Modern Crystallography 1

Modern Crystallography 1

Springer-Verlag Berlin Heidelberg GmbH

Modern Crystallography

Volume 1: Fundamentals of Crystals Symmetry, and Methods of Structural Crystallography

Volume 2: Structure of Crystals

Volume 3: Crystal Growth

Volume 4: Physical Properties of Crystals

Boris K. Vainsthein

Fundamentals of Crystals Symmetry, and Methods of Structural Crystallography

Second, Enlarged Edition

With 330 Figures and 4 Plates in Color

, Springer

Professor Dr. Boris K. Vainsthein Institute of Crystallography, Russian Academy of Sciences, Leninsky prospect 59, 117333 Moscow, Russia

2nd enlarged Edition 1994 Corr. Printing 1996

ISBN 978-3-642-08153-8

Cip-data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Modern crystallography: [in 4 vo!.]/[ed. board: B.K. Vainsthein (ed.-in-chief)]

Einheitssacht.: Sovremennaja kristallografija <engl.> NE: Vajnstejn. Boris K. [Hrsg.]; EST VO!.1. Vajnstejn. Boris K.: Fundamentals of crystals.-2 .• en!. ed .• 2. print. - 1996 Vajnstejn, Boris K.: Fundamentals of crystals : symmetry and methods of structural crystallography / Boris K. Vainshtein. - 2 .• en!. ed .• 2. print.

(Modern crystallography; VoI. 1)

1. Aufl. u.d.T.: Vajnstejn. Boris K.: Symmetry of crystals. methods of structural crystallography ISBN 978-3-642-08153-8 ISBN 978-3-662-02975-6 (eBook) DOI 10.1007/978-3-662-02975-6

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Modern Crystallography in Four Volumes·

1 Fundamentals of Crystals Symmetry, and Methods of Structural Crystallography

2 Structure of Crystals

3 Crystal Growth

4 Physical Properties of Crystals

Editorial Board: B. K. Vainshtein (Editor-in-Chief) A. A. Chernov L. A. Shuvalov

Foreword

Crystallography - the science of crystals - has undergone many changes in the course of its development. Although crystals have intrigued mankind since ancient times, crystallography as an independent branch of science began to take shape only in the 17th-18th centuries, when the principal laws governing crystal habits were found, and the birefringence of light in crystals was dis­covered. From its very origin crystallography was intimately connected with mineralogy, whose most perfect objects of investigation were crystals. Later, crystallography became associated more closely with chemistry, because it was apparent that the habit depends directly on the composition of crystals and can only be explained on the basis of atomic-molecular concepts. In the 20th century crystallography also became more oriented towards physics, which found an ever-increasing number of new optical, electrical, and mechanical phenomena inherent in crystals. Mathematical methods began to be used in crystallography, particularly the theory of symmetry (which achieved its classical completion in space-group theory at the end of the 19th century) and the calculus of tensors (for crystal physics).

* English editions have originally been published in Springer Series in Solid-State Sciences. Vois. 15 (I). 21 (II). 36 (III) and 37 (IV).

VI Foreword

Early in this century, the newly discovered x-ray diffraction by crystals caused a complete change in crystallography and in the whole science of the atomic structure of matter, thus giving a new impetus to the development of solid-state physics. Crystallographic methods, primarily x-ray diffraction analysis, pene­trated into materials sciences, molecular physics, and chemistry, and also into may other branches of science. Later, electron and neutron diffraction structure analyses became important since they not only complement x-ray data, but also supply new information on the atomic and the real structure of crystals. Electron microscopy and other modern methods of investigating matter -optical, electronic paramagnetic, nuclear magnetic, and other resonance tech­niques - yield a large amount of information on the atomic, electronic, and real crystal structures.

Crystal physics has also undergone vigorous development. Many remark­able phenomena have been discovered in crystals and then found various practical applications.

