37 Springer Series in Solid-State Sciences Edited by Hans-Joachim Queisser

Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde K. von Klitzing H.-J. Queisser

40 Semiconductor Physics An Introduction 3rd Edition By K. Seeger

41 The LMTO Method Muffin-Tin Orbitals and Electronic Structure By H. L. Skriver

42 Crystal Optics with Spatial Dispersion, and Excitons By V.M. Agranovich and V.L. Ginzburg

43 Resonant NonIiuear Interactions of Light with Matter By V. S. Butylkin, A. E. Kaplan, Yu. G. Khronopulo, and E.I. Yakubovich

44 Elastic Media with Microstructure II Three-Dimensional Models By I. A. Kunin

45 Electronic Properties of Doped Semicouductors By B. I. Shklovskii and A. L. Efros

46 Topological Disorder in Condensed Matter Editors: F. Yonezawa and T. Ninomiya

47 Statics and Dynamics of Nonlinear Systems Editors: G. Benedek, H. Bilz, and R. Zeyher

48 Magnetic Phase Transitions Editors: M. Ausloos and R. J. Elliott

49 Organic Molecular Aggregates, Electronic Excitation and Interaction Processes Editors: P. Reineker, H. Haken, and H.C. Wolf

50 Multiple Diffraction of X-Rays in Crystals By Shih-Lin Chang

51 Phonon Scattering in Condensed Matter Editors: W. Eisenmenger, K. LaBmann, and S. D6ttinger

52 Superconductivity in Magnetic and Exotic Materials Editors: T. Matsubara and A. Kotani

53 Two-Dimensional Systems, Heterostructures, and Superlattices Editors: G. Bauer, F. Kuchar, and H. Heinrich

54 Magnetic Excitations and Fluctuations Editors: S. Lovesey, U. Balucani, F. Borsa, and V. Tognetti

55 The Theory of Magnetism n Thermodynamics and Statistical Mechanics ByD.C. Mattis

56 Spin Fluctuations in Itinerant Electron Magnetism By T. Moriya

57 Polycrystalline Semiconductors, Physical Properties and Applications Editor: G. Harbeke

Volumes 1-39 are listed on the back inside cover

58 The Recursion Method and Its Applications Editors: D. Pettifor and D. L. Weaire

59 Dynamical Processes and Ordering on Solid Surfaces Editors: A. Yoshimori and M. Tsukada

60 Excitonic Processes in Solids By M. Ueta, H. Kanzaki, K. Kobayashi, Y. Toyozawa, and E. Hanamura

61 Localization, Interaction, and Transport Phenomena Editors: B. Kramer, G. Bergmann, and Y. B ruynseraede

62 Theory of Heavy Fermions and Valence Fluctuations Editors: T. Kasuya and T. Saso

63 Electronic Properties of Polymers and Related Compounds Editors: H. Kuzmany, M. Mehring, and S. Roth

64 Symmetries in Physics: Group Theory Applied to Physical Problems By W. Ludwig and C. Falter

65 Phonons: Theory and Experiments n Experiments and Interpretation of Experimental Results By P. Briiesch

66 Phonons: Theory and Experiments III Phenomena Related to Phonons By P. Briiesch

67 Two-Dimensional Systems: Physics and New Devices Editors: G. Bauer, F. Kuchar, and H. Heinrich

68 Phonon Scattering in Condensed Matter V Editors: A. C. Anderson and J. P. Wolfe

69 Nonlinearity in Condensed Matter Editors: A. R. Bishop, D. K. Campbell, P. Kumar and S.E. Trullinger

70 From Hamiltonians to Phase Diagrams The Electronic and Statistical-Mechanical Theory of sp-Bonded Metals and Alloys By J. Hafner

