Multipole Theory in Electromagnetism Classical, quantum, and symmetry aspects, with applications R. E. RAAB О. L. DE LANGE School of Chemical and Physical Sciences, University of Natal, Pietermaritzburg, South Africa CLARENDON PRESS-OXFORD 2005
CONTENTS Classical multipole theory 1 1.1 Multipole expansion for the potential of a finite static charge distribution 1 1.2 Dependence of electric multipole moments on origin 5 1.3 Permanent and induced multipole moments 6 1.4 Force and torque in an external electrostatic field 7 1.5 Potential energy of a charge distribution in an electrostatic field 8 1.6 Multipole expansion for the vector potential of a finite distribution of steady current 10 1.7 Dependence of magnetic multipole moments on origin 12 1.8 Force and torque in an external magnetostatic field 13 1.9 Potential energy of a current distribution in a magnetostatic field 14 1.10 Multipole expansions for the dynamic scalar and vector potentials 15 1.11 The far- and near-zone limits 17 1.12 Macroscopic media 18 1.13 Maxwell's macroscopic equations: multipole forms for D and H 23 1.14 Discussion 25 1.15 Primitive moments versus traceless moments 27 1.15.1 A charge distribution 27 1.15.2 Macroscopic media 28 References 29 Quantum theory of multipole moments and polarizabilities 32 2.1 Semi-classical quantum mechanics 32 2.2 Electrostatic perturbation 33 2.3 Buckingham's derivation of electrostatic multipole moments 37 2.4 Magnetostatic perturbation 38 2.5 Time-dependent fields: standard gauge 40 2.6 Time-dependent fields: the Barron-Gray gauge 45 2.7 Polarizabilities for harmonic plane wave fields 47 2.8 Absorption of radiation 50 2.9 Additional static magnetic polarizabilities 51 2.10 Symmetries 52 ix
X CONTENTS 2.11 Macroscopic multipole moment and polarizability densities 53 2.12 Phenomenology of the wave-matter interaction 54 References 56 Space and time properties 59 3.1 Coordinate transformations 59 3.2 Vectors 60 3.3 Cartesian tensors 62 3.4 Time reversal 65 3.5 The space and time nature of various tensors 67 3.6 Symmetry and property tensors 70 3.7 Origin dependence of polarizability tensors 75 3.8 A pictorial determination of symmetry conditions 79 3.9 Discussion 82 References 82 Linear constitutive relations from multipole theory 84 4.1 Constitutive relations 84 4.2 Origin independence 86 4.3 Symmetries 86 4.4 The "Post constraint" 90 4.5 Comparison with direct multipole results 92 4.5.1 Electric dipole order 92 4.5.2 Electric quadrupole-magnetic dipole order 93 4.5.3 Electric octopole-magnetic quadrupole order 94 4.6 Discussion 95 References 98 Transmission and scattering effects: direct multipole results 100 5.1 The wave equation 100 5.2 Intrinsic Faraday rotation in a ferromagnetic crystal 103 5.3 Natural optical activity 106 5.4 Time-odd linear birefringence in magnetic cubics 110 5.5 Optical properties in the Jones calculus 111 5.6 Gyrotropic birefringence 112 5.7 Linear birefringence in non-magnetic cubic crystals (Lorentz birefringence) 115 5.8 Intrinsic Faraday rotation in magnetic cubics 118 5.9 The Kerr effect in an ideal gas 120 5.10 Forward scattering theory of the Kerr effect 124 5.11 Birefringence induced in a gas by an electric field gradient: forward scattering theory 127 5.11.1 Forward scattering by a molecule 128
CONTENTS XI 5.11.2 Induced moments 129 5.11.3 Forward scattering by a lamina 130 5.11.4 The electrostatic field 131 5.11.5 Radiated field for linearly polarized light 132 5.11.6 Field-gradient-induced birefringence 133 5.11.7 Comparison between theory and experiment 135 5.12 Birefringence induced in a gas by an electric field gradient: wave theory 136 5.13 Discussion 140 References 141 6 Reflection effects: direct multipole results 145 6.1 Reflection and the reflection matrix 145 6.2 The principle of reciprocity 147 6.3 Equations of continuity 150 6.4 Matching conditions in multipole theory 153 6.5 The reflection matrix for non-magnetic uniaxial and cubic crystals 156 6.6 Solutions of the wave equation 162 6.7 Reflection coefficients 165 6.8 Tests of translational and time-reversal invariance 168 6.9 Discussion 169 References 170 7 Transformations of the response fields and the constitutive tensor 172 7.1 Gauge transformations of the 4-vector potential 172 7.2 "Gauge transformations" of response fields 173 7.3 Faraday transformations 174 7.4 Transformations of linear constitutive relations in multipole theory 174 References 177 8 Applications of the gauge and Faraday transformations 178 8.1 Electric dipole order 178 8.2 Electric quadrupole-magnetic dipole order, non-magnetic medium 180 8.3 Electric quadrupole-magnetic dipole order, magnetic medium 183 8.4 Discussion 186 References 189 9 Transmission and reflection effects: transformed multipole results 191 9.1 The wave equation and transmission 191
Xll CONTENTS 9.2 Reflection from non-magnetic uniaxial and cubic crystals 192 9.3 Explicit results for non-magnetic uniaxial crystals 195 9.4 Explicit results for non-magnetic cubic crystals 197 9.5 Tests of translational and time-reversal invariance 198 9.6 Reflection from antiferromagnetic СГ2О3: first configuration 199 9.7 Reflection from antiferromagnetic СГ2О3: second configuration 202 9.8 Comparison with experiment for СГ2О3 205 9.9 Uniqueness of fields 206 9.10 Summary 206 References 210 A Transformations involving J 211 В Magnetostatic field 213 С Magnetostatic force 214 D Magnetostatic torque 215 E Integral transformations 216 F Origin dependence of a polarizability tensor 218 G Invariance of transformed tensors 220 Glossary of symbols 221 Index 229