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PROPERTIES OF MATERIALS This page intentionally left blank Properties of Materials Anisotropy, Symmetry, Structure ROBERT E. NEWNHAM Pennsylvania State University 1 3 Great Clarendon Street, Oxford OX26DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2005 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2005 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0-19-852075-1 (hbk) ISBN 0-19-852076-x (pbk) 10987654321 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Antony Rowe, Chippenham Preface This book is about anisotropy and structure–property relationships. Tensors and matrices provide the mathematical framework, and symmetry is helpful in determining which coefficients are absent, and which must be equal, but they say nothing about the sizes of the property coefficients. Magnitudes depend more on atomistic arguments. I have tried to point out some of the crystallochem- ical parameters (such as bond lengths, coordination numbers, and electronic structure) that correlate with property coefficients. These relationships provide a qualitative understanding of the molecular mechanisms which underlie the choice of materials for various engineering applications. The book contains 32 chapters and about 370 pages. It covers a wide range of topics and is suitable as an introduction to the physical properties of materials. The use of tensors and matrices provides a common theme which ties the topics together. I have taught the course at the advanced undergraduate level and to beginning graduate students. Instructors can select topics for a one semester course or use the entire book for a two-semester course. The only prerequisites for the course are college-level physics and chemistry. Such courses are commonly offered during the first year or two at American Universities. No special training in tensor or matrix algebra is required, but knowledge of basic crystallography is helpful. In teaching the course and in writing the book, I had the following questions in mind. How do physical properties depend on direction? How are these properties described mathematically, and what do the geometric representations look like? How do matrix representations differ from the tensor descriptions? How do polar tensor properties differ from axial tensor properties? And what determines the tensor rank? How are the property coefficients measured and how are they influenced by measurement conditions such as temperature, frequency, pressure, and external fields? How does symmetry influence the physical properties? Which coefficients are zero by symmetry, which are equal, and how many measurements are required to specify the property? How is anisotropy related to crystal structure and texture? Are there chains or layers in the structure that correlate with the directional properties? How do the magnitudes of the tensor components depend on bond lengths, bond strengths, and chemical composition? vi Preface What are the most important engineering applications? What combination of properties are involved in the “figure of merit”? How might future improvements be made in the properties? The goal of this process—a process some call “Molecular Engineering”—is the optimization of materials and devices through an understanding of structure– property relationships. Many, many colleagues and friends helped me prepare this book: The faculty, staff, and students at the Pennsylvania State University and the Hong Kong Polytechnic University were especially helpful. The sustained support of the Office of Naval Research is also gratefully acknowledged. I dedicate this book to my wife Patricia, the mother of our two wonderful children, and my lifelong companion in the House of Love and Good Suppers. Contents 1 Introduction 1 1.1 Outline 1 1.2 Structure–property relationships 3 1.3 Symmetry of physical properties 4 1.4 Atomistic arguments: Density 5 2 Transformations 9 2.1 Why transformations? 