71 High Magnetic Fields in Semiconductor Physics Editor: G. Landwehr

72 One-Dimensional Conductors By S. Kagoshima, T. Sambongi, and H. Nagasawa

73 Quantum Solid-State Physics Editors: S. V. Vonsovsky and M. I. Katsnelson

74 Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems Editor: M. Suzuki

Modem Crystallography IV Physical Properties of Crystals

Editor: L. A. Shuvalov

With Contributions by L. A. Shuvalov, A. A. Urusovskaya, 1. S. Zheludev, A. V. Zalessky, S. A. Semiletov, B. N. Grechushnikov, I. G. Chistyakov and S. A. Pikin

With 270 Figures and 65 Tables

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Professor Dr. L. A. Shuvalov Professor Dr. l. S. Zheludev Professor Dr. S. A. Semiletov Dr. l. G. Chistyakov

Dr. A. A. Urusovskaya Dr. A. V. Zalessky Professor Dr. B. N. Grechushnikov Dr. S. A. Pikin

Institute of Crystallography, Academy of Sciences of the USSR, 59 Leninsky prospect, SU-117333 Moscow, USSR

Series Editors:

Professor Dr. Manuel Cardona Professor Dr. Peter Fulde Professor Dr. Klaus von Klitzing Professor Dr. Hans-Joachim Queisser

Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1 D-7000 Stuttgart 80, Fed. Rep. of Germany

Title of the original Russian edition: Sovremennaja kristallografija © by "Nauka" Publishing House, Moscow 1981

Library of Congress Cataloging-in-Publication Data. (Revised for volume 4) Modern crystallography. (Springer series in solid-state sciences; 15, 21, 36- 37) Translation of Sovremennaja kristallografija. Includes bibliographies and indexes. Contents: 1. Valnshteln, B. K. Symmetry of crystals. Methods of structural crystallography. - 2. Valnshteln, B. K., Fridkin, V. M., Indenbom, V. L., Structure of crystals. - [etc.] - 4. Physical properties of crystals I L. A. Shuvalov. 1. Crystallography. 1. Valnshteln, B. K. (Boris Konstantinovich), 1921- . II. Series: Springer series in solid-state sciences; 15, etc. QD905.2.S6813 548 80-17797 ISBN-13: 978-3-642-81840-0 e-ISBN-13: 978-3-642-81838-7 001: 10.1007/978-3-642-81838-7

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this pUblication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin, Heidelberg 1988 Softcover reprint ofthe hardcover 1st edition 1988

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Modern Crystallography in Four Volumes *

I Symmetry of Crystals. Methods of Structural Crystallography

II Structure of Crystals

III Crystal Growth

IV 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).

Early in this century, the newly discovered x-ray diffraction by crystals made a complete change in crystallography and in the whole science of the

* Published in Springer Series in Solid-State Sciences, I: Vol. 15; II: Vol. 21; III: Vol. 36; IV: Vol. 37

VI Foreword

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 many 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. Man-made 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 radio engineering 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 of the 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 (of particles and quasi particles)

Crystal physics (electrical, mechan I 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 is 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.

VIII Foreword

Crystal physics deals mainly with the electrical, optical, and mechanical properties of crystals closely related to their stucture and symmetry, and 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 crystal lattices.

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 authors 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 of 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 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 higher technical colleges studying crystallography, solid-state physics, and related subjects.

Modern Crystallography is written by a large group of authors 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 authors 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

Preface

This fourth volume of Modern Crystallography deals with the physical prop­erties of crystals. It completes the presentation of the science of crystallo­graphy, the other components of which - the theory of symmetry of crystals, their structure, and crystallization - are elucidated in the preceding three volumes. The present volume contains a systematic expose of the modern conceptions of the mechanical, electrical, magnetic, and optical properties of crystals, and of transport phenomena in them.

In distinction to textbooks and monographs on solid-state physics, the authors attempted to render the material in the crystallographic key. In partic­ular, they took into account the symmetry (including the space symmetry) in analyzing the properties of crystals and the associated anisotropy of the crystalline substance, and also paid considerable attention to the interrelation of the properties with the atomic and real structure of particular crystals and with the conditions of their growth. The basic notions of the electron and phonon spectra of crystals are covered by the second volume; therefore, the present book treats the relevant aspects within a relatively narrow scope.