Other important factors promoting the development of crystallography were the elaboration of the theory of crystal growth (which brought crystallography closer to thermodynamics and physical chemistry) and the development of the various methods of growing synthetic crystals dictated by practical needs. Man­made crystals became increasingly important for physical investigations, and they rapidly invaded technology. The production of synthetic crystals made a tremendous impact on the traditional branches: the mechanical treatment of materials, precision instrument making, and the jewelry industry. Later it considerably influenced the development of such vital branches of science and industry as radiotechnics and electronics, semiconductor and quantum elec­tronics, optics, including nonlinear optics, acoustics, etc. The search for crystals with valuable physical properties, study of their structure, and development of new techniques for their synthesis constitute one of the basic lines of contempor­ary science and are important factors of progress in technology.

The investigation of the structure, growth, and properties of crystals should be regarded as a single problem. These three intimately connected aspects of modern crystallography complement each other. The study, not only ofthe ideal atomic structure, but also of the real defect structure of crystals makes it possible to conduct a purposeful search for new crystals with valuable properties and to improve the technology of their synthesis by using various techniques for controlling their composition and real structure. The theory of real crystals and the physics of crystals are based on their atomic structure as well as on the theoretical and experimental investigations of elementary and macroscopic processes of crystal growth. This approach to the problem of the structure, growth, and properties of crystals has an enormous number of aspects, and determines the features of modern crystallography.

The branches of crystallography and their relation to adjacent fields can be represented as a diagram showing a system of interpenetrating branches which have no strict boundaries. The arrows show the relationship between the branches, indicating which branch influences the activity of the other, although, in fact, they are usually interdependent.

Computational mathematics

Chemistry

/ Crystal chemistry I

Foreword VII

Solid - state physics

Electronic properties Phonon spectrum

Interaction of properties lof particles and quasI particles)

Crystal physics lelectr I cal, mechani cal,

optical and magnetic properties

Materials

Optics

Acoustics

Branches of crystallography and its relation to other sciences

Crystallography proper occupies the central part of the diagram. It includes the theory of symmetry, the investigation of the structure of crystals (together with diffraction methods and crystal chemistry), and the study of the real structure of crystals, their growth and synthesis, and crystal physics.

The theoretical basis of crystallography in the theory of symmetry, which has been intensively developed in recent years.

The study of the atomic structure has been extended to extremely complic­ated crystals containing hundreds and thousands of atoms in the unit cell. The investigation of the real structure of crystals with various disturbances of the ideal crystal lattices has been gaining in importance. At the same time, the general approach to the atomic structure of matter and the similarity of the various diffraction techniques make crystallography a science not only of the structure of crystals themselves, but also of the condensed state in general.

The specific applications of crystallographic theories and methods allow the utilization of structural crystallography in physical metallurgy, materials science, mineralogy, organic chemistry, polymer chemistry, molecular biology, and the investigation of amorphous solids, liquids and gases. Experimental and theoretical investigations of crystal growth and nucleation processes and their development draw on advances in chemistry and physical chemistry and, in turn, contribute to these areas of science.

Crystal physics deals mainly with the electrical, optical, and mechanical properties of crystals closely related to their structure and symmetry, and

VIII Foreword

adjoins solid-state physics. which concentrates its attention on the analysis of laws defining the general physical properties of crystals and the energy spectra of crysta1lattices.

The first two volumes are devoted to the structure of crystals, and the last two, to the growth of crystals and their physical properties. The contributors present the material in such a way that the reader can find the basic information on all important problems of crystallography. Due to the limitation in space the exposition of some sections is concise, otherwise many chapters would have become separate monographs. Fortunately, such books on a number of crystal­lographic subjects are already available.

The purpose of such an approach is to describe all the branches of crystallography in their interrelation, thus presenting crystallography as a unified science to elucidate the physical meaning of the unity and variety of crystal structures. The physico-chemical processes and the phenomena taking place in the course of crystal growth and in the crystals themselves are described, from a crystallographic point of view, and the relationship of properties of crystals with their structure and conditions of growth is elucidated.

