Diffraction physics

NORTH-HOLLAND PERSONAL LIBRARY

DIFFRACTION PHYSICS

THIRD REVISED EDITION

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DIFFRACTION PHYSICS THIRD REVISED EDITION

JOHN M. COWLEY Regents'Professor emeritus Arizona State University

1995

ELSEVIER Amsterdam

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ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 EO. Box 211,1000 AE Amsterdam, The Netherlands

First edition Second revised edition First impression (hardbound) Second impression (paperback) Third impression (paperback) Fourth impression (paperback) Third revised edition First impression (paperback)

1975 1981 1984 1986 1990 1995

Library oF C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n

C o w l e y , J. M. (John M a x w e l l ) , 1923Diffraction p h # s i c s / John M. C o w l e # . - - 3 r d r e v . p. cm. - - ( N o r t h - H o l l a n d personal library) Includes bibliographical references and i n d e x . ISBN 0 - 4 4 4 - 8 2 2 1 8 - 6 ( a l k . paper) 1. D i f f r a c t i o n . I. Title. QC415.C75 1995 535'.4--dc20

Data

ed.

95-38709 CIP

ISBN: 0-444-82218-6 9 1995 ELSEVIER SCIENCE B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

PREFACE TO THE FIRST EDITION This b o o k has evolved, over the last ten years, from lecture notes for classes in p h y s i c a l optics, d i f f r a c t i o n p h y s i c s and e l e c t r o n m i c r o s c o p y g i v e n to a d v a n c e d u n d e r graduate and graduate students. It reflects my own particular interests in e l e c t r o n diffraction and diffraction from d i s o r d e r e d or i m p e r f e c t c r y s t a l s and e m p l o y s an a p p r o a c h w h i c h is p a r t i c u l a r l y s u i t e d to the t r e a t m e n t of t h e s e topics. This approach, using the F o u r i e r t r a n s f o r m from the b e g i n n i n g i n s t e a d of as an e x t e n s i o n of a F o u r i e r series treatment, is not only more s a t i s f y i n g from a conceptual and t h e o r e t i c a l b a s i s but it p r o v i d e s the p o s s i b i l i t y for a u n i f i e d t r e a t m e n t of all the different b r a n c h e s of d i f f r a c tion physics, employing electrons, X-rays or neutrons. Because this approach has been adopted, the generation of the f a m i l i a r ideas of d i f f r a c t i o n of radiation by solids is slow and detailed. Bragg's Law does not appear until Chapter 6 and then only as a c o r o l l a r y to the Ewald sphere construction. This m a y create d i f f i c u l t i e s for readers or students u n f a m i l i a r with the more c o n v e n t i o n a l approaches. Hence this book is p r o b a b l y more appropriate for those who have had one e l e m e n t a r y course in d i f f r a c t i o n or who are a l r e a d y e n g a g e d in some area of diffraction work. Much of the content relating to e l e c t r o n diffraction has been g e n e r a t e d in c o l l a b o r a t i o n with A.F. Moodie, of the D i v i s i o n of Chemical Physics, C.S.I.R.O., Australia, who has for m a n y years b e e n my m e n t o r in m a t t e r s of d i f f r a c t i o n theory. M y h e a r t f e l t g r a t i t u d e goes to h i m and to all m y colleagues in the C.S.I.R.O., the U n i v e r s i t y of Melbourne and Arizona State U n i v e r s i t y whose valuable and friendly collaboration has made this work p o s s i b l e and who have a l l o w e d me to use their results and their p i c t u r e s to i l l u s t r a t e the text. I wish to thank Drs. Kato, Borrmann and Beauvillain for permission to reproduce Figs. 9.10, 14.4, 14.5 respectively.

PREFACE TO THE SECOND EDITION In p r e p a r i n g the s e c o n d e d i t i o n of this book, I have b e e n g r e a t l y a s s i s t e d by my s t u d e n t s and others who have p o i n t e d out a n u m b e r of e r r o r s in the first edition. Some v a l u a b l e c o m m e n t s came from P r o f e s s o r Z.G. P i n s k e r and c o l l e a g u e s who p r e p a r e d the R u s s i a n language edition. I am very g r a t e f u l to these p e o p l e and also to Dr. Peter G o o d m a n for d i s c u s s i o n s on some of the a d d i t i o n a l sections which have been included. M o s t of the c h a n g e s w h i c h have b e e n m a d e to the first e d i t i o n come w i t h i n the final few c h a p t e r s w h i c h deal w i t h a p p l i c a t i o n s of the d i f f r a c t i o n m e t h o d s since it is in t h e s e areas that progress has b e e n m o s t rapid. In fact, the expansion of the f i e l d has b e e n so g r e a t that it is impossible in a b o o k of this size to p r o v i d e any c o h e r e n t a c c o u n t of the m a n y i n t e r e s t i n g recent r e s e a r c h d e v e l o p m e n t s . However, it is h o p e d that the m e n t i o n of some of t h e s e topics, t o g e t h e r with the a p p r o p r i a t e references, will l e a d the r e a d e r t o w a r d s m o r e c o m p l e t e i n f o r m a t i o n on t o p i c s of a more s p e c i a l i z e d interest.

PREFACE TO THE THIRD EDITION In this t h i r d e d i t i o n an a t t e m p t has b e e n m a d e to m a k e this book more consistent with the present status of d i f f r a c t i o n a c t i v i t y w h i c h has u n d e r g o n e i m p o r t a n t d e v e l o p ments in the last f i f t e e n years. It is not f e a s i b l e to a t t e m p t an a d e q u a t e d e s c r i p t i o n of current t e c h n i q u e s or the most recent theoretical developments, but it is h o p e d that s u f f i c i e n t r e f e r e n c e has b e e n m a d e to t h e s e things to a l l o w t h e m to be p l a c e d in the c o n t e x t of the e l e m e n t a r y t h e o r e t ical ideas w h i c h f o r m the b a s i s for this book. In p r e p a r i n g this E d i t i o n , I have b e n e f i t t e d greatly from interactions with, and a s s i s t a n c e from, the faculty, r e s e a r c h a s s o c i a t e s , staff and students associated w i t h the C e n t e r for H i g h R e s o l u t i o n E l e c t r o n M i c r o s c o p y at A r i z o n a State U n i v e r s i t y . In p a r t i c u l a r , I w o u l d like to e x p r e s s my g r a t i t u d e to K a y e F r i x for her p a i n s t a k i n g w o r k on p r e p a r i n g the c a m e r a - r e a d y copy.

CONTENTS

Preface Preface Preface

m m m

the first edition the second edition the third edition

V VI VI

Section I - PHYSICAL OPTICS

Chapter 1. Fresnel and Fraunhofer diffraction i.I. Introduction 1.2. Wave equations and waves 1.2.1. Wave functions 1.2.2. Electromagnetic waves 1.2.3. Particle waves 1.3. Superposition and coherence 1.3.1. Superposition 1.3.2. Independent point sources I. 4. Huygen' s principle 1.4.1. Kirchhoff's formulation 1.4.2. A p p l i c a t i o n of the Kirchhoff formula 1.5. Scattering theory approach 1.5.1. Integral form of wave equation 1.5.2. Born series 1.6. R e c i p r o c i t y 1.7. Fresnel diffraction 1.7.1. Small angle approximation 1.7.2. Fresnel integrals 1.7.3. Periodic objects - "Fourier images" 1.8. Fraunhofer diffraction Problems

3 5 5 6 7 8 8 9 Ii Ii 12 13 13 14 16 16 16 17 19 21 24

Chapter 2. Fourier transform and convolutions

25

2.1. Preliminaries 2.1.i. Delta-functions and discontinuities 2.1.2. Convolutions 2.1.3. Examples of convolutions 2.2. Fourier transforms- general 2.2.1. Definitions

25 25 26 27 30 30

viii

CONTENTS

2.2.2 9 Properties of Fourier transforms 2.2.3. M u l t i p l i c a t i o n and convolution 2.2.4. Space and time 2.3. Fourier transforms and diffraction: examples 2.3.1. Point source or point aperture 2.3.2. A plane wave- the inverse of 2.3.1. 2.3.3. Translation of an object 2.3 .4. Slit function 2.3 .5. Slit function-alternative 6. Straight edge 2.3 9 2.3 .7. Rectangular aperture 9 Circular aperture 2.3 8. 2.3 .9. Two very narrow slits 2.3 .10. Two slits of appreciable width 2.3 .ii. Finite wave train 2.3 .12. Periodic array of narrow slits 2.3 .13. A r b i t r a r y periodic function 9 D i f f r a c t i o n grating- thin slits 2.3 14. 2.3 .15. D i f f r a c t i o n grating: general 9 Gaussian function 2.3 16. 2.3 .17. Row of circular holes 9 C o m p l e m e n t a r y objects-Babinet' s principle 2.3 18. 2.3 .19. Total intensities- Parseval's theorem Problems

31 33 34 35 35 35 36 36 37 37 38 39 39 4O 4O 42 43 43 44 45 46 47 48 48

Chapter 3. Imaging and diffraction

51

3.1. Wave theory of imaging 3.1.I. Coherent wave optics 3.1.2. Incoherent wave imaging 3.2. Abbe theory 3.3. Small angle approximation 3.4. Phase contrast 3.4.1 9 Phase and amplitude objects 3.4.2. Out-of-focus contrast 3.4.3. Aperture limitation 3.4.4. Zernike phase contrast 3.5. H o l o g r a p h y 3.6. M u l t i - c o m p o n e n t systems 3.7. Partial coherence Problems

51 51 54 55 56 59 59 60 62 62 63 67 69 72

Section II - KINEMATICAL D I F F R A C T I O N

Chapter 4. Radiations and their scattering by matter

77

4.1. X-rays 4.1.i. 4.1.2.

77 77 78

X-ray sources Scattering by electrons

CONTENTS

ix

4.1.3. Scattering by atoms 4.1.4. Dispersion corrections 4.2. Electrons 4.2.1. Sources of electrons 4.2.2. Atom scattering amplitudes 4.2.3. Phase object approximation 4.2.4. Failure of first Born approximation 4.2. 5. "Absorption" effects 4.3. Neut rons 4.3. I. Atomic scattering factors 4.3. 2. Nuclear spin scattering 4.3. 3. Isotopic disorder 4.3. 4. Thermal and magnetic scattering Problems

80 81 81 81 82 84 85 86 88 88 89 90 91 92

Chapter 5. Scattering from assemblies of atoms

93

5.1. The kinematical approximation 5.2. Real and reciprocal space 5.2.1. Reciprocal space distribution 5.2.2. The reciprocal lattice 5.2.3. Friedel's law and the phase problem 5.3. The generalized Patterson function 5.4. Examples of correlation functions 5.4.1. Finite volume limitations 5.4.2. Finite crystals 5.5. Correlation in space and time 5.5.1. Four-dimensional Patterson 5.5.2. Special cases 5.5.3. Ideal monatomic gas or liquid 5.5.4. Real monatomic gases and liquids 5.5.5. The hydrogen atom 5.6. Diffraction geometry and intensities 5.7. Practical considerations 5.7.1. Finite sources and detectors 5.7.2. Wavelength spread 5.7.3. Integrated intensities 5.8. Sections and projections Problems

93 95 95 96 97 98 i01 i01 102 104 104 105 106 109 112 113 115 115 117 118 119 121

Chapter 6. Diffractionfrom crystals

123

6.1. Ideal crystals 6.2. Diffraction geometry 6.2.1. Laue and Bragg diffraction conditions 6.2.2. Shape transforms 6.2.3. Special cases for electron diffraction

123 126 126 127 128

x

CONTENTS

6.3. Crystal structure analysis 6.3.1. The phase p r o b l e m 6.3.2. Supplementary information 6.4. Structure analysis methods 6.4.1. Trial and error 6.4.2. Patterson function 6.4.3. H e a v y - a t o m & isomorphous replacement 6.4.4. Direct methods 6.5. Neutron diffraction structure analysis 6.5.1. Nuclear scattering 6.5.2. Magnetic scattering 6.6. Electron diffraction structure analysis Problems

methods

Chapter 7. Diffractionfrom imperfect crystals

145

7.1. F o r m u l a t i o n of the diffraction p r o b l e m 7.1.i. Types of defects 7.1.2. General diffraction formulation 7.2. Patterson function approach 7.2.1. Patterson with average periodic structure 7.2.2. Patterson with no average structure 7.3. Deviations from an average lattice 7.3.1. Random vacancies- no relaxation 7.3.2. C l u s t e r e d vacancies 7.3.3. Lattice relaxation 7.3.4. Thermal vibrations - Einstein model 7.4. Imperfect crystals with no average lattice 7.4.1. Uneven separation of lattice planes 7.4.2. D i s o r d e r e d orientations Problems

Section

III-

DYNAMICAL

131 131 133 134 134 135 135 136 138 138 139 141 142

145 145 146 147 147 149 150 150 152 154 156 157 157 160 163

SCATTERING

Chapter 8. Diffraction by perfect crystals 8.1. Multiple coherent scattering 8.2. Theoretical approaches 8.3. Bethe theory 8.3.1. The dispersion equations 8.3.2. Solutions of the equations 8.3.3. Boundary conditions 8.4. T w o - b e a m approximation 8.4.1. Bloch waves and dispersion surfaces 8.4.2. Conduction e l e c t r o n s - e n e r g y representation 8.4.3. X-ray diffraction; p o l a r i z a t i o n

167 167 168 170 170 172 173 175 175 177 178

CONTENTS 8.5. The Laue (transmission) case 8.5.1. Electron diffraction for a thin crystal 8.5.2. Small angle approximation 8.5.3. Laue case with absorption 8.6. Bethe potentials 8.7. The Bragg case

Chapter 9. Dynamical diffraction effects 9.1. Thickness fringes, rocking curves-electron diffraction 9.1.1. Intensity formulas 9.1.2. Real space picture 9.1.3. Rocking curves 9.1.4. Extinction contours 9.1.5. Convergent b e a m diffraction 9.1.6. D i f f r a c t i o n and imaging of crystal wedges 9.1.7. Absorption effects for wedges 9.2. Dynamical effects of X-ray and neutron diffraction 9.2.1. Techniques for X-ray diffraction 9.2.2. Energy flow 9.2.3. D i s p e r s i o n surface picture 9.2.4. Neutron diffraction 9.3. Borrmann effect Problems

Chapter 10. Extension to many beams i0.I. Dynamical n-beam diffraction 10.2. Extension of Bethe theory - transmission 10.2.1. M a t r i x formulation 10.2.2. Small angle approximation 10.2.3. Bloch waves and boundary conditions 10.2.4. The scattering matrix 10.2.5. Derivation of the two-beam approximation 10.3. The D a r w i n - t y p e approach 10.4. Special cases - b e a m reduction 10.5. C o m p u t i n g methods 10.6. Column approximation Problems

Chapter 11. Multi-slice approaches ii.i.

P r o p a g a t i o n of electrons in crystals Ii.I.i. Transmission through thin slices ii.I.2. Three-dimensional objects II.I.3. D i f f r a c t i o n by a crystal 11.1.4. General expression; excitation errors

xi 178 178 181 182 182 184

189 189 189 189 191 191 193 195 197 199 199 202 203 204 205 207 209 209 211 211 213 214 216 218 220 222 224 227 229 231 231 231 233 234 236

xu

CONTENTS

.o

11.2. Multiple-scattering series 11.2.1. Zero-order scattering 11.2.2. Single scattering-kinematical approximation 11.2.3. Multiple scattering 11.3. General double-summation solution 11.3.1. General series solution 11.3.2. Phase grating approximation 11.4. Computing methods 11.4.1. "Slice method" calculations 11.4.2. Steps in a computation 11.4.3. Possible errors 11.4.4. Consistency tests 11.4.5. Computing times 11.5. Intensities from non-periodic objects 11.6. Real-space formulations 11.6.1. High-energy approximation 11.6.2. Useful approximations 11.6.3. A real-space basis for computing Problem Section I V -

APPLICATIONS

237 237 238 238 240 240 241 243 243 245 246 247 248 248 250 250 252 253 254

TO SELECTED TOPICS

Chapter 12. Diffuse scattering and absorption effects

257

12.1. Thermal diffuse scattering 12.1.I. Phonons and vibrational waves 12.1.2. Scattering for a longitudinal wave 12.1.3. Diffuse scattering component 12.1.4. Dispersion curves 12. i. 5. Three-dimensional generalizations 12.2. Static atom displacements 12.2.1. Relaxation around point defects 12.2.2. Diffraction intensities for displaced atoms 12.2.3. The Bragg peaks 12.2.4. The diffuse scattering 12.3. Electron excitations 12.3.1. Inelastic X-ray scattering 12.3.2. Electron excitation by electrons-plasmons 12.3.3. Single-electron excitations 12.4. Dynamical effects in diffuse scattering 12.4.1. Scattering and re-scattering 12.4.2. Coherent and incoherent scattering 12.4.3. Analysis of diffuse scattering 12.5. Absorption effects 12.5.1. The nature of absorption parameters 12.5.2. Absorption of X-rays and neutrons 12.5.3. "Absorption" for electrons 12.5.4. Absorption due to thermal vibrations 12.5.5. Absorption from electron excitations 12.5.6. Values of absorption coefficients

257 257 258 259 261 262 262 262 263 265 266 269 269 270 272 274 274 276 278 279 279 280 281 282 284 285

CONTENTS

Chapter 13. Electron microscope imaging

xiii 287

13.1. Electron microscopes 13.1.1. Conventional transmission e. m. 13.1.2. Scanning transmission electron microscopes 13.2. Image formation 13.3. Contrast for thin specimens 13.3.1. Phase-object approximation 13.3.2. Weak-phase object approximation 13.3.3. Failure of weak-phase object approximation 13.3.4. Dark-field images 13.4. The imaging of crystals 13.4.1. Imaging of thin crystals; structure images 13.4.2. Calculation of images of crystals" envelope 13.4.3. Imaging of crystals - inelastic scattering 13.4.4. Lattice fringe imaging 13.4.5. Crystal imaging without lattice resolution 13.5. Image contrast in STEM 13.5.1. STEM imaging of thin crystals 13.5.2. STEM imaging of thicker crystals 13.6. Electron holography 13.7. Combining high-resolution imaging with diffraction Problems

287 287 289 292 294 294 296 299 300 301 301 305 307 309 312 313 313 317 318 326 328

Chapter 14. K-line patterns and channelling

329

14.1. Kossel lines 14.1.1. Geometry of Kossel lines 14.1.2. Dynamical theory of Kossel intensities 14.1.3. Kossel lines with limited resolution 14.2. Kikuchi lines 14.3. External sources of divergent radiation 14.4. Information from K-line patterns 14.5. Channelling 14.6. Secondary radiations

329 329 330 332 335 339 341 343 346

Chapter 15. Application of dynamical effects in single crystals

349

15.1. 15.2. 15.3. 15.4.

Dependence of dynamical effects on crystal parameters X-ray interferometry .-beam and 2-beam dynamical diffraction Accurate determinations of structure amplitudes 15.4.1. Measurements of thickness fringes 15.4.2. Structure amplitudes from rocking curves 15.4.3. Convergent beam electron diffraction method 15.4.4. The use of critical voltages 15.4.5. Intersecting K-lines 15.5. The determination of crystal symmetries 15.6. Coherent convergent-beam electron diffraction

349 349 351 354 355 357 358 359 362 363 367

xiv

CONTENTS

Chapter 16. Mosaic crystals and polycrystalline materials

369

16.1. General 16.2. Mosaic crystals 16.2.1. The mosaic crystal model 16.2.2. Kinematical integrated intensities 16.2.3. Extinction effects 16.2.4. Dynamical electron diffraction intensities 16.3. Polycrystalline material 16.3.1. Idealized models 16.3.2. Kinematical diffraction intensities 16.3.3. Line profile analysis 16.3.4. Rietveld refinements 16.3.5. Dynamical diffraction intensities 16.3.6. n-beam diffraction effects

369 370 370 370 372 374 374 374 377 377 379 380 383

Chapter 17. Ordering of atoms in crystals

385

17.1. The nature and description of disordered states 17.2. Order parameters 17.2.1. Short-range order 17.2.2. Long-range order 17.3. Patterson function 17.4. Size effects 17.5. Kinematical diffraction 17.5.1. Diffraction with ordering only 17.5.2. Diffraction with ordering and size effects 17.6. Relationship with ordering energies 17.7. Dynamical scattering from disordered crystals 17.7.1. Dynamical effects in diffuse scattering 17.7.2. Calculations of diffuse scattering 17.7.3. Strong scattering, multi-atom correlations 17.7.4. High resolution imaging disordered crystals 17.8. Out-of-phase domains 17.8.1. Ordered out-of-phase superlattices 17.8.2. Out-of-phase domains in disordered alloys 17.8.3. Modulated structures Problems

385 387 387 389 389 390 392 392 395 400 401 401 402 403 404 405 405 407 408 410

Chapter 18. Extended defects

411

18.1. Introduction 18.2. Stacking faults-statistical, kinematical theory 18.2.1. Patterson method for a simple case 18.2.2. A general treatment 18.2.3. Faults in close-packed structures 18.3. Dynamical diffraction by stacking faults 18.4. Dislocations 18.4.1. Diffraction effects

411 412 412 414 420 422 424 424

CONTENTS 18.4.2. The imaging of dislocations 18.4.3. Averaging over angles of incidence 18.4.4. n-beam diffraction effects

Chapter 19. Diffraction from surfaces 19.1. Introduction 19.2. Surface imaging and diffraction with electrons 19.2.1. Phase-contrast imaging 19.2.2. Crystal terminations and superlattices 19.2.3. Structure analysis of surface superlattices 19.2.4. Crystal profile imaging 19.3. Reflection from surfaces: grazing incidence 19.3.1. Kinematical approximation: x-rays, neutrons 19.3.2. Standing wave techniques 19.3.3. RHEED and REM 19.4. Reflection at normal incidence: LEED 19.5. Diffraction of emitted electrons

References Index

xv

426 427 427 433 433 435 435 436 438 439 440 440 443 444 451 453 457 477

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Section I

PHYSICAL OPTICS

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CHAPTER 1

Fresnel and Fraunhofer diffraction 1 .I.

Introduction

In attempting a unified treatment of the field of X-ray, neutron and electron diffraction, including electron m i c r o s c o p y and other imaging techniques based on diffraction, we are faced with the task of bringing together a number of theoretical treatments developed for convenience in handling particular topics. Often the "popular" treatment used by experimentalists is not just a simplification of the more rigorous methods of the theorist. It may have a different basis which is more conceptually pleasing but less tractable m a t h e m a t i c a l l y or is easier to visualize as in the geometric optics a p p r o x i m a t i o n for imaging theory. What we hope to do is to provide the experimentalist with a more coherent account of the basic theoretical ideas of this field, using arguments which may be followed with a certain m i n i m u m of mathematics. For both theorist and experimentalist we hope to provide an appreciation of how the physical observations using different radiations and different types of sample may be knit together with a common thread of theory so that ideas and understanding in one area may be related and transfered to other areas. For most people using X-ray diffraction for crystal structure analysis, diffraction is a three-dimensional extension of simple Fraunhofer diffraction theory applied to the idealized case of infinite periodic objects with sharply defined d i f f r a c t e d b e a m directions and a lattice of weighted points in reciprocal space. The basic mathematical tool is the Fourier series. With greater sophistication, the concept of a Fourier t r a n s f o r m is introduced to deal with cases of finite or imperfect crystals in the same single-scattering, kinematic approximation. This is far removed from the theoretical treatment of X-ray d i f f r a c t i o n by Ewald [1916] or von Laue [1931] or even from the simpler, intuitive approach of Darwin [1914]. Yet these are the sources which must be relied upon for discussion of the limitations of the assumption of the simple practical theory. Also, for the u n d e r s t a n d i n g and interpretation of important new techniques such as X-ray topography and X-ray interferometry and the older methods using Kossel lines, there is the same basic diffraction theory but the simplifications for practical experimental use are d e v e l o p e d along different lines.

4

FRESNEL AND F R A U N H O F E R D I F F R A C T I O N

CH.

1

Neutron diffraction was developed first by nuclear physicists with a jargon of differential scattering cross sections rather than atomic scattering amplitudes. Then it diversified, with structure analysis people t r a n s f e r r i n g the X-ray d i f f r a c t i o n ideas and solid-state physicists d e s c r i b i n g their experiments in terms of k vectors, Brillouin zones and so on. The added complication of inelastic scattering studies for time-dependent processes, including phonons and magnons, brought m a i n l y an elaboration of this solid-state physics approach rather than an extension of the Fourier t r a n s f o r m methods. The development of electron diffraction p a r a l l e l e d that of X-ray d i f f r a c t i o n in that the simple kinematic, F r a u n h o f e r diffraction, approximation was used where possible and the dynamical theory of Bethe [1928], analogous to von Laue's Xray theory, was used where necessary. The difference from the X-ray case came from the greater importance of dynamical s c a t t e r i n g effects and the greater complication of the dynamical effects which did occur. Consequently there was a need for relatively simple approximations for practical use, and several were developed to deal with different types of experiment. An additional complication to the electron d i f f r a c t i o n scene arose from the derivation of atomic scattering factors from scattering theory, the domain of nuclear and atomic physicists, with the attendant jargon. A legacy of this origin remains in that those engaged in electron d i f f r a c t i o n from gases consider atomic scattering factors (in a unit of A) as a function of the variable s = 4Kl -I sin 8 rather than the variable ~-i sin 8 used in X-ray diffraction or electron d i f f r a c t i o n of the solid state, or the reciprocal lattice distance u = 21 -I sin 8. The theory of image formation in the electron m i c r o s c o p e naturally follows that of light microscopy, but this must be combined with the theory of the scattering or diffraction of electrons in the object. These two aspects of electron m i c r o s c o p y are brought together naturally by the unifying concept of the Fourier transform which has a basic significance for electron diffraction as for X-ray d i f f r a c t i o n in the simple kinematical approximation and has also formed the basis for recent developments in imaging theory, introduced by Duffieux [1946]. The fact that the basic dynamical theories of X-ray d i f f r a c t i o n and electron diffraction have been d e v e l o p e d in terms of differential equations rather than in terms of integrals such as the Fourier t r a n s f o r m then suggests a major cleavage of the necessary theory into two parts, formally related but very different in practice. However this gap has been bridged, at least for electron diffraction, by development of dynamical diffraction theories formulated in

I. 1

INTRODUCT ION

5

terms of integrals, with the Fourier t r a n s f o r m playing a major part. A further major component can be recognized as related to the elementary physics text book treatment of Fresnel diffraction. With this in mind, we start our discussion of diffraction with a reminder to the reader of the elements of physical optics and introduce in this way the description of diffraction, scattering and imaging in terms of the Fourier t r a n s f o r m integral and the important associated integral, the convolution. It would p r o b a b l y be rather too indigestible for most p r a c t i s i n g electron microscopists and diffractionists if we attempted a complete treatment of diffraction along these lines. Instead we relate our treatment where appropriate, to the more familiar ground of the differential equation methods. However we would like to suggest that these may be considered as parallels drawn at places to an underlying unified line of development and introduced in order to satisfy the needs of convenience or convention. For ease of understanding, we have avoided over-rigorous arguments or m a t h e m a t i c a l complexity to the point where the purist may well criticize the logic of the development. In many cases we have d u p l i c a t e d m a t h e m a t i c a l statements with statements in words which are frequently less precise but may be of assistance for the less m a t h e m a t i c a l l y - m i n d e d reader. To start with, we assume the reader is familiar with the use of a complex exponential to represent a wave function, the use of Fourier series to represent a periodic function and the elements of integration and differentiation. For those not having this b a c k g r o u n d we recommend any one of a large number of text books of physical optics such as Ditchburn [1976] or N u s s b a u m and Phillips [1976]. Excellent treatments at a slightly more advanced level are given by Lipson and Lipson [1969], Goodman [1968], Petykiewicz [1992] and Born and Wolf [1989].

1.2.

Wave 1.2.1.

equations Wave

and

waves

functions

As a means for obtaining the intensities of d i f f r a c t e d beams of radiation, we make use of the convenient conventional idea of a wave function. For none of the electromagnetic radiations and particle beams we will be considering is it possible to observe any oscillatory wave motion. The wave function, a complex function of spatial coordinates which we write ~(r), is a convenient m a t h e m a t i c a l device for obtaining the observable quantity, the intensity or energy transfer in an experiment, given by i~(r) 12. By analogy with

6

FRESNEL AND F R A U N H O F E R D I F F R A C T I O N

CH.

1

water waves or waves in a string, we can visualize a wave function and appreciate the concepts of the wavelength ~, the wave vector k which indicates the direction of p r o p a g a t i o n and is of magnitude l/l, the frequency v or angular f r e q u e n c y in radians per second, ~, the phase velocity of the wave and the group velocity. If we consider them as particles, the X-ray photons, electrons and neutrons have very different properties as shown by their collisions with other particles or, more practically, their generation and detection. However if we consider only their p r o p a g a t i o n through space and their scattering by matter or fields with no appreciable loss of energy all these radiations may be considered as waves, d e s c r i b e d by wave functions which are solutions of the same type of differential equation, the wave equation. Thus we can deal with a relatively simple semi-classical wave mechanics, rather than the full q u a n t u m mechanics needed for interactions of quanta involving changes of energy. The p r a c t i c a l differences in experimental techniques and interpretation of m e a s u r e d intensities for the different radiations derive from the d i f f e r i n g values of the parameters in the wave equation. 1.2.2.

Electromagnetic

waves

For electromagnetic waves, the wave equation is usually written for the electric of magnetic field vector E or H; for example, ~2E

~'-~+

~)2E ~)2E ~2E ---. = V2E = e~ 9 3y2 + ~gz 2 ~t 2 '

(I)

where ~ is the p e r m i t t i v i t y or dielectric constant and ~ is the magnetic p e r m e a b i l i t y of the medium. The velocity of the waves in free space is r = (s For most purposes the vector nature of the wave amplitude will not be important. For simple scattering experiments with u n p o l a r i z e d incident radiation, for example, the only consequence of the vector properties is m u l t i p l i c a t i o n of the scattered intensity by a p o l a r i z a t i o n factor, dependent only on the angle of scattering. Hence we may usually simplify our considerations by assuming that we are dealing with a scalar amplitude function ~(r) which is a solution to the wave equation, ~2~ V 2 V = ep %t2 "

(2)

Except in special situations we may assume ~ = I. The p e r m i t tivity ~, the refractive index n = (s or the velocity = c/n, is a function of position in space c o r r e s p o n d i n g to the variation of electron density.

1.2

WAVE E Q U A T I O N S

AND WAVES

7

For p u r e l y e l a s t i c scattering, we n e e d c o n s i d e r only a single frequency, 0~. As we will see later, if more than one f r e q u e n c y is present, the i n t e n s i t y m a y be c a l c u l a t e d for each w a v e l e n g t h s e p a r a t e l y and then these i n t e n s i t i e s are added. Hence we can assume that the w a v e - f u n c t i o n is of the form U = Us exp { i~t} = Us exp {2~ivt} . Then,

since ~2~/~12 = _~2~, V2U

= -E~t(o2U

the wave e q u a t i o n V2U

or

+ 4~2k2u-

(2) b e c o m e s 0,

(3)

where 2~k is (g~) 1/20) or s the wave number. Here we use the convention, now s t a n d a r d in the e l e c t r o n d i f f r a c t i o n l i t e r a t u r e (Cowley 1992) that the m o d u l u s of the wave vector k has m a g n i t u d e Ikl = l/l, rather than the convention, current in m u c h of s o l i d - s t a t e physics, that lkl = 2~/~. For X-rays the d i e l e c t r i c constant and refractive index are very close to unity.

1.2.3.

Particle

waves

For e l e c t r o n s or other particles, the c o r r e s p o n d i n g wave e q u a t i o n is the S c h r o e d i n g e r e q u a t i o n which may be written

h2

~U

-2-~ V2U + e~(r)u-- -ih-~,

(4)

where U is a wave function such that IUI 2 represents the p r o b a b i l i t y of an e l e c t r o n b e i n g p r e s e n t in unit volume and ~(r) is the e l e c t r o s t a t i c p o t e n t i a l function w h i c h can be a s s u m e d to vary s u f f i c i e n t l y slowly for our p u r p o s e s even though the v a r i a t i o n is c a u s e d by the d i s t r i b u t i o n of p o s i t i v e and n e g a t i v e charges in the atoms and ions in matter. If, as before, we c o n s i d e r only a single f r e q u e n c y (0, and put ho~ = eE, where eE is the kinetic e n e r g y for ~ = 0 i.e. in f i e l d - f r e e space, then the time i n d e p e n d e n t wave e q u a t i o n is V2U+

2me

h2

{E + ~(r) } U =

0,

where the b r a c k e t m u l t i p l i e d by e gives the sum of kinetic and p o t e n t i a l energies of the electron. This is of e x a c t l y the same form as (3), with

k 2 = ( 2 m e / h 2)

{E + ~p(r) } .

(5)

8

F R E S N E L AND F R A U N H O F E R D I F F R A C T I O N

CH.

1

C o r r e s p o n d i n g l y the r e f r a c t i v e index for e l e c t r o n s h a v i n g kinetic (or incident) e n e r g y eE in a p o t e n t i a l field ~(r) is n=

k/co (k / (O)o

(E + q~(r))I/2

The c o n v e n t i o n

(6)

for the sign of ~ u s e d here is such that

~/2E is p o s i t i v e for an e l e c t r o n in a p o s i t i v e field w h i c h

a c c e l e r a t e s it. Hence the r e f r a c t i v e index of m a t t e r for e l e c t r o n s is s l i g h t l y g r e a t e r than unity. The simple, n o n - t r i v i a l s o l u t i o n of the wave e q u a t i o n (3) or (5), which can be v e r i f i e d by substitution, is (7)

= ~0 exp {-2Kik-r}, r e p r e s e n t i n g a plane wave p r o c e e d i n g in the d i r e c t i o n s p e c i f i e d by the v e c t o r k which has a m a g n i t u d e Ikl = k = i/l. A d d i n g the time dependence, this becomes ~t = ~t o e x p { 2 g i ( v t

-

k-r) }.

