Quantum Plasmadynamics: Magnetized Plasmas

D B Melrose Quantum plasmadynamics Magnetized plasmas SPIN Springer’s internal project number, if known – Monograph – ...

Author: Donald Melrose

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D B Melrose

Quantum plasmadynamics Magnetized plasmas SPIN Springer’s internal project number, if known

– Monograph – May 21, 2010

Springer

Preface

Volume II of this book extends the theory developed for unmagnetized plasmas in Volume I to unmagnetized plasmas. (Volume I was published by Springer early in 2008.) The writing of this second volume has taken me longer than originally anticipated. I had hoped to complete it in the first half of 2008. However, it turned out that too many gaps needed to be filled in for me to give the logical development of the theory I had planned. I have been filling in these gaps but am now suspending the writing of volume II to concentrate on other research for the second half of 2008. At this time, chapters 1 to 6 are essentially complete, with a few gaps in chapters 4 and 6. There are significant gaps in chapters 7 and 8, and at least half of the planned chapters 9 remains unwritten. Chapter 10 is little more than a collection of notes. References have not yet been included systematically. Don Melrose July 2008

Contents

1

Covariant fluid models for magnetized plasmas . . . . . . . . . . . . 1.1 Covariant description of a magnetostatic field . . . . . . . . . . . . . . . 1.1.1 Maxwell tensor for a magnetostatic field . . . . . . . . . . . . . . µν 1.1.2 Projection tensors g⊥ and gkµν . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The 4-tensor B µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Components for B along the 3-axis . . . . . . . . . . . . . . . . . . 1.1.5 Basis 4-vectors for the magnetized vacuum . . . . . . . . . . . 1.1.6 Basis 4-vectors for a magnetized medium . . . . . . . . . . . . . 1.1.7 Onsager relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Classification of static fields . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Covariant cold plasma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fluid description of a cold plasma . . . . . . . . . . . . . . . . . . . 1.2.2 Weak-turbulence expansion . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Fourier transform of the fluid equations . . . . . . . . . . . . . . 1.2.4 Expansion in powers of the 4-potential . . . . . . . . . . . . . . . 1.2.5 First order current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Tensor τ µν (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Alternative forms for the Π µν (k) . . . . . . . . . . . . . . . . . . . . 1.2.8 Quadratic response tensor for a cold plasma . . . . . . . . . . 1.3 Inclusion of motions along B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Lorentz transformation to streaming frame . . . . . . . . . . . 1.3.2 Π ij (k) for streaming cold plasma . . . . . . . . . . . . . . . . . . . . 1.3.3 Streaming cold distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Multiple streaming cold components . . . . . . . . . . . . . . . . . 1.3.5 Dielectric tensor for a streaming distribution . . . . . . . . . . 1.4 Relativistic magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Covariant form of the MHD equations . . . . . . . . . . . . . . . 1.4.2 Energy-momentum tensor for the fluid . . . . . . . . . . . . . . . 1.4.3 Electromagnetic energy-momentum tensor . . . . . . . . . . . . 1.4.4 Derivation from kinetic theory . . . . . . . . . . . . . . . . . . . . . .

1 3 3 4 4 5 5 6 7 8 9 9 9 10 11 11 12 13 14 15 15 16 16 17 18 20 20 20 21 22

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1.4.5 1.4.6 1.4.7 1.4.8 References 2

Generalized Ohm’s law and infinite conductivity . . . . . . . Two-fluid model for a pair plasma . . . . . . . . . . . . . . . . . . . Lagrangian density for relativistic MHD . . . . . . . . . . . . . . MHD wave modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................

23 24 25 25 27

Response tensors for magnetized plasmas . . . . . . . . . . . . . . . . . . 2.1 Perturbation theory for a spiraling charge . . . . . . . . . . . . . . . . . . 2.1.1 Orbit of a spiraling charge . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Properties of tµν (τ ), t˙µν (τ ) . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Characteristic response due to a spiraling charge . . . . . . 2.1.4 Expansion in Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Gyroresonance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Perturbation expansion of the 4-current . . . . . . . . . . . . . . 2.1.7 Expansion of the 4-current . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 First order current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 Small gyroradius limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General forms for the linear response 4-tensor . . . . . . . . . . . . . . . 2.2.1 Forward-scattering method for a magnetized plasma . . . 2.2.2 Forward-scattering form summed over gyroharmonics . . 2.2.3 Vlasov method for a magnetized plasma . . . . . . . . . . . . . . 2.2.4 Linearized Vlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Vlasov form for the linear response tensor . . . . . . . . . . . . 2.2.6 Vlasov form summed over gyroharmonics . . . . . . . . . . . . . 2.2.7 Explicit forms for the components of Π µν (k) . . . . . . . . . . 2.2.8 Response 3-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Response of a relativistic thermal plasma . . . . . . . . . . . . . . . . . . . 2.3.1 Trubnikov’s response tensor for a magnetized plasma . . 2.3.2 Manifestly gauge-invariant form . . . . . . . . . . . . . . . . . . . . . 2.3.3 Forward-scattering form of Trubnikov’s tensor . . . . . . . . 2.3.4 Relativistic plasma dispersion functions (RPDFs) . . . . . . 2.3.5 Alternative form of Π µν (k) for a J¨ uttner distribution . . 2.3.6 Strictly-parallel distribution . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Strictly-perpendicular thermal distribution . . . . . . . . . . . 2.4 Weakly relativistic thermal plasma . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Nonrelativistic magnetized thermal plasma . . . . . . . . . . . 2.4.2 Response tensor for a Maxwellian distribution . . . . . . . . . 2.4.3 Cold plasma limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Thermal corrections to longitudinal response tensor . . . . 2.4.5 MHD-like limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Nonrelativistic limit of Trubnikov’s tensor . . . . . . . . . . . . 2.4.7 Shkarofsky’s response 3-tensor . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Shkarofsky functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9 Dnestrovskii functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.10 Weakly relativistic dispersion for kz 6= 0 . . . . . . . . . . . . . .

29 30 30 31 32 33 34 35 35 36 37 38 38 39 41 42 42 44 44 44 47 47 49 50 51 52 53 54 56 56 58 58 59 59 60 61 62 63 64

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2.4.11 Comparison of approximate and exact results . . . . . . . . . 2.5 Response tensor for a synchrotron-emitting gas . . . . . . . . . . . . . . 2.5.1 Method of stationary phase . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Synchrotron approximation to the response tensor . . . . . 2.5.3 Expansion about a point of stationary phase . . . . . . . . . . 2.5.4 Transverse components of the response tensor . . . . . . . . . 2.5.5 Extreme relativistic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Airy integral approximation . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 Response at high frequencies . . . . . . . . . . . . . . . . . . . . . . . . 2.5.8 Power-law distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.9 Highly relativistic thermal distribution . . . . . . . . . . . . . . . 2.6 Response tensor for a pulsar plasma . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Properties of pulsar plasma . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Cold pair-plasma model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Cold counter-streaming electrons and positrons . . . . . . . 2.6.4 Dispersion in intrinsically relativistic pulsar plasmas . . . 2.7 Nonlinear response tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Higher order currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Nonlinear response tensor for a magnetized plasma . . . . 2.7.3 Cold plasma limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4 Quadratic response with one slow disturbance . . . . . . . . . 2.7.5 Cubic response with one slow disturbance . . . . . . . . . . . . 2.7.6 Effective cubic response . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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65 66 66 67 68 69 70 71 72 72 73 75 75 76 79 83 87 87 87 88 89 89 89 89

Wave dispersion in relativistic magnetized plasma . . . . . . . . . 91 3.1 Dispersion in cold magnetized plasma . . . . . . . . . . . . . . . . . . . . . . 92 3.1.1 Invariant dispersion equation . . . . . . . . . . . . . . . . . . . . . . . 92 3.1.2 3-tensor form of the wave equation . . . . . . . . . . . . . . . . . . 93 3.1.3 Dielectric tensor in the rest frame of a cold plasma . . . . 94 3.1.4 Polarization vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.5 Solution of the homogeneous wave equation . . . . . . . . . . . 95 3.1.6 Cold plasma wave modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.2 Waves in cold plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.1 Magnetoionic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.2 Parallel and perpendicular propagation . . . . . . . . . . . . . . . 99 3.2.3 Cutoffs and resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2.4 Dispersion equation for streaming cold plasma . . . . . . . . 103 3.2.5 Magnetoionic waves in a streaming plasma . . . . . . . . . . . 104 3.2.6 Transformation of the polarization vector . . . . . . . . . . . . . 105 3.2.7 Pair plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2.8 High-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2.9 Low-frequency waves in cold plasma . . . . . . . . . . . . . . . . . 108 3.3 Waves in weakly relativistic thermal plasmas . . . . . . . . . . . . . . . . 111 3.3.1 Wave modes for perpendicular propagation . . . . . . . . . . . 111

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3.3.2 Cyclotron harmonic modes in a nonrelativistic plasma . 112 3.3.3 Weakly relativistic modifications . . . . . . . . . . . . . . . . . . . . 115 3.3.4 Ordinary modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.5 Smoothing of the cyclotron resonance . . . . . . . . . . . . . . . . 117 3.4 Waves in pulsar plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.4.1 Low-frequency waves in pulsar plasmas . . . . . . . . . . . . . . . 119 3.4.2 Relativistic plasma emission . . . . . . . . . . . . . . . . . . . . . . . . 122 3.5 Weak-anisotropy approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.5.1 Projection onto the transverse plane . . . . . . . . . . . . . . . . . 126 3.5.2 Modes of a synchrotron emitting gas . . . . . . . . . . . . . . . . . 127 3.5.3 Weak-anisotropy approximation for a pulsar plasma . . . 128 3.5.4 Mode coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4

