated symm etry elementand merits a detailed description. For the operation of a d-glide planeperpendicular to b there are required two planes, Px and P2, which areplaced at the levels y = | and y = , respectively. For each initial point thereare two separate operations generating two new points. The first operationis reflection in P1 followed by a displacement ^c + â, and the second areflection in P2 followed by a disp lace me nt^ + â. If we begin with a point(x, y, z) then thefirstoperation generates a point an equal distance from theplane y = on the opposite side with x and z coordinates increased by , i.e.the point (x + ^ , | y,z + ^). The second operation, involving the planeP 2, similarly generates a point (x , f y, z + {-). These points, and allsubsequent newpoints, may be subjected to the same operations and it willbe left as an exercise for the reader to confirm that the following set of eightpoints is generated:

x y z | + x l-y i + z4 - 4 - v 4- v X A- 7 Y 4-4- v 4- - I - 7

i-y2 yl-y


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1.9 Space groups 23These coordinates show that there are two other glide planes at y = fand y = | associated with displacements | c + a and ^c â, respectively.

Rotation axes (2, 3, 4, 6)The modes of operation of rotation axes are shown in fig. 1.7; the newfeature which arises for a repeated pattern is the generation of subsidiaryaxes of symmetry other than those put in initially. This may be seen in fig.1.25(d) which shows a projected view of a tetragonal unit cell down thetetrad axis. The point Ax is operated on by the tetrad axis through O to giveA2, A3 and A and this pattern is repeated about every equivalent tetradaxis. It is clear that A l9 A'2, A^ and A' are related by a tetrad axis,non-equivalent to the one through 0, through the centre of the cell. Asystem of diad axes also occurs and is indicated in the figure.Screw axes (2X; 3 1 ? 3 2 ; 4 1? 42? 4 3 ; 61? 6 2 , 63 , 6 4 , 65)These symmetry elements, like glide planes, play no part in the macroscopicstructure of crystals since they depend on the existence of a repeat distance.The behaviour of a 2X axis parallel to a is shown infig.1.25(e). The point A xisfirstrotated by an angle n round the axis and then displaced by a to giveA2. The same operation repeated on A2 gives A\ which is the equivalentpoint to A x in the next cell. Thus the operation of the symmetry element 2Xis entirely consistent with the repeat nature of the structural pattern.The actions of the symmetry elements 3j and 3 2 are illustrated in fig.1.25(/). The point Ax is rotated by 2rc/3 about the axis and then displaced byâ to give A2. Two further operations give A3 and A\, the latter point beingdisplaced by a from Av The difference between 3X and 3 2 can either beconsidered as due to different directions of rotation or, alternatively, as dueto having the same rotation sense but displacements of â and fa,respectively. The two arrangements produced by these symmetry elementsare enantiomorphic (i.e. in mirror-image relationship).In general, the symmetry element RD along the a direction involves arotation 2n/R followed by a displacement (D/R)a.

Inversion axes (3, 4, 6)The action of the inversion axis R is to rotate the point abo ut the axis by anangle 2n/R and then invert through a point contained in the axis. Since Tand 2 are equivalent to a centre of symmetry and mirror p lane, respectively,they are not included here as inversion axes.There is given in table 1.2 a list of symmetry elements and the graphicalsymbols used to represent them.1.9 Space groups

Symmetry elements can be combined in groups and it can be shown that230 distinctive arrangements are possible. Each of these arrangements iscalled a space group and they are all listed and described in volume A of theInternational Tables for Crystallography. Before describing a few of the 230


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24 The geometry of the crystalline stateTable 1.2.Type of symm etry element W ritten sym bol G raphical symbolCentre of symmetry

Perpen dicular In plane ofto paper paperMirror planeGlide planes

Rotation axes

Screw axes

Inversion axes

mab

n

23462,

346

c

3 24 2 , 4 362> 6 3 , 6 4 , 65

1 v1 \glide in plane arrow showsof pa per glide directionglide out ofplane of paper

71AA-A * * * -*A

space groups we shall look at two-dimensional space groups (sometimescalled plane groups) which are the possible arrangements of symmetryelements in two dimensions. There are only 17 of these, reflecting thesmaller num ber of possible systems, lattices and symmetry elements. Thusthere are:four crystal systems - oblique, rectangular, square and hexagonal;two types of lattice - primitive (p) and centred (c); andsymmetry elements -rotation axes 2, 3, 4 and 6mirror line mglide line g.


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1.9 Space groups 25We shall now look at four two-dimensional space groups whichillustrate all possible features.

Oblique p2This is illustrated in fig. 1.26 in the form given in the International Tables.The twofold axis is at the origin of the cell and it will reproduce one of thestructural units, represented by an open circle, in the way shown. Theright-hand diagram shows the symmetry elements; the twofold axis mani-fests itself in two dimensions as a centre of symmetry. It will be seen thatthree other centres of symmetry are generated - at the points (x, y) = (\, 0),(0,^) and ({,\). The four centres of symmetry are all different in that thestructural arrangement is different as seen from each of them.Fig. 1.26.The two-dimensionalspace group p2 as itappears in InternationalTables for X-rayCrystallography.