This four-volume edition is intended for researchers working in the fields of crystallography, physics, chemistry, and mineralogy, for scientists studying the structure, properties, and formation of various materials, for engineers and those engaged in materials science and technology, particularly in the synthesis of crystals and their use in various technical devices. We hope that this work will also be useful for undergraduate and graduate students at universities and institutions of technology studying crystallography, solid-state physics, and related subjects.

Modern Crystallography has been written by a large group of researchers from the Institute of Crystallography of the USSR Academy of Sciences, who benefited from the assistance and advice of many other colleagues. The English edition of all four volumes of Modern Crystallography is being published almost simultaneously with the Russian edition. The contributors have included in the English edition some of the most recent data. In several instances some additions and improvements have been made.

B. K. Vainshtein

Foreword to the Second Editions

The 4-volume treatment Modern Crystallography was first published in the early eighties. Crystallography is centuries-old, and its basic concepts and laws are well established. However, the rapid progress in all the branches of science at the end of our century cannot by-pass- crystallography. Our knowledge on the atomic structure of matter, on the formation and growth of crystals and their physical properties are becoming ever deeper, and the experimental methods are being constantly improved. Therefore, to justify the series title -Modern Crystallography - we had to enrich the treatment and complement it with new data.

Most ofthe First Edition has largely has been preserved, but several sections were up-dated, the text somewhere improved and complemented with some new illustrations. At the same time, numerous new results that could not be neglected have been summarized in the updating chapter concluding each volume. Indeed, we could not leave out quasicrystaIs, the high-temperature superconductors, discovered in the eighties, the development in molecular-beam epitaxy, surface melting, improper ferroelectrics, incommensurate phases, and many other topics. We have also presented several novel techniques which became widely used in crystallography - e. g., tunneling microscopy, EX AFS, position-sensitive detectors for x-rays, etc. The list of references has also been revised and supplied with new publications.

The Second Editions of Modern Crystallography were prepared mainly by the contributors of the first editions. In addition, we have been assisted by other colleagues with their notes, new figures and references. We take this opportunity to sincerely thank all of them.

B. K. Vainshtein, A. A. Chernov, L. A. Shuvalov

Preface to Modern Crystallography 1

Modem Crystallography 1 describes the general characteristics of the crystalline state of matter, considers crystal symmetry, and describes the methods for investigating the crystal structure.

The introductory chapter deals with the basic concepts of crystallography and the characteristics of the crystalline state of matter. It studies macroscopic features of a crystalline substance: homogeneity, anisotropy, and symmetry of properties; it also considers crystal habit, the basic regularities of the micro­scopic atomic structure of crystals, and differences between the structures of crystals and of other condensed media.

Chapter 2, which encompasses almost half of the volume, is devoted to a systematic presentation of the symmetry of crystals. The theory of symmetry penetrates all the crystallography, and without it one can neither study nor understand the structure and properties of crystals. The axiomatics of the theory of symmetry is given with group theory as its foundation; the basic concepts are treated geometrically. Point one-dimensional, plane, and space groups are considered, as well as generalizations of symmetry - anti symmetry and color symmetry.

Chapter 3 treats the theory of the geometric description of crystal habit and the geometric theory of a crystal lattice.

Chapter 4 is devoted to experimental methods for studying the atomic structure of crystals. Main attention is given to x-ray diffraction analysis, which is the most important tool for studying structures. This chapter discusses the general diffraction theory, experimental technique, and the fundamentals of the theory and methods of using diffraction analysis to determine the atomic structures of crystals.

The chapter also describes two other related methods - electron and neutron diffraction structure analysis, their specifics, potentialities, and limitations. It gives a brief exposition of other new methods for analysis of the structure of matter: M6ssbauer diffraction and channeling particles in crystals. The final section covers electron microscopy.

Almost all of the volume was written by B. K. Vainshtein, Chapter 3 in co-operation with M. O. Kliya, and Section 4.3, with Z. G. Pinsker; Sections 4.5 and 4.6 were prepared by D. M. Kheiker. Many essential suggestions for presenting the material of Chapter 2 were made by V. A. Koptsik, who coauthored Sections 2.6.6 and 2.9. A number of valuable comments and

XII Preface to Modem Crystallography I

refinements were introduced by R. V. Galiulin. The author expresses his sincere gratitude to these colleagues. He also thanks L. A. Feigin, V. V. Udalova, L. I. Man, and many others who helped with the manuscript, the compilation of literature, and the preparation of the figures.