(8)

The choice of the p o s i t i v e sign of the exponent is arbitrary. A n e g a t i v e sign before the i is e q u a l l y valid. The c h o i c e as to w h e t h e r the cot or the k.r is given the m i n u s sign is a m a t t e r of convention. It is important only to be c o n s i s t e n t in the choice. The form of (8) implies that for l constant the phase d e c r e a s e s for i n c r e a s i n g p o s i t i v e d i s t a n c e in the d i r e c t i o n of k and for a f i x e d p o s i t i o n the phase i n c r e a s e s with time. A further solution of importance, c o r r e s p o n d i n g to a s p h e r i c a l wave r a d i a t i n g from a point source, taken to be the origin, is

~=

~o

exp{2~i(vt

r

- kr)}

(9)

where r = Irl. Since the d i r e c t i o n of p r o p a g a t i o n is always radial, k.r = kr. This form is c o n s i s t e n t with the inverse square law, since the e n e r g y per unit area, p r o p o r t i o n a l to ~n~*, will be p r o p o r t i o n a l to r-2. 1.3.

Superposition

1.3.1.

and

coherence

Superposition

We have seen that the wave functions for both e l e c t r o m a g n e t i c waves and p a r t i c l e s are solutions of linear d i f f e r e n t i a l e q u a t i o n s such as (3). A f u n d a m e n t a l p r o p e r t y of the solutions of such e q u a t i o n s is that the sum of any two s o l u t i o n s is also a solution. In terms of waves, we can state

1.3

SUPERPOSITION

AND C O H E R E N C E

9

that if any n u m b e r of waves, r e p r e s e n t e d by the w a v e - f u n c t i o n s ~n, c o e x i s t in space, the resultant d i s t u r b a n c e will be r e p r e s e n t e d by the w a v e - f u n c t i o n ~ = ~n~n. This p r o p e r t y m a y be r e f e r r e d to as the P r i n c i p l e of Superposition. This is f u n d a m e n t a l to all c o n s i d e r a t i o n s of diffraction. In general, waves may be c o n s i d e r e d to o r i g i n a t e from sources h a v i n g a p p r o x i m a t e l y the d i m e n s i o n s of atoms. X-rays arise when e l e c t r o n s jump from one e n e r g y level to another in an atom, or when an incident e l e c t r o n collides with an atom. E l e c t r o n s m a y be c o n s i d e r e d to be e m i t t e d from the region of high e l e c t r o n d e n s i t y s u r r o u n d i n g an atom. H o w e v e r the " u n c e r t a i n t y p r i n c i p l e " implies that the origin of a p h o t o n or e l e c t r o n can not be s p e c i f i e d with a p r e c i s i o n of b e t t e r than about half a wavelength, so that "point sources" must be of at least this size. For all our present purposes, half a w a v e l e n g t h is s u f f i c i e n t l y small to be n e g l e c t e d in c o m p a r i s o n with all other d i m e n s i o n s c o n s i d e r e d so that we may think of the radiation as coming from one or more i n d e p e n d e n t l y - e m i t t i n g point sources. Hence, first of all, we must c o n s i d e r how we may treat the usual case of radiation e m i t t e d from a large number of such i n d e p e n d e n t sources.

1.3.2. Independent point sources We start by c o n s i d e r i n g two sources at point QI, a~d Q2At some point of observation, P, the wave function is ~p = ~iP + ~2P = A1 exp{2Ei(vl/ - kl'rl)} + A 2 exp { 2~i [ (v2/ - k2" r2) + (x(t) ] }, where the phase factor ~(l) has been added to suggest that the phase of ~2 relative to xVI varies in a r a n d o m m a n n e r with time as separate, i n d e p e n d e n t quanta of r a d i a t i o n are g e n e r a t e d at the two points. The a m p l i t u d e s AI, A2 are a s s u m e d real. The i n t e n s i t y at P at any one time is

l(t)

=

I ~ l P + W2P 12

= A21+A22+2AIA2cos2x{ (vl-v2)t - (k1"rl-k2"r2) -e(t) }.

(10)

The o b s e r v e d i n t e n s i t y is the time average of l(t) since the f r e q u e n c i e s i n v o l v e d and the rate of f l u c t u a t i o n s of the phase factor ~(l) are both u n o b s e r v a b l y high; so that

lob s = < / ( t ) > ,

=t

+ t

+ 2 < A i A 2 c o s 2 ~ { (vl-v2)t - (k1"rl-k2"r2) -u(t) }>t.

(II)

I0

FRESNEL

AND F R A U N H O F E R

DIFFRACTION

The time-average, i n d i c a t e d by the a n g u l a r <--->t m a y be defined, m o r e precisely, as lob(r)

= <1(r,f)>t = T - ~

CH.

1

bracket

l(r.l)dt

For d i f f e r e n t frequencies, the last t e r m f l u c t u a t e s with a f r e q u e n c y v I - v 2 and so a v e r a g e s out to zero even if A 1 = A2, rI = r2 and ~(f) = 0, i.e. if the two waves come f r o m one source. If the f r e q u e n c i e s are the same for two s e p a r a t e sources, the phase t e r m ~(t) varies in a r a n d o m m a n n e r so that the c o s i n e t e r m has p o s i t i v e and n e g a t i v e values with equal p r o b a b i l i t y and so, again, a v e r a g e s to zero. Hence

1ob~- , + t-I~ +12,

(12)

where I1 and 12 are the i n t e n s i t i e s at the point of o b s e r v a t i o n due to the two sources separately. F r o m this rather rough, n o n - r i g o r o u s argument we m a y c o n c l u d e that for any single source the o b s e r v e d i n t e n s i t y will be the sum of the i n t e n s i t i e s for d i f f e r e n t f r e q u e n c i e s for two or more i n d e p e n d e n t sources h a v i n g the same or d i f f e r e n t frequencies, the o b s e r v e d i n t e n s i t y will be the sum of the i n t e n s i t i e s g i v e n by the i n d i v i d u a l sources separately. Hence we are j u s t i f i e d in f o l l o w i n g the usual p r o c e d u r e of c o n s i d e r i n g d i f f r a c t i o n p r o b l e m s for the ideal case of a m o n o c h r o m a t i c point source e i t h e r at a finite d i s t a n c e or at i n f i n i t y (parallel, c o h e r e n t illumination) and then g e n e r a l i z i n g , if necessary, to take account of the finite source size or f r e q u e n c y range. A source of finite size m a y be c o n s i d e r e d as made up of a c o l l e c t i o n of very small, i n d e p e n d e n t l y - e m i t t i n g points. For each one of these p o i n t s the i n t e n s i t y at a point of o b s e r v a t i o n is c a l c u l a t e d t a k i n g account of the relative i n t e n s i t y of the source at that point. Then the i n t e n s i t i e s are a d d e d for all source points. If there is a finite range of frequencies, the i n t e n s i t y is c a l c u l a t e d for each f r e q u e n c y and then the s u m m a t i o n is m a d e for all frequencies. The actual i n t e n s i t y m e a s u r e m e n t r e c o r d e d depends on the c h a r a c t e r i s t i c s of the detector, i n c l u d i n g the v a r i a t i o n of its r e s p o n s e with the f r e q u e n c y of the incident r a d i a t i o n and also with the p o s i t i o n and d i r e c t i o n of the radiation. Most, if not all, d e t e c t o r s may be c o n s i d e r e d i n c o h e r e n t in that they are not s e n s i t i v e to phase r e l a t i o n s h i p s of the incident radiation. They r e c o r d only incident energies. We are j u s t i f i e d in c o n s i d e r i n g the i n t e n s i t y at each point of the d e t e c t o r and for each w a v e l e n g t h s e p a r a t e l y and then a d d i n g i n t e n s i t i e s for all points of the d e t e c t o r and all

1.4

HUYGENS ' PRINCIPLE

ii

wavelengths, taking the response functions of the detector into account, to find the actual measured intensity.

1.4.

Huygens' i. 4. i.

principle

Kirchhoff's

formulation

The original concept of Huygens pictures the propagation of waves through space as involving the generation of secondary waves at each point of a wavefront so that the envelope of the secondary waves becomes a new wave-front. This is a simple intuitive picture which permits an "understanding" or interpretation of the formula of Kirchhoff which may be derived directly from the wave equation by application of Green's theorem. This derivation is a standard one, reproduced in many physics texts and will not be repeated here. The Kirchhoff formula may be written thus; the disturbance at a point P due to any wave-field u, representing a solution of the wave equation, is given by integrating over any closed surface containing P as | I Up = 4--~

(exp{-2Kikr}'grad u-u'grad ['exP!-:'K~r}I) d~"

r

(13)

If we apply this to the wave field u due to a point source of unit strength at Q, we obtain up = 4-~I f

x

+ 2 =ik

exp{-2x~%r}r . exp{-2mkrqrq}

os n ^ r -

+ 2=ik

cos

n^

rq dS,

where n^r and n^rq are the angles between the surface normal n and the vectors r and ~ to the points P and Q, as in Fig. i.I the usual convention in this case being that all distances are measured from the surface. Making the approximation that all the dimensions involved are much greater than the wavelength, r-I and rq-I can be neglected in comparison with k and we obtain

Up = ~i ~ exp{-2~ikrqrq} " exp{-2K~%r}r { c~ This may be interpreted

-

c~

}dS"

(14)

in terms of the Huygens concept

12

FRESNEL

AND

FRAUNHOFER n

DIFFRACTION

CH.

1

r. Q

Fig. i.I. Diagram illustrating Kirchhoff's derivation of the amplitude at P due to a source point Q. as i m p l y i n g that the s e c o n d a r y s p h e r i c a l waves e m i t t e d f r o m each surface element, r-I e x p { - 2 K i ~ } d S , have relative a m p l i t u d e p r o p o r t i o n a l to the a m p l i t u d e of the incident wave from Q, rq-I e x p { - 2 ~ i ~ q } . There is a change of phase of K/2, r e p r e s e n t e d by m u l t i p l y i n g by i, a scale factor of l-l, and an o b l i q u i t y factor, 89 n^r - cos n ^ ~ } , w h i c h ensures that the waves add up with m a x i m u m a m p l i t u d e in the f o r w a r d d i r e c t i o n and are not p r o p a g a t e d backwards. Thus the e x p r e s s i o n (14) m a y be c o n s i d e r e d as a m a t h e m a t i c a l way of w r i t i n g a s l i g h t l y modified, more accurate, v e r s i o n of Huygens' Principle.

1.4.2.

Application

of the K i r c h h o f f

formula

The m a i n area of u s e f u l n e s s of the K i r c h h o f f formula is not for p r o p a g a t i o n through free space, as a s s u m e d for (14), but rather for cases in which the wave function ~(x.y) r e p r e s e n t s the wave m o d i f i e d by the p r e s e n c e of an object and the surface of i n t e g r a t i o n is c o n v e n i e n t l y the exit surface of the object. We neglect the m a n y c o m p l i c a t i o n s which m a y arise, p a r t i c u l a r l y for w a v e l e n g t h s c o m p a r a b l e with the d i m e n s i o n s of the s t r u c t u r e of the object, with c o n f i d e n c e that these c o m p l i c a t i o n s need not concern us. For the simple, i d e a l i z e d case of a planar, two d i m e n s i o n a l object p l a c e d b e t w e e n P and Q, we may d e f i n e a " t r a n s m i s s i o n function" q(X.Y) which is m u l t i p l i e d by the incident wave function to represent the effect of the object on the a m p l i t u d e and phase of the incident wave. Then, for a point source of radiation, the wave incident on the object is rq-I exp{-2~ikrq}, the wave e m e r g i n g from the object is q(X.

Y)rq -1 e x p { - 2 ~ i ~ q } a plane

and the w a v e - f u n c t i o n

of o b s e r v a t i o n

is given by

~(X,y)

at a point

on

1.4

HUYGENS 'PRINCIPLE

13

i ~ exp{-2rcikrq} ---

"q

X q (X, Y) exp{-2~r } { cosZ^r + cosZ^r q } dX dY, r

(15)

where X,Y and Z are coordinates referred to orthogonal axes. The surface S is considered to be the X,Y plane plus a closing surface at infinity, we follow the convention by which the Z axis is the direction of propagation. Equations such as (15) may be applied directly to the elastic interaction of fast electrons (energies greater than about 20 keV) with matter since the wavelengths are less than I0 -I A and so considerably smaller than atomic dimensions. They may also be applied to the scattering of X-rays or thermal neutrons (wavelength of the order of 1 ~) by accumulations of matter which are considerably larger than the wavelength i.e. for small-angle scattering experiments. However these equations can not be used for the consideration of the scattering of X-rays by electrons or of neutrons by nuclei. For such purposes we must rely on scattering theory, which follows a somewhat different line in its development from the wave equation, or from the fundamental postulates of quantum mechanics. 1.5.

Scattering

theory

I. 5. I. Integral

approach

form of wave equation

Again, for simplicity, we consider only a scalar wave, neglecting the complications of treating vector quantities, and so deal with scattering theory as developed for the scattering of particles by a potential field. (See, for example, Wu and Ohumara [1962].) The wave equation (5) may be written [V 2 + 4~ 2k~ + ~q) (t) ] ~ = O,

(16)

where k 0 represents the wave number for the incident wave in free space and ~ is a parameter which specifies the strength of the interaction with the potential field. As an alternative, most useful for scattering theory, we may write the equivalent integral equation by making use of the Green's function G(r, /). For scattering radiation from a potential field, G(r, /) represents the amplitude at a point of observation r, due to a point of unit scattering strength at / in the field, (see Fig. 1.2):

14

FRESNEL

AND

FRAUNHOFER

DIFFRACTION

CH.

1

r-r'

exp { - ~ ik0. r}

Fig. 1.2. Definition of distances for scattering problems.

(r) = ~(0) (r) + la ; G (r, d) ~ (/) ~ (/) d/

(17)

where ~(0)(r) represents the wave incident on the s c a t t e r i n g field and the integral represents the s c a t t e r e d radiation. The a p p r o p r i a t e form of the Green's function is

exp{2rfiklr- r'l} G(r,

F)

=

4rdr-r'i

(18) '

which from (9) is the amplitude at r of a spherical wave e m i t t e d from the point /. Thus the e x p r e s s i o n (17) may be c o m p a r e d with the e x p r e s s i o n (15) d e r i v e d from the K i r c h h o f f integral. We m a y interpret it as i n d i c a t i n g that each point of the s c a t t e r i n g field gives off a spherical wave (18), and the strength of this wave depends on the value of the s c a t t e r i n g p o t e n t i a l ~(r') and the wave function ~(r/) . We w o u l d have the exact t h r e e - d i m e n s i o n a l e q u i v a l e n t of (15) if we could say that the a m p l i t u d e of the s c a t t e r e d wave was p r o p o r t i o n a l to the incident wave amplitude ~(0)(r), but this is not p o s s i b l e in general b e c a u s e the s c a t t e r e d radiation itself forms part of the wave function ~(r). Hence we have an integral e q u a t i o n which is much more difficult to solve. 1.5.2.

Born

series

If the amplitude of the s c a t t e r e d wave is much less than the incident wave amplitude, we may assume as a first a p p r o x i m a t i o n that ~(r) in t h e integral m a y he r e p l a c e d by ~(0) (r), the incident wave amplitude. This is the "First Born A p p r o x i m a t i o n " . Higher order Born a p p r o x i m a t i o n s are found by iteration. Thus the second order a p p r o x i m a t i o n is given by r e p l a c i n g ~(r) in the integral by ~(0)(r) + ~(I)(r) and so on. For a plane incident wave ~(0)(r) = exp{-2~ik0-r}, the first Born a p p r o x i m a t i o n gives

1.5

SCATTERING THEORY APPROACH ~(o) (r) + ~(i)(r)

+ Ix [

= exp{-2Kik0.r}

e x p { - 2 r d k l r - r'l}

4rc J

Ir-r'l

15

(19)

exp{-2~ik~

~(g)

Under the usual conditions of a scattering experiment we may assume that the point of observation is at r = R where R is very large compared with the dimensions of the scattering field. Then, w r i t i n g q = k-ko, we obtain the asymptotic form, ~;(0) (r) + ~;(I)(r)

= exp{-2~ik0-R}

+ I.t exp {-2rdkoR } [ 4~ R d ~ (/) exp { -2~iq- / } dr'.

(2 0 )

The characteristic scattering from the potential field may then be sorted out from this by defining a scattering amplitude f(q) such that the right-hand side of (20) is exp { -2~ik 0 .R } + R -I exp {-2~ik0R }

f(q).

f(q) =4~; ~(r) exp{-2~iq.r}dr.

Then

(21)

This is the first Born approximation for the scattering amplitude, given by assuming that the incident wave amplitude is equal to the total wave amplitude in the scattering field: i.e. that the scattered amplitude is negligibly small. The scattered wave is made up of contributions scattered directly from the incident wave. Hence this is a single-scattering approximation. In general this first Born approximation is very good and useful for weakly scattering fields or objects. For stronger scattering, further terms may be calculated in the Born series

~(r)

= ~(0)(r)

by use of the recurrence

+ ~(i)(r)

"

+ ...

relationship

~ f exp{-2niklr-r'l} ~t (n) (r) =

+ ~(2)(r)

Ir-r'l

(/) ~;(n-l) (/) d/.

(22)

However the convergence of this Born series is usually poor if the first order approximation fails. The addition of the second order term improves the approximation over a rather limited range of scattering strengths and is sometimes useful in suggesting the nature of the modifications needed when the first order approximation fails. But the higher order terms become rapidly more complicated and difficult to calculate and it is not often useful to evaluate them.

16 I. 6.

FRESNEL AND FRAUNHOFER D I F F R A C T I O N

CH.

1

Reciprocity

One point of p a r t i c u l a r interest for the practical application of these m a t h e m a t i c a l treatments is i l l u s t r a t e d by the formulas we have given. In the expressions d e r i v e d from the Kirchhoff formula relating to radiation from a point Q and a point of observation P it is seen that the expressions are symmetrical with respect to P and Q. If the source were at P and the point of observation were at Q, so that the vectors r and ~ . were interchanged, the same amplitude would result. The expression (20) for single scattering, and the more general series (17) relate an incident plane wave defined by k 0 and an out-going plane wave k, c o r r e s p o n d i n g to source and observation points at infinity. Again, if the source and observation points were interchanged there would be no change in the resulting amplitude. These observations are consistent with the general R e c i p r o c i t y theorem of wave optics i n t r o d u c e d by Helmholtz [1886]. This may be restated as follows : The amplitude of the disturbance at a point P due to radiation from a point Q which has traversed any system involving elastic scattering processes only, is the same as the amplitude of the disturbance which would be observed at Q if the point source were placed at P. The first application of this theorem to diffraction phenomena was made by von Laue [1935] who employed it to simplify the theoretical treatment of Kossel lines due to d i f f r a c t i o n of radiation originating from point sources within a crystal. More recently Pogany and Turner [1968] showed its general applicability and usefulness in electron d i f f r a c t i o n and electron m i c r o s c o p y under n-beam dynamical d i f f r a c t i o n conditions. They further showed that the same relationship applies for intensities (but not amplitudes) if inelastic scattering is involved, p r o v i d e d that the energy loss in the inelastic scattering process is negligible. The applications of reciprocity have recently m u l t i p l i e d in number and importance. We will meet several in later chapters.

I. 7.

Fresnel

diffraction

i. 7. I. Small angle approximation From the general Kirchhoff formula it is possible to derive relatively simple forms appropriate to the conditions which typify particular classes of diffraction effects. "Fresnel diffraction" refers usually to, although not

1 .7

FRESNEL

DIFFRACTION

17

e x c l u s i v e l y , the p h e n o m e n a o b s e r v e d close to a twod i m e n s i o n a l o b j e c t i l l u m i n a t e d by p l a n e p a r a l l e l i n c i d e n t light. If the o b j e c t p l a n e is p e r p e n d i c u l a r to the d i r e c t i o n of i n c i d e n c e we m a y r e p l a c e the i n c i d e n t r a d i a t i o n in (15) by ~0 = i, w h i c h r e p r e s e n t s a p l a n e wave of unit a m p l i t u d e h a v i n g zero p h a s e at Z = 0. Then the a m p l i t u d e on any p l a n e of o b s e r v a t i o n a d i s t a n c e R b e y o n d the object is

~t1(x,y)

i ~ = ~-~ q(X,Y)

exp{-2rdkr} r (1 + cosZ^r)dX dY

(23)

where

r2 = (x - X) 2 + (y _ y) 2 + R 2. If, as was a s s u m e d earlier, the w a v e l e n g t h is small c o m p a r e d w i t h the d i m e n s i o n s of the object the angles of d e f l e c t i o n of the r a d i a t i o n will be small a n d we m a y m a k e a small a n g l e a p p r o x i m a t i o n , p u t t i n g cos Z ^ r - i, r = R in the d e n o m i n a t o r and, in the exponent, r =

{R 2 +

-- R +

(x -

2R

V(x,y)

exp

(y _

(x - X) 2 + (y _ y) 2

so that

X [[.q(X,Y)

X) 2 +

=

y) 2}1/2

,

(24)

i exp{-2rfik. R) R~

{-2rci[(x- X) 2 + ( y _

y)

2] ~dX

RI

dY.

(25)

For the s p e c i a l case that the object has a t r a n s m i s s i o n f u n c t i o n q(X) w h i c h varies in one d i m e n s i o n only, as in the i d e a l i z e d cases of s t r a i g h t edges, slits and so on, the i n t e g r a l over Y can be c a r r i e d out to give oo

exp, so that ~/(x)

<~j

exp {-2=ik-R }

-.

q (X) exp

RI

dX.

1.7.2. Fresnel integrals For the s p e c i a l cases of o b j e c t s w h i c h are e i t h e r c o m p l e t e l y o p a q u e or t r a n s p a r e n t so that q(X) = 1 in some

(26)

18

FRESNEL AND FRAUNHOFER DIFFRACTION

parts and q(X) = 0 elsewhere, of integrals of the form

this integral

~ exp ,-gi(x- X) a R~ These integrals Fresnel integrals

CH.

1

reduces to a sum

dX.

can be expressed

in terms of the

X

C(X) = I COS(~KU 2)du, 0 X

S (x) = ; sin ( 89 2) du. 0

For example let us consider a single slit of width a for which the transmission function in one dimension is

q (x) = Then the integral

of

{

1 if

Ixl
0 if

Ixl >a/2

(26) is

a2 {ex,

- n i ( x - X)

-a/2

Putting u 2 - 2 ( x - X ) 2/R~,

R ~,

21

,:IX.

this becomes

i cos( 89 2)du - i i sin( 89 2)du

where

A

A 2 = 2 (x +

A

89

and

B 2 = 2 (x-

~a)2/Rl;

and this is equal to {C(B)

- C(A) } - i{S(B)

- S(A) }.

The Fresnel integrals may be evaluated numerically by reference to tabulations of these functions. Alternatively the amplitudes may be found graphically by use of the very ingenious device of the Cornu Spiral. These methods and results for the most usual applications are described in detail in most text books of optics and so will not be reproduced here. The typical Fresnel fringes

1 .7

FRESNEL D IFFRACTION

19

f o r m e d by d i f f r a c t i o n at a straight edge are well known in both v i s i b l e - l i g h t optics and e l e c t r o n optics and are u s e d in e l e c t r o n m i c r o s c o p y as an aid to focussing.

i. 7.3. Periodic

objects

- "Fourier images"

There are very few functions q(X) for which the integral (26) can be e v a l u a t e d analytically, but the few favorable cases include the important one of p e r i o d i c functions which we c o n s i d e r in some detail here b e c a u s e of its i n t e r e s t i n g i m p l i c a t i o n s for the e l e c t r o n o p t i c a l imaging of crystals. C o n s i d e r a p l a n a r object with t r a n s m i s s i o n function q(X) = cos(2~X/a). This w o u l d be d i f f i c u l t to achieve in practice, but not i m p o s s i b l e since the n e g a t i v e sign r e q u i r e d could be o b t a i n e d by use of a "half-wave plate" which changes the phase of the incident r a d i a t i o n by ~. The integral (26) becomes ~/(x) =

exp { -2~ik2 }

cos

exp.

-xi(x-X) 2 } RX , dX,

(27)

moo

or, p u t t i n g X = x - W,

~;(X)

k ~ ) exp{-2=ik2} cos

X

cos

expl" R 2L

dW.

--oo

The c o r r e s p o n d i n g sine t e r m is an integral over an odd f u n c t i o n and so is zero. M a k i n g use of the s t a n d a r d integral form exp {-a2x 2 } cos bx dx = --oo

/I;1/2 a

exp { -b 2 / 4a 2 },

(28 )

we obtain ~/(x) = exp{-ikR} so that the i n t e n s i t y o b s e r v a t i o n is l(x)

=

e x p { i ~ R X / a 2} cos(2Kx/a),

distribution cos

2 (2xx/a)

.

on the plane

(29)

of

(30)

This is i n d e p e n d e n t of R. Hence on any plane of o b s e r v a t i o n at any d i s t a n c e from the object, the i n t e n s i t y d i s t r i b u t i o n will be e x a c t l y the same as at the exit face of the object. If such an object, i l l u m i n a t e d by plane p a r a l l e l

20

FRESNEL AND FRAUNHOFER DIFFRACTION

CH. 1

light, were imaged, the image could never be made to go out of focus. For a general periodic object, we write

Then ~;(X) =

q(X) = ~ Fh exp{2Kihxla}. h (26) becomes exp { -2~ik-R }

h

Fh

exp { 2~i hx } exp a

RI

x}

dX

The integral over X becomes [ R~ '} ; exp I R~ Making use of

~exp{2Ki(~+~)X

}dX

(28) then gives

~;(x) = e x p { - 2 ~ i k - R } ~

h

F h exp{KiRlh2/a 2 } exp{2Kihx/a}.

(31)

This is again a periodic distribution, having the same periodicity as the object but with relative phases of the Fourier coefficients which vary with the distance R. For special values of R given by R=2na2/l, the exponent in (31) is 2N~ where N = n h 2 is an integer. Then ~;(x) = e x p { - 2 K i k R } E Fh exp{2Kihx/a} h

(32)

and the intensity distribution is exactly the same as at the exit surface of the object. For the values R = na2/l where n is odd, the exponent will be (2N + I)~, so that, since h is odd if nh 2 is odd, ~; (x) = exp {-2KiAR } ~ h

Fh exp {-i~h} exp { 2Kihx/a }

= exp{-2KikR } ~ F h exp{ 2Ki h (x 4" a/2) } h a

(33)

Thus again the intensity distribution is exactly the same as for R = 0 except that it is translated by half the periodicity. Thus for parallel incident illumination the intensity distribution is exactly repeated at intervals of R = na2/l. This phenomenon was first reported by Talbot [1836], was partially explained by Rayleigh [1881], explored further by Weisel [1910] and Wolfke [1913] and "rediscovered" and

1 .7

FRESNEL

DIFFRACTION

21

e x a m i n e d in d e t a i l for p o s s i b l e a p p l i c a t i o n s to e l e c t r o n d i f f r a c t i o n and e l e c t r o n m i c r o s c o p y by C o w l e y and M o o d i e [1957a, b, c; 1960] who n a m e d these s e l f - i m a g e s of a p e r i o d i c object " F o u r i e r Images". A n u m b e r of d i f f e r e n t aspects of the p h e n o m e n o n will be e x p l o r e d in various c o n t e x t s in later pages. Here we m e n t i o n only the case that the incident r a d i a t i o n comes from a point source at a finite distance, say RI, from the p e r i o d i c object. Then (26) is r e p l a c e d by

gt(x)

=

C

RI~

exp

q(X) exp

R3,

,

dX,

(34)

where the first e x p o n e n t i a l f u n c t i o n r e p r e s e n t s a s p h e r i c a l wave from the point source incident of the object h a v i n g t r a n s m i s s i o n f u n c t i o n q(X) . It is r e a d i l y shown that F o u r i e r images will by m a g n i f i e d by a factor (R + R I)/R 1 and will occur at p o s i t i o n s given by

] l --+ . . . R R1

.

(35)

na2 "

The real F o u r i e r images occur at s u c c e s s i v e i nte r v a l s in R up to some m a x i m u m p o s i t i v e value

larger of n, b e y o n d

which I/na 2 is less than I/R I. Then there are virtual F o u r i e r images on the source side of the object c o r r e s p o n d i n g to n e g a t i v e values of n and to values from plus i n f i n i t y down to the n value for which R 1 = na2/l, as s u g g e s t e d in Fig. 1.3. I. 8.

Fraunhofer

diffraction

The a p p r o x i m a t i o n to the g e n e r a l K i r c h h o f f formula which t y p i f i e s the F r a u n h o f e r d i f f r a c t i o n c o n d i t i o n is that the o v e r a l l d i m e n s i o n s of the object s h o u l d be very much smaller than the d i s t a n c e s to source or point of observation; or, in the more usual parlance, the source and point of o b s e r v a t i o n are e f f e c t i v e l y at infinity. Thus we assume an incident plane wave of unit a m p l i t u d e and write

1011 t,

~ ,

"~--R~

n=o

1

~

1

~

z

3

I J-'""~

/

-"~.

Fig. 1.3. Positions of the Fourier images of a periodic object, n = 0, illuminated by a point source. The Fourier image, n = 5, is at infinity.

22

FRESNEL

AND F R A U N H O F E R

DIFFRACTION

CH.

1

r

Fig. 1.4. Coordinate systems for the description of Fraunhofer diffraction.

~ (x, y)

= -~

(1+co)ss 2

q (X, Y)

exo r

,- dX d Y .

(36)

The o b l i q u i t y factor is taken outside the integral since it will not v a r y a p p r e c i a b l y for the small range of X and Y considered. It m a y be i n c l u d e d with the other c o n s t a n t s and terms of m o d u l u s u n i t y which are c o m b i n e d into a factor C and u s u a l l y i g n o r e d when only relative, and not absolute intensities, are of interest. The d i s t a n c e r is c o m p a r e d to the d i s t a n c e r0 from the origin of c o o r d i n a t e s in the object to the point of observation, as s u g g e s t e d in Fig. 1.4. Then r = =

{R 2 +

(x-X)

2 +

(y_y)

2}1/2

{ (R 2 + x 2 + y 2 ) _ 2 (xX + yY) } 1/2 =

ro

-

~Xx ro

-

---YY. ro

Putting

x/ro

= l=

y/ro

= m = sin~y,

sin~x,

where ~x and ~y are the c o m p o n e n t s of the s c a t t e r i n g angle, have the a m p l i t u d e as a function of the a n g u l a r variables;

we

(x)

~(l.m) : C ; ; q(X,Y) exp{2~ik(IX + mY)}dX dY and C =

(I + cos~) iexp{-2gKr~ 2rok

(37)

i. 8

FRAUNHOFER DIFFRACTION

k0

23

i

Fig. 1.5. Definition of scattering vectors. F r o m this formula, all the w e l l - k n o w n results on the F r a u n h o f e r d i f f r a c t i o n from one and two d i m e n s i o n a l objects, such as slits, apertures, gratings etc., m a y be derived. However we note at this stage that the integral in (37) has the form of a F o u r i e r t r a n s f o r m integral. In the next chapter we outline the p r o p e r t i e s of the F o u r i e r t r a n s f o r m and work out the form of the d i f f r a c t i o n p a t t e r n for a number of simple objects as examples in the use of this type of t r a n s f o r m which represents the basic tool for most k i n e m a t i cal d i f f r a c t i o n t h e o r y and a great deal of d y n a m i c a l theory. At this stage we note the e q u i v a l e n c e of (37) and the e x p r e s s i o n for the d i f f r a c t e d a m p l i t u d e in the First Born a p p r o x i m a t i o n of s c a t t e r i n g t h e o r y (20). If we consider the t r a n s m i s s i o n f u n c t i o n q(X.Y) to be d e r i v e d from a p l a n a r d i s t r i b u t i o n of s c a t t e r i n g p o t e n t i a l V(/), the integral of (20) is c o n f i n e d to values of F lying in the X.Y plane. As seen from Fig. 1.5, the m a g n i t u d e of q is ]q[ = 2k sin(~/2), so that,

for the X direction, q-/ = 2k/ sin(~x/2)

for example,

cos(~x/2)

= kr singx = kIX.

Hence the integrals of (20) and (37) are equivalent. The integral of (20) has the a d v a n t a g e that it may be i m m e d i a t e l y a p p l i e d to t h r e e - d i m e n s i o n a l d i s t r i b u t i o n s p r o v i d e d that the n e c e s s a r y c o n d i t i o n s of weak s c a t t e r i n g apply. The form (37) n e c e s s a r i l y deals with objects which can be c o n s i d e r e d twod i m e n s i o n a l and is e s s e n t i a l l y a s i n g l e - s c a t t e r i n g form, since no d i s t a n c e can exist b e t w e e n double s c a t t e r i n g events, but there is no r e s t r i c t i o n on the strength of the scattering.

24

FRESNEL AND F R A U N H O F E R D I F F R A C T I O N

CH.

1

The presence of the obliquity factor in (37) is a consequence of the assumption of a planar distribution of the s c a t t e r i n g function. It does not exist in (20) because no such restriction is placed on V ( / ) . The manner in which (37) may be used to develop a treatment for the scattering from extended t h r e e - d i m e n s i o n a l distributions will be d i s c u s s e d in a later chapter. In this way we provide an additional approach to the p r o b l e m of the m a n y - b e a m dynamical theory for the diffraction from strongly scattering crystals.

Problems I. Starting from the expression (34) show that for a point source of radiation Fourier images of a periodic object will occur at positions given by equation (35). Find the magnification of the images. [Note- make use of the standard integral, (28) .] 2. Make use of a Cornu spiral to plot, to scale, the intensity d i s t r i b u t i o n in a plane of observation 5000 A b e h i n d an opaque straight edge illuminated by a parallel b e a m of electrons of wavelength 0.04 A (and energy 87 keV). [Note: good Cornu spirals are given in many text books including Stone, Radiation and Optics and Joos, Theoretical Physics and in Jahnke and Emde, Tables of Functions.] Give qualitative arguments to suggest how this intensity d i s t r i b u t i o n would be affected if (a) the plane of this Fresnel diffraction pattern is imaged by an electron microscope having a resolution of I0 A, or, (b) the incident electron beam comes from a finite source which subtends an angle of 10 -3 radians at the straight edge. 3. A thin object of u n i f o r m small thickness may be approxim a t e d by a pure phase object for transmission of high energy electrons. Derive an expression for the intensity in an outof-focus image (i.e. the intensity in the Fresnel d i f f r a c t i o n pattern) of a straight edge of such an object (e.g. the edge of a thin crystal of u n i f o r m thickness) in terms of the Fresnel integrals. Show that, if the phase change in the object, relative to that in vacuum is K, the intensity d i s t r i b u t i o n can be described in terms of the square of the distance from one point in the Cornu spiral diagram to a point on the Cornu spiral. (This exercise may be e x t e n d e d to treat phase objects having arbitrary phase changes and also to m i x e d phase- and amplitude-objects, if desired).