No. 2

O/o

P2\

O

2 Oblique

O IOrigin at 2

Rectangular cmThis rectangular space group is based on a centred cell and has a mirror lineperpendicular to the y axis. Infig.1.27 the centring of the cell is seen in tha tfor each structural unit with coordinate (x, y) there is another at ( + x, \ + y) .In addition, the mirror line is shown relating empty open circles to thosewith commas within them. The significance of the comma is tha t it indicatesa structural unit which is an enantiomorph of the one without a comma.The right-hand diagram infig.1.27 shows the symmetry elements in theunit cell and mirror lines are indicated at y = 0 and y = \. What is also

Fig. 1.27.The two-dimension al ^space group cm as itappears in InternationalTables for X-rayCrystallography. 0O

0O

No. 5

oO

c l m l

OO

ooOrigin on m

m Rectangula r


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apparen t, although it was not a part of the description of the two-dimensionalspace group, is the existence of a set of glide lines interleaving the mirrorlines. The operation of a glide line involves reflection in a line followed by atranslation ^a. Because of the reflection part of the operation, the relatedstructural units are enantiomorphs.Square pAgThis two-dimensional space group is illustrated infig.1.28 and shows thefourfold axes, two sets of glide lines at an angle TI/4 to each other and a set ofmirror lines at n/4 to the edges of the cell. Starting with a single structuralunit there are generated seven others; the resultan t eight structural units arethe contents of the square cell. Wherever a pair of structural units arerelated by either a mirror line or a glide line the enantiomorphic relationshipis shown by the presence of a comma in one of them.

Fig. 1.28.The two-dimensionalspace group p4 g as itappears in InternationalTables for X-rayCrystallography.

S q u a r e 4mm

oo

o

oo

o

o0 oo

oo

o

p4 g mO

O

No. 12

OOO

Origin at 4

Hexagonal p6As the name of this two-dimensional space group suggests it is based on ahexagonal cell, which is a rhombus with an angle 2it/3 between the axes. Ascan be seen infig.1.29 the sixfold axis generates six stru ctura l units abouteach origin of the cell. A pair of threefold axes within the cell is also

Fig. 1.29.The two-dimensionalspace group p6 as itappears in InternationalTables for X-rayCrystallography.

Hexagonal 6 No. 16

oOrigin at 6


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1.9 Space groups 27

generated. The complete arrangement of symmetry elements is shown in theright-hand diagram offig.1.29.Having established some general characteristics of space groups by thestudy of some relatively simple two-dimensional examples we shall nowlook at five three-dimensional space groups where the third dimensionintroduces complications not found in two dimensions.Triclinic PIThis space group is based on a triclinic primitive cell which has a centre ofsymmetry. The representation ofthis space group, as given in the InternationalTables, is reproduced infig.1.30. The cell is shown in projection and thethird coordinate (out of the plane of the paper), with respect to an origin at acentre of symmetry, is indicated by the signs associated with the open-circlesymbols. This convention is interpreted as meaning that if one coordinate is+ then the other is t. The comma within the open-circle symbolindicates that if the symmetry operation is carried out on a group of objectsand not just on a point, then the groups represented by O and 0 areenantiomorphically related. The diagram on the right-hand side shows thedistribution of symmetry elements.The information which headsfigs.1.30-1.35 is taken from the InternationalTables and, reading across the page, is (i) the crystal system, (ii) the pointgroup, (iii) symmetry associated with a, b and c axes (where appropriate),(iv) an assigned space-group number and (v) the space-group nameaccording to the Hermann-Mauguin notation with, underneath, the olderand somewhat outmoded Schoenflies notation.

Fig. 1.30.The operation of thespace group P i asshown in theInternational Tables forX-ray Crystallography.

Triclinic 1

/

7+Monoclinic

0

/

Cm

PI- 0

1 / o t /- / /O +

Origin at T

No. 2

o o

PI

This space group is based on a monoclinic C-face centred cell with themirror plane perpendicular to the unique axis. The unique axis formonoclinic space groups is the one perpendicular to the other two and, byconvention, this is taken as the b axis. The letters shown infig.1.31 do notappear in the International Tables but they assist in a description of thegeneration of the complete pattern starting with a single unit.We start with the structural unit Ax and generate A2 from it by theoperation of the C-face centring. The mirror plane gives A3 from A2 and the


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28Fig. 1.31.The operation of thespace group Cm asshown in theInternational Tables forX-ray Crystallography.The letters have beenadded.