The crystallographic literature is enormous. In this volume and the following ones the references are divided into two categories. The Bibliography consists of basic monographs, review articles, and important original papers relating to the subject ofthe volume. The References consist of publications on separate special problems touched upon in the text, and also the works from which illustrations were borrowed. We also list the basic crystallographic journals and periodicals. Some original photographs were made available specially for this edition. Their authors are acknowledged in the captions. The author thanks all of them sincerely, as weli as those who kindly gave permission for reproduction of pictures from their original papers or books.

Moscow, December 1980 B. K. Vainshtein

Preface to the Second Edition

Fundamentals of Crystals, Symmetry, and Methods of Structural Crystallography represents the second edition of the original text of Modern Crystallography 1. It is supplied with new data on the principles of the structure ofthe crystalline state and methods of its investigation. Several chapters have been revised and supplemented with new material. Two sections of Chapter 4 have been substan­tially revised:

4.8 Electron Diffraction written by B. K. Vainshtein and B. B. Zvyagin, 4.9 Electron Microscopy written by B. K. Vainshtein, N. A. Kiselev, and M. B.

Sherman, 4.10 Scanning Tunneling Microscopy (a new section) written by L. M. Blhiov.

We have also prepared a new, additional chapter reviewing important crystallographic achievements and methods developed in recent years. This chapter 5 consists of five sections:

5.1 Quasicrystals written by V. E. Dmitrienko and B. K. Vainshtein, 5.2 Incommensurately Modulated Structures written by V. E. Dmitrienko, 5.3 Development of Experimental Technique of X-Ray Structure Analysis by

D. M. Kheiker and B. K. Vainshtein, 5.4 X-Ray Studies of Crystal Surface by A. Yu. Kazimirov, 5.5 Methods for the Analysis of Powder Diffraction Patterns by A. A. Loshmanov, 5.6 EXAFS Spectroscopy by A.N. Popov.

The author would like to express his sincere gratitude to V. V. Udalova, I. L. Tolstova, L. I. Man, L. A. Antonova for the great amount of technical work they have done in the preparation of the present volume.

Moscow, November 1993 B. K. Vainshtein

Contents

1. Crystalline State 1.1 Macroscopic Characteristics of Crystals ................. 1

1.1.1 Crystals and Crystalline Matter .................. 1 1.1.2 Homogeneity of a Crystalline Substance . . . . . . . . . . . . 5 1.1.3 Anisotropy of a Crystalline Substance ............. 6 1.1.4 Symmetry.................................... 9 1.1.5 Crystal Habit ................................ 11

1.2 Microstructure of a Crystalline Substance. . . . . . . . . . . . . . . . 12 1.2.1 Space Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Experimental Evidence for the Existence of the Crystal

Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Reasons for the Microperiodicity Principle. . . . . . . . . . 18

1.3 Structures with Distortions of the Three-Dimensional Periodicity. Quasicrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Structural Characteristics of Condensed Phases. . . . . . . . . . . . 24

2. Fundamentals of the Theory of Symmetry 2.1 The Concept of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.1 Definition of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.2 Symmetry Operations. . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Space Transformations .............................. 31 2.2.1 Space, an Object in It, Points of Space. . . . . . . . . . . . . 31 2.2.2 Basic Isometric Transformations of Space .......... 32 2.2.3 Analytical Expression for Symmetry Transformations. . 39 2.2.4 Relationships and Differences Between Operations of

the First and Second Kind .......... . . . . . . . . . . . . 41 2.3 Fundamentals of Group Theory ....................... 44