Gyromagnetic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.1 Gyromagnetic emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.1.1 Probability of emission for periodic motion . . . . . . . . . . . 136 4.1.2 Probability of gyromagnetic emission . . . . . . . . . . . . . . . . 137 4.1.3 Gyroresonance condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.1.4 Resonance ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.1.5 Differential changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.1.6 Quasilinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2 Gyromagnetic emission in vacuo . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.2.1 Gyromagnetic emission of transverse waves . . . . . . . . . . . 142 4.2.2 Power emitted in gyromagnetic emission in vacuo . . . . . . 143 4.2.3 Angular distribution of gyromagnetic emission . . . . . . . . 144 4.2.4 The quasilinear coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.5 Emission by accelerated particles . . . . . . . . . . . . . . . . . . . . 146 4.2.6 The radiation reaction 4-force . . . . . . . . . . . . . . . . . . . . . . . 147 4.3 Relativistic effects in cyclotron emission . . . . . . . . . . . . . . . . . . . . 149 4.3.1 Emissivity in the magnetoionic modes . . . . . . . . . . . . . . . . 149 4.3.2 Gyromagnetic emission by thermal particles . . . . . . . . . . 150 4.3.3 Semirelativistic approximation . . . . . . . . . . . . . . . . . . . . . . 152 4.3.4 Integration around the resonant semicircle . . . . . . . . . . . . 152 4.3.5 Relativistic frequency downshift and line broadening . . . 154 4.3.6 Electron cyclotron maser emission . . . . . . . . . . . . . . . . . . . 155 4.4 Gyrosynchrotron emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4.1 Synchrotron approximation . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4.2 Average over pitch-angle distribution . . . . . . . . . . . . . . . . 158 4.4.3 Gyromagnetic absorption coefficient . . . . . . . . . . . . . . . . . 160 4.4.4 Approximations to Bessel functions . . . . . . . . . . . . . . . . . . 161 4.4.5 Gyrosynchrotron emissivity . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.6 Gyrosynchrotron emission: thermal . . . . . . . . . . . . . . . . . . 163 4.4.7 Gyrosynchrotron emission: power-law . . . . . . . . . . . . . . . . 165 4.5 Synchrotron emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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4.5.1 Strong and weak Faraday rotation . . . . . . . . . . . . . . . . . . . 167 4.5.2 Synchrotron emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.5.3 Synchrotron emission: power-law . . . . . . . . . . . . . . . . . . . . 170 4.5.4 Synchrotron absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.5.5 Razin suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.5.6 Possibility of maser synchrotron emission . . . . . . . . . . . . . 172 4.5.7 Synchrotron absorption: thermal . . . . . . . . . . . . . . . . . . . . 173 4.5.8 Synchrotron absorption: Trubnikov’s form . . . . . . . . . . . . 174 4.6 Thomson scattering in a magnetic field . . . . . . . . . . . . . . . . . . . . . 176 4.6.1 Scattering by a magnetized particle . . . . . . . . . . . . . . . . . . 176 4.6.2 Quasilinear equations for scattering . . . . . . . . . . . . . . . . . . 177 4.6.3 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.6.4 Magnetized and unmagnetized particles . . . . . . . . . . . . . . 179 4.6.5 Scattering of magnetoionic waves . . . . . . . . . . . . . . . . . . . . 180 4.6.6 Scattering of high-frequency waves . . . . . . . . . . . . . . . . . . . 181 4.6.7 Resonant Thomson scattering . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5

Magnetized Dirac electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.1 Dirac wave functions in a magnetostatic field . . . . . . . . . . . . . . . 186 5.1.1 Review of Dirac’s equation for B = 0 . . . . . . . . . . . . . . . . 186 5.1.2 Dirac’s equation in a magnetostatic field . . . . . . . . . . . . . 187 5.1.3 Construction of the wave functions . . . . . . . . . . . . . . . . . . 188 5.1.4 Magnetic moment of the electron . . . . . . . . . . . . . . . . . . . . 191 5.1.5 Johnson-Lippmann wave functions . . . . . . . . . . . . . . . . . . . 191 5.1.6 Orthogonality and completeness relations . . . . . . . . . . . . . 192 5.1.7 Normalization of the wave function . . . . . . . . . . . . . . . . . . 193 5.2 Specific spin eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.2.1 Helicity eigenstates in a magnetic field . . . . . . . . . . . . . . . 194 5.2.2 Magnetic moment eigenstates . . . . . . . . . . . . . . . . . . . . . . . 195 5.2.3 Eigenstates in the cylindrical gauge . . . . . . . . . . . . . . . . . . 197 5.3 Electron propagator in a magnetostatic field . . . . . . . . . . . . . . . . 199 5.3.1 Statistically averaged electron propagator . . . . . . . . . . . . 199 5.3.2 Gauge-independent form for the propagator . . . . . . . . . . . 200 5.3.3 G´eh´eniau form for the electron propagator . . . . . . . . . . . . 200 5.3.4 Alternative derivation of the propagator . . . . . . . . . . . . . . 202 5.3.5 Spin projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.4 Vertex function in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.4.1 Definition of the vertex function . . . . . . . . . . . . . . . . . . . . . 205 5.4.2 Symmetry properties of the vertex function . . . . . . . . . . . 206 5.4.3 Landau and cylindrical gauges . . . . . . . . . . . . . . . . . . . . . . 206 5.4.4 Gauge-dependent factor along an electron line . . . . . . . . 207 5.4.5 Vertex function for arbitrary spin states . . . . . . . . . . . . . . 208 5.4.6 Gauge-invariance condition . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.4.7 Sum over initial and final spin states . . . . . . . . . . . . . . . . . 210

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5.5 Ritus method and the vertex formalism . . . . . . . . . . . . . . . . . . . . 212 5.5.1 Factorization of Dirac’s equation . . . . . . . . . . . . . . . . . . . . 212 5.5.2 Gauge-dependent part of the wave function . . . . . . . . . . . 213 5.5.3 Reduced form of Dirac’s equation . . . . . . . . . . . . . . . . . . . 214 5.5.4 Reduced wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.5.5 Reduced propagator in the Ritus method . . . . . . . . . . . . . 215 5.5.6 Propagator in the vertex formalism . . . . . . . . . . . . . . . . . . 216 5.5.7 Vertex matrix in the Ritus method . . . . . . . . . . . . . . . . . . 217 5.5.8 Calculation of traces using the Ritus method . . . . . . . . . 219 5.6 Feynman rules for QPD in a magnetized plasma . . . . . . . . . . . . . 221 5.6.1 Transition rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.6.2 Average over the position of the gyrocenter . . . . . . . . . . . 222 5.6.3 Reduced density of final states . . . . . . . . . . . . . . . . . . . . . . 223 5.6.4 Neglect of gauge-dependent factors . . . . . . . . . . . . . . . . . . 223 5.6.5 Rules for an unmagnetized system . . . . . . . . . . . . . . . . . . . 225 5.6.6 Modified rules for magnetized systems . . . . . . . . . . . . . . . 227 5.6.7 Probability of a process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.6.8 Probabilities for second-order processes . . . . . . . . . . . . . . 233 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6

Quantum theory of gyromagnetic processes . . . . . . . . . . . . . . . . 235 6.1 Gyromagnetic emission and pair creation . . . . . . . . . . . . . . . . . . . 236 6.1.1 Probability for first-order processes . . . . . . . . . . . . . . . . . . 236 6.1.2 Unpolarized particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.1.3 Reduction to the nonquantum limit . . . . . . . . . . . . . . . . . . 237 6.1.4 Kinematics of gyromagnetic processes . . . . . . . . . . . . . . . . 239 6.1.5 Allowed resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6.1.6 Graphical solutions of resonance condition . . . . . . . . . . . . 241 6.1.7 Kinetic equations for gyromagnetic emission . . . . . . . . . . 242 6.1.8 Kinetic equations for one-photon pair creation . . . . . . . . 243 6.1.9 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.1.10 Quantum oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 6.2 Quantum theory of cyclotron emission . . . . . . . . . . . . . . . . . . . . . 248 6.2.1 Cyclotron approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.2.2 Cyclotron approximation to the vertex function . . . . . . . 249 6.2.3 Multipole expansion of the vertex function . . . . . . . . . . . . 250 6.2.4 Probability for cyclotron emission . . . . . . . . . . . . . . . . . . . 251 6.2.5 Spontaneous gyromagnetic emission in vacuo . . . . . . . . . . 252 6.2.6 Lorentz transformation to the laboratory frame . . . . . . . 253 6.2.7 Spontaneous cyclotron emission in vacuo . . . . . . . . . . . . . 254 6.2.8 Cyclotron resonance condition . . . . . . . . . . . . . . . . . . . . . . 256 6.2.9 Cyclotron emission and absorption coefficients . . . . . . . . 257 6.2.10 Bi-Maxwellian distribution . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.2.11 Cyclotron absorption in a completely degenerate plasma 258 6.3 Quantum theory of synchrotron emission . . . . . . . . . . . . . . . . . . . 259

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6.3.1 Semi-quantitative discussion of quantum effects . . . . . . . 259 6.3.2 Quantum synchrotron parameter . . . . . . . . . . . . . . . . . . . . 261 6.3.3 Sum over final states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.3.4 Synchrotron approximation to the resonance condition . 262 6.3.5 Airy-integral approximation to the functions Jνn (x) . . . . 263 6.3.6 Approximation to the vertex function . . . . . . . . . . . . . . . . 265 6.3.7 Transition rate for synchrotron emission . . . . . . . . . . . . . . 266 6.3.8 Power emitted for unpolarized electrons . . . . . . . . . . . . . . 267 6.3.9 Change in the spin during synchrotron emission . . . . . . . 269 6.3.10 Gyromagnetic emission in supercritical fields . . . . . . . . . . 270 6.4 One-photon pair creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 6.4.1 Probability for pair creation and decay . . . . . . . . . . . . . . . 272 6.4.2 Rate of pair creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 6.4.3 Rate of pair production near threshold . . . . . . . . . . . . . . . 274 6.4.4 Relativistic pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.4.5 Airy-integral approximation . . . . . . . . . . . . . . . . . . . . . . . . 278 6.4.6 Relativistic approximation to the vertex function . . . . . . 279 6.4.7 Spin- and polarization-dependent decay rates . . . . . . . . . 279 6.4.8 Spin-summed decay rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 6.4.9 Asymptotic approximation . . . . . . . . . . . . . . . . . . . . . . . . . 282 6.4.10 Energy distribution of the pairs . . . . . . . . . . . . . . . . . . . . . 283 6.4.11 Lorentz transformation to an arbitrary frame . . . . . . . . . 283 6.4.12 One-photon pair annihilation . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7