The introductory chapter describes the tensor apparatus of crystal physics and considers the general problems of symmetry of the physical properties of crystals. The second chapter is devoted to the mechanical properties of crys­tals. Along with the usual description of the elastic properties of crystals, it provides detailed data on their plastic deformation, mechanical twinning, and failure.

The electrical and electromechanical properties of crystals are considered in the third chapter. Special attention is given to ferroelectrics - the most important class of dielectric crystals - and to their properties. The fourth chapter studies the magnetic properties of crystals. There, magnetically ordered crystals are investigated comprehensively, mainly from the crystallo­graphical viewpoint. The physical properties of semiconductor crystals, as well as the operating principles of semiconductor diodes, transistors, and lasers, are described in the fifth chapter.

The sixth chapter is dedicated to transport phenomena; in addition to elec­trical and thermal conductivity, it discusses various thermoelectrical, galvano­and thermomagnetic properties of crystals. The optical properties of crystals are described in the seventh chapter. Together with a nonconventional presen­tation of the traditional and new branches of crystal optics, it elucidates the

X Preface

problems of optical and radiospectroscopy and the generation of coherent radiation. The last chapter deals with the properties of liquid crystals, a rapidly developing branch of crystal physics.

Despite the considerable effort taken in editing, the great number of con­tributors to this volume proved an obstacle in avoiding some divergency in style, depth of presentation, and even symbols in the different chapters. We hope, nevertheless, that the book will provide a sufficiently comprehensive picture of the present-day state of the art in crystal physics and, taken together with the other volumes, of the entire crystallographical science.

In the first three volumes the general bibliography (monographs, surveys, and the most important works) is given for the volume as a whole, while the special literature (original papers) is referred to the separate chapters. In view of the numerous general sources concerning the properties of crystals cited in the present volume, we refer to them also in the respective chapters.

The book has been written by a group of research workers at the Institute of Crystallography of the U.S.S.R. Academy of Sciences: L.A. Shuvalov, A.A. Urusovskaya, I.S. Zheludev, A.V. Zalessky, S.A. Semiletov, B.N. Orechushnikov, 1.0. Chistyakov, S.A. Pikin. The authors deeply appreciate the kind help of V.A. Koptsik, I.M. Silvestrova, M. V. Klassen-Nekhlyu­dova, V.L. Indenbom, D.O. Sannikov, A.P. Levanyuk, A.F. Konstan­tinova, T. F. Veremeichik, I. N. Kalinkina, O. V. Kachalov, L. Li, and others, who took part in the writing and discussion of various chapters.

Moscow, September 1987 L. A. Shuva/ov

Contents

1. Fundamentals of Tensor and Symmetry Description of the Physical Properties of Crystals (L. A. Shuvalov) .......................... 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Crystal as a Continuous Homogeneous Anisotropic Medium.......................................... 1

1.1.2 Cartesian Coordinate System and Its Transformations ... 2 1.2 Tensors and Their Transformations ......................... 4

1.2.1 Scalars, Pseudoscalars, Vectors, and Tensors . . . . . . . . . . . 4 1.2.2 Transformation of the Components of Vectors and

Second-Rank Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Tensors of Different Ranks .......................... 7 1.2.4 Pseudotensors (Axial Tensors) ....................... 8 1.2.5 Symmetric and Antisymmetric Tensors. Interior

Symmetry of Tensors ............................... 9 1.2.6 Reciprocal Tensors ................................. 12 1.2.7 Matrix Notation ................................... 13

1.3 Geometric Interpretation of Tensors. Indicatrice Surfaces ...... 14 1.3.1 Characteristic Surface for a Symmetric Second-Rank