2.3.1 Interaction of Operations ......... . . . . . . . . . . . . . . 44 2.3.2 Group Axioms ............................... 44 2.3.3 Principal Properties of Groups . . . . . . . . . . . . . . . . . . . 46 2.3.4 Cyclic Groups, Generators. . . . . . . . . . . . . . . . . . . . . . . 47 2.3.5 Subgroup.................................... 48 2.3.6 Cosets, Conjugates, Classes, Expansion with Respect

to a Subgroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.7 Group Products .............................. 49 2.3.8 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . 51

XVI Contents

2.4 Types of Symmetry Groups and Their Properties . . . . . . . . . . 53 2.4.1 Homogeneity, Inhomogeneity, and Discreteness

of Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.2 Types of Symmetry Groups and Their Periodicity .... 55 2.4.3 One-Dimensional Groups G1 .................... 57 2.4.4 Two-Dimensional Groups G2 • . • . . . . • • • . . . • • • . • • • 58 2.4.5 Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.6 Three-Dimensional Groups G3 .. . . . . . . . . . . . . . . . . . 61

2.5 Geometric Properties of Symmetry Groups. . . . . . . . . . . . . . . 64 2.5.1 Symmetry Elements ........................... 64 2.5.2 Summary and Nomenclature of Symmetry Elements. . . 66 2.5.3 Polarity..................................... 71 2.5.4 Regular Point Systems ......................... 71 2.5.5 Independent Region ........................... 73 2.5.6 Description of a Symmetric Object by Groups of

Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.5.7 Enantiomorphism............................. 79

2.6 Point Symmetry Groups ........... . . . . . . . . . . . . . . . . . . 82 2.6.1 Description and Representation of Point Groups . . . . . 82 2.6.2 On Derivation of Three-Dimensional

Point Groups G6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6.3 Point Group Families . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.6.4 Classification of Point Groups ................... 95 2.6.5 Isomorphism of Groups K ...................... 100 2.6.6 Representations of Point Groups K ............... 101 2.6.7 Group Representations and Proper Functions ....... 106

2.7 Symmetry Groups Gi, G~, Gt, G~ ..... . . . . . . . . . . . . . . . .. 107 2.7.1 Symmetry Groups Gi of Borders ................. 107 2.7.2 Plane Twice-Periodic Groups G~ ................. 108 2.7.3 Cylindrical (Helical) Groups Gt. . . . . . . . . . . . . . . . . .. 110 2.7.4 Layer Groups G~. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 116

2.8 Space Groups of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . .. 120 2.8.1 Three-Dimensional Lattice ...................... 120 2.8.2 Syngonies.................................... 122 2.8.3 Bravais Groups .. : . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123 2.8.4 Homomorphism of Space and Point Groups ........ 129 2.8.5 Geometric Rules for Performing Operations

and for Mutual Orientation of Symmetry Elements in Groups cPo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130

2.8.6 Principles of Derivation of Space Groups. Symmorphous Groups cPs . . . . . . . . . . . . . . . . . . . . . .. 131

2.8.7 Nonsymmorphous Groups cPo. . . . . . . . . . . . . . . . . . .. 135 2.8.8 Number of Fedorov Groups. . . . . . . . . . . . . . . . . . . .. 138 2.8.9 Nomenclature of Fedorov Groups ................ 139 2.8.10 Subgroups of Fedorov Groups .. . . . . . . . . . . . . . . . .. 144

Contents XVII

2.8.11 Regular Point Systems of Space Groups. . . . . . . . . . .. 145 2.8.12 Relationship Between the Chemical Formula of a

Crystal and Its Space Symmetry. . . . . . . . . . . . . . . . .. 146 2.8.13 Local Condition of Space Symmetry. . . . . . . . . . . . . .. 147 2.8.14 Division of Space ............................. 149 2.8.15 Irreducible Representations of Groups r/J. • . . . . . . . . .. 155

2.9 Generalized Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156 2.9.1 On the Extension of the Symmetry Concept. . . . . . . .. 156 2.9.2 Antisymmetry and Color Symmetry. . . . . . . . . . . . . .. 157 2.9.3 Antisymmetry Point Groups. . . . . . . . . . . . . . . . . . . .. 159 2.9.4 Point Groups of Color Symmetry. . . . . . . . . . . . . . . .. 168 2.9.5 Space and Other Groups of Antisymmetry and