Second order gyromagnetic processes . . . . . . . . . . . . . . . . . . . . . . 289 7.1 General properties of Compton scattering . . . . . . . . . . . . . . . . . . 290 7.1.1 Probability for Compton scattering . . . . . . . . . . . . . . . . . . 290 7.1.2 Compton cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.1.3 Kinematics of Compton scattering . . . . . . . . . . . . . . . . . . . 292 7.1.4 Kinetic equations for Compton scattering . . . . . . . . . . . . . 293 7.1.5 Sum over intermediate states: vertex formalism . . . . . . . . 293 7.1.6 Sum over intermediate states: Ritus method . . . . . . . . . . 296 7.2 Compton scattering by an electron with n = 0 . . . . . . . . . . . . . . 299 7.2.1 Scattering probability for n = 0 . . . . . . . . . . . . . . . . . . . . . 299 7.2.2 Allowed transitions n = 0 → n′ ≥ 0 . . . . . . . . . . . . . . . . . . 300 7.2.3 Scattering for n, n′ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 7.2.4 Scattering n = 0 → n′ ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.2.5 Resonant Compton scattering . . . . . . . . . . . . . . . . . . . . . . . 303 7.2.6 Resonant scattering versus absorption plus emission . . . 304 7.3 Scattering in the cyclotron and synchrotron limits . . . . . . . . . . . 305 7.3.1 Cyclotron-like approximation . . . . . . . . . . . . . . . . . . . . . . . 305 7.3.2 Scattering in the birefringent vacuum . . . . . . . . . . . . . . . . 306 7.3.3 Inverse Compton emission . . . . . . . . . . . . . . . . . . . . . . . . . . 307 7.4 Two-photon processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

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7.4.1 Kinetic equations for double gyromagnetic emission . . . . 309 7.4.2 Kinetic equations for two-photon pair creation . . . . . . . . 309 7.4.3 Double cyclotron emission . . . . . . . . . . . . . . . . . . . . . . . . . . 310 7.4.4 Two-photon pair creation . . . . . . . . . . . . . . . . . . . . . . . . . . 311 7.5 Electron-ion and electron-electron scattering . . . . . . . . . . . . . . . . 313 7.5.1 Collisional excitation by a classical ion . . . . . . . . . . . . . . . 313 7.5.2 Probability for electron-ion bremsstrahlung . . . . . . . . . . . 314 7.5.3 Electron-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . 314 7.5.4 Trident process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8

Magnetized vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.1 Linear response of the magnetized vacuum . . . . . . . . . . . . . . . . . . 320 8.1.1 Unregularized tensor: G´eh´eniau form . . . . . . . . . . . . . . . . . 320 8.1.2 Invariant components of the tensor . . . . . . . . . . . . . . . . . . 322 8.1.3 Regularization of the response tensor . . . . . . . . . . . . . . . . 323 8.1.4 Weak-field and strong-field limits . . . . . . . . . . . . . . . . . . . . 323 8.1.5 Unregularized tensor: vertex formalism . . . . . . . . . . . . . . . 324 8.1.6 Unregularized invariant components: vertex formalism . 324 8.1.7 Antihermitian part of the response tensor . . . . . . . . . . . . 325 8.1.8 Regularization: vertex formalism . . . . . . . . . . . . . . . . . . . . 326 8.1.9 Long-wavelength limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.1.10 Strong-B limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.2 Schwinger’s proper-time technique . . . . . . . . . . . . . . . . . . . . . . . . . 328 8.2.1 Proper-time method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 8.2.2 Propagator in a static magnetic field . . . . . . . . . . . . . . . . . 330 8.2.3 Propagator for an electromagnetic wrench . . . . . . . . . . . . 330 8.2.4 Nonlinear terms in the Lagrangian density . . . . . . . . . . . . 331 8.2.5 Weisskopf’s Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.3 Vacuum with a homogeneous electrostatic field . . . . . . . . . . . . . . 335 8.3.1 Response tensor for an electromagnetic wrench . . . . . . . . 335 8.3.2 Linear and nonlinear response tensors for ω ≪ m . . . . . . 337 8.3.3 Derivation from the Heisenbeg-Euler Lagrangian . . . . . . 338 8.3.4 Spontaneous pair creation . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8.4 Waves in superstrong fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8.4.1 Weak-field, weak-dispersion limit . . . . . . . . . . . . . . . . . . . . 343 8.4.2 Wave properties in an arbitrarily strong field . . . . . . . . . . 343 8.4.3 Weak field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.4.4 Strong field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 8.4.5 Dispersion in an electromagnetic wrench . . . . . . . . . . . . . 345 8.4.6 High-frequency, weak-field limit . . . . . . . . . . . . . . . . . . . . . 346 8.5 Mass operator for a magnetized electron . . . . . . . . . . . . . . . . . . . . 348 8.5.1 Qualitative consequences of g 6= 2 . . . . . . . . . . . . . . . . . . . 348 8.5.2 Reduced mass operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.5.3 Unrenormalized form of reduced mass operator . . . . . . . . 350

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Energy states in a magnetic field . . . . . . . . . . . . . . . . . . . . 351 Weak-field correction to the magnetic moment . . . . . . . . 352 Correction to the ground state energy . . . . . . . . . . . . . . . . 352 Transitions between split levels . . . . . . . . . . . . . . . . . . . . . . 352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Response of magnetized systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 9.1 Response of a magnetized electron gas . . . . . . . . . . . . . . . . . . . . . 356 9.1.1 Calculation of the response tensor . . . . . . . . . . . . . . . . . . . 356 9.1.2 Explicit forms for Π µν (k) . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.1.3 Antihermitian part of the response tensor . . . . . . . . . . . . 360 9.1.4 Kramers-Kronig relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 9.1.5 Gyrotropic and nongyrotropic parts . . . . . . . . . . . . . . . . . . 362 9.2 Explicit 4-tensor form for Π µν (k) . . . . . . . . . . . . . . . . . . . . . . . . . . 363 9.2.1 Spin-dependent occupation number . . . . . . . . . . . . . . . . . . 363 9.2.2 Explicit form for given nǫns (pz ) . . . . . . . . . . . . . . . . . . . . . . 363 9.2.3 Sums over s′ , s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 9.2.4 Separation into nongyrotropic and gyrotropic parts . . . . 365 9.2.5 Sum over electron and positron states . . . . . . . . . . . . . . . . 366 9.2.6 Rationalized resonant denominator . . . . . . . . . . . . . . . . . . 367 9.2.7 Spin-dependent electron gas . . . . . . . . . . . . . . . . . . . . . . . . 368 9.3 Special and limiting cases of the response tensor . . . . . . . . . . . . . 370 9.3.1 Response tensor for parallel propagation . . . . . . . . . . . . . 370 9.3.2 One-dimensional electron gas . . . . . . . . . . . . . . . . . . . . . . . 371 9.3.3 Nonquantum limit: one-dimensional case . . . . . . . . . . . . . 372 9.3.4 Nonquantum limit: Vlasov form . . . . . . . . . . . . . . . . . . . . . 373 9.3.5 Nonquantum limit: forward-scattering form . . . . . . . . . . . 374 9.3.6 First quantum corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 375 9.3.7 Long wavelength limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 9.3.8 Response tensor for a nonrelativistic quantum gas . . . . . 377 9.4 Magnetized thermal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 9.4.1 Reduction of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 9.4.2 Plasma dispersion functions . . . . . . . . . . . . . . . . . . . . . . . . 381 9.4.3 TBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.4.4 Fermi-Dirac distribution for magnetized electrons . . . . . . 382 9.4.5 Plasma dispersion functions . . . . . . . . . . . . . . . . . . . . . . . . 383 9.4.6 Evaluation of specific integrals . . . . . . . . . . . . . . . . . . . . . . 385 9.4.7 Parallel propagation: nondegenerate limit . . . . . . . . . . . . . 385 9.4.8 Complete response tensor: nondegenerate limit . . . . . . . . 385 9.4.9 Antihermitian part: nondegenerate limit . . . . . . . . . . . . . . 385 9.5 Completely degenerate electron gas . . . . . . . . . . . . . . . . . . . . . . . . 386 9.5.1 Completely degenerate limit . . . . . . . . . . . . . . . . . . . . . . . . 386 9.5.2 Logarithmic plasma dispersion functions . . . . . . . . . . . . . 387 9.5.3 Complete response tensor: degenerate limit . . . . . . . . . . . 388 9.5.4 Parallel propagation: degenerate limit . . . . . . . . . . . . . . . . 388

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9.5.5 Antihermitian part: degenerate limit . . . . . . . . . . . . . . . . . 388 9.6 Wave dispersion in relativistic quantum magnetized plasma . . . 389 9.6.1 Dispersion relations: parallel propagation . . . . . . . . . . . . . 389 9.6.2 Gyromagnetic absorption modes . . . . . . . . . . . . . . . . . . . . . 389 9.6.3 Pair modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.7 Nonlinear response tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 9.7.1 Closed loop diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 9.7.2 General expression for the nth order response tensor . . . 393 9.7.3 Quadratic and cubic response tensors . . . . . . . . . . . . . . . . 393 9.7.4 Sum over spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 9.7.5 Alternative form for the quadratic response tensor . . . . . 395 9.7.6 Quadratic response tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.7.7 Cubic response tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 9.8 Photon splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 9.8.1 Kinematics of photon splitting . . . . . . . . . . . . . . . . . . . . . . 399 9.8.2 Three-wave matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 9.8.3 Probability for photon splitting . . . . . . . . . . . . . . . . . . . . . 400 9.8.4 Photon splitting in the weak-field limit . . . . . . . . . . . . . . . 401 9.8.5 CP invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 9.8.6 Decay rates in the weak-field approximation . . . . . . . . . . 402 9.8.7 Kinematic restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 9.8.8 Low-frequencies and arbitrary field strengths . . . . . . . . . . 404 9.8.9 S-matrix approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 9.8.10 Photon splitting in an electromagnetic wrench . . . . . . . . 406 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 A

Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A.1 Bessel functions and J-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A.1.1 Ordinary Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A.1.2 Modified Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 A.1.3 Macdonald functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 A.1.4 Properties of Kν (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 A.1.5 Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 A.1.6 J-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 A.2 Plasma dispersion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 A.2.1 Relativistic thermal function T (z, ρ) . . . . . . . . . . . . . . . . . 415 A.2.2 Trubnikov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 A.2.3 Shkarofsky and Dnestrovskii functions . . . . . . . . . . . . . . . 417 A.2.4 Recursion relations and differential equations . . . . . . . . . 417 A.2.5 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 A.2.6 Half-integer q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 A.3 Dirac algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 A.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 A.3.2 Standard representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 A.3.3 Dirac matrices σ µν and γ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 420

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A.3.4 Basic set of Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . 421 A.3.5 Traces of products of γ-matrices . . . . . . . . . . . . . . . . . . . . . 421 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

1 Covariant fluid models for magnetized plasmas

The generalization of the classical kinetic theory of plasmas to include a background magnetostatic field involves a substantial increase in the physical richness of the phenomena to be described, and a corresponding increase in the algebraic complexity needed to describe these phenomena. The response of a plasma is described in terms of the linear response tensor, and a hierarchy of nonlinear response tensors, with only the quadratic and cubic responses being relevant in applications. In the absence of a magnetostatic field, the response tensors may be calculated using three different but equivalent methods. The conventional method is based on the Vlasov equation, and an alternative is based on forward scattering. In volume 1, covariant forms of these two methods are referred to as the Vlasov method and the forward-scattering method, respectively. The third approach is based on a cold fluid description of each species of particle in the plasma. In a cold fluid model, the thermal (and nonthermal random) motions are neglected in calculating the response tensor. When the calculation is performed covariantly, it is trivial to split the plasma into any countable number of components with arbitrary flow speeds, and to sum over the contributions. By taking the continuum limit of such a sum with appropriate weighting, one can use this method to calculate the response tensor for an arbitrary distribution. When a magnetostatic field is included, this third approach is less general than the Vlasov and forward-scattering approaches, because it cannot be used to include the spiraling motion of particles about the magnetic field, Nevertheless, it may be used to derive the response tensor for an arbitrary distribution of particles in the limit where the spiraling motion is unimportant, referred to as the small-gyroradius limit. In this chapter, a covariant version of the cold plasma model is developed and used to treat the small gyroradius limit. Magnetohydroynamics (MHD) is the conventional fluid description of a plasma. It differs from a cold plasma model in that the plasma is treated as a single fluid, rather than separate fluids for each species, and in that with thermal motions taken into account through the plasma pressure. A covariant version of MHD is discussed at the end of this chapter.

2

1 Covariant fluid models for magnetized plasmas

A covariant description of a magnetostatic field required as the basis for a covariant description of the response. The Maxwell 4-tensor, F µν , for the magnetostatic field is the basis for any covariant description. Given F µν and the wave 4-vector, k µ , one may construct a set of basis 4-vectors. The linear and nonlinear response tensors for the vacuum and for an arbitrary plasma medium can be represented in terms of components along these basis 4-vectors, with these components being functions of invariants. Given, in addition, the 4-velocity, u ˜µ of the rest frame of the medium, one can construct alternative sets of basis vectors that are more convenient when describing the response of the medium. The covariant description of of a magnetic field and the identification of basis 4-vectors for representation of the response tensor are introduced in §1.1. The covariant description of the orbit of a particle is identified, and used to find the response of a cold plasma using both the forward-scattering and the fluid methods, in §1.2. Wave dispersion in a cold plasma is discussed in §3.1. The cold plasma model is generalized to include the effect of motions along the field lines (in the small gyroradius limit) in §1.3. A covariant form of MHD theory for a relativistic plasma is introduced in §1.4 and used to derive the properties of the MHD modes. SI units are used in $1.1.1 in introducing the theory, before reverting to natural units for the formal development of the theory; except where indicated otherwise, other formulae are in natural units,

1.1 Covariant description of a magnetostatic field

3

1.1 Covariant description of a magnetostatic field By definition, an electromagnetic field is static if there exists a frame in which it does not depend on time. Similarly, a static electromagnetic field is said to be a magnetostatic field if there exists an inertial frame in which there is a magnetic field but no electric field. In this section, the Maxwell 4-tensor is written down for an arbitrary static electromagnetic field, and then specialized to a magnetostatic field. The latter Maxwell tensor is used to separate spacetime into two subspaces, one containing the time-axis and the direction of the magnetic field, and the other perpendicular to the magnetic field. 1.1.1 Maxwell tensor for a magnetostatic field The Maxwell tensor for any electromagnetic field, corresponding to an electric field, E and a magnetic field, B, has components (SI units)   0 −E 1 /c −E 2 /c −E 3 /c 1 3 2 0 −B B  E /c  F µν =  2 (1.1.1) . E /c B3 0 −B 1 E 3 /c −B 2 B1 0

The contravariant components E i , B i , with i = 1, 2, 3 correspond to the respective cartesian components of the corresponding 3-vectors. The dual of the Maxwell tensor has components (SI units)   0 −B 1 −B 2 −B 3 0 E 3 /c −E 2 /c   B1 ∗ µν F (x) =  2 (1.1.2) . 3 B −E /c 0 E 1 /c B 3 E 2 /c −E 1 /c 0

Two independent invariants that can be constructed from F µν (SI units): F µν Fµν = −2 (E 2 /c2 − B 2 ),

F µν ∗ Fµν = −4 E · B/c.

(1.1.3)

An invariant definition of a magnetostatic field is that it is a static electromagnetic field with F µν Fµν>0 and F µν ∗ Fµν = 0. It is convenient to denote the magnetostatic field by F0µν , and to write 1/2 . (1.1.4) F0µν = Bf µν , B = 12 F0µν F0µν Equation (1.1.4) defines the invariant B, interpreted as the magnetic field strength, and introduces a dimensionless 4-tensor f µν . A second dimensionless 4-tensor is the dual of f µν : φµν = ∗ f µν = 12 εµναβ fαβ , µναβ

∗

F µν = B φµν ,

(1.1.5) 0123

where ε is the completely antisymmetric tensor with ε = 1. The tensors f µν and φµν are both antisymmetric and they are orthogonal to each other: f µν = −f νµ , φµν = −φνµ , f µα φα ν = 0. (1.1.6)

4

1 Covariant fluid models for magnetized plasmas

µν 1.1.2 Projection tensors g⊥ and gkµν µν The tensors f µν and φµν allow one to construct projection tensors g⊥ and µν gk : µν (1.1.7) g⊥ = −f µ α f αν , gkµν = −φµ α φαν ,

µν The tensors g⊥ and gkµν span the 2-dimensional perpendicular and timeparallel subspaces, respectively. They correspond to a separation of the metric tensor, which is diagonal with components 1, −1, −1, −1 in any frame, into metric tensors for these to subspaces: µν g µν = gkµν + g⊥ .

(1.1.8)

In a frame in which the magnetostatic field is along the 3-axis, gkµν is diagonal µν 1, 0, 0, −1 and g⊥ is diagonal 0, −1, −1, 0. The projection tensors allow one to separate any 4-vector, aµ say, into a sum of two orthogonal 4-vectors, aµ = aµ⊥ + aµk ; µν aµ⊥ = g⊥ aν ,

aµk = gkµν aν .

(1.1.9)

Similarly, for a single 4-vector, aµ , the invariant a2 is written in the form a2 = (a2 )⊥ +(a2 )k , and for two 4-vectors, aµ and bµ , the invariant ab is written in the form ab = (ab)⊥ + (ab)k . These separations are made by writing µν (a2 )⊥ = g⊥ aµ aν ,

µν (ab)⊥ = g⊥ aµ b ν ,

(ab)k = gkµν aµ bν .

(a2 )k = gkµν aµ aν ,

(1.1.10)

Note that in terms of the parallel and perpendicular components, kk , k⊥ of 2 the wave 3-vector, one has (k 2 )k = ω 2 − kk2 , (k 2 )⊥ = −k⊥ . 1.1.3 The 4-tensor B µ The description of the magnetostatic field in terms of F0µν applies in any frame. There is an alternative description of a magnetostatic field in terms of a 4vector, B µ , that is available if there is some preferred frame. In the presence of a medium there is a preferred frame: the rest frame of the medium. Let u ˜µ be the 4-velocity of the rest frame, relative to an arbitrarily chosen frame, with u ˜2 = u ˜µ u ˜µ = 1. One has u ˜ = [1.0] in the rest frame. The 4-tensor is B µ = ∗ F0µν u ˜ ν = B bµ , µ

2

(1.1.11)

µ

One has B Bµ = −B and hence b bµ = −1. In the rest 4-tensor one may write bµ = [0, b], where b is a unit vector along the direction of the magnetic field. An alternative definition of gkµν is gkµν = u ˜µ u ˜ ν − bµ bν .

(1.1.12)

Despite appearances, gkµν is independent of u˜, that is, it is not dependent on the choice of the preferred frame.