CHAPTER 2

Fourier transforms and convolutions 2. I.

Preliminaries

2.1.i. Delta-functions and discontinuities In t h i s c h a p t e r we p r o v i d e the m a t h e m a t i c a l b a c k g r o u n d for m u c h of w h a t f o l l o w s . M o s t c o n s i d e r a t i o n s of k i n e m a t i c a l d i f f r a c t i o n i n v o l v e the use of the F o u r i e r t r a n s f o r m in one f o r m or a n o t h e r . One of the m o s t i m p o r t a n t p r o p e r t i e s of the F o u r i e r t r a n s f o r m i n v o l v e s the c o n c e p t of a C o n v o l u t i o n , or C o n v o l u t i o n i n t e g r a l , a n d for b o t h t h i s a n d the F o u r i e r t r a n s f o r m it is c o n v e n i e n t to use d e l t a - f u n c t i o n s . H e n c e we d e f i n e a n d d i s c u s s t h i s l a t t e r f u n c t i o n first so that we m a y m a k e use of it later. In this, as in all s u b s e q u e n t m a t h e m a t i c a l e x p o s i t i o n , we do not s t r i v e for m a t h e m a t i c a l rigor. We a s s u m e that all f u n c t i o n s c o n s i d e r e d are s u f f i c i e n t l y w e l l - b e h a v e d in a m a t h e m a t i c a l s e n s e to a l l o w us to u s e t h e m in a w a y w h i c h m a k e s p h y s i c a l sense. W h e n we m a k e use of d e l t a - f u n c t i o n s a n d o t h e r d i s c o n t i n u o u s f u n c t i o n s we do so as a m a t t e r of c o n v e n i e n c e . W h e n we w i s h to r e p r e s e n t real s i t u a t i o n s , in w h i c h t h e r e c a n be no d i s c o n t i n u i t i e s , we use the d i s c o n t i n u o u s m a t h e m a t i c a l f u n c t i o n s as a s h o r t - h a n d n o t a t i o n for t h e c o n t i n u o u s f u n c t i o n s of p h y s i c a l r e a l i t y to w h i c h they approximate. F o r e x a m p l e , a D i r a c d e l t a f u n c t i o n at x = a is d e f i n e d by 0 for X ~ a

8(x-

a) = {

for

X =

a

and

i 5(x-

a)dx =

The d e l t a f u n c t i o n at x = 0, ~(x), can be c o n s i d e r e d the l i m i t of a set of real c o n t i n u o u s f u n c t i o n s , s u c h as gaussians: ~(x)

=

(I)

1.

lim [aK -I/2 e x p { - a 2 x 2}]. a-9~

as

(2)

26

FOURIER

TRANSFORMS

AND C O N V O L U T I O N S

CH.

2

As a tends to i n f i n i t y the g a u s s i a n f u n c t i o n has a m a x i m u m value t e n d i n g to i n f i n i t y and a h a l f - w i d t h (l/a) t e n d i n g to zero but the integral over the f u n c t i o n is always unity. Then we can use a delta f u n c t i o n as a c o n v e n i e n t n o t a t i o n for any f u n c t i o n of integral u n i t y in the f o r m of a sharp peak h a v i n g a width so small that it is not e x p e r i m e n t a l l y significant. S i m i l a r l y a w e i g h t e d delta f u n c t i o n c.~(x) is u s e d to i n d i c a t e a sharp peak for which the i n t e g r a l is c. The d e f i n i t i o n of the f u n c t i o n as the limit of a series of f u n c t i o n s can be c o n v e n i e n t to c l a r i f y or prove v a r i o u s r e l a t i o n s h i p s . For example,

6(bx)

which

=

lira

a-..)~

exp{-a2b2x

[ag -1/2

2 } ] = b-l.8(x)

.

We m a y define a delta f u n c t i o n in two dimensions, is zero except at x = 2 = 0 and for which

~

~(x,y)

~(x,y)dxdy = 1.

S i m i l a r l y for any n u m b e r of d i m e n s i o n s we m a y define ~(r) or where r and a are v e c t o r s in the n - d i m e n s i o n a l space. we note that in two d i m e n s i o n s 8(x) represents a line and in three d i m e n s i o n s 8(x) r e p r e s e n t s a plane. For a d e t a i l e d d i s c u s s i o n of delta functions see, for example, L i g h t h i l l [1960] or Arsac [1966] or Shmueli [1992]. We note in p a s s i n g the i m p o r t a n t d e f i n i t i o n of a d e l t a function,

~(r-a)

5(x) =

f exp{2~ixT)dy,

(3)

--oo

which will (equations

occur later in c o n n e c t i o n (33) to (36)).

with F o u r i e r

transforms

2.1.2. Convolutions In one dimension, the C o n v o l u t i o n integral (or, c o n v o l u t i o n product, or convolution, or faltung, or folding) of two functions f(x) and g(x) is d e f i n e d as

C (x) = f ( x )

, g (x) -

i --oo

f(X) g (x - X) dX.

(4)

2 .1

PRELIMINARIES

By simple

change

of variable,

f (x) , g (x) =

i

27

we find that

g (X) f (x - X) dX = g (x) , f (x) .

(5)

--oo

For two or more dimensions

f (r)

Examples

form,

. g (r) = ~ f (R) g (r - R) d R .

The identity operation delta function-

2.1.3.

we may use the vector

is the convolution f (x)

.

8 (x)

f (x)

.

5 (x

(6)

with the Dirac

= f (x) , -

a)

= f (x

-

a) .

(7)

of c o n v o l u t i o n s

The convolution integral (4) or (6) appears with great frequency in many areas of scientific work, being fundamental to the interpretation of most experimental measurements and an essential component of many sophisticated theoretical developments such as the Green's function methods of theoretical physics. To get a clearer understanding of what is involved we analyse the integral (4) in detail. It may be written thus; the function f(X) is m u l t i p l i e d by the function g(X) which has been shifted to an origin at X = x and inverted to give g(x - X). The value of the product of f(X) and g(x - X) is integrated over X and the result plotted as a function of x to give C(x). This is exactly the process involved, for example, in m e a s u r i n g the intensity of a spectral line by scanning it with a detector having a finite slit as input aperture, as suggested in Fig. 2.1. The coordinate X may represent the angle of scattering of light by a p r i s m or diffraction grating and the intensity distribution l(X) shows the spectral lines of interest. The intensity distribution in the spectrum is m e a s u r e d by recording the intensity passing through a slit whose transmission function is

+ixl : {

0

for

IXl>__a/2,

1

for

IXl
28

FOURIER

TRANSFORMS

AND C O N V O L U T I O N S

CH.

2

g(x-X) C{x) = ](x) 9 g ( x )

l(x).g{x-X) .

.

.

.

Fig. 2.1. A convolution operation. An intensity function I(X) representing a spectral line, is multiplied by the transmission function g(x-X) of a slit, centered at X = x ; the product of the two functions is integrated to give the measured intensity lobs(X). i.e. the slit t r a n s m i t s all r a d i a t i o n w i t h i n a range of X of w idt h a and e x c l u d e s all other radiation. When the slit is p l a c e d at X = x, the t r a n s m i t t e d i n t e n s i t y as a f u n c t i o n of X is l(X)g(x - X). What is r e c o r d e d is the total t r a n s m i t t e d intensity. If this is p l o t t e d as a f u n c t i o n of x, the p o s i t i o n of the slit, we get lobs = l ( x ) A single very intensity of unity width a;

lob s = 8(X)

. g(x)

=

J l(X) g(x

- X) dX.

sharp s p e c t r a l line gives an o b s e r v e d for a range of x values equal to the slit

* g(x)

=

~ 8(X) g ( x - X) dX = g ( x ) . m ~

For a g e n e r a l i n t e n s i t y distribution, I(X), each sharp s p e c t r a l line or each part of a b r o a d e r spectral line will be " s p r e a d out" by the " s p r e a d - f u n c t i o n " g(X), so that the r e c o r d e d i n t e n s i t y lob s is less sharply p e a k e d or less well r e s o l v e d than the o r i g i n a l spectrum. Likewise, the b l u r r i n g of the image due to the i m p e r f e c t i o n s of a camera lens m a y be d e s c r i b e d in terms of c o n v o l u t i o n of the i d e a l l y p e r f e c t image i n t e n s i t y with some f u n c t i o n g(x,y). For a point source of light, the ideal image w o u l d be a delta function. The s p r e a d i n g gives

lob s -- g (x, y)

* 8 (x, y)

= g (x, y) .

(8)

2 .1

PRELIMINARIES

29

For a general object, consisting of a large number of i n d e p e n d e n t l y emitting point sources, having an ideal image,

lo(x,y),

lobs = g ( x . y )

(9)

* lo(x.y).

Thus each point of the original intensity distribution is spread into a disc of intensity and the overlapping of these discs gives a blurring and loss of resolution of the image. This illustrates the concept that the "spread function" represents the response of the system to a delta function input, in this case a point source. This is the basis for the Green's function m e t h o d useful in scattering theory and many other fields of physics and also for the analysis of the properties of an electronic circuit by m e a s u r i n g its response to a sharp voltage or current pulse. An excellent example of a convolution is provided by Huygens' Principle as expressed by the Kirchhoff formulas. Each point on a wave front is considered to give rise to a spherical secondary wave having an initial amplitude proportional to the amplitude of the incident wave. Then the amplitudes of the secondary waves are added to give the amplitude on a plane of observation. Thus the amplitude function, q(xoy), on the original wave front is spread out by a function which represents the spherical secondary wave emitted by a single point on the wave front. We have written this explicitly as a convolution integral in the equation (1.25) for Fresnel diffraction in the small angle approximation. This may then be rewritten

~(x.y)

= q(x,y)

.

[iexp{-2~ikR}exp{-2~(x2+y2)}] , RX 2R

,

(10)

and the function in square brackets may be referred to as the "propagation function", or the wave function obtained for a point source, q(x,y) = 8(x.y). Likewise the First Born approximation result given in eq. (1.19) may be rewritten in terms of a convolution. The integral giving the singly scattered amplitude becomes, V(r)

exp{-2Kik0-r } ,

exp {-2~ikr} r

.

(ii)

The first function here is the incident wave m o d i f i e d by the potential field V(r). This is c o n v o l u t e d with the amplitude due to a point source, namely the amplitude of a spherical wave from the origin. Thus the equation (1.19) or (ii) simply states that the observed amplitude is the sum of the amplitudes of spherical waves from all points of the scatterer, and the amplitude of scattering from each point is proportional to the product of the incident wave amplitude and the value of the potential function, V(r) at the point.

30 2.2.

FOURIER TRANSFORMS Fourier

transforms-

AND C O N V O L U T I O N S

CH.

2

general

2.2.1. Definitions The F o u r i e r

defined

as

t r a n s f o r m of a o n e - d i m e n s i o n a l

$~(x)] The inverse

-

function f(x) is

F(u) = i f(x) e x p { 2 K i u x } d x .

transform,

~-I,

(12)

is d e f i n e d so that

f(x) = 5-115{f(x)}] =

I F(u)

exp{-2Kiux}du.

(13)

Here we follow the c o n v e n t i o n of i n c l u d i n g 2~ in the exponent. This is the c o n v e n t i o n c o m m o n l y used in c o n s i d e r a t i o n s of d i f f r a c t i o n and is c o n v e n i e n t in that it avoids the n e c e s s i t y of adding a constant m u l t i p l i e r in e i t h e r (12) or (13) . In other conventions, often u s e d in s o l i d - s t a t e physics, the 2~ is o m i t t e d from the exponent. It must then be i n c l u d e d as a constant; the integral in e i t h e r (12) or (13) is m u l t i p l i e d by (2~) -I or both integrals are m u l t i p l i e d by (2~) -I/2. Also some authors choose to d e f i n e the F o u r i e r t r a n s f o r m with a n e g a t i v e sign in the e x p o n e n t i a l and the inverse t r a n s f o r m with a p o s i t i v e sign. It is necessary, therefore, to use c o n s i d e r a b l e caution in m a k i n g use of F o u r i e r t r a n s f o r m e x p r e s s i o n s d e r i v e d from various sources. For more than one d i m e n s i o n we m a y use the v e c t o r form of (12)

F(u) = [ f(r) exp{ 2Ei,-r}dr. d

(14)

The vector u may be c o n s i d e r e d as a vector in " F o u r i e r t r a n s f o r m space". For the t h r e e - d i m e n s i o n a l case, for example, the vector r m a y be c o n s i d e r e d to have c o o r d i n a t e s x,y,z and u m a y be c o n s i d e r e d to have c o o r d i n a t e s u,v,w. Then the scalar p r o d u c t is u-r = ux + vy + wz and oo

F(u,v.w) = ; ;; f(x,y,z) exp{2~i(ux+vy+wz)}dxdydz,

(15a)

and

f(x,y,z)

= f f f F(u,v,w)

exp{-2~i(ux+vy+wz)}dudydw

(15b)

2 .2

FOURIER

TRANSFORMS-

GENERAL

31

We h a v e seen in C h a p t e r 1 that the a m p l i t u d e of s c a t t e r i n g f r o m an o b j e c t in the F r a u n h o f e r d i f f r a c t i o n a p p r o x i m a t i o n , w h e t h e r d e r i v e d f r o m the K i r c h h o f f f o r m u l a t i o n or f r o m s c a t t e r i n g theory, is d e s c r i b e d by a F o u r i e r t r a n s f o r m integral. In (1.37) for example, we put u = I/l, v = m / ~ to get the t w o - d i m e n s i o n a l f o r m of (15b). We m a y thus d e s c r i b e the d i f f r a c t i o n a m p l i t u d e in terms of a d i s t r i b u t i o n in F o u r i e r t r a n s f o r m space, o f t e n r e f e r r e d to, as we shall see, as " R e c i p r o c a l space". This will be our m o s t c o m m o n means for d e r i v i n g or u n d e r s t a n d i n g d i f f r a c t i o n effects, so we now p r o c e e d to d e s c r i b e and i l l u s t r a t e the most i m p o r t a n t p r o p e r t i e s a n d b e h a v i o r of the F o u r i e r t r a n s f o r m .

2.2.2. (12)

Properties

Instead as

of u s i n g

of Fourier transforms

the

complex

exponential,

we m a y w r i t e

F(u) = i f(x) cos(2Kux)dx + i i f(x) sin(2Kux)dx.

(16)

If the f u n c t i o n f(x) is real and an even function, so that f(-x) = f(x), the sine i n t e g r a l is zero so that

F (u) = i f ( x )

cos (2rcux) dx = 2 i f ( x )

--oa

and the

cos (2rcux) dx,

(17)

0

F(u) is a real function. If f(x) is a real odd f u n c t i o n so that f(-x) = -f(x) t h e n cosine

F(u)

integral

= i 5

is zero

and

f(x) sin (2Kux)dx = 2i i f(x) sin (2~ux)dx.

--~

and the f u n c t i o n F(u) is pure imaginary. Since any real f u n c t i o n can be w r i t t e n even and an odd function,

f(x) =

89

+

as a s u m of an

f(-x) } + ~{f(x) - f(-x) } = fe(X) + fo(x),

we m a y w r i t e F(u)

where A(u)

(18)

0

a n d B(u)

are

A(u)

= A(u)

real

7

functions

= 2 | fe(X) 0

+ iB(u)

given

cos(2~ux) dx,

by

(19)

32

FOURIER TRANSFORMS

and

AND CONVOLUTIONS

7

B (u) = 2 | f0 (x) sin (2~ux)dx. 0

CH. 2

(20)

It is these cosine and sine integrals that are tabulated, for the most part, in the lists of Fourier integrals given, for example, in the compilations of Erdeyli [1954] and Bracewell [1986]. Photographic representations of Fourier transforms, obtained by optical diffraction, are given by Harburn et al. [1975]. A number of general relationships may be written for any function f(x), real or complex, thus;

Real space

Fourier transform space

f(x)

F(u)

(21)

f (-x)

F (-u)

(22)

F* (-u)

(23)

1F (u/a)

(24)

(x)

f(ax)

a

f(x) + g(x)

F(u) + G(u)

f(x-

exp

a)

{2~iau} F(u)

(25) (26)

df(x)/dx

(-2~iu) F (u)

(27)

dnf(x)/ci~

( - 2 ~ i u ) n F (u) .

(28)

These relationships may be readily proved by writing out the relevant integrals. For (24)" f f(ax) exp{2Kiux}dx

1 f f(X) exp { 2~iuX/a }dX

a =

1F(u/a) a

.

2.2 For

FOURIER TRANSFORMS"

GENERAL

33

(26)-

S

f(x - a) exp { 2Kiux} dx

--oo

= ~ f(x) = F(u) For

(27) :

d

exp{2Kiua}.

f -~ f(x)

exp { 2gi~r }dx

=

~

= f

(-2~iv) E(v) f

= The relation (27).

e x p { 2 K i ( u X + u a ) }dX

[F(v) exp{-2~iux}dv]

(-2~iu)

F(u)

exp{2~iux}dx

exp{2~i(,-v) x}dvdx

since

f exp{2~i(u-v)x}dx = ~(u-v).

(28) follows by repetition of the derivation of

2.2.3. Multiplication and convolution We add the two important M u l t i p l i c a t i o n theorem

relationships,

the

~3 If(x) -g (x) ] = F (u) . G ( u ) , i.e. the Fourier transform of a product of two functions the convolution of their Fourier transforms, and the Convolution theorem, ~3 [/'(x) . g (x) ] = F ( u ) - G ( u ) ,

(29) is

(30)

i.e. the Fourier transform of the convolution of two functions is the product of their Fourier transforms. Here we have followed the convention that functions in real space are represented by small letters and the Fourier transforms are represented by the corresponding capital letters. These theorems are again easily proved, using manipulations which are in general non-rigorous but are valid for the types of functions which we will be employing. For example, to derive (30), putting x - X = y in the left side of the integral form gives,

34

FOURIER TRANSFORMS ~

AND CONVOLUTIONS

CH.

2

f (X) g (X - X) dX- exp { 2Kiux }dx

= f f(X) g (y) e x p { 2 ~ i u (X+y) } dXdy = f f(X)exp{2KiuX}dX}f =

g(y)exp{2xiuy}dy

F(u).G(u).

2.2.4. Space and time In addition to providing the relationship between spatial distributions f(r) and the diffraction amplitudes, P(u) the Fourier t r a n s f o r m also relates the variation of a function in time, f(t), and the corresponding frequency distribution. Thus we may write

e x p { 2 x i v t } dt,

(31)

= "f F ( v ) e x p { - 2 ~ i v t } d v ,

(32)

F (v) = i f(t) and

f(t)

where we use the frequency V, rather than the angular frequency ~. To make the analogy complete it is necessary to introduce the artifice of negative frequencies. While this is inconsistent with con~non parlance, it is readily seen that a negative frequency can correspond to a negative p r o g r e s s i o n of the phase of a wave in time, i.e. to a wave going "backwards". For a function of both space and time, f(r, t), a Fourier t r a n s f o r m may be made with respect to any one or all of the coordinates or with respect to time, or with respect to both spatial coordinates and time. Hence where any ambiguity is possible it is necessary to specify the variable or variables involved in the Fourier transform, possibly by use of a subscript on ~. For example,

~3x, t [f(x, y,z,t) ] = F(u, y,z,v) oo

m ~

2 .3

FOURIER TRANSFORMS : GENERAL

2.3.

Fourier

transforms

and

diffraction"

35

examples

We n o w p r o v i d e a series of e x a m p l e s to serve the two f u n c t i o n s of f a m i l i a r i z i n g the r e a d e r with c o m m o n F o u r i e r t r a n s f o r m s a n d d e m o n s t r a t i n g the a p p l i c a t i o n of F o u r i e r t r a n s f o r m s to k i n e m a t i c a l d i f f r a c t i o n . We refer for the most part to d i f f r a c t i o n f r o m simple one- or t w o - d i m e n s i o n a l objects. 2.3.1.

Point

source

or point

aperture

The a m p l i t u d e d i s t r i b u t i o n of a v e r y small source or the t r a n s m i s s i o n t h r o u g h a v e r y small a p e r t u r e (or slit) in one d i m e n s i o n m a y be d e s c r i b e d as 8(x), or by 8(x - a) when it is not at the origin. The F o u r i e r t r a n s f o r m u s e d to d e r i v e the F r a u n h o f e r d i f f r a c t i o n p a t t e r n is

5 8 (x) = I,

(33)

8(x - a) = exp{2Kiua}. To show this

we write

the i n t e g r a l

i 8(X - a) exp{2Kiux}clr. The i n t e g r a n d

be w r i t t e n

is zero except

for x = a. Hence the i n t e g r a l m a y

oo

exp{2~iua} f 8(x - a)dx = exp{2Kiua}. --oo

The a m p l i t u d e

of a d i f f r a c t i o n p a t t e r n is p r o p o r t i o n a l to where u = I/l. The i n t e n s i t y o b s e r v e d is then p r o p o r t i o n a l to IF (u) 12 = I. Thus, as is well known, the F r a u n h o f e r d i f f r a c t i o n p a t t e r n f r o m a point source has u n i f o r m intensity, apart f r o m the f a c t o r I / R 2 and the o b l i q u i t y factor (if applicable) w h i c h have b e e n omitted.

F(u)

=

~ 8(x - a)

2.3.2.

A plane

wave"

the

inverse

F o u r i e r t r a n s f o r m of a plane respect to t only gives St e x p { 2 K i ( v l

wave

of 2.3.1

(equation

1.8)

l - x / l I) } = 5(v + v I) exp{-2Kix/ll},

with (34)

36

FOURIER TRANSFORMS

AND C O N V O L U T I O N S

CH.

2

i.e. a delta function in f r e q u e n c y and a plane wave in real space. F o u r i e r t r a n s f o r m with respect to x gives Sx exp{2~i(vl

= exp{2~ivl l} ~(u - I/~i),

t - X/ll)}

i.e. a s i n u s o i d a l r e c i p r o c a l space. gives,

(35)

v a r i a t i o n in time and a delta f u n c t i o n in T r a n s f o r m i n g with respect to both v a r i a b l e s

Sx, t e x p { 2 ~ i ( v I I - x / l I ) } = 8 (v + v I) .8 (u - I/~ I),

(36)

which is a delta function in both f r e q u e n c y and r e c i p r o c a l space. If we take the F o u r i e r t r a n s f o r m of all p o s s i b l e waves which can exist in a given m e d i u m with respect to both x and t we obtain a set of points d e f i n i n g the r e l a t i o n s h i p b e t w e e n the f r e q u e n c y v and ~-I (or b e t w e e n the a n g u l a r f r e q u e n c y and 2~k) which is known as the d i s p e r s i o n relation for that type of wave and the p a r t i c u l a r medium. 2.3.3. ~3f (x -

Translation

a)

= ~3 I f ( x )

.

o f an 8 (x -

object

a) ] = F (u)

exp {2Kiua},

(37)

where we have used (33) and the c o n v o l u t i o n t h e o r e m (30). Thus t r a n s l a t i o n of the object in real space has the effect of m u l t i p l y i n g the a m p l i t u d e in reciprocal space by a c o m p l e x exponential. The i n t e n s i t y d i s t r i b u t i o n of the F r a u n h o f e r d i f f r a c t i o n p a t t e r n is given by IF(u) 12, which is i n d e p e n d e n t of the translation. 2.3.4.

Slit

function

The t r a n s m i s s i o n function opaque screen is given by g(X)

The F o u r i e r

=

{

of a slit of width a in an

0

if

Ix[>a/2,

1

if

Ixl<_a/2.

t r a n s f o r m of this

is

a/2 F(u)

=

] exp{2~iux}dx

= sin(~au)/(~u).

(38)

-a/2

P u t t i n g u = I/~, we obtain diffraction pattern

the w e l l - k n o w n

F (1) = a sin (xla/l) / (~la/l),

form of the (39)

2.3 with

FOURIER an

TRANSFORMS

intensity

AND DIFFRACTION:

EXAMPLES

37

distribution

I(1)

(40)

a 2 s i n 2 ( ~ l a / l ) / ( ~ l a / l ) 2,

=

w h i c h h a s a c e n t r a l m a x i m u m v a l u e a 2 f o r I = 0, a n d has decreasing subsidiary maxima with increasing Ill w i t h z e r o v a l u e s f o r I = nl/a for , ~ 0 (see Fig. 2.2).

2.3.5. Slit function-alternative the

To i l l u s t r a t e t h e a p p l i c a t i o n of s l i t f u n c t i o n d e f i n e d in 2.3.4,

g(x) as s u g g e s t e d

- dq(x)/dx

in Fig.

2.2.

G (u) = e x p {-Kiau} But,

since

from

that,

89

+

we n o t e

-

-

8(x

that

89

Then

- e x p {Kiau}

= -2i

s i n (~au).

(27) G (u)

it f o l l o w s

= 8(x

(27),

as

=

{-2~iu}

in e q u a t i o n

F(u)

=

sin

F (u),

(38),

(xau) / (xu) .

2.3.6. Straight edge The

transmission

function

is

q(x) [F(u)l

I

!

d q(x) g(x) = (-~ Fig. 2.2. Derivation of the Fourier transform of a slit function by consideration of its differential.

for

38

FOURIER

f(x) Using

the

=

TRANSFORMS

{

0

if

X<0,

1

if

x_>0.

same p r o c e d u r e

as in 2.3.5

= df(x)/dx

g(x)

Then

AND C O N V O L U T I O N S

CH.

2

we put

= 8(x)

G(u) = 1

and

F(u)

But an i n d e t e r m i n a t e i n t e g r a t i o n . We note

=

(2~iu)-i

constant that

t e r m has b e e n

i r189

omitted

in the

= o.

This s u g g e s t s that the m i s s i n g c o n s t a n t ing this g i v e s the c o r r e c t result,

F(u) = + 89

-

term

is i/2.

Insert-

(2~iu) -I.

(41)

2.3.7. Rectangular aperture In the two d i m e n s i o n a l f o r m of 2.3.4 we d e f i n e t r a n s m i s s i o n f u n c t i o n of a r e c t a n g u l a r a p e r t u r e as

f(x, y) = Then

{

lxl
I

if

0

elsewhere.

al2

E(u,v) =

f -al2

the

bl2

exp{2Eiux}dx

I

exp{2~ivy}dy

-bl2

(42)

= ab s i n (nau) / (Tcau) s i n (Tcbv) / (Toby), so that, for d i f f r a c t i o n f r o m a r e c t a n g u l a r i n t e n s i t y d i s t r i b u t i o n is

l (u, v)

aperture

= a2b 2 s i n 2 (~au) / (nau) 2 s i n 2 (nbv) / (nbv) 2.

the (43)

The m a x i m u m i n t e n s i t y at u = v = O is a2b2. Zeros o c c u r at i n t e r v a l s of a -I a l o n g the u direction, p a r a l l e l to the x axis and b -I a l o n g the v direction, as s u g g e s t e d in Fig. 2.3. Thus

2.3

FOURIER

TRANSFORMS

AND D I F F R A C T I O N :

EXAMPLES

39

the i n t e n s i t y f u n c t i o n falls off t h r o u g h d e c r e a s i n g o s c i l l a tions in each direction. The d i m e n s i o n s of the d i s t r i b u t i o n in r e c i p r o c a l space are i n v e r s e l y p r o p o r t i o n a l to the d i m e n s i o n s in real space. I n t e g r a t i n g over the whole peak gives an i n t e g r a t e d i n t e n s i t y equal to ab, the area of the aperture, as expected.

2.3.8.

Circular

aperture

The t r a n s m i s s i o n written Scx,

:

{

function 1

0

if

for a c i r c u l a r

aperture

is

(x2+y2)1/2
elsewhere.

The F o u r i e r t r a n s f o r m is best c a r r i e d out by w r i t i n g the F o u r i e r i n t e g r a l in plane polar coordinates. The result is

v

q(x,y)

0

0

F(u,v) Fig. 2.3. Diagram suggesting the form of the Fourier transform of a rectangular aperture, equation (2.43) .

F(u)

(44)

= (~:a2/2) J1 (~au) I (~au) ,

where u is a radial c o o r d i n a t e and Jl(X) is the first order Bessel function. The function Jl(X)/x is similar in form to (sinx)/x but has a somewhat b r o a d e r central m a x i m u m with the first zero at 1.22a -I i n s t e a d of a -I.

2.3.9.

Two v e r y n a r r o w slits

For two slits a d i s t a n c e A apart h a l f - w a y b e t w e e n t h e m and write

f(x)

= s(x+A/2)

we take the

+ s(x-A/2),

origin

40

FOURIER

TRANSFORMS

AND C O N V O L U T I O N S

CH.

2

where s(x) is the t r a n s m i s s i o n f u n c t i o n for one slit of width a as in 2.3.4. For very n a r r o w slits we let a go to zero. But to keep the i n t e n s i t y finite we imagine the incident a m p l i t u d e to be p r o p o r t i o n a l to I/a. Then s(x) becomes 8(x). We put

f(x) = 8(x+A/2)

and

+ 8(x-A~2),

F (u) = exp { -IriAu } + exp { rriAu } = 2 cos (~Au),

and the d i f f r a c t i o n

pattern

(45)

intensity

is

I(1) = 4 cos 2 (~Al/l).

(46)

Thus we get simple s i n u s o i d a l fringes of u n i f o r m amplitude. C o m p a r i s o n with the F r e s n e l d i f f r a c t i o n p a t t e r n shows that for this p a r t i c u l a r object the d i f f r a c t i o n p a t t e r n is i n d e p e n d e n t of the a p p r o x i m a t i o n made to the g e n e r a l K i r c h h o f f i n t e g r a l (see P r o b l e m 2).

write

2.3.10.

Two slits

For two

slits

f(x)

of a p p r e c i a b l e

of width

width

a, a d i s t a n c e

= s(x).[5(x+A/2)

A apart,

we m a y

+ 6(x-A~2)],

where s(x) is the t r a n s m i s s i o n f u n c t i o n for a single slit d e f i n e d as in 2.3.4. U s i n g the c o n v o l u t i o n t h e o r e m and the results (39) and (45) F(u)

=

cos ( m 4 u ) .sin ( x a u ) / ( x a u ) .

2a

(47)

The i n t e n s i t y of the d i f f r a c t i o n p a t t e r n is then as shown in Fig. 2.4; cos 2 fringes of p e r i o d I/2A are m o d u l a t e d by a (sin2x)/x 2 f u n c t i o n which goes to zero for u = a -I. This is a d e s c r i p t i o n of the result of the Young's fringe experiment, well known as of f u n d a m e n t a l s i g n i f i c a n c e for the d e v e l o p m e n t of p h y s i c a l optics.

2.3.11. The

Finite

inverse S{2

wave

train

of 2.3.9 cosm4x}

gives

= 8(u+A/2)

+ 8(u-A/2).

If the wave train r e p r e s e n t e d by the cosine function is cut to a finite length by m u l t i p l y i n g it by a slit f u n c t i o n of width B we have

2 .3

FOURIER

TRANSFORMS

AND D I F F R A C T I O N :

EXAMPLES

41

f(x) = 2 cos(ru4x)sB(x), so that,

using

the m u l t i p l i c a t i o n

theorem,

F(u) = {8(u+A/2) +8(u-A~2) } , B sin(rd~u) l(~Bu) .

(48)

Thus the a m p l i t u d e d i s t r i b u t i o n in r e c i p r o c a l space is the s u m of two f u n c t i o n s of the f o r m S(u) = ~sB(x) c e n t e r e d on u = +_,4/2 (Fig. 2.5) . If B is m u c h g r e a t e r t h a n the p e r i o d i c i t y 2/A of the w a v e train, the two p e a k s in F(u) are m u c h n a r r o w e r than t h e i r s e p a r a t i o n a n d t h e y do not o v e r l a p a p p r e c i a b l y Then the i n t e n s i t y d i s t r i b u t i o n IF(u)]2 is v e r y n e a r l y g i v e n by

;(u) = B2 sin2(~Bu)/(~Bu)2,

{5(u+A/2) +5(u-A~2) }.

This is a u s e f u l a p p r o x i m a t i o n for m a n y p u r p o s e s , but it can not be u s e d if the l e n g t h of the wave t r a i n B is o n l y a few times the p e r i o d i c i t y .

I ~-

-

A-------I

_I! I ! il I~- a - q

I-- a ~I

q(x)

IF'
A k/

M/

o

F{u)

"-tl~C

Fig. 2.5. A wave train cut off by multiplying by a slit function and its Fourier transform, equation (2.48).

42

FOURIER 2.3.12.

and

TRANSFORMS

Periodic

array

AND C O N V O L U T I O N S of narrow

CH.

2

slits

We a s s u m e a p e r i o d i c a r r a y of slits h a v i n g zero w i d t h r e p e a t e d at r e g u l a r intervals, a i n d e f i n i t e l y . T h e n

f(x) = The F o u r i e r

transform

Z

is,

8(x - ha). using

(33),

oo

F(u) = Z

exp{2~iuna}.

This s u m m a t i o n of a F o u r i e r unity gives a well-known result e l e m e n t a r y m e t h o d s as follows. Since oo

Z

xn =

(i

0 we m a y

s e r i e s w i t h all c o e f f i c i e n t s w h i c h we can d e r i v e by

- x) -I,

write E(u)

= Z

[exp{2Eiua}]n

+ Z

0

=

[exp{-2Eiua}]n

- 1

0

[i - e x p { 2 K i u a } ] - I

+

[i - e x p { - 2 K i u a } ] - i

_ 1

= 0, except

that F(u)

i.e.

= ~

for

if 2Kua = 2h~ w h e r e

exp{2Kiua}

= I,

h is an integer,

F(u) = a -1 Z

6 ( u - h/a),

or u = h/a.

Then (49)

h

w h e r e the f a c t o r a -I g i v e s the d e l t a f u n c t i o n s the c o r r e c t weight. H e n c e the F o u r i e r t r a n s f o r m is a set of e q u a l l y s p a c e d d e l t a f u n c t i o n s of p e r i o d a -I in r e c i p r o c a l space.

2.3

FOURIER TRANSFORMS

AND DIFFRACTION:

EXAMPLES

43

2.3.13. Arbitrary periodic function For an object with an a r b i t r a r y p e r i o d i c f u n c t i o n we write

transmission

oo

f (x) = Z Then

From

{ C hx

r,u, (7) this

(50)

Fh exp{-2~ihx/a}.

Z rh f exp

is F (u)

= Z

----+ ux

dx.