Monoclinic m C 1 m lThe geometry of the crystalline state

No. 8 w+ 0 O + + 0 O +

o ++ O O + + O O +

Origin on plane m; unique axis bcentring gives 4 4 from A3. This constitutes the entire pattern. The + signsagainst each symbol tell us that the units are all at the same level and thecommas within the open circles indicate the enantiom orphic relationships.It may be seen that this combination of C-face centring and mirrorplanes produces a set of a-glide planes.Monoclinic P2JcThis space group is based on a primitive, monoclinic unit cell with a 2 l axisalong b and a oglide plane perpendicular to it. In fig. 1.32(a) these basicsymmetry elements are shown together with the general structural patternproduced by them. It can be found by inspection that other symmetryelements arise; Ax is related to AA and A2 to A3 by glide planes whichinterleave the original set. The pairs of units A 4, A2 and A l9 A3 are relatedby a centre of symmetry at a d istance \c out of the plane of the paper and awhole set of centres of symmetry m ay be found which are related as thoseshown in fig. 1.25(a).The International Tables gives this space group with the unit-cell originat a centre of symmetry and the structure pattern and complete set ofsymmetry elements appears in fig. 1.32(6). If a space group is developedfrom first principles, as has been done here, then the emergence of newsymmetry elements, particularly centres of symmetry, often suggests analternative and preferable choice of origin.Orthorhombic P212121This space group is based on a primitive orthorhom bic cell and has screwaxes along the three cell-edge directions. The name does not appear todefine completely the d isposition of the symmetry elements as it seems thatthere may be a number of ways of arranging the screw axes with respect toeach other.As was noted in 1.4 in some point groups certain symmetry elementsappear automatically due to the combination of two others. If this occurs inthe point group it must also be so for any space group based on the pointgroup . If we start with two sets of intersecting screws axes and generate thestructural pattern from first principles we end up with the arrangementshown in fig. 1.33(


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1.9 Space groups 29Fig. 1.32.(a) The developmentfrom first principles ofthe structural patternfor the space groupP2Jc.(b) The description ofP2Jc as given in theInternational Tables forX-ray Crystallography.

. i-o o +

. ii-Q O +

o- i-oi + o

(a)o - i - o

o +

o +

Monoclinic 2/m

- oP1 2Jc 1

-o 0 +

oi- -oO + i + 0

oi- -o

o +i -

o +Origin at T; unique axis b

(b)

No. 14 P2JcC52b

0 (

Fig. 1.33.(a) The development ofthe structural pattern fortwo sets of 2 t axeswhich intersect. Thisgives the space groupK.2 ,2 .(b) The development ofthe structural pattern forthree sets of 2 t axeswhich intersect. Thisgives the space group7222. (From InternationalTables for X-rayCrystallography.)

P 2 x 2 x 2

+ P_

+ O

N o. 18 P 2 ,2 , 2 222 Orthorhombic+ Oo +

- o o -+ oo +

Orthorhombic 222

+ O O -- o

+ o

o+

o -i + oi - o o i -o i+

- o

+ o

o +

O +Origin at 112 in plane of 2,2 ,

(a)7222

+ o o -o +

- O O + -O OOrigin at 222

(b)

N o . 23 7222

i


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30 The geometry of the crystalline stategives the structural arrangement shown infig.1.34 which is the space group

One can have three sets of screw axes in a different arrangement. Forexample, if one s tarts with three sets of intersecting screw axes onefinds hatthree sets of diad axes are also generated and that the unit cell is non-primitive. This space group is the one shown in fig. 1.33(b)-/222.

Fig. 1.34.The space groupP 2 1 2 1 2 1 as shown in theInternational Tables forX-ray Crystallography.

Orthorhombic 222o i -

i + O!o +

- o i + oo i -

Origin halfway between three pairs of non-intersecting screw axesOrthorhombic Aba2The symbols tell us that there is A-face centring, afo-glideplane perpendicularto a, an a-glide plane perpendicular to b and a diad axis along c. Thediagrammatic representation of this space group, as given in the InternationalTables, is shown in fig. 1.35. We should notice that the diad axis isautomatically generated by the other two symmetry elements.

Fig. 1.35.The space group Aba!as shown in theInternational Tables forX-ray Crystallography.

Orthorhombic mml

i+ oo +i+ o+ oo +

N o . 4 1 Abal

+ oo++ Q

i+o +

+ot I t If I

Abal

o +O H

o+ ^ ' *Origin on 2The determ ination of a crystal structu re is usually a major undertaking andthe first task of the crystallographer is to determine the space group(chapter 7) and to familiarize himself with its characteristics. Some spacegroups occur frequently, for example P2Jc and P212121 are well known bymost crystallographers; other space groups occur much more rarely andthese would usually be studied, as required, on an ad hoc basis.

1.10 Space group and crystal classIn 1.4 it was illustrated for a two-dimensional square unit cell howsymmetry w ithin the cell influences the way in which cells associate to form


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Problems to Chapter 1 31the com plete crystal. The formation of the faces of a crystal in the process ofcrystallization takes place in such a way that the crystal has a configurationof minimum potential energy. If the unit cell contains a diad axis thenclearly, by symm etry, any association of cells giving a par ticu lar face will bematched by an associated face related by a crystal diad axis. However ascrew axis in the unit cell will also, in the macroscopic aspect of the wholecrystal, give rise to a crystal diad axis since th e external appearance of thecrystal will not be affected by atomic-scale displacements due to a screwaxis. Similarly, mirror planes in the crystal are formed in response to bothmirror planes and glide planes in the unit cell.In th e International Tables th e point group is given for each of the listedspace groups. The space groups described in this chapter, with the cor-responding point groups, are:

Space groupP\CmP2JcP2 12121Aba!

Point grouprm2/m222mm !A study of the crystal symmetry can be an important first step in thedetermination of a space group as for a particular point group a limitednumber of associated space groups are possible. However the art ofexamining crystals by optical goniometry is now largely ignored by the

modern X-ray crystallographer who tends to use only diffraction informationif possible.