1.1 Covariant description of a magnetostatic field

5

1.1.4 Components for B along the 3-axis In any frame in which F0µν corresponds to E = 0, one is free to orient the axes such that B is along a specific axis. Choosing B = Bb with b = (0, 0, 1) along the 3-axis, the tensors f µν and φµν have components     0 0 0 0 0 0 0 −1  0 0 −1 0  0 0 0 0   f µν =  φµν =  (1.1.13) 0 1 0 0, 0 0 0 0. 0 0 0 0 1 0 0 0 The projection tensors (1.1.7) then have components     1 0 0 0 0 0 0 0 0 0 0 0  0 −1 0 0  µν   gkµν =  g⊥ = 0 0 0 0.  0 0 −1 0  , 0 0 0 −1 0 0 0 0

(1.1.14)

1.1.5 Basis 4-vectors for the magnetized vacuum Consider the response of a medium described by the response tensor Π µν (k). The tensor indices can depend only on the available 4-vectors and 4-tensors. In a magnetized medium the available 4-tensors and 4-vectors are f µν , k µ , and with the exception of the magnetized vacuum, the 4-velocity, u ˜µ , of the rest frame of the medium. It is possible to construct several different sets of basis 4-vectors that may be used to represent Π µν (k) in terms of invariant components with respect to the chosen basis. For the magnetized vacuum one needs a set that does not involve u ˜µ . A set that involves u˜µ is more convenient for a material medium. Consider the case where u ˜µ is not used. One can construct four mutually µ orthogonal 4-vectors from k and the Maxwell tensor for the magnetostatic field. A convenient choice is kkµ = gkµν kν = (ω, 0, 0, kz ),

µ kG = −f µν kν = (0, 0, k⊥ , 0),

µ µν k⊥ = g⊥ kν = (0, k⊥ , 0, 0), µ kD = φµν kν = (kz , 0, 0, ω),

(1.1.15)

where the components apply in a frame in which the magnetic field is along the 3-axis, and k is in the 1-3 plane. In terms of this choice of basis 4-vectors, the 4-tensors (1.1.13), (1.1.14) become f

µν

kµ kν − kµ kν = − ⊥ G 2 G ⊥, k⊥

gkµν =

µ ν kkµ kkν − kD kD

(k 2 )k

,

µν

φ

=−

µν g⊥ =−

µ ν ν kkµ kD − kD kk

(k 2 )k

µ ν µ ν k⊥ k⊥ + kG kG , 2 k⊥

, (1.1.16)

6

1 Covariant fluid models for magnetized plasmas

µ 2 with (k 2 )k = kkµ kkµ = ω 2 − kz2 , (k 2 )⊥ = k⊥ k⊥µ = −k⊥ . Writing the response 4-tensor in terms of invariant components along the 4-vectors (1.1.15) gives X µ µ µ ν 2 2 ΠAB (k) kA kB , ΠAB (k) = kA kB Πµν (k)/kA kB , (1.1.17) Π µν (k) = A,B

with A, B =k, ⊥, D, G. The charge-continuity and gauge-invariance relations imply that the invariant components Π AB (k) in (1.1.17) satisfy 2 (k 2 )k ΠkB (k) = k⊥ Π⊥B (k),

2 (k 2 )k ΠAk (k) = k⊥ ΠA⊥ (k),

(1.1.18)

with A, B =k, ⊥, D, G. A related choice of basis 4-vectors, made by Shabad [1], that includes the 4-vector k µ is µ bµ1 = kG ,

µ bµ2 = kD ,

µ 2 bµ3 = k⊥ + k µ k⊥ /k 2 ,

bµ4 = k µ .

(1.1.19)

In a form analogous to (1.1.17), the response tensor Π µν (k) =

3 X

ΠA′ B ′ (k) bµA′ bµB ′

(1.1.20)

A′ ,B ′ =1

involves only six invariant components, with the counterpart of (1.1.18) becoming Π4B ′ (k) = 0 = ΠA′ 4 (k). The set of basic vectors bµA with A = 1, 2, 3 may be replaced by any linear combination of them that preserves the orthogonality condition. In particular, one can choose a linear combination that separates longitudinal and transverse parts. Let this choice be denoted bµA′ , with A′ = 1′ , 2′ , 3′ . One requires that two of these satisfy bA′ ·k = 0, and that the third has its 3-vector part parallel to k. The combination kz bµ2 +ωbµ3 has its 3-vector part proportional to k. Thus the choice of (unnormalized) basis 4-vectors bµ1′ = bµ1 ,

2 µ bµ2′ ∝ ωk⊥ b2 − k 2 kz bµ3 ,

bµ3′ ∝ kz bµ2 + ωbµ3 ,

(1.1.21)

allows one to project onto the transverse plane, due to bµ1′ , bµ2′ being transverse and bµ3′ being longitudinal. (This procedure breaks down for the special case k 2 = 0.) The normalization of these 4-vectors is problematic. Only one of the 4vectors can be timelike, and this 4-vector is to be normalized to unity with the others three 4-vectors normalized to minus unity. However, which of them is timelike depends on the signs of the invariants k 2 and (k 2 )k = ω 2 − kz2 . The choice (1.1.21) is not used in the following. 1.1.6 Basis 4-vectors for a magnetized medium In the presence of a medium, the 4-vector, u ˜µ , of the rest frame of the medium may be chosen as one of the basis 4-vectors, and it is then the only time-like

1.1 Covariant description of a magnetostatic field

7

basis 4-vector. One needs three independent space-like 3-vectors. In the rest frame of the medium one possible choice can be constructed from the unit 3-vectors κ = k/|k| and b along the wave 3-vector and the magnetostatic field, respectively, with the vector product of these giving a third 3-vector. One choice of orthonormal 3-vectors in this frame is κµ = [0, κ], aµ = [0, a], tµ = [0, t], with a = −κ × b/|κ × b|,

t = −κ × a.

(1.1.22)

A covariant form of these definitions is κµ =

k µ − k˜ u u˜µ , [(k˜ u)2 − k 2 ]1/2

aµ = −

bµ1 , k⊥

tµ = ǫµαβγ u˜α κβ aτ ,

(1.1.23)

with the fourth basis 4-vector being u ˜µ . To interpret these basis vectors further, it is helpful to express other quantities that appear in the rest frame in terms of invariants. Let the angle between the 3-vector k and B in the rest frame be θ. One has ω = k˜ u, sin2 θ =

2 −k⊥ , (k˜ u)2 − k 2

|k|2 = (k˜ u)2 − k 2 , cos2 θ =

(k˜ u)2 − (k 2 )k . (k˜ u)2 − k 2

(1.1.24)

Using these relations one may rewrite many of the formulae in the noncovariant theory in a covariant form. The 4-vectors (1.1.23) may be written in terms of the angles (1.1.24) in the rest frame of the medium. With the 3-axis along the magnetic field and k in the 1-3 plane, one finds κµ = (0, sin θ, 0, cos θ),

aµ = (0, 0, 1, 0),

tµ = (0, cos θ, 0, − sin θ),

u˜µ = (1, 0, 0, 0).

(1.1.25)

The 4-vectors tµ , aµ span the 2-dimensional transverse space, and may be chosen as basis vectors for the representation of transverse polarization states. An alternative choice of basis 4-vectors, closely related to the set (1.1.25), is the set u˜µ = (1, 0, 0, 0),

eµ1 = (0, 1, 0, 0),

eµ2 = (0, 0, 1, 0),

bµ = (0, 0, 0, 1).

(1.1.26)

µ /k⊥ , Covariant definitions of the last three of these correspond to eµ1 = k⊥ µ µ µ µ µ e2 = kG /k⊥ , b = B /B, with B defined by (1.1.11).

1.1.7 Onsager relations The choice of basis 4-vectors (1.1.25) is particularly convenient for expressing the Onsager relations. Time-reversal invariance requires

8

1 Covariant fluid models for magnetized plasmas

Π 00 (ω, −k) |−B 0 = Π 00 (ω, k) |B 0 ,

Π 0i (ω, −k) |−B 0 = −Π i0 (ω, k) |B 0 ,

Π ij (ω, −k) |−B 0 = Π ji (ω, k) |B 0 ,

(1.1.27)

where the reversal of the sign of any external magnetostatic field is noted explicitly. With the choice of coordinate axes in (1.1.25), the Onsager relations (1.1.27) imply Π 01 (ω, k) = Π 10 (ω, k), Π 03 (ω, k) = Π 30 (ω, k), Π 12 (ω, k) = −Π 21 (ω, k),

Π 02 (ω, k) = −Π 20 (ω, k), Π 13 (ω, k) = Π 31 (ω, k),

Π 23 (ω, k) = −Π 32 (ω, k).

(1.1.28)

An important implication for the polarization vector of a wave in a magnetized medium is that its component along the 2-axis (the a-axis) is out of phase with the other components. Thus, one can choose an overall phase factor such that the 2-component (a-component) is imaginary and the 0-, 1-, 3-components (˜ u-, κ-, t-components) are all real. In terms of the invariant components introduced in (1.1.17), the Onsager relations imply Π AB (k) = Π BA (k),

A, B =k, ⊥, D;

Π AG (k) = −Π GA (k),

µ with kG = [0, k⊥ a] in the frame defined by (1.1.25).

A =k, ⊥, D, (1.1.29)

1.1.8 Classification of static fields An arbitrary static homogeneous electromagnetic field may be classified as a magnetostatic field, and electrostatic field, or an electromagnetic wrench. This classification is based on the fact that there are two invariants (1.1.3). It is convenient to write (SI units) S = − 41 F µν Fµν = 12 (E 2 /c2 − B 2 ),

P = − 41 F µν Fµν = E · B/c. (1.1.30)

An arbitrary static electromagnetic field is (a) a magnetostatic field for S < 0, P = 0, (b) an electrostatic field for S > 0, P = 0, or (c) an electromagnetic wrench for P 6= 0. This classification follows from the fact that in these three cases, one can always choose an inertial frame such that (a) there is no electric field and the magnetic field is along a chosen axis, (b) there is no magnetic field and the electric field is along a chosen axis, and (c) the electric and magnetic fields are parallel and along a chosen axis. The most general form for a static electromagnetic field is an electromagnetic wrench. Such a field may be written in the form (SI units) 1/2 F µν = Bf µν + (E/c)φµν , E/c, B = (S 2 + P 2 )1.2 ± S . (1.1.31) The tensors f µν , φµν have the forms (1.1.13) in the frame in which the electric and magnetic fields are parallel and along the z axis.