(51)

Fh 8 (u - h / a ) .

--oo

Hence the d i f f r a c t e d amplitude is r e p r e s e n t e d by a set of delta f u n c t i o n s e q u a l l y spaced with s e p a r a t i o n a -I in u, each delta f u n c t i o n h a v i n g the "weight" F h equal to the c o r r e s p o n d i n g F o u r i e r c o e f f i c i e n t of (50). This result and those which follow form the basis for much of our c o n s i d e r a t i o n of d i f f r a c t i o n of X-rays and e l e c t r o n s by crystals.

2.3.14. Diffraction grating: thin slits A set of N parallel, e q u a l l y spaced thin slits c o n s t i t u t e s a p r i m i t i v e form of d i f f r a c t i o n grating. t r a n s m i s s i o n function may be written; (N-I)/2 f (x)

=

The

(52)

Z ~ (x - h a ) . -(N-l)/2

Then F(u)

=

-(N-I)/2

exp{2~iuna}

= exp { -~iu (N - i) a }~ 0 =

exp { -~iu

(N

-

exp { 2~iuna }

1) a } exp{2niuNa} - 1

exp{2niua}- 1 i.e.

F(u)

= sin (ENau)/sin (~au) .

(53)

44 write

FOURIER

TRANSFORMS

AND C O N V O L U T I O N S

On the o t h e r hand, we m a y use the the t r a n s m i s s i o n f u n c t i o n as

result

CH. of

(49)

2

and

OO

f(x)

= s (x) E

8 (x - ha) .

(54)

w h e r e s(x) is a slit f u n c t i o n of w i d t h Na, w h i c h cuts off the t r a n s m i s s i o n of all but N of the t h i n slits. Then, as in (48) we m a y w r i t e

F(u)

= E h

5(u-

h/a)

, Na s i n ( r d V a u ) / ( r d V a u ) .

(55)

The f u n c t i o n s (53) a n d (55) are s i m i l a r (see Fig. 2.6). For e a c h t h e r e are s h a r p p e a k s w i t h side ripples, of the f o r m (sinx)/x, at i n t e r v a l s u = a -I. The w i d t h of the peak, the d i s t a n c e f r o m the m a x i m u m to the first zero, is I/Na. It is not o b v i o u s f r o m the f o r m of the f u n c t i o n s (53) a n d (55) that t h e y are identical, but since the f u n c t i o n s in real space, (52) and (54) are identical, the t r a n s f o r m s m u s t be also.

2.3.15. Diffraction grating" general The i n d i v i d u a l lines of a d i f f r a c t i o n grating, u s e d in t r a n s m i s s i o n , m a y be slits of f i n i t e width, r u l i n g s on glass, r e p l i c a s in p l a s t i c of r u l i n g s on m e t a l a n d so on. The t r a n s m i s s i o n f u n c t i o n of the w h o l e g r a t i n g is g i v e n by the c o n v o l u t i o n of the t r a n s m i s s i o n f u n c t i o n of a s i n g l e r u l i n g g(x) w i t h a set of N d e l t a functions, ( 5 2 ) or (54) (N-1)/2

f (x)

=

~ 5 (x - ha) -(N-l)/2

, g (x)

(56)

so that E(u)

= O (u) sin (~Vau)/sin (Eau) ,

(57)

l

I. . . . .

Na

"1

l[ll[lll[{I

Na Fig. 2.6. The amplitude distribution for diffraction from a large number of parallel very fine slits, as given by equation (2.53) or (2.55).

2.3

FOURIER

TRANSFORMS

AND D I F F R A C T I O N :

EXAMPLES

45

or

F (u)

[ ~_, 5 (u - h / a )

= G (u)

. Na s i n (mVau) / (rdVau) ]

h

(58)

Thus the height of each of the d i f f r a c t i o n m a x i m a is p r o p o r t i o n a l to the value of the F o u r i e r t r a n s f o r m of g(x) at that u value. A l t e r n a t i v e l y we c o u l d say that the d i f f r a c t i o n g r a t i n g may be r e p r e s e n t e d by c u t t i n g off a p e r i o d i c t r a n s m i s s i o n fun c t i o n by m u l t i p l y i n g it with a slit f u n c t i o n $(x) of width Na. This is not the same as (56) since in this case we may be using s(x) to cut off a c o n t i n u o u s function. C o r r e s p o n d i n g l y , the d i f f r a c t i o n amplitude, w r i t t e n from (51) as

F(u)

= ~_, Fh 8(u - h / a )

. Na

sin(~dVau)/(TtNau).

(59)

h

is not e x a c t l y

2.3.16.

the

same

as

(57) or

(58) .

Gaussian function

A G a u s s i a n f u n c t i o n rarely occurs in p r a c t i c e as the t r a n s m i s s i o n f u n c t i o n of an object but is f r e q u e n t l y used as an a p p r o x i m a t i o n , to "round-off" a discontinuity, or to achieve a s u i t a b l e c o n v e r g e n c e of an an a l y t i c function, b e c a u s e of its d e s i r a b l e p r o p e r t i e s in relation to F o u r i e r t r a n s f o r m and convolution. Putting

f(x) = exp { -a2x 2 } , we make

use of the

standard

integral

F (u) = (/r 1/2/a)

(1.28)

(60) to obtain

exp {-lr2u 2/a 2 }.

(61)

Hence, if f(x) is a G a u s s i a n of h a l f - w i d t h a -I then its F o u r i e r t r a n s f o r m is also a G a u s s i a n h a v i n g h a l f - w i d t h a/K. We note also that, if f(x)

= exp(-a2x 2) , exp{-b2x 2},

F (u)

and,

applying

f(x) i.e.

the

=

(~/ab)

inverse

exp {-lr2u 2 (a2+b 2) / (a2b2) }, Fourier

transform

= (/1;/(a2+b 2) )1/2 e x p { _ (a2b2/(a2+b 2) )x2},

the c o n v o l u t i o n

of two G a u s s i a n s

is again

a Gaussian.

(62)

46 2.3.17.

FOURIER

TRANSFORMS

Row

circular

of

AND CONVOLUTIONS

CH.

2

holes

The e x t e n s i o n to two d i m e n s i o n s a l l o w s the a b o v e r e s u l t s to be a p p l i e d to c a l c u l a t e the d i f f r a c t i o n p a t t e r n s f r o m m a n y s i m p l e t w o - d i m e n s i o n a l d i s t r i b u t i o n s . However, in u s i n g d e l t a f u n c t i o n s a n d c o n v o l u t i o n s it is i m p o r t a n t to d i s t i n g u i s h c a r e f u l l y the d e l t a functions, a n d c o n v o l u t i o n s in one a n d two d i m e n s i o n s . We t a k e as e x a m p l e the case of a l i n e a r row of e q u a l l y s p a c e d c i r c u l a r h o l e s in an o p a q u e screen. For this the t r a n s m i s s i o n f u n c t i o n m a y be w r i t t e n

f (x, y) = 2

(63)

8 ( x - ha, y) * O (x, y)

n

x,y

where

we h a v e u s e d 5 ( x - ha, 2) to i n d i c a t e a d e l t a f u n c t i o n at y = 0, a n d O(x,y) is the t r a n s m i s s i o n f u n c t i o n for a c i r c u l a r a p e r t u r e as d e f i n e d in 2.3.8, above. See Fig. 2.7. The F o u r i e r t r a n s f o r m a t i o n of the set of d e l t a f u n c t i o n s w i t h r e s p e c t to x is

x = ha,

8(u

-

h/a)

h

, i i I,

\

00000

\

1.22

A

/ '

I

I

'

"

/

l

I/a Fig. 2.7. The diffraction pattern from a row of circular holes equally spaced in an opaque screen. The widths of the lines are made to suggest their intensities.

2.3

F O U R I E R TRANSFORMS AND DIFFRACTION:

EXAMPLES

47

and the Fourier t r a n s f o r m with respect to y is unity. Hence the Fourier t r a n s f o r m in x and y is a set of straight lines parallel to the v axis and equally spaced at intervals of a -I in the u direction. This distribution will be m o d u l a t e d by m u l t i p l y i n g by the Fourier t r a n s f o r m of O(x,y), namely (ru42/2) J1 (nAU) / (nAU) , where A is the diameter of the holes and U = (u2+v2) I/2 . This gives the set of lines of varying intem~ity ' as suggested by Fig. 2.7 where the width of the line has been used as an indication of relative intensity. This example has some relevance in radio astronomy. A common form of radio interferometer used for detecting the radio waves from distant radio stars, consists of an equally spaced line of paraboloidal antennas or "dishes" each of which sums the incident amplitude over the circular aperture of the dish. It can be seen, by application of the reciprocity t h e o r e m that the amplitude obtained by summing the amplitudes from all dishes coherently is exactly the same as the amplitude which would be observed at the distant radio star if the apertures of the dishes were to be illuminated from b e h i n d by a plane parallel incident wave. Thus, as a radio star moves across the sky, the amplitude m e a s u r e d by the interferometer corresponds to a linear trace across the d i a g r a m of the diffraction pattern in Fig. 2.7. 2.3.18.

Complementary

objects

- Babinet

's P r i n c i p l e

Consider two objects for which the transmission functions are c o m p l e m e n t a r y so that ql (x,y)

+ q2 ( x , y )

-

1.

(64)

For black and transparent objects, this means that the black areas of the first are the transparent parts of the second and vice versa. Then, for the second object q2(x,Y)

=

1

-

ql(x,Y).

The d i f f r a c t i o n pattern amplitude

Q2(u,v)

=

8(u,v)

-

is

Q1(u,v),

so that the intensity in the diffraction pattern is p r o p o r t i o n a l to IQ2(u.v) 12 which is equal to IQl(U,V) 12 except at the origin. This exception, the exclusion of the origin point, was overlooked in the original formulation, known as Babinet's Principle or Babinet's Theorem.

48

FOURIER TRANSFORMS

2.3.19. In implies is equal object,

AND C O N V O L U T I O N S

Total intensities:

Parseval's

CH.

2

Theorem

a d i f f r a c t i o n experiment, c o n s e r v a t i o n of e n e r g y that the total i n t e n s i t y in the d i f f r a c t i o n p a t t e r n to the total i n t e n s i t y at the exit surface of the i.e. (65) --oo

--oo

This is a special case of a g e n e r a l Fourier transform theory

relationship

If(r) 12dr = f IF(u) 12du,

of

(66)

which in turn is a special case of the more general P a r s e v a l ' s T h e o r e m r e l a t i n g to any two functions f(r) and g(r);

f f(r) g* (r) dr = f F (u) G* (u) du.

(67)

F r o m this t h e o r e m we see that

f f ( R ) f ( r + R ) d R = ~ IF(u)12 exp{2Kiu.r}du which is a special t h e o r e m (29) ;

case of the inverse

~3[F(u)F*(u) ] = fir)

(68)

of the m u l t i p l i c a t i o n

, f(-r) .

(69)

This c o n v o l u t i o n of a function with the same f u n c t i o n i n v e r t e d is the a u t o c o r r e l a t i o n function which has c o n s i d e r a b l e a p p l i c a t i o n in m a n y fields of science and, as we will see later, has special s i g n i f i c a n c e for d i f f r a c t i o n under the title of a " g e n e r a l i z e d P a t t e r s o n function". Problems

i. Write down an e x p r e s s i o n for, and sketch the form of, the d i f f r a c t i o n p a t t e r n given when an incident plane wave is d i f f r a c t e d by" (I) a row of N c i r c u l a r holes, e q u a l l y spaced in an opaque screen, (2) a s i m i l a r row of opaque c i r c u l a r discs, (3) a similar row of holes in an opaque screen which are a l t e r n a t e l y c i r c u l a r and square,

2.3

F O U R I E R TRANSFORMS

AND DIFFRACTION:

EXAMPLES

49

(4) a row of circular holes in an opaque screen when each second one is covered by a half-wave plate (phase change ~), (5) two parallel rows of circular holes. 2. Find the Fresnel diffraction pattern given by a pair of parallel, very fine slits. Compare this with the F r a u n h o f e r d i f f r a c t i o n pattern. 3. A r a d i o - i n t e r f e r o m e t e r consists of two perpendicular, intersecting lines of regularly spaced circular dishes (paraboloidal antennas) forming an equi-armed cross (known as a "Chris-cross"). If the signals from all dishes are added in phase, how does the response of the interferometer vary with the position of a radio source in the sky? By what manipulation of the electronic signals could the interferometer be "pointed" to receive m a x i m u m radiation from a particular direction? 4. Compare the response of the radio interferometer d e s c r i b e d in 3 with that of a complete two-dimensional square array of N X N paraboloidal antennas. 5. In a Young's fringe experiment, two fine parallel slits, separation a, are illuminated by m o n o c h r o m a t i c light from a distant, parallel slit of width d, which may be regarded as an incoherent source, so that intensities due to each point of the source are added on the plane of observation. Find the visibility (defined as (~ax - ~ m i n ) / ( ~ a x + ~in)) of the interference fringes produced as a function of g and d. Taking the visibility of the fringes as a measure of the degree of coherence of the illumination at the two slits, verify the Zernike-van Cittart theorem which states that the degree of coherence is given by the Fourier t r a n s f o r m of the intensity d i s t r i b u t i o n of the source.

This Page Intentionally Left Blank

CHAPTER 3

Imaging and diffraction 3.1.

Wave

theory

of

imaging

3. I.I. Coherent wave optics while, as we shall see, lenses may be used to obtain either diffraction patterns or images, the major development of the theory of lens action has been directed towards imaging, we give some brief account of this development here but prefer a somewhat different approach which allows the diffraction patterns and images to be treated with equal facility and stresses the relationship between them. This has considerable practical importance, especially in electron microscopy. For many years, imaging was considered in terms of the approximations and from the point of view of geometric optics. The wave nature of light was taken into account only as a refinement when resolution was affected by diffraction effects due to finite aperture size. The formulation of imaging theory completely in terms of waves was introduced by Duffieux [1946] and subsequently developed by many authors including Hopkins [1953, 1950], Fellgett and Linfoot [1955] and Linfoot [1955]. An outline of the development has been given by Born and Wolf [1989]. This approach to imaging theory is b a s e d essentially on the Fourier transform. Here we describe first of all a more graphic version of the theory and then a more formal and elegant form. We consider first the system suggested by Fig. 3.1. Incident radiation, passing through a small object or a small part of an object gives an amplitude distribution which, over a limited region in one dimension, is given by the function f(x). Fraunhofer diffraction then gives rise to a distribution F(u) on a spherical reference surface at the entrance to a lens system. The coordinate u m e a s u r e d along the spherical surface is ~/l, where ~ is the scattering angle, or s/R where s is the distance m e a s u r e d on the sphere. The function of the lens is to transfer the distribution F(u) into the distribution F'(u') on a spherical reference surface centered on the part of the image, ~'(x), corresponding to f(x). Following a common convention, quantities

52

I M A G I N G AND D I F F R A C T I O N

F(u)

CH.

3

F'~u')

,

Fig. 3.1. Diagram suggesting the imaging process in a optical system. The complex amplitude distribution on a spherical reference surface in object space is converted into an amplitude distribution on a spherical reference surface in image space. r e f e r r i n g to the image space are primed. If the t r a n s f e r of F to F' were perfect, with no p e r t u r b a t i o n of the f u n c t i o n or a p e r t u r e limitation, we w o u l d have E'(s) = F(x). F r o m the g e o m e t r y of the situation, s=uR=u'R"

so that,

for this case, F' (u)

(I)

= F (u'R'/R) .

The d i s t r i b u t i o n F(u) is $f(x). By s y m m e t r y we put F'(u') = ~'(x). The t r a n s f e r from F' to ~' might be thought of as an inverse t r a n s f o r m but since the d i r e c t i o n of p r o p a g a t i o n from ~' to F' is o p p o s i t e to that in object space, the sign of the e x p o n e n t is c h a n g e d and a p o s i t i v e sign in the e x p o n e n t i a l of the t r a n s f o r m of F' to ~' makes it a direct F o u r i e r t r a n s f o r m . Hence ~" (x)

= ~3 F ( R ' u ' / R )

= ; ~ f(X) = f f(X)

d

exp{2~i

(u'R'/R)X}dX

exp{2Kiu'x}du'

~(X + Rx/R') dX = f(-Rx/R'),

(2)

i.e., the image is a r e c r e a t i o n of f ( x ) , i n v e r t e d and m a g n i f i e d by a factor R'/R. In practice, of course, the t r a n s f e r from F(u) to F'(u') is not perfect. There is always a l i m i t a t i o n due to the finite a p e r t u r e size. In addition, the lens a b e r r a t i o n s give rise to phase changes which vary as a function of u'. C l a s s i cally, this phase change is d e s c r i b e d by a power series in u' or, in two dimensions, by a p o w e r series in p (proportional to (u2+v2) I/2) and the p o l a r angle ~. In this way the t r a d i t i o n a l a b e r r a t i o n c o e f f i c i e n t s are generated, the most i m p o r t a n t for our p u r p o s e s b e i n g the T h i r d Order S p h e r i c a l A b e r r a t i o n c o e f f i c i e n t which derives from the c o e f f i c i e n t of the p4 term.

3.1

WAVE THEORY OF IMAGING

53

In the w a v e - t h e o r y treatment, the changes of amplitude and phase due to the limitations of the lens are r e p r e s e n t e d by a Transfer Function T(u'), characteristic of the lens. Then F'(u') = F(R'u'/R) Correspondingly by

T(u') .

the m o d i f i c a t i o n ~t" (x) = f (-Rx/R')

(3)

of the image is represented , t (x) ,

(4 )

where t(x), the Fourier t r a n s f o r m of T(u), is the spread function. For the simplest case, in two dimensions, of limitation by a circular aperture only, we have that, as in (2.44) ,

T(u'. v')

= {

so that

t (x. y) and the intensity

l'(x.y)

1

if

(u,2+v,2)1/2
0

elsewhere.

- t (r) = J1 (rcuor)/uor,

distribution

in the image is

= ~r~* = I f ( ( - R / R z ) x , ( - R / R 1 ) y )

. J1 (rcuor) / (rcr) ] 2,

(5)

which represents a loss of resolution due to a blurring of the amplitude distribution. When, in addition to the aperture limitation, the aberrations of the lens are appreciable, the transfer function includes the phase change term so that (3) may be written

F" (u')

= F (R'u'/R)

T a (u')

ex p { i ~ (u') },

where T a(u') is the aperture function and ~)(u) represents the change of phase, which may be expanded as a power series in p and ~ to introduce the aberration coefficients. Then the observed intensity is given by

r(x.y)

= [ f ( ( - R / R 1 ) x , ( - R / R 1 ) y ) . J l (rcuor) / ( r c r ) * s [ e x p { i ~ ( u )

The effects of the apparent. They may cases but a simple will discuss these treatment.

}] [ 2.

added convolution here are not immediately be calculated in detail for p a r t i c u l a r intuitive u n d e r s t a n d i n g is difficult. We effects later in terms of the Abbe

54

IMAGING AND D I F F R A C T I O N

3.1.2.

Incoherent

CH.

3

wave i m a g i n g

The same sort of treatment applies if we consider incoherent imaging, i.e., imaging of an object which is selfluminous or which is illuminated by incoherent incident light or scatters the incident light incoherently. Then we c o n s i d e r the imaging of light from each point of the object s e p a r a t e l y and add the resulting intensities. R e f e r r i n g to (4), the amplitude distribution for a point source on the axis of the lens is t(x) and the intensity d i s t r i b u t i o n is It(x)12. Then the image intensity is given by

l'(x,y)

= I( (-R/R1)x,

(-R/R1)y)

.

It(x)12

.

(6)

The spread function It(x)[2 can be referred to a Contrast Transfer Function which characterizes the lens; O(u')

= T(u')

.

T*(-u')

.

(7)

For the simple case of a circular aperture, the image intensity is c o n v o l u t e d by the spread function It(x,y)

12 =

J2(rcu~ (rl:r)2

If we then apply the Rayleigh criterion for the resolution of two adjacent point sources, that the images can be resolved if the m a x i m u m of one point image falls at the first m i n i m u m of the other, i.e., at r = 1.22/u0, we obtain the least resolvable distance for the lens as =

1.221/2~o

=

1.22t/0~

(8)

where ~ is the angular aperture subtended by the lens at the object. For convenience in considerations of m i c r o s c o p y we have t r a n s f e r r e d the dimensions back into object space. The more formal description of the imaging process eliminates the need for the picture of a transfer from one reference surface to another as suggested in Fig. 3.1. The spread function, t(x,y) for amplitudes for coherent imaging, or ll(x,y)12 for the incoherent case, is defined as the response to a point source in the object. The Fourier transform, T(u,v) or O(u,v) = T(u,v) , ~ (-u,-v), is the appropriate transfer function which is characteristic of the lens. This contrast transfer function may be derived, i n d e p e n d e n t l y of any aberration theory, by studying the image intensity distribution for p a r t i c u l a r test objects; for example, the contrast of the images of objects with t r a n s m i s s i o n function 1 + cos(2Kax) give the amplitudes and

3.2

ABBE THEORY

55

phases of the transfer for particular values of the "spatial frequency", a. From the contrast transfer function the spread function is derived by Fourier transform. 3.2.

Abbe

theory

For our purposes, the formal use of Fourier transforms in imaging theory is interesting but limited, in the form d e s c r i b e d in the previous section, since we are interested in the concrete embodiment of the Fourier t r a n s f o r m in the form of the d i f f r a c t i o n pattern as well as in the image. We turn therefore to the imaging theory originating from the ideas of Abbe. Although this is essentially a wave theory, it is c o n v e n i e n t l y illustrated by reference to the geometric-optics diagram of Fig. 3.2. Parallel light is incident on an object having transmission function /(x.y) which is imaged by a lens having focal length, /. Light scattered by a point A of the object is brought to a focus at A' in the image plane. From the geometry of the d i a g r a m the image is inverted and m a g n i f i e d by a factor R'/R so that the image is

W(x,y) = f ( (-R/R1)x, (-R/R1)y)

.

Also it may be noted that all light scattered through an a n g l e ~ is brought to a focus at one point in the back-focal plane. This is equivalent to interference at a point at infinity. Hence the amplitude distribution on the back-focal plane is that of the Fraunhofer diffraction pattern given by

~'(xy) f(xy)

F E)

Fig. 3.2. Geometric optics ray diagram used to suggest the plausibility of the wave-optics description of the imaging process in the Abbe theory.

56

IMAGING AND DIFFRACTION

CH. 3

the Fourier transform function, F(u,v). In this case u = (sin~x)/l and if ~ is not too large we may write u = x//l,

v =

y/:l.

Thus the imaging process may be described in terms of two Fourier transforms- The scattered radiation from the object interferes on the back-focal plane to give the Fraunhofer diffraction pattern described by a Fourier transform: then the radiation from the back-focal plane again forms an interference pattern on the image plane which is effectively at infinity so that the amplitude distribution in the image is given by Fourier transform of that in the backfocal plane. Two immediate consequences are of interest. The first is that lenses may be used for light or electrons to give the Fraunhofer diffraction pattern of an object at a convenient location and on a convenient scale, depending on the focal length of the lens. Secondly, the limitations of the lens system in reproducing the object transmission function on the image plane may be described in terms of modifications of the amplitude and phase of the distribution on the back-focal plane. Apart from their direct interest for consideration of optical and electron-optical imaging and diffraction, these points are of importance for X-ray diffraction because of the very graphic use made of the optical diffractometer by Taylor and Lipson [1964] to simulate the production of X-ray diffraction patterns and the reconstruction of crystal structures from diffraction amplitudes. 3.3.

Small

angle

approximation

Although it is strictly limited in its validity, the small angle approximation which we introduced for the discussion of Fresnel Diffraction forms a very convenient basis for describing the essential behavior of imaging systems. It provides a model which reproduces all important features of the properties of optical systems with a relative mathematical simplicity and wide versatility. All objects are considered to be composed of planar distributions having transmission functions q,(x,y). Propagation through a medium of constant refractive index is given by convolution with a propagation function which, in the small angle approximation, is (i/R l) exp{-~i (x2+y2)/R ~} . we introduce the concept of an ideal thin lens, which is a planar object having a transmission function exp{~i(/2+y 2)/~I}. It is readily confirmed that exp{~i (x2+22)//l}

, exp{-~i (x2+22)//l} = ~(x, 2)

(9)

3.3

SMALL ANGLE A P P R O X I M A T I O N

i.e. if a plane wave, amplitude ideal thin lens, p r o p a g a t i o n of length f gives a d e l t a - f u n c t i o n S i m i l a r l y a point source p l a c e d thin lens gives a plane wave:

57

unity, passes t h r o u g h the the wave through the focal or a point cross-over. a distance f before the ideal

[~(x.y),exp.-hi(x2f~,+y2)}]exp,

~:i(xz +

fz

y2)~= 1. J

(10)

In this a p p r o x i m a t i o n the amplitude given on a plane of o b s e r v a t i o n when a plane wave passes through an object of t r a n s m i s s i o n function q(x,y) and then an ideal thin lens, as s u g g e s t e d in Fig. 3.3, is, in one d i m e n s i o n and o m i t t i n g constant multipliers, (11) where the o p e r a t i o n s in the successive brackets represent p r o p a g a t i o n through a distance R, t r a n s m i s s i o n through the lens, and p r o p a g a t i o n through a distance R'. By writing out the c o n v o l u t i o n integrals in detail it is readily shown that if R ' = f, ~/(x) = Q(x/fX), and if (l/R) + (IIR') = 1/f, then ~(x) = q(-Rx/R'). To prove the second case, (II) is written in detail, with d u m m y variables X and Y,

_ [-rl:iY2} 1+ .--~-~)X 1 2} ~(x) - IIq(Y)expl~,R~ .. exp{.~ (--.~ x

Then

if

ex-l'-gix2p' R't ['} expi--f [ni('Y+-~')X} dxdy" [,~-

(l/R) + (I/R') = I// the integral

over X is seen to be

(Y+R x/R') and the integral over Y then gives ~ (x) = Cq (-Rx/R'),

q(

i~,,s

~(x,y)

Fig. 3.3. Diagram defining the imaging system described by equation (3.11).

58

IMAGING AND DIFFRACTION

CH. 3

where C has modulus unity and may be absorbed with other similar terms omitted from the propagation functions. The proof of the first case, for R' = ~ is left as an exercise for the reader. Thus the properties of producing diffraction patterns and images are reproduced. Obviously the action of any combination of sources, object and lenses may be reproduced by writing down the appropriate series of operations of convolution with a propagation function and multiplication with a transmission function. For example, if there is a point source at x = X a distance R0 before the object of Fig. 3.3, the amplitude distribution on the plane of observation is

X expI~ix21[ f),. J ]*exP{ -~:ix2R'~}" ,

(12)

By evaluation of the integrals it can be shown that the diffraction pattern is given for (I/R') = and the image is given, as before, (l/R) + (I/R') = i//. The effect on the diffraction pattern or on the image of incomplete coherence of the incident radiation, i.e., of illumination of the object by an incoherent source of appreciable extent, is given by summing the intensities for each point of the source separately. Thus the intensity for the source point x = X, given by I~X(X) 12, from (12), is calculated and multiplied by the intensity of the source point, 10(X). Then the observed intensity distribution is derived by integrating over X. The effect of the limitation of the aperture of the lens on the image is investigated both experimentally and theoretically by placing the aperture in the back-focal plane to multiply the diffraction pattern by the transmission function of the aperture. The result is exactly the same as expressed in equations (5) and (6). Excellent examples of the effects on the image of the limitation of the diffraction pattern by apertures of various sizes and shapes, produced by use of the optical diffractometer, are given by Taylor and Lipson [1964] (their Plates 43 to 46). The effect of defocussing of the lens is readily deduced. If the plane of observation is a distance ~' away from the plane of the in-focus image, which has amplitude distribution ~0(x), then,

I/(Ro+R) +

(X) = ~0(X)

* exp{-~ix 2/A~}.

i/~,

(13)

3.3

SMALL ANGLE A P P R O X I M A T I O N

59

A l t e r n a t i v e l y we m a y say that the a m p l i t u d e on a plane a d i s t a n c e A away f r o m the object is brought to a focus on the plane of o b s e r v a t i o n so that the d i s t r i b u t i o n which is i m a g e d is q(x) . exp{-Kix2/A~} (14) The a m p l i t u d e d i s t r i b u t i o n in the b a c k - f o c a l F o u r i e r transform, is then Q(u)

plane,

given by

e x p { - ~ i ~ u 2} .

Thus the effect of defocus can be c o n s i d e r e d as e q u i v a l e n t to the a d d i t i o n of a s e c o n d order phase t e r m in the b a c k - f o c a l plane. H i g h e r order terms in the exponent are i n t r o d u c e d by a b e r r a t i o n s of the lens. The t h i r d - o r d e r spherical aberration, for example, adds a t e r m p r o p o r t i o n a l to u 4. 3.4.

Phase

contrast

3.4.1. Phase and amplitude objects A pure p h a s e object is an i d e a l i z e d concept. It is e s s e n t i a l l y a t w o - d i m e n s i o n a l object which changes the phase but not the a m p l i t u d e of the incident wave. We may use it for c o n v e n i e n c e to d e s c r i b e a thin object h a v i n g a v a r y i n g refractive index when the angles of r e f r a c t i o n of the incident waves are so small that the lateral spread of the waves within the object thickness can be ignored. In the oned i m e n s i o n a l f o r m which we use for convenience, the t r a n s m i s s i o n function is of the form q(x) = exp{i~(x) }

(15)

where the phase change ~(x) depends on the thickness and refractive index of the material. If a plane incident wave falls on this object, the t r a n s m i t t e d i n t e n s i t y is lq(x)12 -- I. Similarly, an ideally perfect lens w o u l d give an exact r e - c r e a t i o n of q(x) so that the i n t e n s i t y d i s t r i b u t i o n of the image w o u l d be I~(x)

12 =

Iq(-Rx/R')12

=

1.

(16)

Thus there is no contrast in the image c o r r e s p o n d i n g to the structure of the object. For most objects some a b s o r p t i o n and m u l t i p l e s c a t t e r i n g takes place so that the object, if thin enough, must be r e g a r d e d as a m i x e d phase and amplitude object with a t r a n s m i s s i o n function which m a y be w r i t t e n q (x) = exp { i 9 (x) - ~ (x) },

(17)

60

IMAGING AND DIFFRACTION

CH.

Iq(x) 12 = exp{-211(x) }.

3 (18)

However for a large and important class of objects the pure phase object forms a reasonable approximation. Most thin biological samples are very nearly phase objects for visible light. All electron microscope specimens of less than a certain thickness are essentially phase objects for electrons. The refractive index for electrons, given by equation (1.6), varies with the value of the electrostatic potential. While for both light and electrons staining techniques have been developed to increase the effective absorption and so provide contrast in the in-focus image, these techniques are of limited use for some purposes and may introduce artifacts which complicate the image interpretation. We therefore consider in some detail the so-called "phasecontrast" techniques for obtaining contrast in the image of a pure phase object. 3.4.2.

Out-of-focus

contrast

This is the most common form of phase contrast, especially in electron microscopy. It is a well-known experimental observation that the contrast is a m i n i m u m near exact focus. Contrast appears off-focus and reverses when one goes through focus. We consider the amplitude distribution on a plane a distance & from the object, written as V(x)

=

q(x)

(19)

* exp{-~ix2/~}.

Instead of the form (15), it is more convenient to work with the Fourier transform function ~(u), defined by q(x) = exp{i~(x) } - I ~>(u) exp{-2~iux}du. The convolution

in

(19) is

; ~>(u) exp{-KiX2/Al}

e x p { - 2 K i u ( x - X ) }dXdu.

Making use of the standard integral ~ ~(U)

(20)

exp{-2Kiux}

Then if A is sufficiently

(1.28) gives us

exp{iKAlu 2}du.

small we may put

exp{i~A~u 2} = 1 + i ~ u

2,

(21)

3.4

PHASE CONTRAST

and we obtain for the image, plane, ~(x) = ~ ~(,)

61

referred back to the object

exp{-2Kiux}

= exp{i~(x) } + i K A l f

(l+i~A,2)du u2~(,)

exp{-2Kiux}da.

(22)

The relation (2.28) suggests that the integral in this expression may correspond to a second differential and, in fact dx 2 [exp{ i~(x) } ] =

j"

(-4K2u2)~(u)

exp{-2Kiax}d,;

but s t r a i g h t - f o r w a r d differentiation gives d2

dx 2 [exp{i~(x) }] = -exp{i~} [{~'(x) }2 + i~"(x)], where the primes Hence (22) is ~(x)

represent differentiation with respect to x.

= exp{i~(x) } [l+(Al/4K)~"(x)

+

(iAl/4K)(~'(x))2].

(23)

Then the intensity of the out-of-focus object is, to first order in the small quantity, Al, I(x)

=

I

+

(Al/2~)

q)"(x) .

(24)

Thus the contrast depends on the second differential of the phase function ~(x), and is reversed when the sign of the defocus, A, is reversed. As we noted previously, going out of focus is equivalent to changing the phase of the amplitude of the diffraction pattern in the back focal plane by a factor proportional to u2. The effect of lens aberrations can be represented by changing the phase of the diffraction pattern by a function in the form of a power series in even orders of u and defocus is included as a first-order aberration, coming from the second-order term in u. The fourth-order terms include thirdorder spherical aberration. When the lens is exactly focussed the phase changes due to these higher order terms in u remain and provide some amplitude contrast. To an even greater extent than with the second-order defocus terms, these phase shifts are small for small u but increase rapidly for large u so that in the image contrast most contributions come from the outer part of the diffraction patterns. Hence contrast fluctuations may be expected where ~(x) changes rapidly.

62

IMAGING AND D I F F R A C T I O N

CH.

3

The treatment given above in equations (20) to (24) may be e x t e n d e d to include higher order terms in u, but in practice the result contains combinations of h i g h e r - o r d e r d i f f e r e n t i a l s of ~(x) which are not so readily v i s u a l i z e d and so are less useful. For out-of-focus images, the phase changes due to spherical aberration are either added to or subtracted from the phase changes for defocus, depending on the direction of defocus, and either assist or hinder the phase contrast imaging. This is an important c o n s i d e r a t i o n for electron m i c r o s c o p y of thin objects and will be discussed in more detail in Chapter 13.