Problems to Chapter 11.1 A unit cell has the form of a cube. Find th e angles between the norm alsto pairs of planes whose Miller indices are:(a) (100) (010); (b) (100) (210); (c) (100) (111); (d) (121) (111).1.2 The diagrams infig.1.36 show a set of equivalent positions in a unit cell.Find th e crystal system an d suggest a name for the space group.1.3 Draw diagrams to show a set of equivalent positions and the set ofsymmetry elements for the following space groups:(a) P91 (b) p4mm; (c) Pm; (d) P2/m; (e) 14.

Fig. 1.36.Diagrams for Problem1.2. o +

o +

+ 0

+ O

o+ Ho

Oi - -0

o+

0 -

- 0

+o(a) (b) (c)


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2 The scattering of X-rays

2.1 A general description of the scattering processTo a greater or lesser extent scattering occurs whenever electromagneticradiation interacts with matter. Perhaps the best-known example is Ray-leigh scattering the results of which are a matter of common everydayobservation. The blue of the sky and the haloes which are seen to surrounddistant lights on a foggy evening are due to the Rayleigh scattering of visiblelight by molecules of gas or particles of dust in the atmosphere.The type of scattering we are going to consider can be thought of as dueto the absorption of incident radiation with subsequent re-emission. Theabsorbed incident radiation may be in the form of a parallel beam but thescattered radiation is re-emitted in all directions. The spatial distribution ofenergy in the scattered beam depends on the type of scattering processwhich is taking place but there a re many general features common to alltypes of scattering.In fig. 2.1 the point O represents a scattering centre. The incidentradiation is in the form of a parallel monochromatic beam and this isrepresented in the figure by the bundle of parallel rays. The intensity at apoint within a beam of radiation is defined as the energy per unit timepassing through unit cross-section perpendicular to the direction ofpropagation of the radiation. Thus for parallel incident radiation theintensity may be described as the pow er per unit cross-section of the beam.However, the scattered radiation emanates in all directions with somespatial distribution about the point O; this is shown in thefigureby drawinga conical bundle of rays with apex at the point O representing the raysscattered w ithin a small solid angle in some particu lar direction. Clearly, in

Fig. 2.1.Representation of theradiation incident onand scattered from apoint scatterer.

32


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2.1 A general description of the scattering process 33this case, the intensity of the scattered radiation will depend on the distancefrom 0 and there will be an inverse-square law fall-off of intensity withdistance. The intensity of the scattered radiation is thus usually described asthe energy scattered per unit time per unit solid angle in a particulardirection and is therefore a measure of what is happening at the scatterer itself.If the incident radiation falling on O is in the form of a simplemonochromatic wave then the variation with time of the displacement y ofthe incident wave can be described by the equation

y = A cos(27ivr), (2.1)where v is the frequency of the radiation and A its amplitude.The scattered wave will have a displacement at the point P (fig. 2.1) whichwill depend on a number of factors:

(i) The distance OP ( = D) will introduce a phase shift w ith respect to thescattered wave at O of inD/X where X is the wavelength of theradiation. This can also be expressed as 2nDv/c where c is thevelocity of propagation of the radiation.(ii) The scattering process itself may introduce a phase shift so that thescattered wave at O will be retarded with respect to the incident wave atO. This quantity as is called the scattering phase shift.(iii) The inverse-square law of reduction of intensity with distance for thescattered radiation causes the fall-off of amplitude to be inverselyproportional to distance D.The displacement at P can now be described by

y(26,D, t) =f26^cos[_2nv(t - D/c) - a j . (2.2)The influence of the factors (i), (ii) and (iii) may readily be seen in equation(2.2). The qua ntity/^ is a constant of proportionality with the dimension oflength which is a function of the scattering angle and will be referred to asthe scattering length. For a particular type of scatterer it will be a function ofthe scattering angle denoted by 28 infig.2.1. In X-ray diffraction theory thescattering angle is conventionally denoted by 29 (and not simply by 6) asthis leads to simplifications in no tation in later developm ents of the theoryand is also associated with the historical development of the subject (see 3.6).It is mathematically convenient to write the equation of a progressivewave in complex form as7 = r o e x p [ 2 7 i i v ( r - x / c ) ]= Yo cos[27iv(r - x/c )] + i Yo sin[27tv(r - x/c)]. (2.3)

In equation (2.3) Yo is the amplitude of the wave, the real part of theexpression is the displacement and the ratio (imaginary part/real part) is thetangent of the phase of the wave motion a t (x, t) with respect to that at theorigin (0,0).With this nomenclature one can express the time dependence of thedisturbance at P in fig. 2.1 as


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34

y{26D, t) =/29^exp[27iiv(t - D/c) - i a j .

The scattering of X-rays

(2.4)

The amplitude of the disturbance at P due to a single scatterer is then given byt!(2d,D)=f2e^ (2.5)

and the phase lag of the disturbance at P behind the incident wave at 0 isaOP = 2nvD/c + as. (2.6)The intensity of the scattered beam in terms of power per unit solid angle

is given by

J2e = ]2

x D2 =f2dKA2

or

where K is the constant relating intensity to (amplitude)2 and / 0 is theintensity of the incident beam on the scatterer. The use of the distinctsymbols to represent the differently defined incident and scattered beamintensities should be noted.

