Tensor Properties of Crystals, Second Edition

This tenth in the series of conferences provides an overview of recent developments in the wide range of fields covered by sensors, & in-depth coverage of the latest work. It demonstrates the range of applications for sensor & actuator technology in aerospace, remote sensing, environmental sensing, medical / health sensing, automotive sensing, energy monitoring & solar energy. It is a valuable addition to the libraries of researchers in all areas of sensor technology & application.

Preface

ix

List of main symbols

xi

Crystals and crystal symmetry

1

(18)

Structure of solids

1

(1)

Close-packing structures

1

(4)

Two-dimensional lattices

5

(2)

Three-dimensional lattices

7

(4)

Crystallographic indices for planes; Miller indices

11

(2)

Crystallographic direction indices

13

(1)

Worked examples on symmetry

14

(5)

Problems

17

(2)

Introducing tensors

19

(17)

Introduction to the notation

19

(1)

Reducing the number of components

20

(3)

Transformation of axes

23

(1)

Transformation of a vector

24

(1)

Transformation of the coordinates of a point

24

(1)

Transformation and definition of a tensor

25

(2)

Second-rank symmetrical tensors; the representation quadric

27

(4)

Neumann's principle

31

(1)

Some examples of tensors

31

(2)

Worked example showing change of axes for a tensor

33

(3)

Problem

35

(1)

Second-rank tensors; conductivity

36

(14)

Thermal conductivity and thermal resistivity

36

(1)

Heat flow in crystal samples

37

(3)

The radius-normal property of the representation quadric

40

(1)

Electrical conductivity and electrical resistivity

41

(1)

Diffusion

42

(1)

Some worked examples of second-rank tensor properties

43

(7)

Problems

48

(2)

Fourth-rank tensors; elasticity

50

(21)

Introduction

50

(1)

Strain

50

(3)

Symmetrical and antisymmetrical tensors

53

(1)

The strain tensor

54

(2)

Stress

56

(1)

Elasticity

57

(1)

The matrix notation

58

(4)

Effect of crystal symmetry; equating components by inspection

62

(4)

Elasticity components in cubic crystals and polycrystalline samples

66

(2)

Elasticity components in other crystal systems

68

(1)

Worked examples on stress, strain and elasticity

69

(2)

Problems

70

(1)

Crystal optics

71

(17)

Introduction

71

(1)

The indicatrix

72

(4)

The wave surface

76

(3)

Biaxial crystals

79

(2)

Double refraction (birefringence) at a boundary

81

(3)

Worked examples on polarisation and birefringence

84

(4)

Problems

70

(18)

Axial tensors

88

(18)

Definition of an axial tensor

88

(1)

Transformation of axial vectors

88

(3)

Transformation of axial tensors

91

(1)

Optical activity

91

(4)

Optical activity in the presence of birefringence

95

(1)

The gyration tensor; second-rank axial tensor

96

(3)

The Hall effect; third-rank axial tensor

99

(2)

The Hall effect; relationship to symmetry

101

(3)

Magnetoresistance and other effects

104

(2)

Problems

104

(2)

Optoelectronic effects

106

(27)

Introduction

106

(1)

The linear electro-optic effect (the Pockels effect)

107

(2)

The Pockels effect in lithium niobate

109

(3)

The Pockels effect in ADP (ammonium dihydrogen phosphate)

112

(2)

The Pockels effect in gallium arsenide

114

(3)

The quadratic electro-optic effect (the Kerr effect)

117

(1)

The Kerr effect in barium titanate

118

(2)

The Kerr effect in ADP and KDP

120

(1)

Optical switching, intensity modulation and phase modulation

120

(3)

Optical beam deflectors

123

(1)

Non-linear optics and phase matching

124

(2)

Worked examples to show magnitudes in actual materials

126

(7)

Problems

129

(4)

Further tensor applications

133

(13)

Introduction

133

(1)

Thermal expansion

133

(2)

The pyroelectric effect

135

(1)

Piezoelectricity

136

(1)

Photoelasticity

137

(2)

Incommensurate modulated structures

139

(5)

Some concluding comments

144

(2)

Problems

144

(2)

Appendices

146

(7)

1 Number of independent coefficients for tensors

146

(2)

2 Diagonalisation of a tensor

148

(2)

3 Table of tensor properties

150

(3)

References and further reading

153

(3)

Answers to problems

156

(5)

Index

161

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