3.4.3. Aperture limitation For a phase object the formation of an image of zero contrast depends on interference of waves from the b a c k - f o c a l plane with exactly the right amplitude and relative phase. Any m o d i f i c a t i o n of the back-focal plane amplitude d i s t r i b u t i o n will upset this balance and produce some contrast. If, for example, a circular aperture is placed c e n t r a l l y in the back-focal plane, the image has the form

l (X, y)

=

I e x p { i ~ (x, y) } . J1 (r~ar) / (Kr) i 2.

(25)

It is not immediately obvious that this represents amplitude contrast. However from qualitative reasoning we may predict that, firstly, since the outer parts of the d i f f r a c t i o n pattern are affected, only the higher-order Fourier components of q(x,y) are modified; hence contrast appears where there are sudden changes in ~(x,2) as at the edges of particles. In fact the image contrast shows some s i m i l a r i t y to the function l~'(x) [. Secondly, since the image is c o n v o l u t e d by a spread function the resolution is not better than the width of the spread function. Hence again, the gain in contrast is made at the expense of resolution. Other forms of aperture limitation include the wellknown case of "Schlieren" optics used for visualizing air flow in wind tunnels. A straight edge is introduced in the b a c k - f o c a l plane to cut off half the diffraction pattern. From the relation (2.41) it is seen that the effect may be r e p r e s e n t e d in part by convolution of the image exp{i~(x)} with a function of the form (2Kix) -I, which gives contrast somewhat resembling ~'(x) .

3.4.4. Zernike phase contrast The form of phase contrast introduced by Zernike is the most effective in that it produces m a x i m u m contrast with no

3.4

PHASE C O N T R A S T

63

loss of r e s o l u t i o n and the contrast is p r o p o r t i o n a l to the phase f u n c t i o n ~(x) itself, rather than to any d i f f e r e n t i a l of this function. It is most e a s i l y u n d e r s t o o d for the case of a small phase change, l~(x) I << I. For this c o n d i t i o n we can assume that all h i g h e r powers of ~(x) are n e g l i g i b l e and write

q(x) = 1 + ig(x). Then in the b a c k - f o c a l Q(u)

(26)

plane the a m p l i t u d e

= ~(u)

+ i~(.),

is (27)

where the delta function represents the d i r e c t l y t r a n s m i t t e d p l a n e - p a r a l l e l incident b e a m and ~(u) is the r a d i a t i o n s c a t t e r e d into the rest of the d i f f r a c t i o n pattern. No contrast results when these two p o r t i o n s are r e - u n i t e d in the image b e c a u s e they are ~/2 out of phase, g i v i n g (x) = 1 + i~(x) and ~ * = 1 to first order terms only. The central b e a m and the rest of the d i f f r a c t i o n p a t t e r n can be b r o u g h t back into phase if the phase of the central b e a m is c h a n g e d by ~/2 by i n s e r t i n g a q u a r t e r - w a v e plate. Then (27) becomes Q'(u) = i[6(u)

and

+ ~(u)]

(x) = i[l + ~(x)],

SO that,

to the first order

in ~(x),

l(x) = ~r~* = 1 + 2~(x).

(28)

This simple d e s c r i p t i o n does not often represent the p r a c t i c a l situation. If the phase change is not small, as in many important cases in both optical and e l e c t r o n microscopy, the t h e o r y b e c o m e s much more complicated. Also it is impossible to change the phase of a d e l t a - f u n c t i o n central b e a m only. The phase plate has finite size and so changes the phase of part of the d i f f r a c t i o n p a t t e r n also. Further, we have a s s u m e d a plane incident wave. This is a r e a s o n a b l e a p p r o x i m a t i o n for the usual conditions of e l e c t r o n m i c r o s c o p y . It is not so in optical m i c r o s c o p y since for a plane wave the i n t e n s i t y of the image is insufficient. Instead, a h o l l o w cone of i l l u m i n a t i o n is u s e d with an annular p h a s e - p l a t e near the b a c k - f o c a l plane. 3.5.

Holography

Gabor (1948,1949) i n t r o d u c e d the concept of h o l o g r a p h y as the basis for a scheme to overcome the l i m i t a t i o n s of r e s o l u t i o n in e l e c t r o n m i c r o s c o p y due to the u n a v o i d a b l e effects of lens

64

IMAGING AND D I F F R A C T I O N

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3

aberrations (see Chapter 13). In order to correct for the phase changes due to lens aberrations, it is necessary to influence the relative phases of the waves c o n t r i b u t i n g to the image, but in the recording of the image intensities, the phases are normally lost. Gabor p r o p o s e d the use of a reference wave which could interfere coherently with the image wave, giving rise to intensity data which c o n t a i n e d the phase information of the image. In his initial proposal, a strong e l e c t r o m a g n e t i c lens is used to form a small crossover close to the plane of a weakly scattering thin object, as in Fig. 3.4(a), so that a highly m a g n i f i e d "shadow image" is formed on a distant plane of observation. The electron source is assumed to be so small that the electron b e a m

Hologram

II

Specimen

II Fig.3.4(a) Hologram Conjugate

Image

Fig. 3.4(b). Fig. 3.4. Diagram suggesting the principles of Gabor holography. In (a a lens is used to form a cross-over close to a thin specimen and a m a g n i f i e d shadow-image or hologram. In (b), the hologram is illuminated by a convergent wave to form the reconstruction of the specimen wave an a conjuqate imaqe.

incident on the specimen is completely coherent. The wave d i r e c t l y t r a n s m i t t e d through the object is taken as the reference wave and the much weaker wave, scattered by the

3.5

HOLOGRAPHY

65

object, is the image signal. In the shadow-image plane, or hologram, these two waves interfere, so that the intensity d i s t r i b u t i o n reflects their relative phases. In a reconstruction process, the recorded h ologram is illuminated by coherent visible light, as in Fig. 3.4(b), and the image wave function of the object is recreated in the equivalent plane. Unfortunately, because it is the intensity of the hologram which is recorded, the reconstructed wave function is a c c o m p a n i e d by a defocussed "conjugate" image, providing an unwanted background. In a simplified, one-dimensional description, we may consider the electron cross-over as a point source created by a lens with no aberrations. The wave incident on the specimen is then exp(-Kix2/Rol). The transmission function for the weakly scattering object may be written q(x) = 1 + e(x), where s is small and, in general, complex. The intensity distribution on the distant hologram is then l~(u) [2 = = ~(,) + E(,)

IQ(,) , exp(KiR0 u2) 12

* exp(-KiRo u2) + E*(,)

, exp(+KiRo,2),

(29)

where the term of second order in E(u), the Fourier transform of ~(x), has been neglected. P e r f o r m i n g the inverse Fourier transform in the reconstruction process gives the wave function (X) = 1 + E(X) .exp(-Kix 2/R0~,) + ~*(x) . e x p ( + K i x 2/R0~,) .

(3O)

If this reconstruction is imaged a distance R 0 out of focus, to give s directly, the image is accompanied by the conjugate image s*(x), imaged a distance -2R 0 out of focus. If the distance R 0 can be made large, and if the desired image consists of small, sharply defined objects, a reasonable representation of the desired object may be achieved as in the original reconstructions from holograms made with visible light (Gabor, 1949) in the absence of suitable bright electron sources at that time. The presence of aberrations in the lens used to form the small electron crossover of Fig. 3.4(a) has the effect of adding additional phase terms to the exp(KiR01u 2) term in (29). If an equivalent aberration phase term is subtracted from the real image term in the reconstruction process of (30), to give an image free of aberration, a doubled, reversed phase term is added to the conjugate image, making it more strongly aberrated and closer to random noise. Realization of the Gabor scheme, using a small electron crossover, and showing an improvement of resolution by correction for aberrations, was not achieved for many years (Lin and Cowley, 1986) when the use of high-brightness electron sources involving field-emission guns became

66

IMAGING AND D I F F R A C T I O N

CH.

3

feasible. Lin and Cowley showed that reconstructions could be c o n v e n i e n t l y made by the computer m a n i p u l a t i o n of d i g i t a l l y recorded holograms and it is possible to remove the contributions of the unwanted conjugate image by, for example, superimposing a number of reconstructions from holograms obtained with lateral shifts of the electron b e a m relative to the object (see also Wang and Cowley, 1995). The great expansion of the application of the holographic principle to the many areas of imaging with visible light and other radiations, including high energy electrons, came with the realization by Leith and Upatnieks (1962) that the reference wave used to define the image wave phases need not be one t r a n s m i t t e d through the object but could be one passing around the object in free space. For visible light it is easy to split a coherent wave into two portions, widely separated in space, by means of prisms or partially transparent mirrors.

ologram

exp {2~iuo.r }

mirrors

~ 8 (u+u0)

8(u-uo)

~ reconstruction Fig. 3.5. Diagrams suggesting the formation of an off-axis hologram with light when a plane reference wave interferes with the wave scatter by an object and the reconstruction of the object wave in the direction 8(u-uo) when the hologram is illuminated by a plane wave. For a system represented as in Fig. 3.5, the reference plane wave exp(2~iu0x) may be added to the image signal,~l (x). At a large distance, the diffraction pattern intensity is then represented by

3.5

HOLOGRAPHY l~(u) 12 = + WI(U)

{6(U) +

(WI(U)

67

* WI*(U))}

, 8(U - U0) + WI*(U)

* ~(U)

(31)

, 8(U + U0).

The d i s t r i b ution around the direction 8(u-u0) may be readily separated from the distributions around the directions 8(,) and ~(u+,0) by use of apertures. Then the inverse Fourier t ransform of ~l(U) gives the desired image wave, ~l(X), directly, without confusion. For visible light, the p o s s i b i l i t y of using large angles between the object wave and the reference wave introduces the possible use of holography for three-dimensional imaging. Applications for electron waves will be discussed further in Chapter 13.

3.6.

Multi-component

systems

The treatment of optical systems in the small angle approximation reproduces most of the properties of real optical systems and is a very good approximation for electron optics of medium- and high-energy systems since the scattering of electrons of these energies by atoms is essentially a smallangle phenomenon. In Section 3.3 we showed how to write down the expressions for the diffraction patterns, images or the phase distributions on any plane in a simple ideal-thin-lens system, using this approximation. We now consider the extension of this treatment to multi-component systems. We consider, for brevity and convenience, only one-dimensional objects. The extension to two dimensions is obvious. Radiation from a source with amplitude distribution qo(x) passes through a series of planar objects having transmission functions qn(x). Propagation over the distance Rn, from the nth to the (n + I) th object is represented by convolution with the p r o p a g a t i o n function pn(X) (see Fig. 3.6). In the smallangle approximation, Pn(X)

=

(i/Rl) 1/2 exp{_2~ikRn } exp{_~ix2/Rl},

and neglecting the exp{-2~ikR n} term, p, (X) is pn(U)

(32)

the Fourier t r a n s f o r m of

= exp{~iRnlU2}.

(33)

Then the amplitude on the plane of observation, which may be regarded as the plane of the (N + i) th object is written

WN+I (x)=qN (x) [... ~q2 (x) ~ql (x) [qo (x) *Po (x) ]*Pl (x) ] *P2 (x) 1... ] *PN (x) N

I

2

3

N

(34) where the brackets have been numbered for clarity. The contents of the N bracket represent the amplitude of the wave

68

IMAGING AND D I F F R A C T I O N

CH.

3

incident on the N th object. This amplitude is m u l t i p l i e d by the t r a n s m i s s i o n function qN (x) and the product is then c o n v o l u t e d with PN (x) . R2

Ro ~i= R!

I.

.po(X)

q -)

q,x)

ix)

<

q

Fig. 3.6. Diagram illustrating the description of wave propagation through a multi-component system. By m a k i n g use of the c o n v o l u t i o n and m u l t i p l i c a t i o n theorems of Fourier transforms, the Fourier t r a n s f o r m of is written

(34)

~N+I (u) = Q N (u) *...[Q2 (u) *[QI (u) *Qo (u)P (u) ]PI (u) ]P2 (u)... ]PN (u), 2 1 1 2 N (35) where again, the m u l t i p l i c a t i o n takes place before the convolution. This represents the F r a u n h o f e r d i f f r a c t i o n p a t t e r n given by the radiation e m e r g i n g from the N objects. In the case of the small angle a p p r o x i m a t i o n using (32) and (33), the e x p r e s s i o n for either the amplitude, (34), or the d i f f r a c t i o n pattern, (35), may be written in terms of any c o m b i n a t i o n of the functions in real space, q,(x), p,(x), or the functions of Fourier t r a n s f o r m space, an(u), Pn(u). This follows from the special p r o p e r t i e s of the complex e x p o n e n tials with respect to Fourier t r a n s f o r m and convolution. Thus we have

q (x)

exp

{ -i0~ 2 } = ~ q (X) exp {-i~ (x - X) 2 }dX

= exp{-i0cr 2} f q(X) d =

e x p { - i ~ X 2} exp{2i05~X}dX

(K/i~)l/2 e x p { - i o ~ 2}

[Q(o~/K)

, e x p { i o ~ 2}] .

(36)

Q(u) , e x p { i ~ u 2} = (~i/~)i/2exp{i~u2} [q(~u/~) * e x p { - i ~ u 2 } ]

(37)

Similarly

Thus any b r a c k e t e d term of (34) or (35) may be c o n v e r t e d to a form involving a instead of q or q instead of Q. This is

3.6

MULTI-COMPONENT

SYSTEMS

69

of p a r t i c u l a r value, for example, for periodic objects for which it is often more convenient to deal with the Fourier transform, a set of weighted delta functions, rather than a continuous function. Another useful relationship is [q(X) exp{-i~x 2}]

= (i=/(u

* exp{-i~/2}

+ ~ ) ) 1 / 2 exp{_i~x2}[Q(~x/K),exp{i~2x2/( ~ + ~ ) } ]

Using the relationship this becomes [q(x) exp{-i0~r 2}]

(38)

(37) to modify the b r a c k e t e d term, , exp{-i~x 2}

= e x p { - i ~ x 2 } [q(~x/ (~ + ~))

, exp{-i~2x2/(~ + ~) } ] .

(39)

Similarly, [Q(,) e x p { i ~ 2}]

= exp{i~.

2}

, exp{i~u 2}

[Q(~u/(~ + ~))

. exp{i~2.2/(~

+ ~)}],

(40)

and so on. Such relationships are of use when a transmission function is m u l t i p l i e d by an incident wave from a point source, exp{-~ix2/Ro~}, or when a transmission function qn(X) modifies the transmission function of an ideal thin lens,

exp {Kix2 /f ~ } .

In the limit that the number of two-dimensional objects tends to infinity and the distance between them tends to zero, expressions of the form (34) or (36) may be used to represent rigorously the scattering form any threedimensional object (Moodie [1972]). To represent the scattering from a three-dimensional object with any desired degree of accuracy, the number of planar objects must be made sufficiently large and the distance between them sufficiently small. An example of this approach will be given in our consideration of the scattering of electrons by crystals (Chapter Ii).

3.7.

Partial

Coherence

In this chapter we have considered the two limiting cases of coherent imaging, when the object is illuminated by a plane wave or radiation from a monochromatic point source, and incoherent imaging which applies to the imaging of a selfluminous source consisting of independently emitting point sources. Most practical situations are intermediate between these limiting, ideal cases. As has been pointed out in

70

IMAGING AND D I F F R A C T I O N

CH.

3

Section 1.3, it is always p o s s i b l e to derive the o b s e r v e d i n t e n s i t y in an e x p e r i m e n t by c o n s i d e r i n g s e p a r a t e l y the i n t e n s i t y given by each point of the source and by each d i f f e r e n t wavelength, and then adding all these intensities. However, it is often c o n v e n i e n t to make use of the concept of a degree of c o h e r e n c e as a m e a s u r e of p a r t i a l coherence. In its most complete form, the concept of a degree of c o h e r e n c e as a function of space and time provides a d e s c r i p t i o n of i m a g i n g or d i f f r a c t i o n p r o c e s s e s in terms of o b s e r v a b l e q u a n t i t i e s so that there is no need to p o s t u l a t e an u n o b s e r v a b l e and hence fictitious wave function. (See, for example, Born and Wolf [1989]). In its simplest form, the use of the degree of c o h e r e n c e or the a p p r o x i m a t i o n s of a " c o h e r e n c e width" and "coherence length" p r o v i d e very c o n v e n i e n t i n d i c a t i o n s of the amount and type of the i n t e r f e r e n c e effects which might be o b s e r v e d in any e x p e r i m e n t a l situation. We will confine our a t t e n t i o n to these latter, less e x a c t i n g aspects. The degree of c o h e r e n c e is not a p r o p e r t y of r a d i a t i o n at any one point. It is a m e a s u r e of the c o r r e l a t i o n w h i c h exists b e t w e e n the phases of the r a d i a t i o n at two points. If there is a c o r r e l a t i o n b e t w e e n the phases at the two points, then light p r o c e e d i n g from those two points can give i n t e r f e r e n c e effects if it is a l l o w e d to overlap. The degree of c o h e r e n c e is d e f i n e d in terms of the strength of these i n t e r f e r e n c e effects. Thus a p r a c t i c a l i n t e r f e r e n c e e x p e r i m e n t can be set up to m e a s u r e the degree of coherence. If the two points are s e p a r a t e d by a vector p e r p e n d i c u l a r to the d i r e c t i o n of p r o p a g a t i o n we can m e a s u r e the "lateral coherence". If they are s e p a r a t e d by a vector p a r a l l e l to the d i r e c t i o n of p r o p a g a t i o n we m e a s u r e the " l o n g i t u d i n a l coherence", also c a l l e d the "chromatic coherence" b e c a u s e the c o r r e l a t i o n of the phases then depends only on the s p r e a d of w a v e l e n g t h s or f r e q u e n c i e s present. To m e a s u r e the lateral c o h e r e n c e b e t w e e n two points we can e n v i s a g e an ideal Young's fringes experiment, in one d i m e n s i o n and with m o n o c h r o m a t i c r a d i a t i o n for simplicity. N a r r o w slits are p l a c e d at the two points and the interference p a t t e r n is o b s e r v e d on some plane b e y o n d them. For p e r f e c t coherence, the i n t e r f e r e n c e p a t t e r n is given as s q u a r e d cosine fringes [equations (1.30) and (2.46)]. For c o m p l e t e incoherence, the i n t e n s i t i e s from the slits are added s e p a r a t e l y and there are no fringes. The degree of c o h e r e n c e b e t w e e n the two points is given by the v i s i b i l i t y of the fringes, d e f i n e d as Visibility

=

(Imax - Imin)/(/max + Imin) 9

For a fringe i n t e n s i t y function A + B cos bx, the v i s i b i l i t y is B/A. As an example, we c o n s i d e r the c o h e r e n c e due to a small but finite source which may be c o n s i d e r e d as ideally i n c o h e r -

3.7

PARTIAL COHERENCE

71

ent, with an i n t e n s i t y d i s t r i b u t i o n 10(x). If this is d i s t a n c e R 0 from the two points, which are s e p a r a t e d d i s t a n c e x I and the i n t e r f e r e n c e p a t t e r n is o b s e r v e d d i s t a n c e R, the o b s e r v e d i n t e n s i t y due to a point in source at x = X is, from (1.30) .

placed a by a at a the (41)

IX(X) = 4 cos 2 (~Xl/~)(x/R+X/Ro). The total i n t e n s i t y ing over the source, as

observed

I (x) = f I0 (X) 4 cos 2 [ (TLXI/~)

= 2 ~ T0 CX) ~

is then given by integrat-

(x/R+X/R O) Xl] dX

+ 2 co~2 {~=:/RX}.; 10 :X) co~ {2=:X/RoX}dX.

So that the v i s i b i l i t y c o h e r e n c e is

-

of the fringes

,oCX,

/

and the degree

,oCX,

X

C42) of (43)

Thus the degree of c o h e r e n c e is given by the n o r m a l i s e d F o u r i e r t r a n s f o r m of the source i n t e n s i t y d i s t r i b u t i o n on a scale of 2Xl/Rol. This, in s i m p l i f i e d form, is the Z e r n i k e - v a n Cittart theorem. For a c i r c u l a r source of u n i f o r m i n t e n s i t y and d i a m e t e r a, for example, the degree of c o h e r e n c e takes the form Jl(Eau)/(~au) where u = xI/R~. This defines the "coherence patch". As a m e a s u r e of the width of this p a t c h we take the radius to the first zero, g i v i n g x I = I/~ where ~ = a/Ro, the angle s u b t e n d e d by the source at the points of interest. This gives a rough i n d i c a t i o n of the d i s t a n c e s b e t w e e n two points for which i n t e r f e r e n c e effects are observable. Conversely, if the i n t e r f e r e n c e fringe v i s i b i l i t y is m e a s u r e d as a f u n c t i o n of the s e p a r a t i o n of two slits, it is p o s s i b l e to deduce ~, the angle s u b t e n d e d by the source. This is the basis for the use of the M i c h e l s o n Stellar I n t e r f e r o m e t e r to m e a s u r e apparent d i a m e t e r s of stars. In an e n t i r e l y analogous fashion it is p o s s i b l e to define the l o n g i t u d i n a l or c h r o m a t i c c o h e r e n c e in terms of the v i s i b i l i t y of fringes p r o d u c e d in a M i c h e l s o n interf e r o m e t e r or any similar i n t e r f e r o m e t e r which takes r a d i a t i o n going in one direction, splits it into two paths of unequal length and recombines it to p r o d u c e i n t e n s i t y f l u c t u a t i o n s as a function of path length difference. For m o n o c h r o m a t i c radiation one gets fringes l(x)

=

cos

(~vx/c),

72

IMAGING AND DIFFRACTION

CH. 3

where v is the frequency, x is the path length difference and c is the velocity. For a small range of frequencies v - v0 + vI around an average frequency v0, the observed intensity is given by adding intensities for all frequencies as I (x) = ; I (vI) cos 2 (nvx/c) dv and the longitudinal coherence is given by the visibility, [ (; 1(v) cos(2Kvx/c)dv)2 +

(; I(v) sin(2Kvx/c)dv)2]l/2 / ] I(v) dv

or ]Jl(v)

e x p (2~l.v/c) dv ] . / ~ I (v) d v .

Thus for a spectral line which is gaussian in form of width v I = a, the longitudinal coherence function is a gaussian of width x = c/~a, which may be called the coherence length. In Chapter 4 we give estimates of the coherence width and coherence lengths commonly encountered in diffraction experiments with the usual radiations and sources. Problems i. A lens is used to image a coarse diffraction g r a t i n g composed of slits in an opaque screen. An aperture is inserted in the back-focal plane of the lens. Find what image will be produced if the aperture lets through (a) the central spot and the first diffracted spot on each side of it, (b) the central spot and the first diffracted spot on one side only, (c) the first diffracted spot on each side but not the central spot. 2. An object having a transmission function A + B cos(2Kx/a) is imaged by an ideal thin lens as defined in Section 3. (a) Find the variation of image intensity as a function of the position of the plane of observation. (b) Determine how the sequence of out-of-focus images will be affected by spherical aberration of the lens, limitation of the lens aperture, and by imaging in white light rather than monochromatic light.

3.7

PARTIAL COHERENCE

73

3. (a) M a k e a simple argument to show that if a lens is used to image a p e r f e c t l y incoherent source, the c o h e r e n c e in the image plane depends only on the aperture of the lens and not at all on the extent of the source (provided that the source is larger than some m i n i m u m size). (b) U s i n g the d e f i n i t i o n of a degree of c o h e r e n c e b e t w e e n two points as the v i s i b i l i t y of i n t e r f e r e n c e fringes r e s u l t i n g when small apertures in an opaque screen are p l a c e d at the two points, derive the result (a) m a t h e m a t i c a l l y . 4. A h i g h - c o n t r a s t (black and transparent) p h o t o g r a p h i c n e g a t i v e of a chess b o a r d (8 squares each way with a white surround) is i m a g e d in an optical system. The t r a n s p a r e n t parts of the n e g a t i v e have a u n i f o r m phase retardation. What is the form of the d i f f r a c t i o n p a t t e r n in the b a c k - f o c a l plane of the imaging lens? If a screen with a slit p a r a l l e l to one edge of the chess b o a r d is used to cut out all the d i f f r a c t i o n m a x i m a except those in a row t h r o u g h the central maximum, how will the image be affected? What image will result if the slit is r o t a t e d t h r o u g h 45~

(Note. The a m p l i t u d e along a line in a d i f f r a c t i o n p a t t e r n is the F o u r i e r t r a n s f o r m of the p r o j e c t i o n of the object t r a n s m i s s i o n function on to a p a r a l l e l line.)

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Section II

KINEMATICAL DIFFRACTION

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CHAPTER 4

Radiations and their scattering by matter 4.1.

X-rays 4.1.i.

X - r a y sources

The X-ray radiation normally used in diffraction experiments is the relatively sharp lines of the characteristic Ks radiation s p e c t r u m emitted when an electron b e a m of sufficient energy strikes the metallic anode of an X-ray tube. The wavelengths vary from 2.28 ~ for chromium to 0.71 for molybdenum, the most commonly used being the copper radiation of wavelength 1.54 A (actually the Ks! and K~2 doublet having wavelengths approximately 1.537 and 1.541 ~). The radiation from a typical X-ray tube contains, in addition to these strong maxima, one or more K~ lines of shorter wavelength, some weak long-wavelength lines of the L series, some weak lines from impurities or contamination of the X-ray tube target, and a continuous b a c k g r o u n d of white radiation starting abruptly at a short wavelength limit (for which the energy of the emitted X-ray, he~l, is equal to the energy of the electrons of the exciting electron beam, eE) passing through a m a x i m u m and then decreasing with increasing wavelength. Confusion of the diffraction patterns by unwanted radiation may be reduced by use of various m o n o c h r o m a t i n g devices, including absorbing filters, crystal monochromators and e n e r g y - d i s c r i m i n a t i n g detectors. The isolated Ks lines are rather broad compared with spectral lines in the visible region, having a relative width, AI/I, of the order of 10 -4 . This limits the chromatic coherence of the radiation so that the coherence length is of the order of I041 or approximately 1 ~m. The more serious complication for many purposes comes from having the two adjacent Ks! and K~2 lines. Recently increasing use has been made of the very intense synchrotron radiation emitted in a comparable wavelength range when electrons of sufficient energy (in the GeV range) are a c c e l e r a t e d by the bending magnets of the synchrotron or by special "wigglers" inserted in the b e a m path. The synchrotron radiation is "white" radiation, i.e. it is emitted with a continuous range of wavelengths. The m a x i m u m intensity, for a circular electron beam orbit, is given for a wavelength ~ = 1.4~c, where the critical wavelength I c = (4~/3) R(E/mc2) -3 where R is the radius of

78

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CH.

4

c u r v a t u r e and E is the e l e c t r o n energy, e.g. m a x i m u m i n t e n s i t y is at I = 1.5 A for E = 5.5 GeV (less with a wiggler). The r a d i a t i o n is h i g h l y p o l a r i z e d and is p u l s e d with the o r b i t a l f r e q u e n c y of the e l e c t r o n bunches in the synchrotron. For r e c e n t l y - c o n s t r u c t e d s y n c h r o t r o n r a d i a t i o n sources, this X-ray i n t e n s i t y o b t a i n e d when a m o n o c h r o m a t o r is u s e d to select a small range of w a v e l e n g t h s is m u c h h i g h e r than that for the c h a r a c t e r i s t i c X - r a y line from a c o n v e n t i o n a l X - r a y tube h a v i n g a c o m p a r a b l e range of w a v e l e n g t h s (Coppens, 1992; M a r g a r i t o n d o , 1988). Special e x p e r i m e n t a l g e o m e t r i e s and e q u i p m e n t are b e i n g d e v e l o p e d to take a d v a n t a g e of the specific p r o p e r t i e s of the s y n c h r o t r o n r a d i a t i o n sources (Winick, 1980; Bachrach, 1992).

4.1.2.

Scattering

by electrons

The s c a t t e r i n g of X-rays by m a t t e r is u s u a l l y c o n s i d e r e d in terms of the s c a t t e r i n g by a single electron, l o c a l i z e d at the origin of c o o r d i n a t e s by some sort of r e s t o r i n g force and so h a v i n g some resonant frequency, v0, a s s o c i a t e d with it. For incident r a d i a t i o n

E = Eo e x p { 2 ~ i (vt - k . r) } h a v i n g a f r e q u e n c y v >> v0, the simple T h o m s o n t h e o r y gives a r a d i a t e d wave of a m p l i t u d e

scattering

Es = -Eo (e2/mc2)R -I s i n V . e x p { 2Ki (vt - k. r) }

(I)

where R is the d i s t a n c e to the point of o b s e r v a t i o n and ~; is the angle b e t w e e n the s c a t t e r e d b e a m and the d i r e c t i o n of the a c c e l e r a t i o n of the electron. If, as in Fig. 4.1 we c o n s i d e r the e l e c t r o n at the origin of o r t h o g o n a l axes and the d i r e c t i o n of p r o p a g a t i o n is the z-axis, then we c o n s i d e r the c o m p o n e n t s of the incident r a d i a t i o n p o l a r i z e d in the x - z and y - z planes. The s c a t t e r e d r a d i a t i o n is in, say, the y - z plane. Then for the c o m p o n e n t p o l a r i z e d in the y - z plane, the d i r e c t i o n of the e l e c t r o n a c c e l e r a t i o n is the y axis and ~ = 90 - ~, where is the s c a t t e r i n g angle. Then

IEI For the p o l a r i z a t i o n so that

IEI

= Eo ( e 2 / m c 2 ) R -1 c o s ~ . in the x - z plane, -- Eo (e21mc 2) I R .

~ = K/2 and cos~ = 1

4.1

X-RAY SOURCES

79

Y ,#

x

AAAA

z

Fig. 4.1. Coordinate system for discussion of scattering of X-rays by an electron. For u n p o l a r i z e d incident r a d i a t i o n i n t e n s i t y is the sum of the intensities p o l a r i z a t i o n s , i.e.

the total s c a t t e r e d for the two

I = I0 (e2/mc 2) 2R-2 (i + cos2~)/2.

(2)

The last t e r m here is the p o l a r i z a t i o n factor. It is c u s t o m a r y to treat the s c a t t e r i n g as for a scalar wave function, omit the p o l a r i z a t i o n factor until the final r e c k o n i n g of i n t e n s i t i e s and take (e2/mc2)R -I as 1 "electron unit" of scattering. The factor m -I in (2) ensures that for s c a t t e r i n g from atoms the s c a t t e r i n g from the nucleus can be ignored. We c o n s i d e r the s c a t t e r i n g from the cloud of e l e c t r o n s a r o u n d the nucleus and use the nucleus only as an origin of coordinates. Then for each e l e c t r o n we may define a d i s t r i b u t i o n function or e l e c t r o n d e n s i t y function pn(r) which gives the p r o b a b i l i t y that the e l e c t r o n s h o u l d be c o n t a i n e d in unit v o l u m e at the p o s i t i o n d e f i n e d by r. As we will see later, in C h a p t e r 7, the p u r e l y elastic s c a t t e r i n g is given by such a t i m e - a v e r a g e d s c a t t e r i n g function. Hence, from e q u a t i o n (1.19) and (1.21), using pn(r) i n s t e a d of -~(r) for the s c a t t e r i n g strength and e x p r e s s i n g the result in e l e c t r o n units in order to e l i m i n a t e the constants, we find the s c a t t e r i n g a m p l i t u d e for the nth e l e c t r o n is

fn(q)

or,

putting

P

= ]pn(r)

exp{-2~iq.r}dr,

(3)

q = g,

fn (u) = ~ Pn (r) exp { 2~ig

9 r } dr.

(4)

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RADIATIONS AND THEIR SCATTERING BY MATTER

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4

The total scattering from the electron is given by (i) and is equal to one electron unit. The difference between the elastic scattering, (4), and the total scattering is then the inelastic scattering, or Compton scattering, corresponding to the case that the X-ray photon collides with the electron, with loss of energy and change of momentum calculated in the usual way. The intensity of the inelastic scattering is then l'

,,~

=

1 -

If.I 2

4.1.3. Scattering by atoms For all the electrons associated with an atom, the elastic scattering is given by scattering from the time averaged electron density function p(r)

= ~

pn(r),

n

so that

f(u) = ~/.(u) n

and the so-called "atomic scattering factor" is f(u)

= ~p(r)

(5)

exp{2Ki,'r}dr

or, if spherical sy~uaetry of the atom is assumed, always be for free atoms,

as it may

f(0) = [ 4~2p (r) sin (2~ur) / (2~ur) dr, 0

(6)

where 0 is half the scattering angle ~. For inelastic scattering, the scattering from the various electrons is incoherent, so we add intensities to give /inel. = ~ a

{i - Ifn[2}

= Z - ~

If.l2,

(7)

n

where Z is the atomic number. Thus the calculation of the Compton scattering is in general more complicated than for elastic scattering since the distribution functions for all electrons and their Fourier transforms must be calculated separately. The elastic atomic scattering factors, calculated from various approximations for the electron density function, are listed in the International Tables for Crystallography, Vol. C (Wilson, 1992).

4.1

X-RAY SOURCES

81

4.1.4. Dispersion corrections The a s s u m p t i o n which forms the basis for e q u a t i o n (I) and all the s u b s e q u e n t derivation, is that the f r e q u e n c y of the incident r a d i a t i o n is much g r e a t e r than the f r e q u e n c y c o r r e s p o n d i n g to any e n e r g y for e x c i t a t i o n of the atom. This is c l e a r l y not the case if, as f r e q u e n t l y happens, the a t o m has an a b s o r p t i o n edge not far from the incident X - r a y frequency. In the p r e s e n c e of absorption, the r e f r a c t i v e index and also the atomic s c a t t e r i n g factor become complex. An i m a g i n a r y part and a small real part are a d d e d to f(u) as d e f i n e d by (5), so we write

f = fo + f '

+ if".

(8)

The v a r i a t i o n of f' and ~' with w a v e l e n g t h are roughly as i n d i c a t e d in Fig. 4.2 with d i s c o n t i n u i t i e s at IK, the w a v e l e n g t h of the K a b s o r p t i o n edge for the atom. Since f' and f" arise from the e x c i t a t i o n of e l e c t r o n s from the inner e l e c t r o n shells, they m a y be c o n s i d e r e d as F o u r i e r t r a n s f o r m s of h i g h l y l o c a l i z e d s c a t t e r i n g d i s t r i b u t i o n s and so d e c r e a s e with s c a t t e r i n g angle much more slowly than f0(u) which is given by F o u r i e r t r a n s f o r m of the total d i s t r i b u t i o n of e l e c t r o n s of the atom.

5 A

-5

Fig. 4.2. The variation of the magnitude of the real and imaginary "anomalous scattering" components f' and f" of the atom scattering factor for X-rays as a function of wavelength. The K absorption edge of the atoms is at l K.