2.2 Scattering from a pair of pointsConsider the situation shown in fig. 2.2 where radiation is incident on twoidentical scattering centres Ox and O2. We shall find the resultant at P, apoint at a distance r from Ox which is very large compared to the distanceOlO2. Under this condition the scattered radiation which arrives at P hasbeen scattered from Ot and O2 through effectively the same angle 26 . Theplanes defined by (i) Ov O2 and the incident beam direction and (ii) OUO2and P are indicated infig.2.2 to emphasize the three-dimensional nature ofthe phenomenon we are considering.

Since the scatterers are identical the scattering phase shift as will be thesame for each. Hence, for the radiation arriving at P the phase difference ofthe radiation scattered at O2 with respect to that scattered at Ox is

Fig. 2.2.Scattering from a pair ofpoint scatterers.0 , So

Incident radiation

Scattered radiation


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Scattering from a pair of points 35

Oio2=--^(CO2 + O2D). (2.8)Two unit vectors 0 and S are now defined which lie, respectively, along thedirections of the incident and scattered beams. If the vector joining O1 to O 2is denoted by r then

CO2 = r-S0, O 2D = - r and thus, from equation (2.8),

(2.9)The bracketed quantity in equation (2.9) may be replaced by an equivalentvector

s = ^ (2 10)giving

aolo2 = 2n T's- (2-n)The vector s is highly significant in describing the scattering process anda geometrical interpretation of it is shown infig.2.3. The vectors So/X andS/X in the incident and scattered directions have equal magnitudes 1/1. Itcan be seen from simple geometry that s is perpendicular to the bisector of

the angle between So and S and that its magnitude is given bys = (2sin0)/A. (2.12)If the displacement due to the incident rad iation at Ox is described byequation (2.1) then the resultant disturbance at P, a distance D from O l9 willbe given byy(209D,t) =/2,-{exp[27iiv(r - D/c) - iaj

+ exp[27iiv(t D/c) ias + 27iir-s]}= / 2e^exp[2itiv(t - D/c) - i a j [ l + exp(2iurs)]. (2.13)

The amplitude of this resultant is

Fig. 2.3.The relationship of s to S()/lSo and S.

S/A


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36

Fig. 2.4.(a) A phase-vectordiagram for a pair ofpoint scatterers with on epoint as phase origin.(b) A phase-vectordiagram for a pair ofpoint scatterers with ageneral point as phaseorigin.

The scattering of X-rayswhich, using equa tion (2.5), may be expressed in terms of the amplitude ofscattering from a single unit as

rj2(26,D) = exp(2iurs)]. (2.14)This equation is interpreted in terms of a phase-vector diagram in fig.2.4(a). The amplitude of the disturbance at P due to scattering at Ox isrepresented by the vector AB and that due to scattering at O2 by the vectorBC. Both these vectors have the same magnitude, rt(29,D), and the anglebetween them equals the difference of phase of the radiation scattered fromO1 and O2, 27irs. The resultant AC has m agnitude rj2(29,D) and differs inphase from the radiation scattered at Ox by the angle 0.However in this description we have given a special role to one of thescatterers, O l9 with respect to which as origin all phases are quoted. Thephase-vector diagram can be drawn with m ore generality if one measures

phases with respect to radiation which would be scattered from somearbitrary point 0 if in fact a scatterer was present there. Then, if thepositions of Ox and O2 with respect to O are given by the vectors rx and r2,equation (2.14) appears asri2(20,D) = f7(20,D)[exp(27tiiys) 4- exp(27iiiyr)]

and the phase-vector diagram appears as in fig. 2.(2.15)

2.3 Scattering from a general distribution of point scatterersLet us now examine the scattering from a system of identical pointscatterers O v O 2 , . . . , On. We are interested in the amplitude ofthe disturbancein some direction corresponding to a scattering vector s at a distance whichis large compared with the extent of the system of scatterers.If the position of the sca tterer at Om is denoted by its vector displacement

0, D) B(a)


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2.4 Thomso n scattering 37rm from some origin point 0 then, by an extension of the treatm ent whichled to equation (2.15) we find

(2.16)That this equation applies to identical scatterers is revealed by the factor

7/(20, D) appearing outside the summation. When the scatterers are non-equivalent the scattering amplitude must be writtenrjn(2e,D)=

= ^ Z (fie)] exp(27iir/s), (2.17)where the scattering length for each of the scatterers now appears within thesumm ation symbol. The phase-vector diagram for non-identical scatterersis shown in fig. 2.5 for the case n = 6. It will be evident that, although werefer to the scatterers as non-equivalent, we have assumed tha t they all havethe same associated values of as. This is the usual situation with X-raydiffraction. However it is sometimes possible to have the scatterers withdiffering phase shifts and, when this happens, useful information may beobtained (8.5).We shall find later that equation (2.17) is the basic equation fordescribing the phenomenon of X-ray diffraction and, when the symmetry ofthe atomic arrangem ents w ithin crystals is taken into account, tha t it mayappear in a number of modified forms.

2.4 Thom son scatteringWe have discussed the results of scattering by distributions of scattererswithout concerning ourselves with the nature of the scatterers or of thescattering process. It turns out that the scatterers of interest to us areelectrons and the theory of the scattering of electromagnetic waves by free(i.e. unbou nd and unrestrained) electrons was first given by J. J. Thomson.