Tensor ........................................... 14 1.3.2 Principal Axes of a Symmetric Second-Rank Tensor ..... 15 1.3.3 Property in a Given Direction. . . . . . . . . . . . . . . . . . . . . . .. 16 1.3.4 Geometric Properties of the Characteristic Surface of a

Symmetric Second-Rank Tensor. . . . . . . . . . . . . . . . . . . . .. 18 1.3.5 Finding the Principal Axes of a Symmetric Second-Rank

Tensor ........................................... 20 1.3.6 Other Indicatrice Surfaces for a Second-Rank Tensor. . .. 21 1.3.7 Indicatrice Surfaces for Higher-Than-Second-Rank

Tensors. . . ... ... . . . ... . . .... ... .. ... ... .. .. ... .... 23 1.4 Proper (Exterior) Symmetry of Tensors ...................... 24

1.4.1 General Notions of the Proper Symmetry of Tensors. . . .. 24 1.4.2 Limiting Groups of Symmetry ....................... 25 1.4.3 Proper Symmetry of Scalars, Pseudoscalars,

and Vectors ....................................... 26 1.4.4 Proper Symmetry of Second-Rank Polar Tensors ....... 27 1.4.5 Proper Symmetry of Second-Rank Pseudotensors ....... 29

XII Contents

1.4.6 Proper Symmetry of Higher-Rank Tensors. . . . . . . . . . . .. 30 1.5 Symmetry of Physical Properties. . . . . . . . . . . . . . . . . . . . . . . . . . .. 30

1.5.1 Material and Field Tensors .......................... 30 1.5.2 Crystallophysical System of Coordinates. . . . . . . . . . . . . .. 31 1.5.3 Relationship Between the Proper Symmetry of Tensors

and the Symmetry of the Physical Properties Described by Them ............................................ 33

1.5.4 Relationship Between the Point Symmetry of a Crystal and the Symmetry of Its Physical Properties ............... 35

1.5.5 Form in Different Coordinate Systems of the Matrices of Tensors Describing the Physical Properties of Crystals of Different Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38

1.5.6 Determining the Number of Independent Components of Tensors Describing the Physical Properties of Crystals ... 43

1.5.7 The Curie Principle and Its Application. . . . . . . . . . . . . . .. 46

2. Mechanical Properties of Crystals (A. A. Urusovskaya) ............ 50

2.1 Elastic Properties of Crystals ............................... 50 2.1.1 Stress ............................................ 50 2.1.2 Characteristic Stress Surface. . . . . . . . . . . . . . . . . . . . . . . .. 54 2.1.3 Strain............................................ 54 2.1.4 Characteristic Surface and the Strain Ellipsoid . . . . . . . . .. 58 2.1.5 Hooke's Law for Crystals ........................... 60 2.1.6 Matrix Notation of Coefficients of Elasticity ........... 61 2.1.7 Effect of the Crystal Symmetry on the Tensor of

Coefficients of Elasticity ............................ 62 2.1.8 Hooke's Law for an Isotropic Body.. .. . ... . ... .... ... 66 2.1.9 Young's Modulus in an Arbitrary Direction and the

Characteristic Surfaces of Elastic Properties . . . . . . . . . . .. 68 2.1.10 Adiabatic and Isothermal Values of Elasticity Coefficients 69 2.1.11 Volume Compressibility of Crystals ................... 71 2.1.12 Born Theory ...................................... 71 2.1.13 Cauchy Ratio ...................................... 72 2.1.14 Elastic Waves in Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . .. 73

2.2 Plastic Glide Deformation ................................. 77 2.2.1 Elastic Deformation in Crystals ...................... 77 2.2.2 Translational Glide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77 2.2.3 Plastic Strain Tensor ............................... 78 2.2.4 Glide Elements .................................... 81 2.2.5 Slip Lines ......................................... 85 2.2.6 Polar Glide ....................................... 87 2.2.7 Critical Shear Stress Law ............................ 87 2.2.8 Independent Glide Systems .......................... 90 2.2.9 Cases of Complex Slipping .......................... 91