4.2.

Electrons 4.2.1.

Sources of electrons

The e l e c t r o n s to be c o n s i d e r e d for e l e c t r o n d i f f r a c t i o n and m i c r o s c o p y have energies m o s t l y in the range I00 to 400 keV a l t h o u g h with the advent of high v o l t a g e e l e c t r o n m i c r o s c o p e s this range needs to be e x t e n d e d to 1 M e V or more. For e l e c t r o n s in the range 10-200 eV used in Low E n e r g y

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Electron D i f f r a c t i o n (LEED) the interactions with m a t t e r are s u f f i c i e n t l y different to require a separate treatment (see Chapter 19). The wavelength of the electron b e a m including the relativistic correction is = h/my = ~0(I + eE/(2m0 C2))-1/2, = 12.2643/(E0

(9)

+ 0.97845x10-6E02) I/2

where E0 is the accelerating voltage in volts and I is in angstroms (A), and 10 is the non-relativistic wavelength. This is 0.0548 A for 50 keV, 0.0370 for I00 keV and 0.00867 for 1 MeV electrons. The electron b e a m is p r o d u c e d in an electron gun in which the effective source size has a diameter of about I0 ~m for a normal "hair-pin" filament, a p p r o x i m a t e l y 1-2 ~m for heated pointed filaments and as little as 20-50 ~ for field emission tips. In combination with electron lenses these sources can give well collimated beams. A divergence of 10 -3 or 10 -4 fads. is used for most purposes but a divergence of 10 -6 or less can be achieved fairly readily. In the i00 keV range, the voltage supplies may have a stability of better than 10 -5 giving sufficient chromatic coherence for most experiments, with a coherence length of the order of 1 ~m or better, often limited only by the thermal energy spread of the electron source.

4.2.2. Atom scattering amplitudes TO an approximation which is sufficiently accurate for all calculations of elastic scattering of electrons, the p r o p e r t y of matter which is of interest is the p o t e n t i a l d i s t r i b u t i o n ~(r). The potential distribution is related to the electron density by Poisson's equation

V29(r)

=

le_/l{pn(r)

Eo

- pe(r) },

(I0)

where Pn is the charge density due to the atomic nuclei and Pe is that due to the electrons. The atom scattering amplitude for electrons is then a p p r o p r i a t e l y d e f i n e d as the Fourier t r a n s f o r m of the potential distribution for the atom, ~(r) m e a s u r e d in volts; re(U)

= ;~(r)

exp{2ziu.r}dr.

(11)

4.2

ELECTRONS

83

The f values are then defined as properties of the atoms, independent of any assumption as to the scattering process or the theoretical approximation used to describe it (Dawson et al. [1974]) . The relationship of this scattering factor to that for Xrays, fx, is d e r ived by inserting the inverse Fourier transforms in Poisson's equation (i0);

V2[Ife(u) exp{-2~iu.r}du]

ffx(u)

= le_/l go

exp{-2~iu.r}du

- le._.[;lZ go

exp{-2=iu.r}d,,

where the final integral is a delta-function of weight Z, the atomic number, due to the positive charge on the nucleus. Writing the left-hand side as (-2~i lul ) 2re (u) exp { -2~i, we equate integrands

9 r} du

to obtain the Mott formula

fe (u) -- lel (Z-fx(u))/4K2Eo u2.

(12)

we note that for scattering at large angles for which fx is small, fe is approximately proportional to Z and decreases with angle as (sin2O)/•2. For (sinO)/~ tending to zero, since for neutral atoms fx tends to Z, the value of fe becomes indeterminate. However the limiting value is obtained from (ii) as fe(0)

P

= J~(r) dr.

(13)

By historical accident, the atomic scattering factor for electrons has been defined as the quantity occurring in the first Born approximation for the theory of the scattering of electrons by atoms, equation (1.21); this gives fVB (U) =

(2nmelh 2) fe (u) =

(all) fe (u) ,

where G is the interaction constant to be defined below. Since this quantity is not a property of the atom itself and since the first Born approximation has a very restricted range of validity for electron scattering, p a r t i c u l a r l y from solids, the use of this basis for a definition is inappropriate. Its use has led to considerable confusion in the literature. However it is fFB(U) measured in A for which tables of values are given, for example in the International Tables for Crystallography, Vol. C (Wilson, 1992).

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4.2.3. Phase object approximation Since a p p r e c i a b l e s c a t t e r i n g takes place t h r o u g h r e l a t i v e l y small angles only, we m a y use the F r a u n h o f e r a p p r o x i m a t i o n of e q u a t i o n (1.37) i n s t e a d of the Born a p p r o x i m a t i o n (1.21) to d e s c r i b e s c a t t e r i n g by atoms. R e g a r d i n g the atom as a q u a s i - t w o - d i m e n s i o n a l object, we d e r i v e the t r a n s m i s s i o n f u n c t i o n qe(X,Y)F r o m (1.6) the r e f r a c t i v e index for the e l e c t r o n wave is 1 + ~(r)/2E. Then the phase d i f f e r e n c e for the wave p a s s i n g t h r o u g h the p o t e n t i a l field ~(r) in the z direction, r e l a t i v e to the wave in v a c u u m is (~/kE) f ~(r)dz. P u t t i n g the p r o j e c t i o n e x p r e s s e d by the integral e q u a l to ~(x,y) and ~ = ~/kE, we have (14)

qe(x,Y) = exp{-io~(x,y) }, and the d i f f r a c t i o n

amplitude

V(U,v)

= 5[exp{-io~(x,y)

With the r e l a t i v i s t i c

0 =

is

( 2 ~ / E ; L ) { 1+ (1-132) 1/2 } - 1

correction, =

(15)

}]. we write

(2rr,moe~,/h2) (1+h2/m02c2~2)

1/2 (16)

where ~ = v/C. This r e l a t i v i s t i c i n t e r a c t i o n factor (; is seen to tend to a constant value as E increases, the limit c o r r e s p o n d i n g to the n o n - r e l a t i v i s t i c value for ~ = 0.02426, the C o m p t o n wavelength, or E = 212 keV. The r e l a t i v i s t i c c o r r e c t i o n for the wavelength, on the other hand, is seen from (9) to make the d e c r e a s e of I with E more rapid. A form very close to (15), but somewhat more a c c u r a t e at h i g h e r angles, is known in n u c l e a r s c a t t e r i n g theory as the "Moli~re h i g h - e n e r g y a p p r o x i m a t i o n " . It is d e r i v e d from the g e n e r a l p a r t i a l wave t h e o r y of s c a t t e r i n g from a central force field by m a k i n g a small angle a p p r o x i m a t i o n (see, for example, Wu and Ohmura [1962]) . It has been shown by Doyle [1969] by d e t a i l e d c a l c u l a t i o n that this a p p r o x i m a t i o n and (15) are g o o d for most atoms over the range of s c a t t e r i n g angles n o r m a l l y u s e d in d i f f r a c t i o n e x p e r i m e n t s with solids. If o~(x.y) is small so that the e x p o n e n t i a l in (15) m a y be w r i t t e n e x p { - i ~ ( x . y ) } -- 1 - i~(x,y), (17) then the s c a t t e r i n g

is given by

~(u,v)

= 5(u,v)

- i6~(U,V) .

(18)

4.2

ELECTRONS

85

The delta function represents the transmitted, undiffracted b e a m and ~(,,v) is equal to a planar section of re(u) d e f i n e d by (II) .

4.2.4. Failure of First Born Approximation Numerical evaluation shows that, except for very light atoms, the assumption that ~(x,y) is much smaller than unity is not justified. Then the First Born Approximation is no longer adequate. The effect of this failure may be estimated by considering the second Born Approximation (Schomaker and Glauber [1952], Glauber and Schomaker [1953]) or by invoking the p a r t i a l - w a v e scattering theory (Hoerni and Ibers [1953]). However, the essential points are seen more immediately and g r a p h i c a l l y by use of (15), which may be written (U.V)

=

S[1

+

{cos6 9(x,y)

= 5(u,v) + 3 [ c o s ~ 9

-

1}]

-

iS{sin~

9(x,y)

- i] - i 3 [ s i n ~ 9].

or, to make the analogy with

}

(19)

(I0) closer

~t(,,v) = ~(,.v) - i [ ~ { s i n ~ }

+ i~{cos~-

(20)

I}],

and the atomic scattering factor /e(U.V) is the complex function in the square brackets, divided by (;. If ~ is not too large it may be useful to expand the sine and cosine terms and take the Fourier transforms so that we get a real part of /e(U.V) equal to 9 (U,V)

-

(~2/3l){~(U,V)

,

~(U,V)

,

O(U,v)}

+

(21)

...,

and an imaginary part equal to ( ( Y / 2 ! ) {~)(U,V)

* dO(U.V) } -

((;314!)

{~,~,~,~}

+

...,

(22)

Thus the real part is equal to ~)(u,v), the planar section of /e(U) given by (II), plus higher order terms. The imaginary term is given approximately by the two-dimensional s e l f - c o n v o l u t i o n of fe(U,v) and so increases more rapidly than fe with atomic number and falls off more slowly than the real part with [,] or (sin0)/l. If the complex atomic scattering factor is written f(") =

[fe(") [ exp{i~(u) },

(23)

then the phase angle ~ increases with l, with atomic number and with scattering angle. For the uranium atom, for example, = 0.29 radians for ( s i n O ) / ~ = 0 a n d ~ = 2.4 rad for (sin0)/~ = 1.15 A -I when E = 39.5 keV.

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The effects of the failure of the first Born a p p r o x i m a t i o n for electron scattering from atoms were first recognized as such in the explanation by Schomaker and Glauber [1952] of the intensities of diffraction from gaseous molecules of UF6. Tables giving the values of the amplitudes and phases of the complex atomic scattering factors, used in the calculation of intensities for gas diffraction have been p r e p a r e d by Ross, Fink & Hilderbrandt (International Tables for Crystallography, Vol. C (Wilson, 1992). It was shown later by Gjznnes [1964] and Bunyan [1963] that it may not be an adequate approximation for some gas molecules to merely replace the first Born approximation atom s c a t t e r i n g factors by the complex scattering factors (23). In terms of the phase object approximation, (14), it is seen that if ~(x, 7) is the p r o j e c t e d potential distribution for a molecule, the values of the real and imaginary components, (21) and (22), will depend on whether or not the atoms overlap in projection. If two atoms overlap, the contributions to ~(u,v) will be m u l t i p l i e d by 2, but the contributions to the second and third order terms will be m u l t i p l i e d by 4 and 8. Bartell [1975] has shown that this effect can be described by an extension of Glauber's theory which provides a convenient basis for calculations. The effects of the o v e r l a p p i n g of atoms in p r o j e c t i o n is much more important in solids. The "pseudo-kinematical" theory p r o p o s e d by Hoerni [1956], in which complex atom scattering factors replace the real ones in kinematical intensity formulas, has a very limited range of validity. It is appropriate only for "solids" consisting of single layers of atoms p e r p e n d i c u l a r to the incident beam. Coherent multiple scattering, or "dynamical scattering", must be taken into account for most experiments by use of the special theoretical treatments to be described in later chapters. 4.2.5.

"Absorption"

effects

As in the case of X-ray scattering, the presence of absorption gives the effect of a complex refractive index and so a complex scattering potential for electrons. In the simple formulation of (14), absorption modifies the transmission function of an atom to give q(x,y)

= exp{-io9(x,y)

scattering

~[exp{-~(x,y) }] =

6(u,v)

= 8(u,v) - M(u,v)

+

(24)

+ i~(x,y)] }.

= exp{-i[~(x,y)

Then the atomic is c o n v o l u t e d by

- gCx, y) }

factor

for elastic

- ~[i

- exp{-~(x,y) }]

89

, M(u,v)}

scattering

+

...

(25)

4.2

ELECTRONS

87

where M(u, v) is the Fourier t r a n s f o r m of ~(x,y). Then, in place of the b r a c k e t t e d part of (20), we obtain

fe(U,V) = S { s i n ~ } -

M(u,v),5{sin~9}

+ iS{cos~-

i}

- iM(u,v)

- iM(u,v)*S{cos~-l}

+ ...,

(26)

It is to be noted that the effect of absorption in rendering the atomic scattering factor complex is essentially different from the effect of the failure of the singlescattering a p p r o x i m a t i o n since for the latter the object remains a pure phase object and no energy is lost. The absorption function ~(x,y), which may be considered as the p r o j e c t i o n of a t h r e e - d i m e n s i o n a l function ~(r), arises from any scattering process for which the scattering is e x p e r i m e n t a l l y d i s t i n g u i s h e d from and not interacting coherently with the scattering which is of interest. For isolated single atoms, the only appreciable contribution to the absorption originates from the excitation of the atomic electrons. Electrons from the incident b e a m which have undergone an inelastic scattering process involving this excitation have lost energy of the order of i0 eV and may be separated from the elastically scattered electrons by use of an energy analyser. Hence the inelastic scattering process may give the effect of an absorption function for elastic scattering. For X-rays the main contribution to the absorption comes from the excitation of electrons from the inner shells of atoms and so, except in the immediate vicinity of the absorption edge, shows very little dependence on the association of atoms in molecules, liquids or solids. For electrons however, the most important inelastic scattering processes involve the outer electron shells and energy losses in the range 0 to 50 eV. The absorption coefficients are thus strongly dependent on the state of bonding or ionization of the atoms. For solids the most important contributions come from the e x c i t a t i o n of collective electron oscillations (plasmons), the production of excited states of the crystal electrons and the excitation of lattice vibrations (phonons). Again, for X-rays, the absorption processes subtract most of the energy from the incident photons involved so that they no longer contribute to m e a s u r e d intensities. This is not the case for electrons. The energy losses of the incident electrons are frequently so small that they can not be d e t e c t e d except by the use of special devices, so that often no d i s t i n c t i o n is made e x p e r i m e n t a l l y between elastic and inelastic scattering. Furthermore, if only one sort of elastic scattering is considered, e.g. only the sharp Bragg scattering, then other elastic scattering as well as the inelastic scattering may be excluded from the measurement and will then contribute to the effective absorption coefficient.

88

RADIATIONS

AND THEIR SCATTERING

BY M A T T E R

CH.

4

Hence the absorption coefficients for electron scattering is strongly dependent not only on the state of a g g r e g a t i o n of the atoms but also on the type of m e a s u r e m e n t made and the equipment used in the measurement. We therefore defer d e t a i l e d discussion of the absorption coefficients for electron scattering until Chapter 12, after we have considered the elastic scattering of electrons from solids in more detail. At this stage we remark only that the imaginary part of the effective complex potential, ~(x,y) in (24) is usually smaller than the real part ~(x,y) by a factor of from 5 to 50, and its effect may often be treated as a p e r t u r b a t i o n on the purely elastic scattering case. 4.3.

Neutrons

4.3.1. Atomic scattering factors The neutrons of interest for diffraction experiments are usually the thermal neutrons which are in thermal e q u i l i b r i u m with the atoms in a nuclear reactor and so have an average energy of about 0.025 eV for which the corresponding w a v e l e n g t h is about 1.5 A. However the spread of energies or w a v e l e n g t h s in the b e a m of neutrons obtained from a reactor is quite b r o a d and for d i f f r a c t i o n experiments a narrow range of w a v e l e n g t h s is usually selected out by use of a crystal m o n o c h r o m a t o r or, especially for long wavelengths, by a timeof-flight chopper device which selects a range of neutron velocities. Time-of-flight energy (and wavelength) analysis may also be used with the p u l s e d spallation sources in which fast neutrons are g e n e r a t e d by b o m b a r d i n g heavy metal targets with high-energy proton beams from an accelerator. Small blocks of m o d e r a t o r material near the target serve to slow the neutrons to thermal velocities. For recent accounts of experimental arrangements for neutron diffraction see Craven [1987] and Majkrzak and W o o d [1992]. Neutrons are not appreciably scattered by electrons. Their main interaction is with the nucleus. We quote here the results derived initially in the context of nuclear physics and retaining something of that flavor. Since the nucleus of an atom is very much smaller than the w a v e l e n g t h of the thermal neutron, the atomic scattering factor for neutrons is isotropic, independent of scattering angle and represented by a s i n g l e - v a l u e d "scattering length", b. Contributions to b include a "potential" scattering t e r m ~, c o r r e s p o n d i n g to scattering from a hard sphere of a p p r o p r i a t e radius and "resonance" scattering terms arising from reactions of the neutron with the nucleus. The B r e i t - W i g n e r formula for scattering from a zero-spin, isolated nucleus gives 1 (r) b = ~ + ~F, /k (27)

+roC,, '

4.3

NEUTRONS X-rays

5

e:o

89 X-rays

" O)/~:o.sk

j

f x I()-12 cn

1

. . . . . . . . . . .

Potential scattering I _

_

O.

I

t.wt.

Fig. 4.3. Variation of the atomic scattering factor for neutrons, b, and the atomic scattering factor for X-rays (at particular values of (sinS)/~) with atomic weight (after Bacon [1975]). where E r is the e n e r g y for resonance, Fn (r) is the width of the r e s o n a n c e for r e - e m i s s i o n of the n e u t r o n with its original e n e r g y and Fa (r) is the width of the resonance for absorption. More u s u a l l y the s c a t t e r i n g is d e s c r i b e d in terms of a s c a t t e r i n g cross-section, if= 4Kb 2 .

(28)

It is seen from (27) that, d e p e n d i n g on the interactions with the nucleus, b m a y be positive or negative, real or complex. In p r a c t i c e ~ is r e l a t i v e l y small, and being p r o p o r t i o n a l to the nuclear radius, increases as the onethird power of the atomic number. The imaginary part is a p p r e c i a b l e for only a few isotopes such as those of boron and c a d m i u m which are strong absorbers of thermal neutrons. For a n u m b e r of isotopes, including the important cases of IH, 7Li, 48Ti, 51V, 53Mn, and 62Ni, b is negative. Tables of the values of b and of s c a t t e r i n g c r o s s - s e c t i o n s are given, for example by Bacon [1975] and the I n t e r n a t i o n a l Tables for Crystallography, Vol. C (Wilson, 1992). The v a r i a t i o n with atomic n u m b e r is c o m p a r e d with that for X-rays in Fig. 4.3.

4.3.2. Nuclear spin scattering For n e u t r o n s c a t t e r i n g the nuclear spin is also of importance. For a nucleus with spin, l, two s c a t t e r i n g lengths b+ and b_ must be d e f i n e d c o r r e s p o n d i n g to the formation of c o m p o u n d nuclei of spin ; + 1/2 and ] - 1/2 with relative p r o b a b i l i t i e s w+ and w_ respectively. For an a s s e m b l y of atoms, or for the time average of the s c a t t e r i n g of neutrons from a single atom, the two states c o n t r i b u t e at random, subject to the p r o b a b i l i t i e s w+ and w_. Then, following the

90

RADIATIONS

AND T H E I R S C A T T E R I N G

BY M A T T E R

CH.

4

concepts to be d e v e l o p e d in C h a p t e r 7, the s c a t t e r i n g c o n s i s t s of "coherent" s c a t t e r i n g from the average of the two states with a c r o s s - s e c t i o n ~c = 4~(w+b+ + w_b_) 2, and " i n c o h e r e n t " scattering, given by the d i f f e r e n c e (y and the average cross-section, as (Yinc = (y - (Yc

~-

(29) between

4~ (w+b+ 2 + w_b_ 2) - 4~ (w+b+ + w_b_) 2 g

(30)

The coherent s c a t t e r i n g c r o s s - s e c t i o n represents the s c a t t e r i n g g i v i n g rise to d i f f r a c t i o n effects. The i n c o h e r e n t scattering cross-section contributes a uniform background with no d i f f r a c t i o n effects. This is p a r t i c u l a r l y i m p o r t a n t for h y d r o g e n for which [ = 1/2, b+ = 1.04 X 10-12cm, b_ = -4.7 x 10-12cm, so that ~ = 81 X i0-24cm 2 and ~c = 2 X 10 -24 cm 2 . Then most of the s c a t t e r i n g is " i n c o h e r e n t " background. In what follows, however, we will ignore this b a c k g r o u n d s c a t t e r i n g and c o n s i d e r only the b and ~c values a p p r o p r i a t e for the "coherent" scattering. In the above, we have used the terms "coherent" and " i n c o h e r e n t " in the incorrect m a n n e r which, unfortunately, has b e c o m e conventional. It must be e m p h a s i z e d that the results (29) and (30) are d ~ r i v e d by c o n s i d e r i n g the c o h e r e n t s c a t t e r i n g of r a d i a t i o n from a d i s o r d e r e d array of spins. C o h e r e n c e refers to the p r o p e r t i e s of the incident radiation. The order or d i s o r d e r of the s c a t t e r i n g m a t e r i a l is p r o p e r l y d e s c r i b e d in terms of c o r r e l a t i o n functions. Thus (29) refers to c o h e r e n t s c a t t e r i n g from the c o r r e l a t e d part of the s c a t t e r i n g m a t e r i a l distribution, and (30) refers to the coherent s c a t t e r i n g from the n o n - c o r r e l a t e d part of the scattering material distribution.

4.3.3.

Isotopic

disorder

For a s s e m b l i e s of atoms a further c o m p l i c a t i o n is that, for most elements the atoms present consist of a r a n d o m a r r a y of the various isotopes, present either with their n a t u r a l a b u n d a n c e or sometimes with a r t i f i c i a l e n r i c h m e n t of p a r t i c u l a r isotopes. Since n e u t r o n s c a t t e r i n g depends on the reactions of the neutrons with the nucleus, the b value m a y vary w i d e l y from one isotope to another. For example, b for 58Ni is 1.44 X 10-12cm, for 60Ni it is 0.30 X 10 -12 cm and for 62Ni it is -0.87 X 10-12cm, g i v i n g an average b for n a t u r a l l y o c c u r r i n g Ni of 1.03 X 10-12cm. F o l l o w i n g the c o n s i d e r a t i o n s of d i s o r d e r e d v a r i a t i o n s c a t t e r i n g factors given in C h a p t e r 7 we see that for d i f f r a c t i o n effects from the c o r r e l a t e d component of the

of

4.3

NEUTRONS

91

material (e.g. for giving Bragg diffraction maxima) the effective scattering factor is given by the average value

b = s

wnb,

(31)

n

where Wn is the relative abundance of the nth isotope. The difference between scattering from this average and the total scattering is the diffuse b a c k g r o u n d scattering which has intensity p r o p o r t i o n a l to /cliff

= ~ot

- Ibl 2

- X =-'s n

'=-

(32)

n

Again, the portions of the scattering represented by (31) and (32) are often, inappropriately, referred to as "coherent" and "incoherent".

4.3.4. Thermal and magnetic scattering Two further aspects of neutron scattering are of considerable importance for practical purposes of solid-state investigations and will be treated in context in later chapters. Firstly, because the energies of the incident neutrons are a p p r o x i m a t e l y the same as the thermal energies of vibration of the scattering atoms, the energy changes involved with the excitation of thermal lattice vibrations, or the creation or annihilation of phonons, are important and can be m e a s u r e d to provide information on phonon energies as well as momentums. Alternatively one can consider that the velocity of the thermal neutrons is so low that the diffraction process in matter is sensitive to the variation of atom positions in time so that the Fraunhofer diffraction pattern gives information concerning the relative positions of atoms in both space and time. Secondly, because of their spin, the neutrons can interact with u npaired electron spins as well as with nuclear spins. For the n e a r - r a n d o m spin arrays of paramagnetic material this gives rise to diffuse b a c k g r o u n d scattering, falling off with increasing scattering angle rather more rapidly than the X-ray scattering from an atom because the scattering is done by the electrons having unpaired spins, which are usually confined to the outer electron shells. When the electron spins are ordered in parallel or antiparallel array as in ferromagnetic and anti-ferromagnetic materials the spin scattering gives rise to appropriate diffraction maxima. Hence neutron diffraction has become a major tool for the study of the magnetic properties of materials.

92

RADIATIONS AND THEIR SCATTERING BY M A T T E R

CH.

Problems

I. Estimate the m a g n i t u d e of the correction to X-ray atomic scattering factors due to the inclusion of m u l t i p l e scattering terms (e.g. the second Born approximation). 2. Will Compton scattering give rise to an absorption effect for X-rays? If so, estimate its magnitude. 3. By use of the Mott formula consider the effect of ionization of an atom on its electron atomic scattering factor as the scattering angle tends to zero. What do the infinities imply? For atoms in solids, the formulas d e r i v e d for isolated atoms do not apply. Suggest how scattering from ions may be treated in this case. (Then see Doyle and Turner [1968].)

4

CHAPTER 5

Scattering from assemblies of atoms 5.1.

The

kinematical

approximation

The first Born approximation for scattering from a threedimensional d i stribution otherwise known as the "kinematical" or "single scattering" approximation, is given in equation (1.20). It is not limited in its application to the scattering from single atoms but may be applied to any collection of scattering matter. Normally we think in terms of assemblies of distinguishable atoms although for X-rays when ~(r) is replaced by the electron density distribution, p(r), the m o d i f i c a t i o n s of the electron distribution due to bonding may make it difficult to give a meaningful assignment of separate components of p(r) to separate atoms. For electrons when the ~(r) of (1.20) becomes the electrostatic potential distribution, ~(r), the assignment to atoms may be even more difficult especially when the scattering involves, in the usual theoretical approximation, excitation of a whole crystal from one state to another i.e. the transfer from one n o n - l o c a l i z e d wave function of the crystal electrons to another. However these limitations are important only for special considerations and will be treated separately when the need arises. We will p r o c e e d under the assumption that the electron density of a collection of atoms may be written

p(r) = ~ i

pi(r)

.

$(r-ri)

(i)

where pi(r) is the electron density associated with the atom centered at r-ri and is not necessarily assumed to be the same as for a free atom. For convenience we use the notation appropriate to X-ray diffraction with the understanding that exactly the same considerations apply for electron and neutron scattering within the limits of applicability of the s i n g l e - s c a t t e r i n g approximation. V a l i d i t y of the single-scattering approximation implies that the amplitude of the single scattered radiation is very small compared with the incident beam amplitude. Then the amplitude of the doubly- and m u l t i p l y - s c a t t e r e d radiation is very small and negligible compared with the singly scattered.

94

SCATTERING

FROM ASSEMBLIES

OF A T O M S

CH.

5

The amplitude of a scattered b e a m in a particular d i r e c t i o n depends to a very great extent on the p o s s i b i l i t y of cooperative scattering from an ordered array of atoms. If, as in the case of X-rays and neutrons, the interaction with atoms is so weak that, in a crystal, the diffracted energy can be c o n c e n t r a t e d into one or two sharply defined directions by three dimensional diffraction before the incident b e a m has lost much energy, then we may consider the p o s s i b i l i t y of multiple scattering of w e l l - d e f i n e d beams. A d i f f r a c t e d b e a m will be d i f f r a c t e d again if it passes through another crystal region which is set at the right angle for Bragg reflection. This condition is always ensured in a large perfect single crystal, but becomes less probable in the presence of crystal defects, grain boundaries and so on. For a perfect crystal and a strong crystalline reflection, multiple scattering becomes appreciable for X-rays for path lengths of the order of 1 ~m. For neutrons the necessary path lengths are several times greater. If the atoms are not s u f f i c i e n t l y ordered to give w e l l - d e f i n e d diffracted beams, the d i f f r a c t e d intensity in any direction is much less and multiple scattering effects are c o r r e s p o n d i n g l y less important. On the other hand, we have seen that for electrons the interaction with atoms is much stronger so that multiple scattering may be important in the scattering from a single heavy atom. W e l l - d e f i n e d d i f f r a c t e d beams are not g e n e r a t e d before the amplitude of scattering is appreciable. Hence m u l t i p l e scattering effects become important within a distance which is of the order of one or two hundred A for light atoms and less for heavy atoms, and are almost as strong for non-crystalline as for crystalline specimens. It is worthwhile for any radiation to draw the d i s t i n c t i o n between "coherent" and "incoherent" multiple scattering. Here the word "coherent" is used, in accordance with a common terminology, to refer not to the incident radiation but to the correlation of the atomic positions and so to the relationships of phases of scattered radiation. It would be more appropriate to refer to coherent multiple scattering from "correlated" and "un-correlated" arrays of atoms. At one extreme, the positions of the atoms in a perfect crystal are well correlated in that they may all be related to the sites of a periodic lattice. Then the relative phases of waves scattered by any atoms may be uniquely defined and the amplitudes of the waves may be added. The c o r r e l a t e d multiple scattering is then referred to as "dynamical" scattering. In the other extreme, the correlation of atomic positions does not extend over the distances required to generate appreciable scattered amplitude. The relative phases of waves scattered by strongly correlated groups of atoms varies at random from one group to another throughout the crystal. Then multiple scattering intensities are added

5.2

REAL AND R E C I P R O C A L

SPACE

95

incoherently. This case has been referred to as "multiple e l a s t i c scattering". O b v i o u s l y in practice any i n t e r m e d i a t e situation, or c o m b i n a t i o n of situations, may exist so that an adequate d e s c r i p t i o n of the s c a t t e r i n g may be very complicated. Fortunately, for m a n y specimens of interest, the s c a t t e r i n g m a y be d e s c r i b e d in terms of one of the two extreme cases, pure k i n e m a t i c a l or pure d y n a m i c a l diffraction, or else in terms of small p e r t u r b a t i o n s of one or the other. For other specimens the p h y s i c a l d i m e n s i o n s or state of c r y s t a l l i n i t y can often be m o d i f i e d to simplify the s c a t t e r i n g theory. 5.2.

Real

5.2.1.

and

reciprocal

Reciprocal

space

space d i s t r i b u t i o n

We have seen that k i n e m a t i c a l s c a t t e r i n g amplitudes may be e x p r e s s e d in terms of the Fourier t r a n s f o r m of a d i s t r i b u t i o n in real or direct space. In real space we consider a p o s i t i o n vector r with coordinates x,y,z. In reciprocal space we c o n s i d e r a p o s i t i o n v e c t o r u with c o o r d i n a t e s u.v,w. Then, in X-ray terminology, a d i s t r i b u t i o n p(r) in real space is related to a d i s t r i b u t i o n F(u) in reciprocal space by the Fourier transform, F(u)

or

= ~ p (r) exp{ 2~iu. r}dr,

F(u.v.w) = ;JJ p(x,y,z)

exp{2~i(ux+vy+wz)}dxdydz.

(2) (3)

If an e l e c t r o n d e n s i t y distribution, p(r), is c o n s i d e r e d to be the sum of the distributions pi(r) a t t r i b u t e d to individual atoms as in equation (i) and if (U) - ; Pi (r) exp { 2Kiu. r} dr, then F o u r i e r

transform F(u)

of = ~

l

(4)

(I) gives ~(u)

e x p { 2 K i u . r i}.

(5)

If it is further a s s u m e d that the electron d e n s i t y pi(r) is s u f f i c i e n t l y close to the e l e c t r o n d e n s i t y of an isolated free atom, the value of F(u) can be found from the tables of atomic s c a t t e r i n g factors given, for example, in the International Tables for Crystallography, Vol. C (Wilson, 1992).

96

SCATTERING FROM ASSEMBLIED OF ATOMS

CH.

5

The form of F(u) c o r r e s p o n d i n g to various forms of the function p(r) may be seen by an extension to t h r e e - d i m e n s i o n s of the relationships and examples of Fourier transforms given in Chapter 2. For example, by extension of (2.38) and (2.42), if p(r) = 1 inside a rectangular box of dimensions a,b,c and zero outside, then F(u)

sin(r~au) sin(nbv) sin(=cw)

abc

=

rr,au

rr,bv

~

(6)

. =cw

This is the three dimensional analogue of (2.42), with a central peak of height abc and falling off through diminishing oscillations along each axis. In this way we define a "shape function" and the "shape transform" often used to describe rectangular volumes of material.

5.2.2. The reciprocal lattice By extension to three dimensions of our results for a d i f f r a c t i o n grating, we see that, corresponding to (2.51) we have that, if oo

p(r)

= s

s n

m

s

8 ( x - na,

y - rob, z - p c ) ,

p

then

s s s h

k

(h/a), v -

(k/b), w -

(l/c) ) .

(7)

l

Thus for a periodic lattice in real space with p e r i o d i c i t i e s a,b,c the corresponding distribution in reciprocal space is a lattice of points with periodicities a-l,b-l,c-I. This is the "reciprocal lattice" for the special case of rectangular axes. C o r r e s p o n d i n g to (2.50) and (2.51) we have the relationship, that, if p(r) is a periodic function e x p r e s s e d by the Fourier series p(r)

= s

s k

F Cu) = E h

E k

then

Fhkl exp{-2gi((hxla)

s

h

+ (kylb)+

(lzlc))},

(8)

(I/c)),

(9)

l

E Fhkl 8 ( u l

(h/a), v - (k/b), w -

which represents a reciprocal lattice with each reciprocal lattice point weighted by the Fourier coefficient.

5.2

REAL AND RECIPROCAL SPACE

97

For a periodic function in real space chopped off by rectangular shape-function having dimensions A,B,C, we have from (2.59), (6) and (9),

F(u) = ~ h I

~ k I

~_~ FhkI ~ ( u l

(h/a), v -

(k/b),

w-

(l/c) ) ,

I

. ABC sin(rtAu) sin(rd3v) sin(~;Cw) xAu xBv ~Cw

(10)

which implies that each point of the weighted reciprocal lattice, (9), is spread out into a continuous distribution given by a shape transform of the form (6). 5.2.3.

F r i e d e l 's l a w a n d the p h a s e p r o b l e m

There are several relations of importance which follow from the definitions of Chapter 2. P r o v i d e d that no absorption effect is important, p(r) may be assumed to be a real function. Then

F(-u)

= Jp(r) =

Jp(r)

exp{2~i(-,).r}dr exp{-2xiu.r}dr

= F* (u),

so that

(11)

IF(-u)

Then it follows from

12 =

I F ( u ) 12.

(2.22) that

I~p(-r)

12 =

ISp(r)12

.

(12)

Since the intensity of diffraction is proportional to IF(u) 12, this is a convenient statement of Friedel's law which implies that inversion of a crystal through a center of symmetry does not change the diffraction intensities in a kinematical approximation. The inversion of the relationship (2) gives

p (r) = J F (u)

exp { -2~i,- r} du.