Color Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171 2.9.6 Symmetry of Similarity. . . . . . . . . . . . . . . . . . . . . . . .. 175 2.9.7 Partial Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177 2.9.8 Statistical Symmetry. Groupoids . . . . . . . . . . . . . . . . .. 177

3. Geometry of the Crystalline Polyhedron and Lattice 3.1 Basic Laws of Geometric Crystallography. . . . . . . . . . . . . . .. 179

3.1.1 Law of Constancy of Angles. . . . . . . . . . . . . . . . . . . .. 179 3.1.2 Law of Rational Parameters. Lattice. . . . . . . . . . . . . .. 180

3.2 Crystalline Polyhedron .............................. 182 3.2.1 Ideal Shape. Bundle of Normals and Edges. . . . . . . . .. 182 3.2.2 Simple Forms ................................ 183 3.2.3 Distribution of Simple Forms Among Classes ....... 189 3.2.4 Holohedry and Hemihedry . . . . . . . . . . . . . . . . . . . . .. 192 3.2.5 Combinations of Simple Forms .................. 192 3.2.6 The Zone Law ............................... 193

3.3 Goniometry....................................... 194 3.3.1 Crystal Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 194 3.3.2 Experimental Technique of Goniometry . . . . . . . . . . .. 198 3.3.3 Goniometric Calculations . . . . . . . . . . . . . . . . . . . . . .. 200

3.4 Lattice Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205 3.4.1 Straight Lines and Planes of the Lattice . . . . . . . . . . .. 205 3.4.2 Properties of Planes ........................... 206 3.4.3 Reciprocal Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 208

3.5 Lattice Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 212 3.5.1 Transformation of Coordinates and Indices in the

Atomic and Reciprocal Lattices .................. 212 3.5.2 Reduction Algorithm .......................... 216 3.5.3 Computation of Angles and Distances in Crystals. . . .. 219

4. Structure Analysis of Crystals 4.1 Fundamentals of Diffraction Theory. . . . . . . . . . . .. . . . . . .. 222

4.1.1 Wave Interference. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 222

XVIII Contents

4.1.2 Scattering Amplitude. . . . . . . . . . . . . . . . . . . . . . . . . .. 224 4.1.3 Electron Density Distribution. Fourier Integral ...... 226 4.1.4 Atomic Amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227 4.1.5 The Temperature Factor. . . . . . . . . . . . . . . . . . . . . . .. 231

4.2 Diffraction from Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 234 4.2.1 Laue Conditions. Reciprocal Lattice . . . . . . . . . . . . . .. 234 4.2.2 Size of Reciprocal Lattice Nodes ................. 237 4.2.3 Reflection Sphere ............................. 239 4.2.4 Structure Amplitude ..... . . . . . . . . . . . . . . . . . . . . .. 241 4.2.5 Intensity of Reflections. . . . . . . . . . . . . . . . . . . . . . . . .. 242 4.2.6 Thermal Diffusion Scattering .................... 244 4.2.7 Symmetry of the Diffraction Pattern and Its

Relation to the Point Symmetry of the Crystal . . . . . .. 244 4.2.8 Manifestation of Space-Symmetry of a Crystal in a

Diffraction Pattern. Extinctions .................. 245 4.3 Intensity of Scattering by a Single Crystal. Kinematic

and Dynamic Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 252 4.3.1 Kinematic Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 252 4.3.2 Integrated Intensity of Reflection in Kinematic

Scattering ................................... 253 4.3.3 Principles of Dynamic Theory. . . . . . . . . . . . . . . . . . .. 256 4.3.4 Darwin's Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . .. 257 4.3.5 Laue-Ewald Treatment . . . . . . . . . . . . . . . . . . . . . . . .. 258 4.3.6 Dynamic Scattering in an Absorbing Crystal.