Contents XIII

2.2.10 Stages of Glide Deformation. . . . . . . . . . . . . . . . . . . . . . . .. 92 2.2.11 Theoretical Shear Strength .......................... 94

2.3 Dislocation Description of Plastic Deformation by Glide ....... 95 2.3.1 Dislocations as the Result of Shear Strain .............. 95 2.3.2 Dislocation Structure of Slip Lines and the Movement of

Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96 2.3.3 Force Acting on a Dislocation. .. . . . . . . . . .. . . .. . . . .. .. 97 2.3.4 Dislocation Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99 2.3.5 Lattice Resistance to the Dislocation Movement ......... 101 2.3.6 Multiplication of Dislocations ........................ 103 2.3.7 Interaction Between Dislocations .... " ............... 106 2.3.8 Dislocation Reactions ............................... 108 2.3.9 Interaction of Dislocations with Point Defects .......... 110 2.3.10 Types ofInteraction of Dislocations with Impurities ..... 112 2.3.11 Precipitation Particles .............................. 116 2.3.12 Internal Friction ................................... 118 2.3.13 Anisotropy of the Internal Friction ................... 120 2.3.14 Theories of Dislocation Internal Friction ............... 121 2.3.15 Mechanisms of Overcoming Impurity Obstacles ......... 123

2.4 Diffusion Mechanisms of Plastic Deformation ................ 128 2.4.1 GorskyEffect ..................................... 128 2.4.2 Diffusion Creep ................................... 129 2.4.3 Climb of the Edge Dislocations ....................... 130 2.4.4 Diffusion-Dislocation Creep ......................... 132 2.4.5 Creep Stages ...................................... 133 2.4.6 Radiation-Induced Diffusion Creep ................... 134 2.4.7 Stress Relaxation ................................... 135

2.5 Mechanical Twinning of Crystals ........................... 136 2.5.1 Twinning of Crystals ............................... 136 2.5.2 Representation of Twinning with a Change in Shape as a

Uniform Shear .................................... 137 2.5.3 Polarity of Twinning ............................... 139 2.5.4 Twinning Elements ................................. 139 2.5.5 Types of Twinning ................................. 140 2.5.6 Transformation of Indices of Planes and Directions in

Twinning ......................................... 142 2.5.7 Energy Conditions for the Formation of Twins Causing a

Change in the Shape of Crystals ...................... 142 2.5.8 Twin Nucleation ................................... 144 2.5.9 Twinning Without a Change in Crystal Shape ........... 146 2.5.10 Behavior of Crystals with Polysynthetic Twins .......... 149 2.5.11 Martensitic Transformations ......................... 151

2.6 Fracture ................................................ 152 2.6.1 Theoretical and Actual s.trength ...................... 152 2.6.2 Cleavage Plane .................................... 153

XIV Contents

2.6.3 Fracture Surface .............. , .................... 154 2.6.4 Law of Critical Normal Stress . . . . . . . . . . . . . . . . . . . . . . .. 156 2.6.5 Role of Plastic Deformation in the Preparation of Failure

Nuclei ............................................ 157 2.6.6 Griffith's and Orowan's Criteria of Growth of Nucleus

Cracks ........................................... 161 2.6.7 "Viscous" and "Brittle" Cracks ...................... 162 2.6.8 Joffe Effect ....................................... 163 2.6.9 Rebinder Effect .................................... 164

2.7 Methods for Studying the Mechanical Properties of Crystals 164 2.7.1 Hardness Measurement ............................. 165 2.7.2 Investigating the Elastic Properties of Crystals .......... 169 2.7.3 Investigating the Plastic Properties of Crystals . . . . . . . . .. 170 2.7.4 Studying the Ability of Crystals to Undergo Failure ...... 173 2.7.5 Long-Term Strength ................................ 176