Fig. 2.5.A phase-vector diagramfor six non-identicalscatterers.2n rh s


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38 The scattering of X-raysThe basic mechanism of Thomson scattering is simple to understand.When an electromagnetic wave impinges on an electron the alternatingelectric-field vector imparts to the electron an alternating acceleration, andclassical electromagnetic theory tells us that an accelerating chargedparticle emits electromagnetic waves. Thus the process m ay be envisaged asthe absorption and re-emission of radiation and, although the incidentradiation is unidirectional, the scattered radiation will be emitted in alldirections. If we have the straightforward case where the incident radiationis a single, continuous and monochrom atic wave then the acceleration ofthe electron will undergo a simple harmonic variation and the incident andemitted radiation will quite obviously have the same frequency.If an electron at 0 , of charge e and mass m, is undergoing an oscillationsuch that the acceleration is periodic with amplitude a (fig. 2.6) then theorytells us that the scattered radiation at P , which is travelling in the direction

OP, has an electric vector of amplitudeE = ea sin (/>4nsorc 2 (2.18)

which is perpendicular to OP and in the plane defined by OP and a. Here, 0is the permittivity of free space.In fig. 2.7 a parallel beam of electromagnetic radia tion travelling alongOX falls upon an electron at O. We wish to determine the nature of thescattered wave at P . The am plitude of the electric vector E of the incidentwave is perpendicular to OX and may be resolved into components E and|l perpendicular to and in the plane OX P. The electron will havecorresponding components of acceleration of amplitudes

Fig. 2.6.The relationship of theelectric vector ofscattered electromagneticradiation at a point P tothe acceleration vector ofan electron at 0. Thevectors are both in theplane of the diagram.Fig. 2.7.The relationship of thecomponents of theelectric vector ofscattered electromagneticradiation at P to thecomponents of theelectric vector of theincident radiation at 0.

Accelerationvector Electricvector

Direction of scatter


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2.4 Thomso n scattering 39

and

Applying equation (2.18) we find the electric vector components of thescattered wave at P as1 4n orc 2m

ande2cos 20

11 4n orc 2mThe quantity e2/4noc2m, which has the dimensions of length and equals2.82 x 10~ 15 m, is referred to in classical electromagnetic theory as theradius of the electron.Although we have been thinking abo ut a simple, continuous, monochro-matic, electromagnetic wave all the theory described above can be appliedwhen the incident radiation is complex in form. A complicated incidentwave may be analysed into simple components (see chapter 4) and theresultant electron acceleration and re-radiation may be found by adding

together the effects of the simple components. Thus E and E^ may bethought of as the components of the amplitude of any arb itrary electromag-netic radiation arriving at O.If the intensity of the incident radiation is Io and if this radiation isunpolarized then= CI0, say. (2.21)

The intensity of the scattered radiation, defined as the power per unit solidangle scattered through an angle 20 is given by1

Af-^Yn^^2\4noc2m (1 + cos2 20)10. (2.22)The factor 1/m2 in equation (2.22) shows why electrons are the onlyeffective scatterers; the lightest nucleus, the p roton, has the same magnitudeof charge as the electron but 1837 times the mass.Thomson scattering is coherent, that is to say there is a definite phaserelationship between the incident and scattered radiation; in the case of afree electron the scattering phase shift is n. In all the processes concernedwith the scattering of X-rays the electrons are bound into atoms and in 2.6we shall investigate the form of the scattering from the to tal assemblage ofelectrons contained in an atom.


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40 The scattering of X-raysIt is instructive to determine the proportion of the power of a beamincident on a material w hich will be scattered. First we calculate the totalpower scattered for each individual electron. In fig. 2.8 the point Orepresents the electron and OX the direction of the incident beam. Thepower d^J scattered into the solid angle dft, defined by the region betweenthe surfaces of the cones of semi-angles y and y + dy, is

and since dQ = 27tsinydy and


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2.4 Thomson scattering 41

c = = 3 \4nsoc2m n. (2.26)

The quantity o is the fraction of the incident radiation scattered per unitlength of path (one metre in SI units).If it is assumed tha t all the electrons in a material are free we can make anestimate of the fraction of the incident radiation which is scattered. Forexample a material of specific gravity 1.2 consisting of light atoms (say ofatomic weight less than 30) would contain about 3 x 1029 electrons m~3.This value of n substituted in equation (2.26) gives a 20. Since the crystalsused in X-ray diffraction usually have dimensions less than 1 mm it will beseen that only 2% or less of the incident X-ray beam is scattered.However for the scattering which normally occurs when X-rays interactwith m atter the electrons are bound with various degrees of strength to thenuclei of atoms. The advent of the laser has made possible the directmeasurement of Thom son scattering by the interaction of light from a rubylaser with an intense electron beam. In an experiment by Fiocco andThompson (1963), illustrated in fig. 2.9, a 75 mA, 2kV electron beam,magnetically focussed to give an electron density within the beam of5 x 10 15 m ~ 3 was crossed by a beam from a ruby laser. This laser gave aburst of 20 joules of light in about 800 fxs and the light measured by thephotomultiplier detector had an intensity abou t 10 ~8 of that of the incidentbeam. The scattered light was of two types - some was scattered by theelectron beam and there was also stray light reflected from the walls of theapparatus. The central feature of this experiment was that it was possible toseparate the stray light from the Thom son-scattered component. Since thescattering was from rapidly moving electrons (speed about one-tenth ofthat of light) there was a Doppler shift in frequency. This shift was about260 A and with a suitable filter the stray radiation, whose wavelength wasunshifted, could be prevented from reaching the detector.This experiment barely detected the presence of Thomson scattering asthere were very few (less than ten) scattered pho tons per pulse. Later workusing higher-current electron beams and m ore powerful lasers has enableda direct confirmation of equation (2.22) to be m ade (see Problem 2.3 at theend of this chapter).