Borrmann Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 262 4.3.7 Experimental Investigations and Applications

of Dynamic Scattering. . . . . . . . . . . . . . . . . . . . . . . . .. 266 4.4. Scattering by Noncrystalline Substances. . . . . . . . . . . . . . . .. 271

4.4.1 General Expression for Intensity of Scattering. Function of Interatomic Distances ................ 271

4.4.2 Spherically Symmetric Systems: Gas, Liquid, and Amorphous Substances. . . . . . . . . . . . . . . . . . . . .. 272

4.4.3 Systems with Cylindrical Symmetry: Polymers and Liquid Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . .. 274

4.4.4 Small-Angle Scattering ......................... 276 4.5 Experimental Technique of X-Ray Structure Analysis of

Single Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 278 4.5.1 Generation and Properties of X-Rays. . . . . . . . . . . . .. 278 4.5.2 Interaction of X-Rays with a Substance ............ 282 4.5.3 Recording of X-Rays. . . . . . . . . . . . . . . . . . . . . . . . . .. 283 4.5.4 Stages of X-Ray Structure Analysis of Single Crystals.. 284 4.5.5 Laue Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285 4.5.6 Crystal Rotation and Oscillation Methods .. . . . . . . .. 287 4.5.7 Moving Crystal and Film Techniques. . . . . . . . . . . . .. 291

Contents XIX

4.5.8 X-Ray Diffractometers for Investigating Single Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295

4.5.9 Diffractometric Determination of the Crystal Orientation, Unit Cell, and Intensities. . . . . . . . . . . . . . . . . . . . . . .. 297

4.6 X-Ray Investigation of Polycrystalline Materials. . . . . . . . . .. 299 4.6.1 Potentialities of the Method '" . . . . . . . . . . . . . . . . .. 299 4.6.2 Cameras for Polycrystalline Specimens. . . . . . . . . . . .. 300 4.6.3 Indexing of Debye Photographs and Intensity of

Their Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 303 4.6.4 Diffractometry of Polycrystalline Specimens . . . . . . . .. 304 4.6.5 Phase Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 305 4.6.6 Investigation of Textures ...................... " 305 4.6.7 Determination of the Sizes of Crystals and Internal

Stresses ..................................... 307 4.7 Determination of'the Atomic Structure of Crystals . . . . . . . .. 307

4.7.1 Preliminary Data on the Structure . . . . . . . . . . . . . . .. 307 4.7.2 Fourier Synthesis. Phase Problem. . . . . . . . . . . . . . . .. 308 4.7.3 The Trial and Error Method. Reliability Factor. . . . .. 312 4.7.4 The Patterson Interatomic-Distance Function ....... 313 4.7.5 Heavy-Atom Method .......................... 319 4.7.6 Direct Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 321 4.7.7 "Statistical-Thermodynamical" Approach to the

Crystal Structure Determination . . . . . . . . . . . . . . . . .. 325 4.7.8 Nonlocal-Search Method. . . . . . . . . . . . . . . . . . . . . . .. 327 4.7.9 Determination of the Absolute Configuration . . . . . . .. 330 4.7.10 Structure Refinement. . . . . . . . . . . . . . . . . . . . . . . . . .. 331 4.7.11 Difference Fourier Syntheses. . . . . . . . . . . . . . . . . . . .. 332 4.7.12 Automation of the Structure Analysis. . . . . . . . . . . . .. 334

4.8 Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 335 4.8.1 Features of the Method. . . . . . . . . . . . . . . . . . . . . . . .. 335 4.8.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . .. 336 4.8.3 Electron Diffraction Structure Analysis. . . . . . . . . . . .. 338 4.8.4 Dynamic Scattering of Electrons. . . . . . . . . . . . . . . . .. 349 4.8.5 Special Methods of Electron Diffraction . . . . . . . . . . .. 350 4.8.6 Low-Energy Electron Diffraction (LEED) . . . . . . . . . .. 355

4.9 Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 357 4.9.1 The Features of the Methods .................... 357 4.9.2 Transmission Electron Microscopy. . . . . . . . . . . . . . .. 358 4.9.3 HREM Imaging of Atomic Structure of Crystals ..... 368 4.9.4 EM in Molecular Biology. Experimental Technique ... 374 4.9.5 Processing of the Images of Biomolecules.