1.2 Covariant cold plasma model

9

1.2 Covariant cold plasma model A covariant formulation of the cold plasma model is used in this section to calculate the linear response tensor for a cold magnetized plasma. 1.2.1 Fluid description of a cold plasma A cold plasma, in which thermal or other random motions are neglected, can be described using fluid equations, with one fluid for each species of particle. Consider particles of species a, with charge qa and mass ma . For simplicity in writing, the affix a to denote the species is suppressed for the present, with the charge and mass denoted by q and m, respectively. The fluid equations in covariant form involve the proper number density, npr (x), and the fluid 4-velocity, uµ (x). The equation of continuity is ∂µ [npr (x)uµ (x)] = 0.

(1.2.1)

The equation of motion for the fluid is uα (x)∂α uµ (x) =

q µν F + ∂ µ Aν (x) − ∂ ν Aµ (x) uν (x), m 0

(1.2.2)

where the contributions of the static field F0µν and of a fluctuating field ∂ µ Aν (x) − ∂ ν Aµ (x) are included separately in the 4-force on the right hand side. The operator uα (x)∂α in (1.2.2) may be interpreted as the total derivative ∂/∂τ (x), where τ (x) is the proper time along the flow lines. However, although it is useful to adopt the proper time as the independent variable in a covariant treatment of particle dynamics, the proper time for a fluid element is not particularly useful in a fluid treatment. The response of a plasma may be defined in terms of the induced 4-current. The 4-current density due to a single fluid is µ Jind (x) = qnpr (x) uµ (x).

(1.2.3)

1.2.2 Weak-turbulence expansion In the weak-turbulence expansion, the induced current is expanded in powers of the fluctuating electromagnetic field, described by Aµ (x). This expansion is carried out after Fourier transforming. The Fourier transform and its inverse for the induced current is Z Z d4 k −ikx µ µ µ µ e Jind (k). (1.2.4) Jind (k) = d4 x eikx Jind (x), Jind (x) = (2π)4 The weak-turbulence expansion is

10

1 Covariant fluid models for magnetized plasmas µ Jind (k)

+

Z

µ

ν

Z

= Π ν (k)A (k) + dλ(2) Π (2)µ νρ (−k, k1 , k2 )Aν (k1 )Aρ (k2 ) Z + dλ(3) Π (3)µ νρσ (−k, k1 , k2 , k3 )Aν (k1 )Aρ (k2 )Aσ (k3 ) + · · ·

dλ(n) Π (n)µ ν1 ν2 ...νn (−k, k1 , k2 , . . . , kn ) Aν1 (k1 )Aν2 (k2 ) . . . Aνn (kn ) +···,

(1.2.5)

where the n-fold convolution integral is defined by dλ(n) =

d4 kn d4 k1 d4 k2 · · · (2π)4 δ 4 (k − k1 − k2 − · · · − kn ), (2π)4 (2π)4 (2π)4

(1.2.6)

with δ 4 (k) = δ(k 0 )δ(k 1 )δ(k 2 )δ(k 3 ). The expansion (9.3.10) defines the linear response tensor Π µν (k) and a hierarchy of nonlinear response tensors, of which only the quadratic response tensor Π (2)µνρ (k0 , k1 , k2 ), with k0 + k1 + k2 = 0, and the cubic response tensor Π (3)µνρσ (k0 , k1 , k2 , k3 ), with k0 +k1 +k2 +k3 = 0, are usually considered when discussing specific weak-turbulence processes. 1.2.3 Fourier transform of the fluid equations The first steps in evaluating the induced current for a cold plasma are to Fourier transform the fluid equations (1.2.1), (1.2.2) and the induced current (1.2.3), and to carry out a perturbation expansion in powers of Aµ . The Fourier transformed form of the continuity equation (1.2.1) is Z dλ(2) npr (k1 ) ku(k2 ) = 0, (1.2.7) where the convolution integral is defined by (1.2.6) with n = 2. The Fourier transform of the equation of fluid motion (1.2.2) is Z q dλ(2) k2 u(k1 ) uµ (k2 ) = i F0µν uν (k) m Z q (1.2.8) dλ(2) k1 u(k2 ) Gµν k1 , u(k2 ) Aν (k1 ), − m with

Gµν (k, u) = g µν −

k µ uν . ku

The induced 4-current (1.2.3) becomes Z µ Jind (k) = dλ(2) npr (k1 ) uµ (k2 ).

(1.2.9)

(1.2.10)

The 4-current has contributions from each species, only one of which is included in (1.2.10).

1.2 Covariant cold plasma model

11

1.2.4 Expansion in powers of the 4-potential In the weak-turbulence expansion for a fluid, the proper number density and the fluid 4-velocity are expanded in powers of A(k). The expansion of the proper number density npr (k) = n ¯ pr (2π)4 δ 4 (k) +

∞ X

) n(N pr (k),

(1.2.11)

N =1

where the zeroth order proper number density is denoted n ¯ pr . In the rest of a cold fluid, all the particle are at rest, and in this frame the proper number density is equal to the actual number density, n ¯ say. The proper number density is an invariant, and we may replace n ¯ pr in (1.2.11) by n ¯ , with the interpretation that n ¯ is the actual number density is the rest frame of the fluid. In any other frame, where the fluid is moving with a Lorentz factor γ¯ say, the actual number density is n ¯ pr γ¯ = n ¯ . In the following, n ¯ is used to describe the number density of the fluid, where n ¯ is the actual number density in the rest frame of the fluid. The expansion of the fluid 4-velocity gives uµ (k) = u˜µ (2π)4 δ 4 (k) +

∞ X

u(N )µ (k).

(1.2.12)

N =1

The zeroth order fluid 4-velocity is necessarily nonzero, being u ˜µ = (1, 0, 0, 0) in the rest frame of the fluid. More generally, the fluid has a nonzero velocity along the direction of the magnetic field, with u ˜µ = γ¯(1, 0, 0, v¯z ), with γ¯ = 2 −1/2 . Fluid flow (for a charged fluid) across the magnetic field is (1 − v¯z ) inconsistent with the assumption that there is no static electric field. The expansion of the induced 4-current (1.2.10) gives µ Jind (k) = qn˜ uµ (2π)4 δ 4 (k) +

∞ X

J (N )µ (k).

(1.2.13)

N =1

The zeroth order current density, qn˜ uµ , associated with a given fluid is nonzero, but is usually ignored. The justification for this is either that the contributions from the different species sum to zero, or that the static fields generated by this current density are otherwise negligible. 1.2.5 First order current The first order term in the expansion (1.2.13) of the current determines the linear response. On substituting (1.2.11) and (1.2.12) into (1.2.13), for N = 1 one has (1.2.14) J (1)µ (k) = q n u(1)µ (k) + n(1) (k) u˜µ ,

where the subscript ‘ind’ is omitted. To first order, the equation of continuity (1.2.7) gives

12

1 Covariant fluid models for magnetized plasmas

k˜ u n(1) (k) = −n ku(1) (k),

(1.2.15)

which determines the first order number density in terms of the first order fluid velocity. The first order fluid velocity is determined by the first order terms in (1.2.8): k¯ u u(1)µ (k) = i

q µν (1) q F u (k) − k¯ u Gµν (k, u ¯)Aν (k). m 0 ν m

(1.2.16)

Let the solution of (1.2.16) be written u(1)µ (k) = −

q µ τ ρ (k˜ u)Gρν (k, u ˜)Aν (k), m

(1.2.17)

where an explicit form for the tensor τ µν (ω) is determined below. On inserting (1.2.15) and (1.2.17) into (1.2.14), one identifies the linear response tensor by writing J (1)µ (k) = Π µν (k)Aν (k). The resulting cold plasma response tensor is Π µν (k) = −

q2 n ¯ αµ G (k, u¯)ταβ (k¯ u)Gβν (k, u¯), m¯ γ

(1.2.18)

where n ¯ /¯ γ is the proper number density, n ¯ pr . 1.2.6 Tensor τ µν (ω) The tensor τ µν (ω), introduced in (1.2.17), is constructed as follows. Write (1.2.16) in the form [k¯ ug µν − iηΩ0 f µν ] u(1) ν (k) = −

q k¯ u Gµν (k, u¯)Aν (k). m

(1.2.19)

with F0µν = Bf µν , η = q/|q|, Ω0 = |q|B/m. In the rest frame of the plasma, k¯ u is the frequency ω, and τ µν (ω) is defined as the inverse of the tensor ωg µν − iηΩ0 f µν on the left hand side of (1.2.19). Specifically, the definition is [ωg µν − iηΩ0 f µν ] τ ν ρ (ω) = ωg µρ .

(1.2.20)

Solving (1.2.20) gives τ µν (ω) = gkµν +

ω2

ω µν ωg⊥ + iηΩ0 f µν . 2 − Ω0

The matrix representation of τ µν (ω) is   1 0 0 0 2   iηΩ0 ω 0 − ω  − 0   2 2 ω 2 − Ω0 ω 2 − Ω0 . τ µν (ω) =    2 ω iηΩ0 ω 0  − 0   ω 2 − Ω02 ω 2 − Ω02 0 0 0 −1

In the unmagnetized limit, Ω0 → 0, τ µν (ω) reduces to g µν .