(13)

If the diffraction amplitudes ~(l,m,n) could be measured so that F(u) could be derived, then the distribution p(r) could be d e d u c e d by numerical evaluation of this integral. However the measurement of wave amplitudes is not possible for the radiations we consider. Only the intensities given by ~n~* can be recorded. Thus information on relative phases of

98

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

5

the d i f f r a c t e d beams is lost and the function p(r) cannot be deduced directly. For most experimental situations complete knowledge of p(r) is neither necessary nor desirable since it would involve knowledge of the relative positions and bonding of all the atoms of the sample and represent a quantity of data which would be difficult to handle. We therefore proceed to investigate the type of information which may be d e r i v e d d i r e c t l y from the observable intensities.

5.3.

The

generalized

Patterson

function

The observable intensity of d i f f r a c t e d radiation is related in a known way to ~r~* and so to IF(u) [2. This function, IF(u) [2, sometimes written J(u), is a function in reciprocal space. It is a real positive function and, from (Ii), is c e n t r o - s y m m e t r i c about the reciprocal space origin. The form of the function for particular cases is readily derived from the relations for F(u) given, for example, in equations (6) to (9) By inverse Fourier t r a n s f o r m we see that the corresponding function in real space is 5 -1

{ IF(u)12}

-

P(r)

= p (r) = ~p(R)

.

p (-r) p(r

(14)

+ R)dR.

This function of real space, P(r), we call the g e n e r a l i z e d Patterson function to distinguish it from the Patterson function employed for crystal structure analysis and usually referring to periodic structures only. When no chance of confusion exists, we will call it simply the "Patterson". It is similar to the Q - f u n c t i o n of Hosemann and Baggchi [1962]. It may also be called a "self-correlation" function, as we shall see. It is the real space function directly derivable from the observable intensities. The greatest value of P(r) occurs at the origin, r = 0, since P(O) = Ip2(R) dR = IIF(u)12 du, (15) from (2.66). It tends to have a relatively large value when r is a vector which connects points having high p(r) values. When p(r) represents the electron density for a collection of atoms, we write, as before,

p(r) = ~ l

pi(r)

, 5(r-

ri).

(16)

5.3

THE G E N E R A L I Z E D

PATTERSON

FUNCTION

99

Then

p(r) . p(-r) = ~

~

i

j

~

[pi(r)

pi(r) .

,

pj(-r)

pj(-r)]

.

,

$(r-

${r

-

ri) (ri

j

-

, rj)

$(r

}

+ rj)

(17)

Since Pi (r) is a p e a k e d function, roughly r e p r e s e n t e d by a gaussian, such that ~pi(r) dr = Zi, the convolution Pi * P], will represent a slightly b r o a d e r g a u s s i a n - l i k e peak of i n t e g r a t e d weight ZiZ]. This is p l a c e d at the end of the i n t e r a t o m i c v e c t o r r i - r] relative to the origin of P(r). There is an identical peak at r] - ri. Thus P(r) is made up of peaks c o r r e s p o n d i n g to all interatomic vectors present. The weight of each peak c o r r e s p o n d s to the sum of all products ZiZ ] for the pairs of atoms h a v i n g that interatomic vector. This is i l l u s t r a t e d for simple two- and three- and f o u r - a t o m objects in Fig. 5.1. The peak at the origin of P(r) c o r r e s p o n d s to all vectors b e t w e e n centers of atoms of length zero and so is

5 (r) , ~

[pi(r)

* Pi(-r)],

l

and has weight

~iZi2.

If there are two vectors of the same length and d i r e c t i o n in the object, as in Fig. 5.1(c), the c o r r e s p o n d i n g P a t t e r s o n peaks are twice as large. Thus the P a t t e r s o n function m a y be r e g a r d e d as a m a p p i n g of the w e i g h t e d p r o b a b i l i t y that two atoms are found a distance r apart, the w e i g h t i n g b e i n g a c c o r d i n g to the atomic numbers. Or we may say that P(r) gives the p r o b a b i l i t y that if any one atom is taken as origin, there will be another atom at a vector distance r from it. Thus it is, in essence, a c o r r e l a t i o n function g i v i n g the spatial c o r r e l a t i o n of e l e c t r o n densities. A n o t h e r way of looking at P(r) is i l l u s t r a t e d in Fig. 5.2, where we have redrawn the P a t t e r s o n of the t h r e e - a t o m structure of Fig 5.1(b) in two ways to emphasize that it may be r e g a r d e d as a s u p e r p o s i t i o n of "images" of p(r) or its inverse, p(-r). We may c o n s i d e r each of the atoms in turn to be p l a c e d at the origin and the resulting "images" added with a w e i g h t i n g factor given by the Z for the atom at the origin. A l t e r n a t i v e l y , we may consider the d i s t r i b u t i o n p(r) to be t r a n s l a t e d so that one of its atoms is placed, in turn, at each of the atom positions of p(-r), the points m a r k e d by P in the diagram.

i00

SCATTERING

FROM

ASSEMBLIES

_ r~.r2 . r l ~ r 2 ~ 2

OF ATOMS

r = r l - rz

0

Z1Z2

CH.

r~O

r = r~- rl

ZIZ2

ZIZ2

+ Z2Z2

2,3 (b)

1

-r3 r, - r 2

--1,2 ~2,2'+ 3,1

2,4

(c)

~1,,"~ 3,2

1,4 2,3

1,3

3,4"~ ~ . . . . ~ 3

_ 1

.~3

2,1"

2

5

2

3,1.

4,1 3,2

4,2

Fig. 5.1. Diagrams to suggest the form of the electron density distribution, p(r), and the Patterson function, P(r) for (a) two atoms, at positions defined by vectors r I and r2; (b) three atoms, and (c) four atoms forming a parallelogram.

P

P

P

Fig. 5.2. Representation of the Patterson function for three atoms, Fig. 5.1(b), as a superposition of images of the object or its inverse. For

systems

composed

of

a very

large

number,

N,

of

atoms, the Patterson function has N 2 - N + 1 peaks (since N peaks are superimposed at t h e o r i g i n ) , a n d so is t o o complicated to be interpreted e x c e p t in a s t a t i s t i c a l sense as a p r o b a b i l i t y - d e n s i t y or self-correlation function. We now consider a few representative c a s e s to i l l u s t r a t e t h e application of t h e s e c o n c e p t s .

5.4

5.4.

EXAMPLES

Examples 5.4.1.

of

OF C O R R E L A T I O N

correlation

Finite

volume

FUNCTIONS

I01

functions

limitations

A d i s t r i b u t i o n of atoms over a finite volume is often d e s c r i b e d in terms of a function r e p r e s e n t i n g an infinite d i s t r i b u t i o n p' (r) m u l t i p l i e d by a s h a p e - f u n c t i o n $(r) i.e. p(r)

In reciprocal

= p ' (r)

s(r).

space, F(u)

= F ' (u)

, S(u).

If s(r) has d i m e n s i o n s which are very large c o m p a r e d to atomic d i m e n s i o n s so that S(u) is a very n a r r o w function c o m p a r e d with the variations of F' (u) which are of interest, then it is a reasonable a p p r o x i m a t i o n to put I F ( u ) 12 =

IF' (u)12

,

IS(u)[2.

(18)

The most obvious error in this a s s u m p t i o n comes where F' (u) changes sign, when IF' (u) , S(u)12 has zero values but IF' (u)12 , IS(u) 12 does not. Also, IF' (,) , S(,)12 has a greater region of near-zero values, but the region a f f e c t e d has a p p r o x i m a t e l y the h a l f - w i d t h of S(u). F o u r i e r t r a n s f o r m of (18) gives the c o r r e s p o n d i n g a s s u m p t i o n for real space, P(r)

= P' (r) Is(r) . s(-r)],

(19)

i.e. the effect of spatial limitation of the sample is to m u l t i p l y the P a t t e r s o n for the infinite crystal (the P a t t e r s o n "per unit volume") by the s e l f - c o n v o l u t i o n of the shape transform. In one dimension, if s(x) is the slit function given by s (x) =

{

Then

if

Ixl<_Al2,

1

if

IxI>A/2.

x + A s(x) ,s(x)

which

0

is the F o u r i e r

if

-A < x < O,

- { A - x if O < x < A , 0 elsewhere,

transform

of

IS(u)12;

see Fig.

5.3.

102

SCATTERING

FROM ASSEMBLIES

OF A T O M S

CH.

5

Fig. 5.3. Diagram of a shape function $(x) and its Patterson function P(x), and the corresponding Fourier transforms. S i m i l a r l y in t h r e e d i m e n s i o n s , if the s a m p l e is a rectangular block limited by planes x = + A/2, y = + B/2, z = + C / 2 , w h e r e the d i m e n s i o n s A,B, C a r e m u c h g r e a t e r t h a n the d i m e n s i o n s of the g r o u p s of atoms of interest, the P a t t e r s o n per unit v o l u m e is m u l t i p l i e d by a f a c t o r A B C = V at the o r i g i n a n d d e c r e a s e s l i n e a r l y to zero in a d i s t a n c e A in the x d i r e c t i o n , B in the y d i r e c t i o n , C in the z d i r e c t i o n , c o r r e s p o n d i n g to the fact that no i n t e r a t o m i c v e c t o r can h a v e x, y, z c o m p o n e n t s g r e a t e r t h a n A,B, C. U s u a l l y , on the a s s u m p t i o n that A , B a n d C are v e r y m u c h g r e a t e r t h a n any i n t e r a t o m i c v e c t o r s of interest, this " s h a p e - c o n v o l u t i o n " f u n c t i o n is i g n o r e d or omitted. 5.4.2.

Finite

crystals

AS a s i m p l e model, we c o n s i d e r a c r y s t a l h a v i n g one a t o m in a r e c t a n g u l a r unit cell w h i c h has axes a,b,c. The c r y s t a l has d i m e n s i o n s A - Nla, B = N2b, C = N3c a n d so can be represented by

[Po (r)

(20)

~_j 8 (r - R n) ] s (r) , n

where

Rn

s(r)

= nla

is the

+ n2b

shape

+ n3C.

function

having

dimensions

A,B,C

and

5.4

EXAMPLES

'!~

OF C O R R E L A T I O N

.,

FUNCTIONS

A=N~a .

103

. ~!

l p(r) '~

A - N, a

P(r)

Fig. 5.4. Representation of a finite crystal as the product of a periodic object and a shape function, the corresponding Patterson function and its Fourier transform. Then, f o l l o w i n g the c o n s i d e r a t i o n s P a t t e r s o n f u n c t i o n m a y be written,

of 5.4.1,

the (21)

n

= Npo(r)

.

Po(-r)

m

.

[Z

n

8(r - Rn) ] [ , ( r )

.

,(-r)],

(22)

since the c o n v o l u t i o n of the two sets of N delta functions gives N times the set of all vectors b e t w e e n lattice points m u l t i p l i e d by the shape convolution, as s u g g e s t e d in the oned i m e n s i o n a l e x a m p l e of Fig. 5.4. Then the F o u r i e r t r a n s f o r m of (25) gives the c o r r e s p o n d ing d i s t r i b u t i o n in reciprocal space as

IF(u) 12 = Nlfo(u)12 s 8(u-(h/a), v-(k/b), w-(l/c) ),lS(u)12 h,k,l (23) which is the r e c i p r o c a l lattice h a v i n g spacings a -1, b -1, c-I with r e c i p r o c a l lattice points w e i g h t e d by the function If0(u) 12 , and with each reciprocal lattice point spread out by c o n v o l u t i o n with the function

104

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

IS (u) 12 = A 2 B 2 C 2 sin2(ru4u) sin2(rt/~v) sin2(nCw) (z4u) 2

(rcCw) 2

(~v) 2

5

(24) "

which

is the square of the function (6). This case represents the most p e r f e c t o r d e r i n g of atoms w i t h i n a finite volume with the c o r r e l a t i o n function of atomic p o s i t i o n s l i m i t e d only by the s h a p e - c o r r e l a t i o n . There are m a n y stages of o r d e r i n g i n t e r m e d i a t e b e t w e e n this and the case of an ideal gas r e p r e s e n t i n g the m i n i m u m of ordering. Several examples of i n t e r m e d i a t e degrees of order, p a r t i c u l a r l y those a p p r o x i m a t i n g to the order of the p e r f e c t crystal, will be c o n s i d e r e d in later chapters. 5.5.

Correlation

in

space

and

5.5. I. F o u r - d i m e n s i o n a l

time

Patterson

The concept of a P a t t e r s o n function, which is a c o r r e l a t i o n function for the d i s t r i b u t i o n of e l e c t r o n d e n s i t y in space, can be e x t e n d e d to include c o r r e l a t i o n s in time. In space we ask; what is the p r o b a b i l i t y that an a t o m will be s e p a r a t e d f r o m a given a t o m by a v e c t o r r? We m a y ask, similarly: if an a t o m is at the point r at time 0, what is the p r o b a b i l i t y that there will be an atom at that point (or at any other point) after a time t? The function which includes such c o r r e l a t i o n s is the f o u r - d i m e n s i o n a l P a t t e r s o n function

P ( r , t) - ~ p ( R , = p ( r , t)

T) p ( r + R, .

p (-r,

-t)

t + T) dRdT (25)

.

F o u r i e r t r a n s f o r m a t i o n gives a function in fourd i m e n s i o n a l r e c i p r o c a l space. The t r a n s f o r m with respect time, f o l l o w i n g e q u a t i o n (2.31), gives a function of f r e q u e n c y so we write

IF(u.v) 12 = ~;P(r, t) e x p { 2 ~ i ( u . r

+ vt) }drdt.

to

(26)

Just as in the spatial relationship, u is r e l a t e d to the v e c t o r q of a d i f f r a c t i o n experiment, g i v i n g the change in the wave v e c t o r k, or in the m o m e n t u m hk, so v is r e l a t e d to the change in f r e q u e n c y from the incident frequency, or the change in e n e r g y hv of the incident photons (or p a r t i c l e s ) . This type of formalism, i n t r o d u c e d by van Hove [1954], is of special value for n e u t r o n d i f f r a c t i o n for which c h a n g e s in both m o m e n t u m and e n e r g y of the incident neutrons m a y be measured. For s c a t t e r i n g from a crystal having t h e r m a l

5.5

CORRELATION

IN SPACE AND TIME

105

v i b r a t i o n of the atoms or, more appropriately, a d i s t r i b u t i o n of phonons, both the m o m e n t u m s and energies of the phonons may be deduced.

5.5.2.

Special

cases

A few special cases will serve to illustrate the p r o p e r t i e s and r e l a t i o n s h i p s of the function P(r, t). For the point r = 0 we have P(0, t) = ~ p(R, T) p(R,

t + T)dTdR.

(27)

The integral over T gives the p r o b a b i l i t y that if there is an atom at R at time 0 there will be an atom at R at time t, and this is a v e r a g e d over all points R. This gives an indication of the rate of d i f f u s i o n of atoms away from their initial positions. For p u r e l y elastic s c a t t e r i n g there is no change of f r e q u e n c y of the incident radiation so that v = 0. Then the reciprocal space function of interest is

IF(u, 0) I2 = ~ P(r,t)

exp{2~i(u.r)}dr~

= ~ [~ P(r, t)dt] e x p { 2 K i u . r } d r .

(28)

This is also equal to the Fourier t r a n s f o r m of the threed i m e n s i o n a l P a t t e r s o n function of the time average of the electron density. Hence the section of the f o u r - d i m e n s i o n a l reciprocal space distribution, v = 0, corresponds to the F o u r i e r t r a n s f o r m of the t i m e - a v e r a g e of the f o u r - d i m e n s i o n a l P a t t e r s o n function. If a m e a s u r e m e n t is made in which the s c a t t e r e d radiation intensities for all energies are added together, the r e c i p r o c a l space function of interest is IF (u. v) 12dr = ~ P (r, t) exp { 2~i (u- r) } 8 (t) drdt = ~P(r, 0)

exp{2~iu.r}dr.

(29)

Hence the total s c a t t e r e d intensity, elastic plus inelastic, is r e l a t e d to e(r, 0), which gives the c o r r e l a t i o n s of atomic p o s i t i o n s for no d i f f e r e n c e in times; c o r r e s p o n d i n g to the sum of all c o r r e l a t i o n functions for i n s t a n t a n e o u s "pictures" of the atomic configuration. E x p r e s s i o n s such as (28), however, are u n s a t i s f a c t o r y as they stand b e c a u s e they lead to infinities. If e(r, /) refers to a s y s t e m which is a constant, or repeats p e r i o d i c a l l y in time, the integral ~ P ( r , t ) ~ is infinite. This u n s a t i s f a c t o r y result must arise because we have not d e s c r i b e d the e x p e r i m e n t a l s y s t e m adequately.

i06

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

5

Measurements of intensities are always made with instruments of finite resolution. The finite energy resolution may be taken into account by replacing IF(u,y)12 by, say,

IF(u,v) 12 ,

(/I;M2)-1/2 e x p { - v 2 / M 2 } ,

where M is a measure of the range of frequencies The Fourier t r a n s f o r m of this function is then

included.

P(r, t) exp{-2/I;2M2t 2 } . Then the m e a s u r e m e n t in the limit M-)oo,

of intensities

giving,

for all y values

by Fourier

transform,

is given

P(r, 0).

In the limit that M tends to zero we return to the ideal case of m e a s u r i n g IF (u, v) 12 and obtaining P (r, t) by Fourier transform. The case where near-elastic scattering only is m e a s u r e d is represented by selecting only a small range of frequency around the origin-

JIF(u,V) 12 exp{-V2/M2}dV. The Fourier t r a n s f o r m of this is e(r, t) ,

(~I/2M)

exp{-2~2M2t 2} .

In the limiting case MoO, c o r r e s p o n d i n g to purely elastic scattering this integral tends to <e(r, t)>t, i.e. the time average of P(r, t), since P(r, t) is c o n v o l u t e d with a function which has a width tending to infinity but an integrated value always equal to unity. In what follows we will not complicate the m a t h e m a t i c s by using these more complete formulations, but will bear in m i n d that the difficulties arising from the o v e r - s i m p l i f i e d expressions may be avoided in this way.

5.5.3.

Ideal monatomic

gas or liquid

We consider the idealised case of N n o n - i n t e r a c t i n g atoms contained in a volume V such that the distribution of atom positions is completely random, i.e. there is no correlation of atom center positions except for the limitations of the finite volume. Each atom has an electron density d i s t r i b u t i o n p0(r). We ignore the usual practical limitation that the electron distributions of separate atoms cannot overlap appreciably.

5.5

CORRELATION

IN SPACE

AND

TIME

107

o

/llVl dv

}'(r,())

Fig. 5.5. Instantaneous correlation monatomic gas and the corresponding

function, P (r, 0), for an ideal distribution in reciprocal space.

The f o r m of the P a t t e r s o n f u n c t i o n P(r, t) can be w r i t t e n down i m m e d i a t e l y . The i n s t a n t a n e o u s c o r r e l a t i o n f u n c t i o n P(r, 0) has a p e a k at the o r i g i n g i v e n by Np0(r) , p0(r). For any n o n - z e r o l e n g t h v e c t o r the p r o b a b i l i t y of f i n d i n g an a t o m at that v e c t o r d i s t a n c e f r o m a g i v e n a t o m has the same v a l u e w h i c h d e p e n d s o n l y on the gas or l i q u i d density, apart f r o m the finite v o l u m e e f f e c t d i s c u s s e d in 5.4.1. Hence we can write, as a r e a s o n a b l e a p p r o x i m a t i o n e x c e p t n e a r r = 0,

P ( r , O) =

(N/V)[. { p o ( r )

= Z 2(N/V)

, po(r)}dr

[s(r)

.

s(-r)].

[s(r)

, s(r)] (30)

The f o r m of P(r, 0) is then as shown in Fig. 5.5. This f u n c t i o n c o u l d be d e r i v e d f r o m the s c a t t e r i n g p o w e r d i s t r i b u t i o n for all e n e r g i e s , ~ IF (u, v)12dr, w h i c h c o n s i s t s of a c e n t r a l p e a k c o m i n g f r o m the F o u r i e r t r a n s f o r m of (30) plus Nlfo(u)12 f r o m the o r i g i n p e a k of P(r, 0). Thus the s c a t t e r i n g is N times that for a s i n g l e atom. The c o r r e l a t i o n P(r, t) for r, t not e q u a l to zero is the same. For r = 0, the value of P(O, t) gives the p r o b a b i l i t y that if an a t o m is at a point at time 0, there will be an a t o m t h e r e at time t. For large times this tends to the r a n d o m value of (30). The rate at w h i c h e(0, t) a p p r o a c h e s this r a n d o m v a l u e d e p e n d s on the a v e r a g e v e l o c i t y of the a t o m or the s e l f - d i f f u s i o n c o e f f i c i e n t . M o r e f o r m a l l y and g e n e r a l l y we m a y d e s c r i b e the e l e c t r o n d e n s i t y f u n c t i o n as

p ( r , t)

= po(r)

, g ( r , t)

where

g ( r , t) = Z n

5{r

- rn(t) } .

(31)

108

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

Thus g(r, t) is a d i s t r i b u t i o n function: a set of delta f u n c t i o n s d e s c r i b i n g the p o s i t i o n s of the a t o m centers space and time. Then P (r, t)

= Po (r)

,

9o ( - r )

,

g (r, t)

,

The c o n v o l u t i o n s are over space and time. d e p e n d on time the first c o n v o l u t i o n is

~J'po(R)

g (-r,

-t).

5

in (32)

Since p0(r)does not

po(r + R)dRdT,

where the n o r m a l i z i n g factor must be i n t r o d u c e d as b e f o r e so that the integral ~dT does not give an infinity. The d i s t r i bution function convolution G ( r , t)

= g(r,

t)

,

g(-r,

-t),

gives the p r o b a b i l i t y that, given an atom c e n t e r e d r = 0 at time t = 0, there will be an a t o m c e n t e r e d at the p o s i t i o n r at time t. P r o v i d e d that we are d e a l i n g with a very large n u m b e r of atoms, G(r, t) can be c o n s i d e r e d to be a c o n t i n u o u s function. Then in a d i f f r a c t i o n e x p e r i m e n t in which all f r e q u e n c i e s are r e c o r d e d with equal efficiency, the o b s e r v a t i o n m e a s u r e s ~ IF (u, v) ]2dv from which one m a y d e d u c e P (r, 0) w h i c h d e p e n d s on G ( r , 0)

=

[ g ( r , t)

- ~g(R,

9 g(-r,-t)]t=o

T) g ( r + R ,

= < g (r, t)

* g (-r,

T) dR d T

- t ) >t

(33)

r and this is the time average of the i n s t a n t a n e o u s d i s t r i b u t i o n Patterson. The p u r e l y e l a s t i c intensity, IF(u, 0) 12, gives JP(r, t) dt, which depends on G ( r , t ) d t = ;; g ( R , T)

g(r + R,

t + T)dtdTdR

= ~ < g ( R , t)>t

< g ( r + R, t ) > t d R

=

t)>t

*

r

and this is the self c o n v o l u t i o n d i s t r i b u t i o n function.

-t)>t,

of the time average

(34) of the

5.5

CORRELATION

5.5.4.

Real

IN SPACE AND

TIME

109

monatomic gases and liquids

An i m p o r t a n t d i f f e r e n c e b e t w e e n (33) a n d (34) a p p e a r s w h e n we i n t r o d u c e the r e s t r i c t i o n that atoms c a n n o t o v e r l a p a p p r e c i a b l y , i.e. that at a p a r t i c u l a r time t the d i s t r i b u t i o n P a t t e r s o n g (r, t) 9 g(-r, -t) a n d t h e r e f o r e in g e n e r a l G(r, 0) c a n n o t c o n t a i n a n y p o i n t s b e t w e e n Irl = 0 a n d Irl = 2r0 w h e r e

r0 is the e f f e c t i v e r a d i u s of an atom. For Irl > 2r0 some f l u c t u a t i o n s in G(r, 0) o c c u r b e c a u s e atoms u s u a l l y have an a t t r a c t i v e i n t e r a c t i o n . T h e r e tends to be an e x c e s s of atoms at the s m a l l e s t p o s s i b l e , " n e a r e s t n e i g h b o r " d i s t a n c e ; t h e n e s p e c i a l l y for liquids, an a p p r e c i a b l e but less m a r k e d t e n d e n c y for a t o m s to o c c u r also at a s e c o n d nearest, t h i r d nearest, etc. n e i g h b o r d i s t a n c e , so that G(r, 0) has a f o r m as suggested, in one d i m e n s i o n , in Fig. 5.6(a), w i t h a d e l t a f u n c t i o n at the origin. The f o r m of e(r, 0) is t h e n as s u g g e s t e d in Fig. 5.6(b) a n d the d i s t r i b u t i o n of s c a t t e r i n g p o w e r o s c i l l a t e s w i t h lul as s u g g e s t e d in Fig. 5.6(c).

On the o t h e r h a n d the p u r e l y e l a s t i c s c a t t e r i n g IF(u00)12 d e r i v e s f r o m the self c o n v o l u t i o n of the time a v e r a g e of the d i s t r i b u t i o n function, t. But, since all atoms are

(a) g(r)~g(-r)

I r l -~

V(r)

I r l -~

(b)

(C)

~_

.

.

.

.

.

i u)i

iu I

Fig. 5.6. Diagrams suggesting the form of, (a) the distribution Patterson, (b) the actual Patterson function and (c) the scattering power distribution for a real monatomic gas or liquid.

II0

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

5

moving and all positions for an atom are equally probable, this averaged d i s t r i b u t i o n function is a constant within the volume of the system. The c o n v o l u t i o n (34) and ] P(r, t)dt are then also constants, m u l t i p l i e d by the convolution of the shape function [$(r) , s(-r)]. Then the purely elastic scattering consists of the single sharp peak IS(u) 12. Thus we deduce that, apart from the sharp central, zero angle peak, all d i f f r a c t e d intensity is inelastic. This is a somewhat academic point for X-ray or electron d i f f r a c t i o n because the rate of decay of P(0, t) with time is c o m m e n s u r a t e with the Boltzmann d i s t r i b u t i o n of atom velocities for normal temperatures and average energies of the atoms of kT(=0.02 eV). Energy changes of the incident radiation of this m a g n i t u d e are not detectable and the normal measurements reflect the Patterson P(r, 0) as given by (33). For neutron d i f f r a c t i o n however such changes in energy, or frequency, can be d e t e c t e d and the situation represented by (34) may be approached. A further stage of complication is added if we consider gases or liquids composed of molecules rather than independent atoms. Then instead of p0(r) for a single atom we must consider p0(r, 0n, ~n) where the angles 0n, ~n specify the o r i e n t a t i o n of the molecule and are functions of time. The total scattered intensity then corresponds to a time average of the instantaneous Pattersons,

,'(r, O, -<

po(r, 0,,, (p,.,,,po(-r, On, n

>,.

(35)

In

If it can be assumed as a first approximation that the relative positions of the centers of all molecules are completely random, all these terms for n ~ m give only a continuous smooth b a c k g r o u n d and the Patterson is the same as for a monatomic gas except that the origin peak is replaced by

< T_., po(r, n

.

),.

For each orientation On, ~n, P(r, 0) contains a set of peaks c o r r e s p o n d i n g to the interatomic vectors within the molecule. The summation over n and the averaging over time reproduces this set of peaks in all possible orientations about the origin with equal probability. Then the radial d i s t r i b u t i o n P(r, 0) contains a set of peaks corresponding to the interatomic vector as suggested in Fig. 5.7. (a) and the intensity function c o r r e s p o n d i n g l y shows fluctuations about the smooth Nlfo(u)12 curve, Fig. 5.7(b).

5.5

C ORRELATION

IN SPACE AND TIME

iii

A more d e tailed and complete discussion of diffraction from liquids along lines related to the above is given by Guinier [1963]. Diffraction from gases, as a means for analysing the structures of gas molecules, has been developed most e x t e n s i v e l y using electron beams. The scattering by the low-density gases is much stronger for electrons than for Xrays or neutrons, and the shorter wavelength of electrons in the 30-40 keV range implies that intensity measurements can be made for much greater values of (sin0)/l. An account of the gas electron diffraction techniques and a survey of the structural information obtained by the m e t h o d have been given by Hargittai and Hargittai (1988), (see also Hargittai, 1992). For an array of atoms, as in an isolated molecule, separated by vectors ri] = ~ - ~, the diffracted intensity given by the time-averaged, spherically symmetrical distribution is given (as in (4.6)) by N

I (s) = (K2Io/R 2) Z

N

Z

fi (s) ~* (s) (sinsri] /srij)

(36)

i=1 /=1

where K = 2~2n~2/h2Eo and, as is conventional for gas diffraction, the angular variable is s = (4Ksin0)/l=2K[ul 9 The distance ri] varies with time because the molecule has thermal vibrations. This complicates the expression (36) somewhat (Hargittai, 1992). The terms with i - j in (36) give the sum of intensities of scattering from all atoms considered separately, Zi[~(s)[2, and this constitutes a smoothly falling b a c k g r o u n d to the oscillations produced by the i ~ j terms, the molecular scattering denoted by [m (s). By Fourier t r a n s f orm of this function, [m (s), obtained e x p e r i m e n t a l l y by subtraction of the smooth background, one obtains the radial distribution function

7 D (r) = | s Im (s) el

sin

(sr) ds

(37)

0

which gives, directly the distribution of probabilities for the occurrence of the interatomic distance r, weighted by the scattering strengths of the contributing atoms i.e. a radial Patterson function. From this, the interatomic distances and hence the structure of the molecule may be derived. Also, in favorable cases, the amplitudes of thermal vibration of the atoms may be deduced.

112

SCATTERING

P(r/

FROM ASSEMBLIES

[r ]-~

OF A T O M S

CH.

i

II,'(u~l 2

5

[u] -~

Fig. 5.7. Diagrams suggesting the Patterson function for a molecular gas, and the corresponding variation of scattering power in reciprocal space.

5.5.5.

The hydrogen atom

W h e n d e a l i n g with q u a n t u m p r o c e s s e s of e x c i t a t i o n of the s c a t t e r i n g object, c o n s i d e r a b l e c a u t i o n is r e q u i r e d and it cannot n e c e s s a r i l y be a s s u m e d that the simple c l a s s i c a l c o n s i d e r a t i o n s we have used can be c a r r i e d over (van Hove [1954]). H o w e v e r a direct a p p l i c a t i o n of the above t h e o r y gives an a d e q u a t e t r e a t m e n t for one case; the s c a t t e r i n g of X-rays by a h y d r o g e n atom, i.e. by the e l e c t r o n of a h y d r o g e n atom. For this e l e c t r o n P(r, 0) is a delta f u n c t i o n at the origin, since all i n s t a n t a n e o u s p i c t u r e s show a single point, one electron. Then the total scattering, in e l e c t r o n units squared, is given by

IF(u,v) For the e l a s t i c

scattering

12dv =

1.

we c o n s i d e r

P (r, t) dt = Pe (r) * Pe (-r) , where pe(r) is the time average d i s t r i b u t i o n of the e l e c t r o n p o s i t i o n or the e l e c t r o n d e n s i t y function for the e l e c t r o n in a h y d r o g e n atom. Then in r e c i p r o c a l space IF(,, 0) ]2 = f2 e (,) .

(38)

The i n e l a s t i c (Compton) s c a t t e r i n g is then given by the d i f f e r e n c e b e t w e e n the total and the elastic s c a t t e r i n g as, 1 - f2 e (u), in a c c o r d a n c e with our result of e q u a t i o n (4.7). The a p p l i c a t i o n of this f o r m a l i s m to the s c a t t e r i n g n e u t r o n s by p h o n o n s will be given in C h a p t e r 12.

(39) of

5.5

5.6.

CORRELATION

Diffraction

IN SPACE AND TIME

geometry

and

113

intensities

We have e s t a b l i s h e d that kinematical, elastic d i f f r a c t i o n a m p l i t u d e s and intensities, o b t a i n e d by the s c a t t e r i n g of Xrays from e l e c t r o n d e n s i t y distributions, m a y be r e l a t e d to the d i s t r i b u t i o n s in reciprocal space given by F o u r i e r t r a n s f o r m of p(r) or P(r). The next step is to show how the a m p l i t u d e s or i n t e n s i t i e s for p a r t i c u l a r e x p e r i m e n t a l a r r a n g e m e n t s m a y be d e r i v e d from the reciprocal space distributions. The argument m a y be c a r r i e d t h r o u g h either for d i f f r a c t i o n a m p l i t u d e s in terms of the reciprocal space d i s t r i b u t i o n F(u) or for d i f f r a c t i o n intensities in terms of the d i s t r i b u t i o n IF(u) J2, which m a y be c a l l e d the d i s t r i b u t i o n of the s c a t t e r i n g power. For the moment we restrict ourselves to the latter as b e i n g more a p p r o p r i a t e to general d i f f r a c tion e x p e r i m e n t s in which intensities or energy fluxes are measured. F r o m e q u a t i o n s (1.20) and (1.21), the s c a t t e r e d amplitude is given in the asymptotic limit of large R, as a f u n c t i o n of q = k - k0;

~(q) = -(B/4K); ~(r) exp{-2~iq-r}dr. In the a p p r o p r i a t e units and p u t t i n g q - -u, we have ~(q) F(u), given by e q u a t i o n (2) and the intensity is

l(q)

= W*

(q) = ~ P(r)

exp{2~iu-r}dr.

(40) (41)

Thus for an incident m o n o c h r o m a t i c b e a m in a d i r e c t i o n d e f i n e d by the wave vector ko, the i n t e n s i t y d i f f r a c t e d e l a s t i c a l l y in a p a r t i c u l a r direction d e f i n e d by wave vector k is equal to the value of the function IF(u) i2 at the p o s i t i o n in r e c i p r o c a l space d e f i n e d by u = (k0 - k). This r e l a t i o n s h i p is e x p r e s s e d by the Ewald sphere c o n s t r u c t i o n in reciprocal space, Fig. 5.8. A vector of length l-i (=Ikol) is drawn to the origin, O, of reciprocal space, in the d i r e c t i o n of k 0 from the point P. A sphere of radius l-i is drawn a r o u n d P as center. Then for any point on the sphere, u, the radial vector (length l-l) from P represents the d i r e c t i o n of the d i f f r a c t e d b e a m k such that u = (k0 - k). The intensity of this d i f f r a c t e d b e a m is IF(u) 12. Thus this Ewald sphere c o n s t r u c t i o n gives the d i r e c t i o n s and intensities for all d i f f r a c t e d beams p r o d u c e d for a given incident b e a m direction. Unless IF(u) J2 is isotropic, its o r i e n t a t i o n in reciprocal space is d e f i n e d in terms of the o r i e n t a t i o n of P(r) in real space. Then a rotation of the sample in real space produces the c o r r e s p o n d i n g rotation of the s c a t t e r i n g power distri-

114

SCATTERING

FROM ASSEMBLIES

OF ATOMS

CH.