9 2.2 Axis transformations 9 2.3 Orthogonality conditions 10 2.4 General rotation (Eulerian angles) 12 3 Symmetry 14 3.1 Symmetry operations 14 3.2 Symmetry elements and stereographic projections 15 3.3 Point groups and their stereograms 17 3.4 Crystallographic nomenclature 20 3.5 Point group populations 20 4 Transformation operators for symmetry elements 23 4.1 Transformation operators for the crystallographic symmetry elements 23 4.2 Transformation operations for the thirty-two crystal classes 25 4.3 Standard settings 26 4.4 Curie group symmetries 26 5 Tensors and physical properties 30 5.1 Physical properties 30 5.2 Polar tensors and tensor properties 31 5.3 Axial tensor properties 32 5.4 Geometric representations 33 5.5 Neumann’s Principle 34 5.6 Analytical form of Neumann’s Principle 34 6 Thermodynamic relationships 37 6.1 Linear systems 37 6.2 Coupled interactions: Maxwell relations 38 6.3 Measurement conditions 40 viii Contents 7 Specific heat and entropy 43 7.1 Heat capacity of solids 43 7.2 Lattice vibrations 46 7.3 Entropy and the magnetocaloric effect 48 8 Pyroelectricity 50 8.1 Pyroelectric and electrocaloric tensors 50 8.2 Symmetry limitations 50 8.3 Polar axes 52 8.4 Geometric representation 53 8.5 Pyroelectric measurements 54 8.6 Primary and secondary pyroelectric effects 54 8.7 Pyroelectric materials 55 8.8 Temperature dependence 55 8.9 Applications 57 9 Dielectric constant 58 9.1 Origins of the dielectric constant 58 9.2 Dielectric tensor 60 9.3 Effect of symmetry 62 9.4 Experimental methods 63 9.5 Geometric representation 67 9.6 Polycrystalline dielectrics 69 9.7 Structure–property relationships 69 10 Stress and strain 72 10.1 Mechanical stress 72 10.2 Stress transformations 74 10.3 Strain tensor 75 10.4 Matrix transformation for strain 77 11 Thermal expansion 79 11.1 Effect of symmetry 79 11.2 Thermal expansion measurements 81 11.3 Structure–property relations 82 11.4 Temperature dependence 85 12 Piezoelectricity 87 12.1 Tensor and matrix formulations 87 12.2 Matrix transformations and Neumann’s Law 89 12.3 Piezoelectric symmetry groups 91 12.4 Experimental techniques 93 12.5 Structure–property relations 94 12.6 Hydrostatic piezoelectric effect 97 12.7 Piezoelectric ceramics 99 12.8 Practical piezoelectrics: Quartz crystals 100 13 Elasticity 103 13.1 Tensor and matrix coefficients 103 Contents ix 13.2 Tensor and matrix transformations 105 13.3 Stiffness-compliance relations 106 13.4 Effect of symmetry 107 13.5 Engineering coefficients and measurement methods 109 13.6 Anisotropy and structure–property relations 110 13.7 Compressibility 113 13.8 Polycrystalline averages 114 13.9 Temperature coefficients 116 13.10 Quartz crystal resonators 118 14 Magnetic phenomena 122 14.1 Basic ideas and units 122 14.2 Magnetic structures and time reversal 124 14.3 Magnetic point groups 125 14.4 Magnetic axial vectors 130 14.5 Saturation magnetization and pyromagnetism 131 14.6 Magnetic susceptibility and permeability 134 14.7 Diamagnetic and paramagnetic crystals 135 14.8 Susceptibility measurements 137 14.9 Magnetoelectricity 138 14.10 Piezomagnetism 142 14.11 Summary 146 15 Nonlinear phenomena 147 15.1 Nonlinear dielectric properties 147 15.2 Nonlinear elastic properties 148 15.3 Electrostriction 151 15.4 Magnetostriction 153 15.5 Modeling magnetostriction 154 15.6 Magnetostrictive actuators 159 15.7 Electromagnetostriction and pseudopiezoelectricity 160 16 Ferroic crystals 162 16.1 Free energy formulation 162 16.2 Ferroelasticity 165 16.3 Ferromagnetism 168 16.4 Magnetic anisotropy 170 16.5 Ferroelectricity 174 16.6 Secondary ferroics: Ferrobielectricity and ferrobimagnetism 177 16.7 Secondary ferroics: Ferrobielasticity and ferroelastoelectricity 179 16.8 Secondary ferroics: Ferromagnetoelectrics and ferromagnetoelastics 182 16.9 Order parameters 183 17 Electrical resistivity 188 17.1 Tensor and matrix relations 188 x Contents 17.2 Resistivity measurements 189 17.3 Electrode metals 191 17.4 Anisotropic conductors 193 17.5 Semiconductors and insulators 194 17.6 Band gap and mobility 196 17.7 Nonlinear behavior: Varistors and thermistors 199 17.8 Quasicrystals 202 18 Thermal conductivity 203 18.1 Tensor nature and experiments 203 18.2 Structure–property relationships 206 18.3 Temperature dependence 208 18.4 Field dependence 210 19 Diffusion and ionic conductivity