(1.2.21)

(1.2.22)

1.2 Covariant cold plasma model

13

1.2.7 Alternative forms for the Π µν (k) The explicit form (1.3.7) for the cold plasma response tensor is written in a concise notation, and it is desirable to have alternative more explicit forms for various applications. The form obtained by using the explicit form (9.2.27) for Gµν (k, u) and (1.2.21) for τ µν (ω) is Π

µν

µ ν ¯µ kkν (k 2 )k u ¯µ u ¯ν q2 n ¯ µν kk u¯ + u + (k) = − gk − m¯ γ k¯ u (k¯ u)2 µ ν µ ν 2 k⊥ u ¯ +u ¯ k⊥ (k 2 )⊥ u¯µ u ¯ν (k¯ u) µν g − + + (k¯ u)2 − Ω02 ⊥ k¯ u (k¯ u)2 iηΩ0 µ ν µν µ ν + , (1.2.23) k¯ u f + k u ¯ − u ¯ k G G (k¯ u)2 − Ω02

µ µ 2 where k⊥ , kG are defined by (1.1.15), and with (k 2 )⊥ = −k⊥ . The space components of (1.3.5) are 2 2 k⊥ vz ω2 q2 n ¯ − bi bj − 2 Π ij (k) = − m¯ γ γ¯ (ω − kz vz )2 (ω − kz vz )2 − Ω 2

ij j i i j (ω − kz vz )2 g⊥ − (ω − kz vz )(k⊥ b + k⊥ b )vz (ω − kz vz )2 − Ω 2 i j i iηΩ (ω − kz vz )f ij + (kG b − bj kG )vz , (1.2.24) + (ω − kz vz )2 − Ω 2

+

where b is a unit vector along the 3-axis, k⊥ and kG are vectors of magniij tude k⊥ along the 1- and 2-axes, respectively, with gkij = −bi bj , g⊥ diagonal ij 12 21 −1, −1, 0, f nonzero only for f = −f = −1, and with Ω = Ω0 /¯ γ . In the absence of a streaming motion, (1.3.6) simplifies further to q2 n ¯ ω q2n ¯ ij ij ij i j (ωg + iηΩf ) − b b τ (ω) = − . (1.2.25) Π ij (k) = − ⊥ m m ω 2 − Ω02 The dielectric tensor is K i j (k) = δji +

ωp2 i 1 i i Π (k) = δ − τ j (ω), j j ε0 ω 2 ω2

(1.2.26)

with ωp2 = q 2 n ¯ /ε0 m and with  iηΩ0 ω ω2 0  ω2 − Ω2 ω2 − Ω2   0 0  i . ω2 τ j (ω) =   − iηΩ0 ω  0  ω2 − Ω2 ω2 − Ω2  0 0 0 0 1 

(1.2.27)

14

1 Covariant fluid models for magnetized plasmas

1.2.8 Quadratic response tensor for a cold plasma The quadratic response tensor characterizes second order term in the expansion of the response of the plasma in powers of the amplitude of an electromagnetic disturbances. The second order terms in the expansion (1.2.13) of the induced current are Z (2)µ (2)µ (2) µ (2) (1) (1)µ J (k) = q n u (k) + n (k) u˜ + dλ n (k1 )u (k2 ) . (1.2.28) The first order terms n(1) (k), u(1)µ (k) are given by (1.2.15), (1.2.17). The second order term n(2) (k) follows from the quadratic terms in (1.2.7) Z (2) k¯ un (k) = − dλ(2) n(1) (k1 ) ku(1) (k2 ). (1.2.29) The second order term u(2)µ (k) follows from the quadratic terms in (1.2.8), which give Z (2)µ µ u (k) = τ ρ (k˜ u) − dλ(2) k2 u(1) (k1 ) u(1)ρ (k2 ) Z q (2) (1) ρν (1) − dλ k1 u (k2 ) G k1 , u (k2 ) Aν (k1 ) , (1.2.30) m The quadratic response tensor follows by writing (1.2.28) in the form defined by the second order term in the weak turbulence expansion (9.3.10). One needs to symmetrize the result over ν, k1 ↔ ρ, k2 , to avoid the result depending on the details of the calculation. It is convenient to write k0 = −k, so that one has k0µ +k1µ +k2µ = 0. The result is then symmetric under µ, k0 ↔ ν, k1 ↔ ρ, k2 . The resulting form is Π µνρ (k0 , k1 , k2 ) = −

q 3 n αµ G (k0 , u ˜)Gβν (k1 , u ˜)Gγρ (k2 , u ˜) fαβγ (k0 , k1 , k2 , u ˜), 2m2 (1.2.31)

with f

αβγ

k1σ ασ k2σ ασ τ (k0 u)τ βγ (k2 u) + τ (k0 u)τ γβ (k1 u) k0 u k0 u k0σ σγ k0σ σβ τ (k1 u)τ αγ (k0 u) + τ (k2 u)τ αβ (k0 u) + k1 u k2 u k1σ σγ k2σ σβ + τ (k2 u)τ αβ (k0 u) + τ (k1 u)τ αγ (k0 u) , (1.2.32) k2 u k1 u

(k0 , k1 , k2 , u) = −

1.3 Inclusion of motions along B

15

1.3 Inclusion of motions along B A covariant formulation of plasma response facilitates the inclusion of bulk motions. The response 4-tensor may be calculated in the frame in which there is no bulk motion, and trivially transformed to the frame where the plasma has the specified bulk motion. In the presence of a magnetic field this applies only to bulk motions along the magnetic field lines. This procedure is used here in three ways. First, it is applied to the plasma as a whole, and used to discuss the wave properties in the presence of streaming. Second, different streaming motions are introduced for different components in the plasma, allowing the existence of instabilities due to counterstreaming motions. Third, it is used to write down the response tensor for an arbitrary one-dimensional, strictlyparallel (p⊥ = 0) distribution of particles. 1.3.1 Lorentz transformation to streaming frame Given the response tensor, Π µν (k) in one inertial frame, one can write it down in any other frame by making the appropriate Lorentz transformation. Let K ′ and K ′ be two inertial frames, and let Lµ µ and its (matrix) inverse Lµ µ′ be the Lorentz transform matrices between K and K ′ . These matrices are defined such that any 4-vector with contravariant components aµ in K and ′ aµ in K ′ satisfies ′

′

aµ = L µ with

′

Lµ µ′ Lµ Given Π

µν

ν

µ

aµ ,

= δνµ ,

′

aµ = Lµ µ′ aµ , ′

′

Lµ

(1.3.1)

µ

Lµ ν ′ = δνµ′ .

(1.3.2)

′

(k) in K, the response tensor in K is ′

′

Π µ ν (L−1 k ′ ),

′

(L−1 k ′ )µ = Lµ µ′ k µ .

(1.3.3)

For the purpose of including a streaming motion in a magnetized plasma one is concerned with a boost in which the axes in K and K ′ are parallel, and K ′ is moving along the 3-axis of K at velocity −vz . In K ′ the bulk velocity of a plasma at rest in K is then vz . The explicit form for the transformation matrices in this case is     γ 0 0 −γvz γ 0 0 γvz ′ 1 0 0  1 0 0   0  0 Lµ µ (vz ) =  Lµ µ′ (vz ) =  , , 0 0 1 0 0 0 1 0 −γvz 0 0 γ γvz 0 0 γ (1.3.4) with γ = (1 − vz2 )−1/2 . It is usually convenient to write the dependence of Π µν (k) on k, in the frame K, in terms of invariants that involve k: the transformation of the dependence on k then involves writing these invariants in terms of k ′ , in the frame K ′ . An example of this procedure is for the cold plasma response tensor.

16

1 Covariant fluid models for magnetized plasmas

1.3.2 Π ij (k) for streaming cold plasma The explicit form (1.3.7) for the cold plasma response tensor can be rewritten in terms of 4-vectors, 4-tensors and invariants. The form obtained by using the explicit form (9.2.27) for Gµν (k, u) and (1.2.21) for τ µν (ω) is µ ν ¯µ kkν (k 2 )k u ¯µ u ¯ν q2 n ¯ µν kk u¯ + u + Π µν (k) = − gk − m¯ γ k¯ u (k¯ u)2 µ ν k⊥ u ¯ν + u ¯µ k⊥ (k 2 )⊥ u¯µ u ¯ν (k¯ u)2 µν g − + + (k¯ u)2 − Ω02 ⊥ k¯ u (k¯ u)2 iηΩ0 µ ν µν µ ν , (1.3.5) + k¯ u f + k u ¯ − u ¯ k G G (k¯ u)2 − Ω02 µ µ µ µ where gkµν , f µν are 4-tensors, u ¯µ , k⊥ , kG are 4-vectors, with k⊥ , kG defined by 2 2 2 (1.1.15), and where k¯ u, (k )k and (k )⊥ = −k⊥ are invariants. The compo′

′

′

′

′

′

µ µ , kG are all either ±1 or 0 and are unchanged from the nents of gkµ ν , f µ ν , k⊥ µ µ corresponding component of gkµν , f µν , k⊥ , kG , respectively. The components ′

of the 4-velocity of the rest frame become u ¯µ = (γ, 0, 0, −γvz ). The invariant ′ ′ ′ k¯ u is equal to ω in K and to γ(ω − kz vz ), where the components k µ are ′ ′ written (ω ′ , k⊥ , 0, kz′ ), with k⊥ = k⊥ . The transformed response tensor in K ′ is that for a plasma that is at rest in K. Simply by now omitting the primes, this response tensor can be re-interpreted as the response tensor in K for a plasma streaming with velocity vz in K. The form of this response tensor is the same at (1.3.5) with u¯ re-interpreted as the bulk 4-velocity of the plasma in K. Note that n ¯ is defined at the number density in the rest frame, and the actual number density in the frame in which there is a bulk streaming motion is n ¯ /γ. With this re-interpretation, the space components of (1.3.5), with u ˜→u for streaming at velocity vz , give 2 2 q2 n ¯ ω2 k⊥ vz ij Π (k) = − bi bj − 2 − mγ γ¯ (ω − kz vz )2 (ω − kz vz )2 − Ω 2 ij j i i j (ω − kz vz )2 g⊥ − (ω − kz vz )(k⊥ b + k⊥ b )vz (ω − kz vz )2 − Ω 2 i j i iηΩ (ω − kz vz )f ij + (kG b − bj kG )vz + , (ω − kz vz )2 − Ω 2

+

(1.3.6)

where b is a unit vector along the 3-axis, k⊥ and kG are vectors of magniij tude k⊥ along the 1- and 2-axes, respectively, with gkij = −bi bj , g⊥ diagonal ij 12 21 −1, −1, 0, f nonzero only for f = −f = −1, and with Ω = Ω0 /γ. 1.3.3 Streaming cold distribution The contribution to the response 4-tensor from a single cold distribution of particles may be written in the concise form (1.3.7) that involves the 4-velocity,

