PHYSICAL PROPERTIES OF

CRYSTALS

THEIR R E P R E S E N T A T I O N B Y

TENSORS A N D MATRICES

By

J. F. N Y E , F.R.S.

CLARENDON PRESS•OXFORD

CONTENTS

N O T A T I O N xiii I N T R O D U C T I O N xv

P A R T 1. G E N E R A L P R I N C I P L E S

I. T H E G R O U N D W O R K O F C R Y S T A L P H Y S I C S

1. Sealars, vectors and tensors of the second rank 3 2. Transformations 8 3. Definition of a tensor 14 4. The representation quadric 16 5. The effect of crystal symmetry on crystal properties 20 6. The magnitude of a property in a given direction 24 7. Geometrical properties of the representation quadric 26 Summary 30

II. TRANSFORMATIONS AND SECOND-RANK TENSORS: FURTHER DEVELOPMENTS 1. Transformation of axes 33 2. Vector produet. Polar and axial vectors 38 3. The prineipal axes of a tensor 41 4. The Mohr circle construetion 43 5. The magnitude ellipsoid 47 Summary 48

P A R T 2. E Q U I L I B R I U M P R O P E R T I E S

I I I . P A R A M A G N E T I C A N D D I A M A G N E T I C S U S C E P T I B I L I T Y

1. General relations 53 2. The energy associated with a magnetized crystal 57 3. Couples and forces 60 4. The suseeptibility of a powder 66 Summary 66

IV. E L E C T R I C P O L A R I Z A T I O N

1. General relations 68 2. Differences between electric polarization and magnetization 70 3. The relations between D, E and P in a parallel plate condenser 72 4. The energy of a polarized crystal 74 5. The force and couple on a crystal in an electric field 74 6. The electrostatic field in a homogeneous anisotropic dielectric 75 Summary of §§ 1-6 77

x CONTENTS

7. Pyroelectricity 78

8. Ferroelectricity 80

Summary of §§ 7, 8 81

V. T H E S T R E S S T E N S O R

1. The notion of stress 82

2. Proof that the <7y form a tensor 87

3. The stress quadric 89

4. Special forms of the stress tensor 90

5. DifTerence between the stress tensor and tensors representing erystal properties 91

Summary 92

V I . T H E S T R A I N T E N S O R A N D T H E R M A L E X P A N S I O N

1. One-dimensional strain 93

2. Two-dimensional strain 94

3. Three-dimensional strain 98

4. Strain and erystal symmetry 104

Summary of §§ 1-4 105

5. Thermal expansion 106

Summary of § 5 109

V I I . P I E Z O E L E C T R I C I T Y . T H I R D - R A N K T E N S O R S

1. The direct piezoelectric effect 110

2. Reduction in the number of-independent moduli. Matrix notation 113

3. The converse piezoelectric effect 115

4. Reduction in the number of independent moduli by erystal symmetry \ 116

5. Results for all the erystal classes 122

6. Representation surfaces 126 Summary 130

V I I I . E L A S T I C I T Y . F O U R T H - R A N K T E N S O R S

1. Hooke's law 131

2. The matrix notation 134

3. The energy of a strained erystal 136

4. The effect of erystal symmetry 137

5. Representation surfaces and Young's Modulus 143

6. Volume and linear compressibility of a erystal 145

7. Relations between the compliances and the stiffnesses 147

8. Numerical values of the elastic coefficienta 147

Summary 148

CONTENTS

I X . T H E M A T R I X M E T H O D

1. The matrix and tensor notations 150

2. Matrix algebra 150

3. Crystal properties in matrix notation 153

4. Two derived matrices 155

5. The magnitude of a second-rank tensor property in an arbitrary direction 157

6. Rotation of axes 157

7. Examples of matrix calculations 158

Summary 168

X. T H E R M O D Y N A M I C S OF E Q U I L I B R I U M P R O P E R T I E S OF C R Y S T A L S

1. The thermal, electrical and mechanical properties of a crystal 170

2. Thermodynamics of thermoelastic behäviour 173

3. Thermodynamics of thermal, electrical and elastic properties 178

4. Relations between coefficients measured linder different conditions 183

Summary 191

PART 3. TRANSPORT PROPERTIES

X L T H E R M A L A N D E L E C T R I C A L C O N D U C T I V I T Y

1. The thermal conductivity and resistivity tensors 1Ö,">

2. Two special cases of steady heat flow 197

3. Steady-state heat flow in general 200

4. Electrical conductivity 204

5. The reciprocal relation fcy = kj, 205

6. Thermodynamical arguments. Onsager's Principle 207

Summary 212

X I I . T H E R M O E L E C T R I C I T Y 1. Thermoelectric effects in isotropic conductors 215 2. Thermoelectric effects in isotropic continuous media 218 3. Thermoelectric effects in crystals 224 Summary 230

P A R T 4. C R Y S T A L O P T I C S

X I I I . N A T U R A L A N D A R T I F I C I A L D O U B L E R E F R A C T I O N . S E C O N D - O R D E R E F F E C T S

1. Double refraction 235

2. The electro-optical and photoelastic effects 241

3. Second-order effects in general 254 Summary 258

xü CONTENTS

X I V . OPTICAL ACTIVITY 1. Introduction 260 2. Optical activity and birefringence 263 3. The principle of superposition 266 4. The size of the effect 268 5. The tensor character of [git] 269 6. The effect of crystal symmetry on the git 270 Summary 273

APPENDIXES A. Summary of vector notation and formulae 275 B. The symmetry of crystals and Conventions for the choice of azes 276 C. Summary of crystal properties 289 D. The number of independent coefficients in the 32 crystal classes 293 E. Matrices for equilibrium properties in the 32 crystal classes 295 F. Magnetic and electrical energy 302 G. The difference between the clamped and free isothermal permittivities 304 H. Proof of the indicatrix properties from Maxwell's equations 306

BIBLIOGRAPHY 310

SUPPLEMENTARY R E F E R E N C E S A N D NOTES (1985) 313

SOLUTIONS TO T H E E X E R C I S E S WITH NOTES 320

INDEX OF NAMES 323

INDEX OF SUBJECTS 324

NOTE Fio. 13.7 is reproduced, by permission, from Hartshorne and Stuart's Crystals and thepolarismg microscope (Edward Arnold (Publishers) Ltd.).