Fig. 2.9.The arrangement ofcomponents of theFiocco and Thompson(1963) experiment. Collector forun scatteredradiation


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42 The scattering of X-rays2.5 Compton scattering

Experimentally it is found that the radiation scattered by materials consistsof two parts. The first part, that associated w ith Thom son scattering, hasthe same wavelength as the incident radiation; the second part has awavelength longer than that of the incident radia tion with the difference ofwavelength depending on the angle of scatter. This latter com ponent is dueto what is known as Compton scattering and it is incoherent with theincident radiation. It is best described in terms of the elastic collision of aphoton with an electron. Infig.2.10(a) the incident ph oton moves along thedirection PO and, after collision with the electron, moves off along OQ whilethe electron recoils along OR. From the conservation of energy in the elasticcollision we find that

he he 1 2T ~ X + dX + 2Vor, making usual approximations,

heT 2' (2.27)In addition to energy, momentum must also be conserved and in fig.

2.10(fr) is shown the a ppropriate momentum vector diagram. It is a validapproximation to ignore the change in magnitude of the momentum of thescattered photon and thus we deduce from simple geometry thatmv = - sin 6.A (2.28)

Fig. 2.10.(a) Velocity vectordiagram for Comptonscattering of a photon byan electron.(b) Momentum vectordiagram for Comptonscattering.

Scattered photonheenergy =

Incident photonheenergy = \ Recoil electronenergy = jm v2(a)

Scattered pho ton _ J7momentum magnitude 6Recoil electronmomentum magnitude = mv

Incident photonmomentum magnitude = hjX(b)


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2.6 The scattering of X-rays by atoms 43Eliminating v from equations (2.27) and (2.28) we have


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Fig. 2.11.An elemental volume forspherical polarcoordinates.

where p represents the charge density in electron units per unit volume. Inthe special case when | *F |, and therefore p, is spherically symmetric we mayrepresent the electron density by p(r). If, for example, we express thepositional parameters in spherical polar coordinates with respect to thecentre of the atom as origin, then the charge associated with a smallelemental volume is p{r)r2 sin ij/dr dij/d(/> (fig. 2.11). Thus, if the scatteringvector is s and if the coordin ate system is so arranged tha t s is parallel to theaxis from which i// is measured, then the total amplitude of coherentlyscattered radiation may be found from equation (2.17) with integrationreplacing summation. We take the am plitude of the scattered wave from thesmall elemental volume as Cs x charge where C s is some constant dependenton the scattering vector s and we obtain the amplitude from the wholeelectron asAs = C s p(r)r2Qxp(2nirs c o s i/f)sin \// dr d ^ dcf)J r = O J\J/ = O J ^ = 0 (2.32)

since r*s = rs cos \jt. Note particularly that the integration limits for \jj and 0do cover the whole of space.If the electron-density distribution is spherically symm etric then it is alsocentrosymmetric and this enables simplifications in equation (2.32) to bemade. For every point P with coordinates (r, ij/, ) there is another point P'with coordinates (r,n ij/,n + ) (see fig. 2.11) and the same electrondensity. The contribution of two elemental volumes round P and P' willgive a resultant, the form of which can be appreciated by adding two termssuch as


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2.6 The scattering of X-rays by atoms 45) + exp[27iirs COS(TI \jj)~]= exp(27iirs cos \j/) + exp( 2nirs cos \j/)= 2cos(2i i rscos^) .

It is clear from this that A s is a real quantity for a centrosymmetricdistribu tion of electron density an d t ha t equ ation (2.32) can be rewritten asAs=cA p(r)r2cos(2nrs cos ^)sin x// Ar d\jj d0 (2.33)The fact tha t p(r) is indep enden t of \\t and 0 enables one to integrate overthese latter variables giving

Fo r a given value of s we shall now find the scattered am plitu de from p(r)as a fraction ps of that amp litude, say (A s)o, which would be given by a po intelectron at the origin. This may be found from equation (2.34) by notingthat a point electron at the origin has an electron density p(r) = d(r). T hedelta function d(r) has the property

S(r) = oo, r = 0,and J


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pJr) = I pfo). (2.37)Th e amplitud e of cohere nt scattering from the total electron density willbe obtained by adding th e amplitud es for the electrons taken individually.W e now define an atomic sca ttering factor fa as the ratio of the amplitude ofthe coherent scattered radiatio n from an ato m to tha t from a single electronsituate d at the ato m ic centre. This is derived from equ atio ns (2.36) and (2.37)and is4\^dr=i>s), (2.38)