3. Electrical Properties of Crystals (I. S. Zheludev) .................. 178 3.1 Polarization, Electrical Conductivity, and Dielectric Losses ..... 178

3.1.1 General ........................................... 178 3.1.2 Main Types of Polarization .......................... 180 3.1.3 Electrical Conductivity .............................. 184 3.1.4 Dielectric Losses ................................... 186

3.2 Pyroelectric Phenomena ................................... 188 3.2.1 General ........................................... 188 3.2.2 Pyroelectric Effect ................................. 189 3.2.3 Electrocaloric Effect ................................ 191

3.3 Piezoelectric Effect and Electrostriction ...................... 192 3.3.1 Piezoelectric Effect ................................. 192 3.3.2 Electromechanical Transformation ................... 194 3.3.3 Piezoelectric Properties of Linear Dielectrics ........... 199 3.3.4 Electrostriction.................................... 203 3.3.5 Piezoelectric Textures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 207

3.4 Domain Structure and Peculiarities of the Electrical Properties of Ferroelectrics and Antiferroelectrics ............. 208 3.4.1 General........................................... 208 3.4.2 Domains.......................................... 209 3.4.3 Polarization, Electrical Conductivity, and Dielectric

Losses ............................................ 212 3.4.4 Piezoelectric Properties ............................. 215 3.4.5 Antiferroelectrics .................................. 216

3.5 Structure and Properties of Some Ferroelectrics and Antiferroelectrics ......................................... 217 3.5.1 Barium Titanate ................................... 217 3.5.2 Potassium Dihydrophosphate . . . . . . . . . . . . . . . . . . . . . . .. 225 3.5.3 Rochelle Salt ...................................... 229

Contents XV

3.5.4 Triglycinesulphate ................................. 234 3.5.5 Antiferroelectrics .................................. 239

3.6 Phase Transition in Ferroelectrics. Fundamentals of Spontaneous Polarization Theory ........................... 241 3.6.1 General ........................................... 241 3.6.2 Thermodynamic Theory of Phase Transitions in

Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242 3.6.3 Macroscopic Models ................................ 248 3.6.4 SublattiCe Polarization and Phase Transitions in

Antiferroelectrics .................................. 251 3.6.5 Crystal Lattice Dynamics and Phase Transitions in

Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 254 3.6.6 Incommensurate Phase Transitions in Ferroelectrics ..... 262

4. Magnetic Properties of Crystals (A. V. Zalessky) .................. 267 4.1 Disordered Magnetics ..................................... 267

4.1.1 Basic Relations Characterizing the Behavior of a Substancein a Magnetic Field ........................ 267

4.1.2 Tensors of Diamagnetic and Paramagnetic Susceptibility . 269 4.1.3 Classification of Magnetic Substances ................. 270 4.1.4 Diamagnetism..................................... 270 4.1.5 Paramagnetism .................................... 272

4.2 Ordered Magnetics ....................................... 275 4.2.1 Different Types of Magnetic Structures in Crystals.

Ferromagnetism, Antiferromagnetism, and Ferrimagnetism ................................ 275

4.2.2 Magnetic Symmetry ................................ 278 4.2.3 Basic Types of Interaction in Ordered Magnetics ........ 281 4.2.4 Molecular Field Theory. The Curie and Neel Points ..... 283

4.3 Domain Structure of Ferromagnetic Crystals and Magnetization Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 289 4.3.1 Change of Symmetry at the Curie Point.

Symmetry Aspects of Splitting of a Ferromagnetic into Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 289

4.3.2 Spontaneous Magnetostriction. Magnetoelastic Energy .. 293 4.3.3 Energy Aspects of Domain Structure. Domain Walls .... 295 4.3.4 Methods for Observing the Domain Structure.