3D Reconstruction ............................ 376 4.9.6 Two-Dimensional Biocrystals .... . . . . . . . . . . . . . . .. 381 4.9.7 TEM of Single Bioparticles . . . . . . . . . . . . . . . . . . . . .. 382

XX Contents

4.9.8 Scanning Electron Microscopy (SEM) of Solids. . . . . .. 383 4.10 Scanning Tunneling Microscopy. . . . . . . . . . . . . . . . . . . . . .. 385

4.10.1 Principle of Operation. . . . . . . . . . . . . . . . . . . . . . . . .. 385 4.10.2 Basic Construction of an STM ............. . . . . .. 387 4.10.3 Specific Features of Scanning Tunneling Microscopy. .. 388 4.10.4 Atomic Force Microscope. . . . . . . . . . . . . . . . . . . . . .. 389 4.10.5 Several Examples of Surface Images . . . . . . . . . . . . . .. 390

4.11 Neutron Diffraction. Mossbauer Diffraction, and Scattering of Nuclear Particles in Crystals. . . . . . . . . . . . . . . . . . . . . . .. 392 4.11.1 Principles and Techniques of the Neutron

Diffraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. 392 4.11.2 Investigation of the Atomic Structure . . . . . . . . . . . . .. 394 4.11.3 Investigation of the Magnetic Structure ............ 397 4.11.4 Other Possibilities Offered by the Neutron-

Diffraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. 400 4.11.5 Diffraction of Mossbauer Radiation ............... 401 4.11.6 Particle Channeling and the Shadow Effect. . . . . . . . .. 403

5. New Developments in Crystallography and its Methods 5.1 Quasicrystals...................................... 405

5.1.1 The Discovery of Quasicrystals . . . . . . . . . . . . . . . . . .. 405 5.1.2 The Non-Traditional Symmetry of Aperiodic Objects.. 407 5.1.3 The One-Dimensional Quasicrystal (d = 1, N = 2) . . .. 409 5.1.4 Fourier Transform of Quasiperiodic Sequences. . . . . .. 410 5.1.5 Two-Dimensional Quasicrystals and Their

Symmetry (d = 2, N = 3, 4, 5, ... ). . . . . . . . . . . . . . . .. 412 5.1.6 Three-Dimensional Quasicrystals (d = 3, N = 4,5,6, ... ) 415 5.1.7 Structure Analysis of Quasicrystals . . . . . . . . . . . . . . .. 419 5.1.8 Order and Disorder in the Structure of Quasicrystals .. 424

5.2 Incommensurately Modulated Structures. . . . . . . . . . . . . . . .. 427 5.3 Development of Experimental Technique for X-Ray

Structure Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 427 5.3.1 Powerful Sources of X-Ray Radiation. . . . . . . . . . . . .. 427 5.3.2 The Synchrotron Radiation Sources . . . . . . . . . . . . . .. 429 5.3.3 Characteristics of Synchrotron Radiation ........... 429 5.3.4 The Primary SR Beam ......................... 431 5.3.5 Laue Method for Crystallographic Data Acquisition

from Single Crystals ........................... 431 5.3.6 Some Results Obtained with Synchrotron

Radiation Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 432 5.3.7 Diffractometers with Two-Dimensional

Position-Sensitive Detectors ..................... 433 5.4 X-Ray Studies of Crystal Surface. . . . . . . . . . . . . . . . . . . . . .. 438

5.4.1 Surface Diffraction ............................ 440

Contents XXI

5.4.2 The Method of X-Ray Standing Waves. . . . . . . . . . . .. 443 5.5 Methods for Analysis of Powder Diffraction Patterns . . . . . .. 446 5.6 EXAFS Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 451 5.6.1 The Fundamentals of the EXAFS

Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 451 5.6.2 Structure Information Extracted from EXAFS

Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 452 5.6.3 Experimental Methods and Apparatus . . . . . . . . . . . .. 454

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 455

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 467

Subject Index . .............................. , . . . . . . . . . . . . .. 475