Atom ic scattering factors are well tabula ted in vol. Il l of the InternationalTables for X-ray Crystallography. Var ious mo dels of atom s have been usedto give electron-density distributions. For light atoms the Hartreeself-consistent field m ethod of com putin g wave functions is usually employed,while for heavy atoms the Thomas-Fermi approximation may be used.We can now investigate the nature of Compton scattering from atoms.From equation (2.36) we can see that the intensity of coherent scatteringfrom an atomic electron is p\^ 2Q and hence, since the total intensity ofscattering as revealed by wave mechanics is J2By the intensity of theCo mp ton scattering must be (1 Ps)^2o- However the Co mp ton scatteringfrom one atom ic electron is incohe rent w ith respect to that scattered by anyother and hence the total intensity from all the electrons is obtained byaddin g the individual intensities from each of the electrons. Thu s we have

^Comp,on = Z {1 -( P S ) , ? }X^2> (2.39)j= 1

andj i ] 2 2 ( , (2.40)

It will be seen from eq ua tion (2.36) th at since for s = 0 (9 = 0) we have/sin(27irs)\V 2 )

then p0 = 1. Hence, for radiation scattered in th e incident-beam directionthere is no incoherent c om pon ent. As 6 increases so p decreases but the ra teof decrease is less for those electrons which are most tightly bound in theatom . Infig.2.12(a) there are given the radial electron-density distributions,4n r2p(r), for the six electrons of the carbon atom calculated from Slater'sappr oxim ate analytic wave functions. In fact there are two Is electrons, two2s electrons and two 2p electrons; when radial symmetry is assumed (andthis is an assumption which, although usually made, is not really justifiedfor 2p electrons) the 2s and 2p electrons give equivalent rad ial electron-densitydistributions.


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2.6 The scattering of X-rays by atoms 47Fig. 2.12.(a) The radialelectron-density functionsof Is and 2s electrons ofa carbon atom asdefined by Slater'sanalytic wave functions.(b) The amplitude ofscattering from the Isand 2s electrons.(c) The atomic scatteringfac tor /a for the carbonatom and the coherentand incoherent scatteredintensities from a singlea tom.

4n r2p(r)

1.0r(a)

2.0

1.0

0.5

Is

2.0


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48 The scattering of X-raysFrom the Slater formulae we find

c 3pls(r) = exp( 2cxr) (2.41)and

(2.42)where, for carbon, c1 = 10.77 A " 1 and c2 = 6.15 A " 1 .Th e much tighter distribution of the Is electrons can be apprec iated fromfig. 2.12(a). For a given value of s(9) if more of the electronic charge isconcentrated at low values of r then more of the contribution of theintegration, equation (2.36), is for regions of space where sin(2nrs)/2nrs isclose to unity and hence ps will also be closer to unity.Fo r the analytical expressions (2.41) and (2.42) we find from equ ation (2.36)

and

A *\ I 0 / f\ \ *** M -M-M -\**v* **/ 1 / / % A ^ \Zx/* I I* PY H I /P VI (\Y r=z - l / di l 'T C i I / CAUI Z.C i / I "" U7 " x ^ TTT \ ^ " * ^ /Jr = O

C2V C2 ~~ 4 7 1 S )yFsns (4 + 4u V ) 4 'In fig. 2.12(fe) the re ar e plo tted the v alue s of (p s)ls and (p s)2s and it ca n be

seen how the tighter distribution of electron density for the Is electron leadsto a slower fall-off in the value of ps.The total coherent scattering amplitude/ a is found by adding togetherthe values of ps for the individual electrons and this is shown infig.2.12(c)together with^f the total coherent scattered intensity from a single atom .Th e incoheren t scattered intensity is found from equ ation (2.39) and is alsoshown plotted infig.2.12(c); it appears that for a single atom the incoh erentscattering is quite appreciable for high values of s. However under theconditions of the diffraction of X-rays from crystals, very large nu mb ers ofatoms co-operatively scatter so that the amplitudes of the coherentscattering from different atoms add together whereas for the incoherentscattering it is the intensities which add. Thus when we consider thediffraction of X-rays by crystals incoherent scattering may generally beignored.

Problems to Chapter 22.1 Fo ur identical coheren t scatterers are placed in a row with a distance 31between neighbours. Radiation of wavelength X falls on the scatterersalong a direction which is normal to the row.

(a ) For scattering angles 26 = 0 to 180, in steps of 20, determine thescattered amplitudes and intensities as fractions of that whichwou ld result if all four scatterers were at one poin t. Plot the results.


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Problems to Chapter 2 49(b) For the same values of 20 find the phases of the resultant scattered

radiation with respect to that from a scatterer at one end of the line.2.2 Eight identical coherent scatterers are placed at the corners of a cube of

side L A parallel beam of radiation of wavelength A, moving along thedirection of a body diagonal of the cube, falls on the scatterers. What isthe ratio of the intensity of the scattering along the direction of one ofthe sides of the cube compared to that which would be obtained from asingle scatterer for the same scattering angle?(Note: there are two directions to be considered, one representingforward and the other backward scatter.)

2.3 In an experiment to investigate scattering from free electrons afield-emission discharge of 2 kV electrons of 5000 A is confined within acylinder of 4 mm diameter. A Q-spoiled ruby laser produces a 10 joulepulse of duration shorter than that of the field emission and this passesin a fine pencil through the centre of, and perpendicular to, the electronbeam. A d

introduction x ray

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