Examples of Domain Structures ...................... 298 4.3.5 Magnetization Processes ............................ 301 4.3.6 Magnetic Hysteresis ................................ 305

4.4 Anisotropy of Ferromagnetic Crystals ....................... 308 4.4.1 Peculiarities of the Tensor Description of Ferromagnetic

Crystals .......................................... 308 4.4.2 Magnetostriction Anisotropy in Ferromagnetics of

Different Symmetry ................................ 311

XVI Contents

4.4.3 Magnetic Anisotropy Energy Corresponding to Zero Strains and Zero Stresses ....................... 319

4.4.4 Equilibrium Directions of Spontaneous Magnetization ... 322 4.4.5 Magnetic Anisotropy Measurement ................... 324

4.5 Structure of Some Magnetically Ordered Crystals and Reorientation Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327 4.5.1 Ferrites ........................................... 327 4.5.2 Weakly Ferromagnetic Crystals

(Canted Antiferromagnetics) ......................... 333 4.5.3 Reorientation Transitions ........................... 340

4.6 Piezomagnetic and Magnetoelectric Effects . . . . . . . . . . . . . . . . . .. 343

5. Semiconducting Crystals (S. A. Semiletov) ....................... 348 5.1 Basic Properties of Semiconductors ......................... 348

5.1.1 Metals, Semiconductors, and Insulators ............... 348 5.1.2 Intrinsic and Extrinsic Conductivity ................... 351 5.1.3 Electrical Conductivity of Semiconductors ............. 353 5.1.4 Hall Effect ........................................ 356 5.1.5 Photoconductivity ................................. 359

5.2 Electron-Hole Junctions ................................... 360 5.2.1 Electron Energy Distribution ........................ 360 5.2.2 Semiconductor-Metal Contact ....................... 363 5.2.3 Semiconductor Diodes with p-n Junctions .............. 366 5.2.4 Semiconductor Triode (Transistor) ................... 369 5.2.5 Semiconductor Laser ............................... 370 5.2.6 Photocells with a p-n Junction ....................... 373

6. Transport Phenomena in Crystals (S. A. Semiletov) ................ 376 6.1 Electrical Conductivity of Crystals .......................... 376

6.1.1 Electrical Conductivity and Resistivity Tensors ......... 376 6.1.2 Crystal Conductivity in a Given Direction .............. 379

6.2 Thermal Conductivity of Crystals ........................... 381 6.2.1 Tensors of the Thermal Conductivity and

Thermal Resistivity Coefficients ... . . . . . . . . . . . . . . . . . .. 381 6.2.2 Onsager's Principle ................................. 382

6.3 Thermoelectric Effects .................................... 385 6.3.1 Definition of Thermal Effects ........................ 385 6.3.2 Thermoelectric Effects in an Isotropic Medium ......... 386 6.3.3 Thermoelectric Effects in Crystals .................... 389 6.3.4 Dependence of the Tensor of the Thermoelectric

Coefficients on the Crystal Symmetry ................. 390 6.3.5 Peltier, Thomson, and Bridgman Effects .............. 390 6.3.6 Thermoelectric Effects in Technology ................. 393

6.4 Galvano- and Thermomagnetic Effects ...................... 393 6.4.1 Onsager's Principle in the Presence of a Magnetic Field .. 393

Contents XVII

6.4.2 Hall, Righi-Leduc, Nernst, and Ettingshausen Effects .. 394 6.4.3 Tensors of the Hall, Righi-Leduc, Nernst, and

Ettingshausen Coefficients and of Magnetoresistance for Crystals of Different Classes ........................ 398

7. Optical Properties of Crystals (B. N. Grechushnikov) .............. 405 7.1 Plane Electromagnetic Waves in an Anisotropic Medium ...... 405

7.1.1 Dielectric Constants of Crystals ..................... 405 7.1.2 Plane Waves in Transparent Crystals ................. 407 7.1.3 Optical Surfaces .................................. 409

7.2 Uniaxial and Biaxial Crystals .............................. 410 7.2.1 UniaxiaICrystals .................................. 410 7.2.2 Biaxial Crystals ................................... 412