5

Fig. 5.8. The Ewald sphere construction. bution IF(u)[2 in the reciprocal space. For a constant incident b e a m direction the d i f f r a c t e d intensities then vary as regions of different scattering power are rotated through the Ewald sphere. Naturally, exactly the same sequence of intensities is p r o d u c e d if the distribution IF(u)12 is kept stationary and the Ewald sphere is rotated in the opposite direction, i.e. if the sample is kept stationary and the direction of the incident b e a m is rotated. The form of the diffraction pattern recorded in p r a c t i c e depends on the geometry of the recording system and the w a v e l e n g t h of the radiation. The function IF(u)12 decreases, on the average, with the square of the atomic scattering factor for an average atom, If(u) 12 . If the mean radius of an atom is taken to be about 0.5 ~, the half-width of the d i s t r i b u t i o n If(u)12 is of the order of 2 ~-I and the range of lul which is normally of interest is several times this: say 5 A -I . For X-ray and neutron diffraction, the wavelengths are of the order of 1 A so that the diameter of the Ewald sphere is 2 A -I. Thus the whole of the intersection of the sphere with the function IF(u)12 is of interest and scattering through all angles from 0 to ~ is normally recorded as suggested by Fig. 5.9(a). Thus for photographic recording it is customary to use a cylindrical film with the sample on the axis. For electronic recording with a photon- or particlecounting detector, a goniometer stage is used which allows the detector to be swung through diffraction angles which are as large as is convenient. For the diffraction of electron having a wavelength of about 0.04 A, the diameter of the Ewald sphere is 50 ~-i. On this sphere, only the small region of radius about 5 A -I around the reciprocal space origin is of interest and the scattering is p r e d o m i n a n t l y through small angles as s u g g e s t e d by Fig. 5.9(b) .

5.6

DIFFRACTION

GEOMETRY

AND

INTENSITIES

115

Fig. 5.9. Comparison of the scales of the Ewald spheres for X-rays, neutrons and electrons, in relation to normal scattering power distributions. The d i f f r a c t i o n p a t t e r n m a y be r e c o r d e d on a flat p l a t e or f i l m p l a c e d p e r p e n d i c u l a r to the i n c i d e n t b e a m at some d i s t a n c e b e h i n d the s p e c i m e n and r e p r e s e n t s an a l m o s t p l a n a r s e c t i o n of the d i s t r i b u t i o n of s c a t t e r i n g p o w e r in r e c i p r o c a l space. In this way it w o u l d s e e m that the i n t e n s i t i e s to be o b s e r v e d for a g i v e n r a d i a t i o n and for a p a r t i c u l a r g e o m e t r y of the e x p e r i m e n t m a y be d e d u c e d for s c a t t e r i n g f r o m any s y s t e m for w h i c h the P a t t e r s o n f u n c t i o n can be d e r i v e d or p o s t u l a t e d . H o w e v e r our d i s c u s s i o n so far has b e e n for the i d e a l i z e d case of p e r f e c t l y p l a n e and m o n o c h r o m a t i c waves. These c o n s i d e r a t i o n s m u s t be b r o a d e n e d in order to m a k e c o n t a c t with e x p e r i m e n t a l s i t u a t i o n s .

5.7.

Practical 5.7.1.

considerations

Finite

sources

and detectors

In X - r a y d i f f r a c t i o n e x p e r i m e n t s , the finite s o u r c e size results in an angle of c o n v e r g e n c e of the i n c i d e n t b e a m at any p o i n t of the sample of s o m e t h i n g like 10 -4 to 10 -3 radians a l t h o u g h with s y n c h r o t r o n r a d i a t i o n m u c h s m a l l e r c o n v e r g e n c e angles m a y be achieved. The i n t e n s i t y is not n e c e s s a r i l y u n i f o r m over this range of angles. In n e u t r o n d i f f r a c t i o n the angle of c o n v e r g e n c e is often made g r e a t e r than this since the source i n t e n s i t y is low.

116

SCATTERING

FROM ASSEMBLIES

P

OF A T O M S

CH.

5

k2

0

Fig. 5.10. The effect of a finite source size in spreading the Ewald sphere into a spherical shell of varying thickness. For e l e c t r o n d i f f r a c t i o n the angle of c o n v e r g e n c e m a y be made much less, a l t h o u g h it may be as high as 10 -3 radians when the specimen is i m m e r s e d in the m a g n e t i c field of the o b j e c t i v e lens in an e l e c t r o n m i c r o s c o p e and the " s e l e c t e d - a r e a e l e c t r o n d i f f r a c t i o n " t e c h n i q u e is used (see C h a p t e r 13). Such a c o n v e r g e n c e of the incident b e a m m o d i f i e s the p i c t u r e of the E w a l d sphere in reciprocal space in a way s u g g e s t e d in Fig. 5.10. D r a w i n g the incident b e a m d i r e c t i o n s as vectors k 0 to the r e c i p r o c a l lattice origin, O, the origin points P are d i s t r i b u t e d over a disc h a v i n g the shape of the source and a relative w e i g h t i n g of the points given by the i n t e n s i t y d i s t r i b u t i o n of the source. C o r r e s p o n d i n g to each point of this disc there is a d i f f e r e n t l y - o r i e n t e d E w a l d sphere, so that we may think of an E w a l d sphere t h i c k e n e d into a s p h e r i c a l shell of thickness which varies with d i s t a n c e from O. For a given d i f f r a c t e d b e a m direction, the vectors k from the source disc at P, each drawn to the c o r r e s p o n d i n g E w a l d sphere, define a d i s c - s h a p e d section of the "Ewald shell". This we m a y call the " s c a t t e r i n g disc". The total i n t e n s i t y s c a t t e r e d in the d i r e c t i o n k is then given by i n t e g r a t i n g IF(u) i2 over this s c a t t e r i n g disc with a w e i g h t i n g factor c o r r e s p o n d i n g to the i n t e n s i t y of the source disc at P. Clearly, the size and shape of the s c a t t e r i n g disc will vary with the angle of scattering, so the effect on the i n t e n s i t i e s cannot be r e p r e s e n t e d by a simple c o n v o l u t i o n of some shape f u n c t i o n with iE(u) i2 unless all s c a t t e r i n g angles are small as in the case of e l e c t r o n diffraction. When the r e c o r d i n g of the d i f f r a c t i o n p a t t e r n is p h o t o g r a p h i c , the r e s o l u t i o n of the p h o t o g r a p h i c plate or film is n o r m a l l y sufficient to allow each d i f f r a c t e d b e a m d i r e c t i o n of interest to be r e c o r d e d separately. For e l e c t r o n i c recording, however, the d e t e c t o r u s u a l l y accepts a finite a n g u l a r range of d i f f r a c t e d beams from each point of the sample. Then for each incident b e a m d i r e c t i o n the d i f f r a c t e d i n t e n s i t y depends on the value of not just for one

5.7

PRACTICAL

CONSIDERATIONS

117

P

0 Fig. 5.11. The combined effects of finite source size and finite detector aperture size to define "scattering volumes" in reciprocal space over which the scattering power is integrated. p o i n t on t h e E w a l d s p h e r e but for a f i n i t e a r e a of the E w a l d sphere. C o m b i n i n g this w i t h the e f f e c t of the f i n i t e s o u r c e size, as s u g g e s t e d b y Fig. 5.11, t h e r e c o r d e d i n t e n s i t y for a n y s e t t i n g of the c r y s t a l a n d the d e t e c t o r c o m e s f r o m a v o l u m e of r e c i p r o c a l space. B o t h the size a n d s h a p e of this " s c a t t e r i n g v o l u m e " v a r y w i t h s c a t t e r i n g angle.

5.7.2. Wavelength spread A f u r t h e r c o m p l i c a t i o n c o m e s f r o m the f i n i t e r a n g e of w a v e l e n g t h s p r e s e n t for a n y real source. F o r X - r a y s the n a t u r a l h a l f - w i d t h s of the c h a r a c t e r i s t i c e m i s s i o n lines are of the o r d e r of 10 -4 ~ or more. F o r n e u t r o n d i f f r a c t i o n or for s y n c h r o t r o n X-rays, s i n c e the r a d i a t i o n u s e d is s e l e c t e d out of a c o n t i n u o u s " w h i t e r a d i a t i o n " d i s t r i b u t i o n , the r a n g e of w a v e l e n g t h s u s e d m a y be m a d e g r e a t e r in o r d e r to i n c r e a s e the t o t a l i n t e n s i t y of i n c i d e n t r a d i a t i o n . F o r e l e c t r o n s the r a d i a t i o n is u s u a l l y m u c h m o r e n e a r l y m o n o c h r o m a t i c w i t h a w i d t h of a b o u t 10 -6 I.

E

ko

0 Fig. 5.12. The effect of a spread of wavelength in producing a spread in values of the Ewald sphere radius.

118

SCATTERING

FROM ASSEMBLIES

OF A T O M S

CH.

5

The spread of wavelengths produces a spread in the values of the Ewald sphere radius, as suggested in Fig. 5.12 for the case of very small incident and diffracted b e a m divergence. This results in a b roadening of the scattering region into a line rather than a disc since the origin points P lie on a line. There is an essential difference from the case illustrated in Fig. 5.10 in that the wavelength spread gives a scattering line which varies in its orientation as well as length, being small and parallel to k 0 in the limit of small scattering angles; of m e d i u m length and roughly p e r p e n d i c u l a r to k0 for a scattering angle like the larger one shown in the figure, and of m a x i m u m length and o p p o s i t e l y d i r e c t e d to k 0 for a scattering angle of ~. Adding this effect to the effects of the divergence of the incident and scattered beams suggested in Fig. 5.11 gives a scattering volume having a shape which is complicated and varies with scattering angle in a complicated way. Thus, in general, the relationship of the observed intensity to the function IF(u) 12 is only to be derived by laborious calculation from a detailed knowledge of the parameters of the experimental arrangement.

5.7.3. Integrated intensities The usual procedure to simplify matters for the m e as u r e m e n t of continuous distributions of scattering power is to ensure that the scattering volume is much smaller than the region over which IF(u) 12 varies significantly. In the case of diffraction from w e l l - c r y s t a l l i z e d materials, when the object is to measure the integrated scattering power contained in sharp maxima separated by a low flat background, the usual scheme is to generate a scattering volume in the shape of a thin disc, e.g. by having a well-collimated, n e a r l y - m o n o c h r o m a t i c beam and a relatively wide angle of acceptance of the detector so that the disc is a portion of the Ewald sphere. Then the disc is swept through the sharp m a x i m u m of scattering power, e.g. by rotating the crystal, and the observed intensity is integrated over time. For each experimental situation involving these simplifying special cases, the variation of the shape and size of the scattering volume and the rate at which it is scanned through reciprocal space are taken into account by m u l t i p l y i n g the observed intensities by an appropriate factor, known as the Lorentz factor, which may usually be taken as a function of the magnitude of the scattering angle only. D e t a i l e d derivations are given, for example, by Warren [1969] or in the International Tables for Crystallography, Vol. C (Wilson, 1992). For the diffraction of electrons with m e d i u m or high energies (greater than 50 keV say) all these considerations of scattering volumes are greatly simplified by the small

5.7

PRACTICAL

CONSIDERATIONS

119

scattering angles involved. The effect of the wavelength spread is negligible. The effect of the finite source size is to convolute the two-dimensional scattering power distribution, IF (u, v) [2, with a source function, say S (u. v) since this may be assumed independent of scattering angle. Then

l(u,v)

= IF ( u , v ) 12 . S ( u , v ) .

(42)

Also if this intensity distribution is observed with a detector of finite aperture or photographic plate of finite resolution with sensitivity represented by a function D(u0v), the m e a s u r e d intensity is given by

Iobs= =

I(u,v)

. D(u,v)

IF(u,v) I2

,

S(u,v)

,

D(u,v).

(43)

The Lorentz factor then derives solely from the way in which intensities are integrated over time when a sample is rotated or, more commonly, integrated over crystal orientation when the sample contains a range of crystal orientations. In neutron diffraction involving measurements of intensity as a function of energy loss as well as diffraction angles, the complications may be c o r r e s p o n d i n g l y greater than for X-ray diffraction. We prefer not to deal with them here.

5.8.

Sections

and

projections

For a number of purposes it is convenient to deal with twoor one-dimensional sections or projections of the threedimensional functions p(r), P(r), F ( u ) and [F(u)] 2. It is much easier to represent and appreciate functions plotted in one or two dimensions and the amount of data required to define them is often very much less. For the case of a radially symmetric function, such as P(r) for a gas or liquid, no new information is gained by going to more dimensions than one. Experimentally, electron diffraction patterns represent, to a first approximation, planar sections of reciprocal space. In X-ray diffraction a planar section of reciprocal space may be obtained by use of the Buerger Precession Camera or similar devices, and the simplest form of operation of the X-ray diffractometer, the 0 - 28 scan, gives a section along a radial straight line. It is therefore appropriate to summarize now the general relationships between sections and projections in real and reciprocal space. We do so with reference to the functions p(r) and F ( u ) for convenience but draw our examples from our broader range of interest. The p r o j e c t i o n of the function p(r) in the direction of the z-axis is p(x,y) =

J p(r)dz.

(44)

120

SCATTERING FROM ASSEMBLIES OF ATOMS

In terms of F(u)

CH. 5

this is

p(x,y) = ~;F(,) e x p { - 2 ~ i ( u x + v y + w z ) }dzdu.

(45)

The integral over z is the delta function ~(w) so that p (x,y) is the inverse Fourier t r a n s f o r m of F(u,v. 0), the section of F(u.v,w) on the plane w = 0. Similarly, a planar section of p(r) on the z - 0 plane is p(x,y, 0) = ~F(,) exp{-2~i(ux+vy) } d u d v d w = J [;E(u,v,w) dw]

exp{-2Ki(ux+vy) }dudv .

(46)

Thus we have the general relationship, that a planar section through the origin in real space corresponds to a p r o j e c t i o n of the reciprocal space distribution on a parallel plane, and vice versa. For example, the electron diffraction pattern, to a first approximation is a planar section of reciprocal space so that Fourier transform of the intensity distribution of the d i f f r a c t i o n pattern gives the projection of the Patterson function P(r) in the b e a m direction. This is the approximation that the object may be treated as a two-dimensional phase- and amplitude-object. We have seen examples in Section (5) above for the case of the four-dimensional distributions in space and time when intensity is m e a s u r e d as a function of scattering angles and frequency change. Thus the section of reciprocal space on the v = 0 plane, appropriate for purely elastic scattering, equation (28), gives the projection of the Patterson function in the time direction or the time-average of the correlation function. The projection of the four-dimensional d i s t r i b u t i o n scattering power in reciprocal space in the v direction, given by the integral over v in equation (29), is the Fourier t r a n s f o r m of the section of the Patterson function P(r, 0) which is the sum of the instantaneous spatial correlations of the object. If the section in real space is not through the origin, a c o r r e s p o n d i n g phase factor is introduced in reciprocal space. Thus the Fourier transform of the section p(x,y,c) gives the m o d u l a t e d projection F(u,v,w)

Extending the considerations projections, we have

to one-dimensional

~3p (X, O, O) = ~ F (u, v, w) dv dw, and

(47)

exp{-2Eiwc}dw. sections

and (48)

5.8

SECTIONS AND PROJECTIONS

S If. p ( x , y , z ) d y d z

= F(u, o, O).

121 (49)

Thus the value of p(r) along the line of the x axis is given by Fourier t r a n s f o r m of the projection of F(u) along both v and w directions on to the u-axis. An example from the four-dimensional situation is that of equation (27), where the function of interest is P(0, t), the c o r r e l a t i o n in time for the position at the origin. This function would be given by p r o j e c t i n g the reciprocal space function [F(u,v)[2 in the u, v and w directions, i.e. by integrating over all scattering directions for each frequency change, v. Problems

i. Sketch the Patterson function in two dimensions for a stationary m o l e c u l e in the form of an equilateral triangle of atoms. Is there a centro-symmetrical object or a group of c e n t r o - s y m m e t r i c objects which gives the same Patterson peaks (apart from the origin peak)? Derive an expression for the c o r r e s p o n d i n g distribution IF(u)[2 in reciprocal space. Repeat the considerations for an ideal gas of such molecules. 2. A diatomic molecule of fixed orientation (along x axis) vibrates about a fixed center of mass with frequency v0. Sketch the correlation function P(x, t) and the corresponding reciprocal space distribution [F(u.v) [2. 3. Derive the Lorentz factor to be applied to X-ray diffraction from a gas and for the intensities of rings in a powder pattern (from crystallites in all possible orientations, each giving the same set of sharp peaks in reciprocal space) .

This Page Intentionally Left Blank

CHAPTER 6

Diffraction from crystals 6.1.

Ideal

crystals

Although, as is well known, real crystals contain many different types of imperfections and faults, including point defects, impurities, dislocations, stacking faults and so on, it is often possible to consider the main diffraction effects as coming from an ideally periodic average crystal, as we shall see in Chapter 7. The discussion of the kinematical diffraction from ideal periodic crystals forms the basis for the important field of crystal structure analysis and so merits some special attention here. An ideal crystal is made up by the repetition in three dimensions of a unit cell containing one or more atoms. In general the unit cell is not rectangular. It is defined by three vectors a,b,c which have length a,b,c and angles between the axes ~,~,THence, writing p0(r) for the contents of the unit cell and s(r) for the shape function,

p(r) =

[po(r)

p(r)

[po(r).~_

, ~, ~ I

~=~ 8{r - (la + mb + #c) } ] s ( r )

m

n

or =

z ~_~ ~.~ l

m

8(x-la, y-rob, z-nc)]

s(r),

(i)

n

where x,y,z are coordinates with respect to the axes, in the directions of the unit vectors x,y,Z parallel to a,b,c. This is a g e n e r a l i z a t i o n of equation (2.56) following (5.8) . For the special case for which a,b,c are at right angles, we saw (equation (5.9)) that Fourier t r a n s f o r m gives a reciprocal lattice of points with spacings a-l,b-l,c -I i.e. that the reciprocal lattice is defined by vectors, a*,b*, c* such that la*l = a -1 , Ib*l = b -1 , I c * I = C-I. For the more general case of n o n - r e c t a n g u l a r axes, we must redefine the reciprocal lattice and put

124

DIFFRACTION

a

,

~

FROM CRYSTALS

b Xc

~

~'-

--

6

b Xc

--,-.--~

a(b Xc) b*

CH.

v

c Xa

V C where

,

=

aXb

V

,

(2)

V is the unit cell volume, a * -b a.a

Then F o u r i e r

a * -c

=

b* .c

=

b. b*

=

c.c

t r a n s f o r m of

F(u) = Fo(u) or

*

=

EEEs h k l

*

so that

=

...

=

1.

-

o,

(3) (4)

(i) gives

.-

(ha* + kb* + lc* ) } . S (u)

F(u.v.w) = Fo(U) Z E E 5(u-ha*, v-kb*, w-lc*).S(u). h k l

(5)

For a crystal h a v i n g the d i m e n s i o n s A , B , C in the d i r e c t i o n s of the three axes, the f u n c t i o n S(u) has the same form as in e q u a t i o n (5.10). A l t e r n a t i v e l y , since the only v a l u e s of Fo(u) of interest are those at the r e c i p r o c a l l a t t i c e points we may write

F(u) = E Z E Fhkl 5{u-(ha*+kb*+Ic*) }.S(u). h k 1

(6)

Then Fhk I is the " s t r u c t u r e factor" or preferably, the " s t r u c t u r e a m p l i t u d e " for the h,k,l r e c i p r o c a l lattice point and is given by abc

Fhkl

=

f~ ooo

p(x,y,z)

e x p { 2 K i ( (hxla)+(ky/b)+(Izlc) ) } d x d y d z

or

F h - Fhk I = ~p(r)

e x p { 2 K i h . r }dr

where the i n t e g r a t i o n is over the unit cell and h is the v e c t o r ha* + kb* + Ic*.

(7)

6.1

IDEAL CRYSTALS

125

An alternative and often useful convention is to use fractional coordinates which we signify for the moment as X.Y.Z so that distances are m e a s u r e d in terms of unit cell parameters ; X = ~a,

Then

Y = ylb,

Z = zlc.

(8)

(7) becomes, I

(9)

FhM - V ; ~ ; p (X, Y, Z) exp { 2xi ( hX + kY + lZ) } dX dY dZ. o

By inverse t r a n s f o r m of

P(x'Y'z)=V-lX X h

X

k

(6) we derive the alternatives to

Fh~ exp{-2~i ( (hxla)+(kylb)+(lzlc) ) }-s(r)

(i);

(10)

l

p(r) = X Fh e x p { - 2 ~ i h . r h

} s(r),

(11)

or

p(X.Y.Z)

= X h

X k

X FhM e x p { - 2 ~ i ( h X + k Y + l Z ) } . ; ( r ) . l

(12)

If the electron density in the unit cell is assumed to be the summation of the electron densities for individual atoms, we write

po(r) = X

(13)

pi(r) . $(r - ri),

l

so that

FhM

= X

'~ exp{2xi(hXi

l

+ kY i + IZi) },

(14)

where Xi, Yi, Zi are the fractional coordinates of the atom at ri and fi is the atomic scattering factor. The d i s t r i bution of scattering power in reciprocal space consists of a sharp peak around each reciprocal lattice point of the form iS(u)12 so we may write, to a good approximation

,.

l**, t.,sc., ,'

- 7__..Z Z h

k

l

Inverse t r a n s f o r m a t i o n

of this then gives,

following

(12),

(15)

126

DIFFRACTION

FROM CRYSTALS

CH.

P (X, Y, Z)

s s h

k

6

(16)

IFhid 12 exp { -2~i ((lucia)+ (kylb) + (Izlc) ) } Is (r) ,s (-r) ],

I

P (r) =

[p0(r)*p0(-r)*s s 1

m

s

~{r-(la+mb+nc) } 1j [ s ( r ) , s ( - r )

].

(17)

n

Thus, apart from the gradual fall-off due to the shape convolution, the Patterson function is periodic, as foreseen in equation (5.17), and is made up by repetition of the selfc o n v o l u t i o n of the contents of the unit cell. For convenience the shape convolution and its t r a n s f o r m are often omitted. Then it must be understood that the periodic functions in real space and the delta functions in reciprocal space are operational abstractions which can be e x p a n d e d into the more realistic descriptions of equations (15) to (17) when necessary.

6.2. Diffraction 6.2.1.

geometry

Laue and B r a g g

diffraction

conditions

From equation (15) we see that for the ideal finite crystal the distribution of scattering power is a sharp peak, IS(u) 12, around each point of the reciprocal lattice d e f i n e d by the vectors a*, b*, c* . Then a sharply defined d i f f r a c t e d b e a m is g e n e r a t e d when the Ewald sphere cuts through one of these sharp peaks of scattering power. From our previous considerations of the Ewald sphere construction we see that the geometric condition to be satisfied is (18)

q - k - ko = h - ha* + kb* + lc*. We may write this in terms of the projections real space axes as q- a

=

h,

q- b

=

k,

q- c

=

1,

of q on the (19)

which represent the well-known "Laue conditions" for diffraction. The condition for diffraction may also be written in terms of the concept of planes of atoms in the crystal. The p e r i o d i c i t y of the crystal ensures that sets of parallel planes may be drawn to pass through atom centers at regular

6 29

DIFFRACTION

GEOMETRY

127

intervals. These sets of planes are denoted by the Miller indices hkl if, when one plane is drawn through an atom at the unit cell origin, the intercepts of the next plane of the set on the axes are a/h. b/ko c/l. It is easy to see in the case of rectangular axes, that the p e r p e n d i c u l a r distance between planes of the hkl set is dhld where lld2hkl = (h2/a 2) + (k2/b 2) + (121c2). (20) But this is exactly the square of the distance from the reciprocal lattice origin to the hkl reciprocal lattice point and the direction of the p e r p e n d i c u l a r to the hkl lattice planes is the direction from the origin to the hkl reciprocal lattice point. This relationship holds also for nonorthogonal axes, although then the relationship (20) becomes more complicated. Hence the condition that a sharp diffraction b e a m should be generated is, from (19) and (20), Iql

or, since Ik01 = 20, we have

Ikl = I/~,

=

lldhk.t,

and the angle between

2dhkl sin0h~

= ~,

k and k0 is (21)

which is Bragg's law. The condition that q should be p e r p e n d i c u l a r to the lattice planes is equivalent to Bragg's concept of a "reflection" in the optical sense from the planes, subject to (21). We refer for convenience to the strong w e l l - d e f i n e d diffracted beams given under these conditions as "Bragg reflections".

6.2.2.

Shape

transforms

The Bragg or Laue conditions refer to the Ewald sphere passing exactly through a reciprocal lattice point. Our more d e t a i l e d discussion of diffraction conditions suggests that, for m o n o c h r o m a t i c incident plane waves, the diffracted intensity for this case would be given by a near-planar section of the distribution of scattering power, IS(u) 12, given a weighting IFhl 2. If, in practice, the divergence of the incident beam, finite acceptance angle of the detector and wavelength spread of the incident radiation together define a scattering volume, as described in the last chapter, which is very much larger in size than IS(U)1 2 r then the observed intensity may correspond to a t h r e e - d i m e n s i o n a l integration over the peak of scattering power and so be proportional to IFhl 2. A l t e r n a t i v e l y a thin "scattering disc"

128

DIFFRACTION

FROM CRYSTALS

CH.

6

may be swept through the peak at a constant rate so that the o b s e r v e d intensity integrated over time is proportional to the integral over iS(u) J21Fhl 2 and is proportional to IFhl 2. For X-rays and neutrons the peaks of scattering power are very sharp. The kinematical scattering approximation m a y apply for perfect crystal regions several thousand times the dimensions of the unit cell so that the dimensions of the IS(u) 12 d i s t r i b u t i o n are several thousand times smaller than the reciprocal lattice unit cell dimensions. Let us take as a representative value a halfwidth of iS(u) 12 of 1/2000 ~-I. An incident b e a m divergence of 10 -3 radians gives the Ewald sphere a thickness of about 1/2000 ~-i for a reciprocal lattice point with i/dh = 0.5 A -I and the wavelength spread of the incident b e a m adds to this. Hence even for a single crystal region of this size and for favorable d i f f r a c t i o n conditions it would not be feasible to see anything of the details of the function iS(u) 12. Only an integrated i n t e n s i t y could be recorded. However with readily available sources of radiation the intensity scattered by such a small region is too small to be observed. The crystals normally used for diffraction under n e a r - k i n e m a t i c a l conditions are imperfect and may be thought of as very large numbers of small regions of this order of size, or a "mosaic" of small crystals, having a spread of orientations of perhaps 10 -2 to 10 -3 radians. Then we may consider the Ewald sphere to be cutting an aggregate of a very large number of slightly rotated reciprocal lattice configurations or, equivalently, we may consider the reciprocal space distribution for a single average crystal cut by a f u r t h e r - b r o a d e n e d Ewald sphere. This makes it even more nearly certain that only an integration over the scattering power peak is seen and no information on individual IS(u) i2 functions is attainable (Chapter 16). 6.2.3.

Special

cases

for e l e c t r o n

diffraction

For electron diffraction the situation is quite different. Crystals giving near-kinematical intensities are normally a few hundred ~ in size in at least the one direction parallel to the incident beam. Sources are sufficiently bright to allow diffraction from single crystals of this size to be observed readily and the m o n o c h r o m a t i z a tion and collimation give a b r o a d e n i n g of the Ewald sphere with an angular spread of as little as 10 -5 radians. Thus for a reflection with I/dh = 0.5 ~-i the extent of the shape

t r a n s f o r m function may be 10 -2 A -I or more while the thickness of the Ewald sphere may be as little as 5 X 10 -6 A -I. Thus n e a r - p l a n a r sections of the scattering power peak are frequently observed. Fig. 6.1 is a portion of a d i f f r a c t i o n pattern from small needle-like crystals of ZnO (Rees and

6.2

DIFFRACTION

GEOMETRY

129

Fig. 6.1. Portion of a diffraction pattern (enlargement of spots on one ring) from small needle-like crystals of ZnO, showing shape-transform modulation of the scattering power (Rees and Spink [1950]). Spink [1950]). The limitation of crystal size in the direction p e r p e n d i c u l a r to the beam gives rise to an extension of the peak of scattering power in the plane of the Ewald sphere. The m o d u l a t i o n of the intensity corresponding to the (sin2x)/x 2 form of IS(u) i2 is clearly seen in the spots from several individual needle crystals. (The intensity variation is usually m o d i f i e d by dynamical effects but for these p a r t i c u l a r cases this is not very obvious.) It follows that in electron diffraction special techniques of specimen preparation or of intensity recording are required in order to obtain integrated intensities proportional, in the kinematical approximation, to IF(h) 12. For example, the techniques for crystal structure analysis by electron d i f f r a c t i o n developed in the USSR (Pinsker [1953], Vainshtein [1964], Vainshtein et al. [1992]) depend m o s t l y on the use of o r i e n t e d p o l y c r y s t a l l i n e specimens having a random d i s t r i b u t i o n of orientations about one axis so that each reciprocal lattice spot is spread into an annular ring and the section of this by the Ewald sphere gives an integrated intensity. Single crystal patterns are often obtained from extended thin sheets of crystal, of the order of i00 ~ thick but p o s s i b l y microns in diameter. Inevitably these thin sheets are often bent. This again provides an integration over the scattering power maximum, although the observed intensities are proportional to IF(h) 12 only if the bending is s u f f i c i e n t l y u n i f o r m or sufficiently random to ensure that all crystal orientations are equally represented. A further consequence of the difference in geometry between the X-ray and electron cases is the difference in the number of d i f f r a c t e d beams p r o d u c e d at any one time. For X-rays, even with the spread of the maxima of scattering

130

DIFFRACTION

FROM CRYSTALS

CH.

6

AI tile Bragg angle for (400) reflection of gold Electrons 80 KV thin crystal plate X--ray Cu -Ks

t

. . l:;wald . . spllere

ko

I I

io

o'

0"60

4o '

----V--

~

I

I

phere

I

Fig. 6.2. Ewald sphere diagram for CuK~ X-rays and for 80 keV electrons for a crystal of gold or aluminum when the Bragg condition is satisfied for the 400 reflection. The crystal dimensions in the beam direction are assumed to be I000 ~ for the X-rays and 50 A for the electrons. p o w e r or the E w a l d sphere w h i c h we have discussed, the p r o b a b i l i t y that a s t r o n g r e f l e c t i o n s h o u l d occur for any p a r t i c u l a r i n c i d e n t b e a m o r i e n t a t i o n is low for c r y s t a l s w i t h small unit cells. If a s t r o n g r e f l e c t i o n does occur, it is u n l i k e l y that a s e c o n d one will be generated. For e l e c t r o n s on the o t h e r hand, the E w a l d sphere n o r m a l l y cuts a n u m b e r of the e x t e n d e d s c a t t e r i n g - p o w e r regions and for p a r t i c u l a r o r i e n t a t i o n s the n u m b e r of d i f f r a c t e d beams m a y be large. This is i l l u s t r a t e d in Fig. 6.2 d r a w n a p p r o x i m a t e l y to scale for the d i f f r a c t i o n of CuK~ X-rays and 80 keV e l e c t r o n s f r o m c r y s t a l s of g o l d or a l u m i n u m with the B r a g g r e f l e c t i o n c o n d i t i o n s a t i s f i e d for the 400 r e c i p r o c a l lattice p o i n t in each case. For the X - r a y s c a t t e r i n g the p e r f e c t c r y s t a l regions are a s s u m e d to be i000 A or m o r e in size. For e l e c t r o n s the c r y s t a l is a s s u m e d to be a thin f i l m 50 thick. It is seen that for electrons, e s p e c i a l l y when the t h i r d d i m e n s i o n is t a k e n into account, the n u m b e r of s i m u l t a n e o u s l y d i f f r a c t e d beams is quite large. If the v o l t a g e of the e l e c t r o n b e a m is i n c r e a s e d and the w a v e l e n g t h is c o r r e s p o n d ingly reduced, the E w a l d sphere b e c o m e s more n e a r l y p l a n a r and the n u m b e r of r e f l e c t i o n s for such an o r i e n t a t i o n i n c r e a s e s rapidly, e s p e c i a l l y for the v o l t a g e range g r e a t e r t h a n about 200 keY for w h i c h r e l a t i v i s t i c e f f e c t s b e c o m e important. If the thin c r y s t a l s u s e d in e l e c t r o n d i f f r a c t i o n are bent, the r o t a t i o n of the r e c i p r o c a l lattice r e l a t i v e to the E w a l d s p h e r e e n s u r e s that m a n y m o r e d i f f r a c t e d beams are p r o d u c e d , g i v i n g the d i f f r a c t i o n p a t t e r n the a p p e a r a n c e of a

6.2

DIFFRACTION

GEOMETRY

131

Fig. 6.3. Electron diffraction pattern from a thin, slightly bent crystal of Mo5014 with the incident beam nearly parallel to the c-axis. c o m p l e t e s e c t i o n of the r e c i p r o c a l lattice as in Fig. 6.3, a d i f f r a c t i o n p a t t e r n o b t a i n e d from a thin bent crystal with the incident b e a m a p p r o x i m a t e l y p e r p e n d i c u l a r to the hkO r e c i p r o c a l lattice plane. O b v i o u s l y this d i f f r a c t i o n p a t t e r n contains i n f o r m a t i o n on the unit cell g e o m e t r y and s y m m e t r y of the crystal. Also, p r o v i d e d that a k i n e m a t i c a l a p p r o x i m a t i o n can be used, the relative i n t e n s i t i e s of the spots should allow values of [Fh[ 2 to be d e r i v e d as a basis for the study of crystal structures. The p o s s i b i l i t i e s in this d i r e c t i o n have been r e v i e w e d by C o w l e y [1967] and Dorset [1995].

6.3.

Crystal

structure

analysis

6.3. I. The phase problem The p r i m a r y object of the analysis structure of a s u b s t a n c e not p r e v i o u s l y

of the crystal s t u d i e d is to find

132

DIFFRACTION

FROM CRYSTALS

CH.

6

the positions of the centers of the atoms, as given by the maxima in the electron density function p(r) and to identify the atoms present from the relative weights of the maxima. As secondary objectives to be achieved by use of more accurate data and more extended analysis, one may