7.3 Birefringence in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 417 7.3.1 Birefringence of Plane Waves at the Interface of

Two Media ....................................... 417 7.3.2 Light Reflection from Transparent Crystals ........... 418

7.4 Light Interference in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 420 7.4.1 Elliptical Polarization ............................. 420 7.4.2 Light Interference in Parallel Beams ................. 423 7.4.3 Conoscopic Patterns in Uniaxial Crystals ............. 425 7.4.4 Conoscopic Patterns in Biaxial Crystals. . . . . . . . . . . . . .. 427 7.4.5 Measurements of Refractive Indices of Crystals ........ 430

7.5 Light Absorption in Crystals .............................. 433 7.6 Optical Activity of Crystals ............................... 436 7.7 Electro-optical Properties of Crystals . . . . . . . . . . . . . . . . . . . . . .. 442 7.8 Magneto-optical Properties of Crystals ..................... 447 7.9 Piezo-optical Properties of Crystals ........................ 449 7.10 Light Scattering in Crystals ............................... 459 7 .11 Nonlinear Optical Properties of Crystals .................... 465 7.12 Essentials ofthe Crystal-Field Theory ...................... 470

7.12.1 Crystal-Field Model ............................... 470 7.12.2 A Single d Electron in a Cubic Field .................. 476 7.12.3 Ion with Configuration 3d2 in a Weak Cubic Field ..... 481 7.12.4 Strong Crystal Field ............................... 484 7.12.5 Calculating the Spin-Orbital Interaction of

Configuration d 2 in a Cubic Field . . . . . . . . . . . . . . . . . . .. 491 7.13 Laser Crystals .......................................... 496 7.14 Polarized Luminescence in Crystals ........................ 501 7.15 Electron Paramagnetic Resonance of Impurity Ions in

Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 506

8. Liquid Crystals (1. G. Chistyakov, S. A. Pikin) .................... 513 8.1 Basic Characteristics of Liquid Crystals ..................... 513 8.2 Texture and Optical Properties of Liquid Crystals ............ 516

XVIII Contents

8.2.1 Smectic Liquid Crystals ............................. 516 8.2.2 Nematic Liquid Crystals ............................ 520 8.2.3 Cholesteric Liquid Crystals .......................... 523 8.2.4 Lyotropic Liquid Crystals ........................... 528

8.3 Theory of Liquid-Crystal State ............................. 529 8.3.1 Thermodynamic States of Liquid Crystals .............. 530 8.3.2 Supercooled Liquid-Crystal Phases ................... 536 8.3.3 Two-Dimensional Ordering and Plastic SLC ........... 538 8.3.4 Orientational Distortions of Liquid Crystals ............ 540 8.3.5 Polarized States of Liquid Crystals .................... 544 8.3.6 Electrohydrodynamic Phenomena in Liquid Crystals .... 550

8.4 Magnetic Properties of Liquid Crystals ...................... 554 8.5 Electrical Properties of Liquid Crystals ...................... 556

8.5.1 Dielectric Properties of Liquid Crystals . . . . . . . . . . . . . . .. 556 8.5.2 Transverse Domains in Nematics with a Negative

Dielectric Anisotropy ............................... 557 8.5.3 Effect of Dynamic Light Scattering ................... 558 8.5.4 Effect of Deformation of a Nematic Liquid-Crystal Layer 561 8.5.5 Longitudinal Domains in Substances with a Negative

Dielectric Anisotropy ..... . . . . . . . . . . . . . . . . . . . . . . . . .. 562 8.5.6 Domains in Liquid Crystals with a Positive Dielectric

Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 563 8.5.7 Behavior of Cholesteric and Smectic Liquid Crystals in an

Electric Field ...................................... 563 8.6 Thermal Properties of Liquid Crystals ....................... 564

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 570

Subject Index .................................................. 579