1.3 Inclusion of motions along B

17

u ¯µ , interpreted as the 4-velocity of the rest frame of the cold distribution. One may include a streaming motion simply by reinterpreting this 4-velocity as the bulk 4-velocity of a streaming distribution of cold particles. Omitting the over-bar, this reinterpretation gives Π µν (k) = −

q 2 npr αµ G (k, u)ταβ (ku)Gβν (k, u). m

(1.3.7)

where the proper number density related to the actual number density, n, in the chosen frame in which the plasma is streaming with Lorentz factor γ by npr = n/γ. An alternative way of writing (1.3.7) is 1 µ q 2 npr [u kα τ αν (ku) τ µν (ku) − Π µν (k) = − m ku uµ uν +kβ τ µβ (ku)uν ] + kα kβ τ αβ (ku) . (1.3.8) (ku)2 A further alternative way of writing (1.3.7) is kµ kν 1 q2 n µν µν − D D2 + Π (k) = − (ku)2 g⊥ 2 2 mγ (ku) (ku) − Ω0 µ ν −ku (k⊥ u

+

ν uµ k⊥ )

−

2 µ ν k⊥ u u

+ iη Ω0 ku f

µν

+

µ ν kG u

−

ν uµ kG

,

(1.3.9)

where the 4-vectors introduced in (1.3.9) are defined by (1.1.15). With the form (1.3.9), the gyrotropic terms, which depend on the sign of the charge, are identified explicitly as the terms proportional to η. The 3-tensor components of the response tensor follow by applying the same re-interpretation to (1.3.6)– (1.2.27). The forms (1.3.7)–(1.3.9) involve only invariants, 4-vectors and 4-tensors, and apply in any frame. Their specific forms in a chosen frame are obtained by writing the invariants, 4-vectors and 4-tensors in terms of quantities in the chosen frame. 1.3.4 Multiple streaming cold components The generalization of a single cold streaming distribution to a plasma consisting of several cold components in relative motion to each other follows by summing over the relevant contributions to the response 4-tensor. Let a label an arbitrary component, which has charge qa = ηa |qa |, mass ma , number density na , cyclotron frequency Ωa and 4-velocity uµa = [γa , γa va b]. With this generalization (1.3.7) implies Π µν (k) =

X a

−

qa2 na αµ (a) G (k, ua )ταβ (kua )Gβν (k, ua ), ma γa

(1.3.10)

18

1 Covariant fluid models for magnetized plasmas

with (1.2.21) translating into τ (a)µν (ω) = gkµν +

ω2

ω µν ωg⊥ + iηa Ωa f µν . 2 − Ωa

The alternative form (1.3.8) gives q 2 na 1 Π µν (k) = − a τ (a)µν (kua ) − [uµ kα τ (a)αν (kua ) ma γa kua a uµ uν +kβ τ (a)µβ (kua )uνa ] + kα kβ τ (a)αβ (kua ) a a2 , (kua )

(1.3.11)

(1.3.12)

The sum over species a may be interpreted as a sum over different species of particles, or of component of a single species of particle with different streaming motions. For example, in the case of an electron gas that consists of two cold counterstreaming electron beams, one can interpret the sum over a as the sum over these two components. 1.3.5 Dielectric tensor for a streaming distribution As in the case of a cold plasma in its rest frame, in solving the dispersion equation for a streaming plasma, it is convenient to use a 3-tensor for the response tensor. The dielectric tensor is defined by K i j (k) = δji +Π i j (k)/ε0 ω 2 in any given frame, and may be written down simply by translating between 4-tensor and 3-tensor notation. In particular, the sum over components in (1.3.9), rewritten in the form (1.3.8), translates into the dielectric tensor i

K j (k) =

δji

2 X ωpa va [bi kr τ (a)r j (kua ) τ (a)i j (kua ) − − 2 γ ω ω − k v a z a a va2 [ω 2 + kr k s τ (a)r s (kua )] bi bj s (a)i +k τ , (1.3.13) s (kua )bj ] + (ω − kz va )2

2 where ωpa = qa2 na /ε0 na defines the plasma frequency for component a. The matrix form of the tensor appearing on the right hand side of (??) is   (ω − kz va )2 iηa (ω − kz va )Ωa /γa  (ω − kz va )2 − Ω 2 /γ 2 (ω − kz va )2 − Ω 2 /γ 2 0   a a a a  (a)  . (1.3.14) (ω − kz va )2 τ (kua ) =   − iηa (ω − kz va )Ωa /γa   (ω − kz va )2 − Ω 2 /γ 2 (ω − kz va )2 − Ω 2 /γ 2 0  a a a a 0 0 1

The resulting matrix form for the dielectric tensor (1.3.13) can be written as a generalization of the form (3.1.15) involving the functions S, D, P , where the dependence on ω is implicit. The components of (1.3.13) are functions of k⊥ , kz , as well as of ω, and this is implicit in writing (1.3.13) in the form

1.3 Inclusion of motions along B





S −iD Q K i j =  iD S −iR  , Q iR P

19

(1.3.15)

with the components identified as

2 X ωpa [γa (ω − kz va )]2 , γa ω 2 [γa (ω − kz va )]2 − Ωa2 a 2 2 2 X ωpa ω2 γa2 k⊥ va P = 1− , − γa ω 2 [γa (ω − kz va )]2 [γa (ω − kz va )]2 − Ωa2 a

S = 1−

D=−

X a

ηa

2 ωpa γa (ω − kz va )Ωa , γa ω 2 [γa (ω − kz va )]2 − Ωa2

2 X ωpa γa2 k⊥ (ω − kz va ) , Q=− 2 γa ω [γa (ω − kz va )]2 − Ωa2 a

R=

X a

ηa

2 ωpa γa k⊥ va Ωa , γa ω 2 [γa (ω − kz va )]2 − Ωa2

(1.3.16)

20

1 Covariant fluid models for magnetized plasmas

1.4 Relativistic magnetohydrodynamics On sufficiently large scales and over sufficiently long times, plasmas may be regarded as fluids, and described using the equations of magnetohydrodynamics (MHD). MHD differs from the fluid theory used to treat the response of a cold plasma in several ways: the fluid is described by a single fluid velocity, rather than by different velocities for different species, the force equation includes a pressure force, which is absent in a cold plasma, and an equation of state is introduced. A relativistic generalization of MHD is presented here. The theory is used to derive the properties of MHD waves in relativistic plasmas. 1.4.1 Covariant form of the MHD equations In principle, the MHD equations can be derived from kinetic theory by taking moments and introducing a closure assumption. In practice, the equations of MHD are postulated, and their formal derivation is of interest only as an indication of their limits of validity (e.g., Lichnerowicz 1967; Anile 1989; Uchida 1997). There are two basic equations that describe the fluid itself: the continuity equation and the equation of motion. Let the proper mass density be ηm (x) and the fluid 4-velocity be uµ (x). The equation of mass continuity is ∂µ [ηm (x)uµ (x)] = 0.

(1.4.1)

The equation of motion depends on the assumed forces on the fluid. Assuming that the only forces are those internal to the system, including electromagnetic forces, the equation of motion can be written in the form of a conservation equation for the energy-momentum tensor: µν µν (x)] = 0, (x) + TEM ∂µ [TM

(1.4.2)

µν µν where TM (x) is the energy-momentum tensor for the matter and TEM (x) is the energy-momentum tensor for the electromagnetic field. The terms in (1.4.2) can be rearranged into the rate of change of the 4-momentum density µν (x), and in the fluid, which arises from the kinetic energy contribution to TM thermal and electromagnetic forces that arise from the thermal contribution µν µν to TM (x) and from TEM (x), respectively.

1.4.2 Energy-momentum tensor for the fluid The energy-momentum tensor for the matter is identified as µν TM = (ηm c2 + E + P )uµ uν − P g µν ,

(1.4.3)

where ηm c2 is the proper rest energy density, E = U/V is the internal energy density and P is the pressure, and where the dependence on x is implicit.

1.4 Relativistic magnetohydrodynamics

21

The form (1.4.3) is well known. It is implicit in the final form (1.4.3) that the internal motions are thermal. The combined first and second laws of thermodynamics imply the familiar relation dU = T dS − P dV , where U is the internal energy, T is the temperature and S is the entropy. Regarding U (S, V ) as the state function, with independent variable S, V , implies T = (∂U/∂S)V , P = −(∂U/∂V )S . In the present case one has U = V E and V ∝ 1/ηm . Making the physical assumption that all changes are adiabatic, that is, at constant entropy, the relation P = −(∂U/∂V )S translates into ηm

∂E = E + P, ∂ηm

(1.4.4)

where constant entropy is implicit. (The combination U +P V is the enthalpy.) Assuming an adiabatic law with an adiabatic index Γ one has ηm

∂E = Γ E, ∂ηm

(1.4.5)

with Γ = 5/3 for a monatomic nonrelativistic ideal gas, Γ = 4/3 for a highly relativistic gas. 1.4.3 Electromagnetic energy-momentum tensor µν The electromagnetic energy-momentum tensor, which satisfies ∂µ TEM = Jµ F µν , has the canonical form

TEM =

1 (F µ α F αν + µ0

1 4

g µν Fαβ F αβ ).

(1.4.6)

The Maxwell tensor, F µν , may be written in terms of the 4-vectors E µ = F µν uν , B µ = ∗ F µν uν , defined for an arbitrary 4-velocity u, identified here as the fluid 4-velocity. One has F µν =

E µ uν − E ν uµ + ǫµναβ uα Bβ , c

∗

F µν = B µ uν − B ν uµ −

ǫµναβ uα Eβ , c (1.4.7)

with Maxwell’s equations taking the form ∂µ F µν = µ0 J ν ,

∂µ ∗ F µν = 0.

(1.4.8)

The energy-momntum tensor (1.4.6) becomes TEM = −

BµBν − ε0 E µ E ν + ( 12 g µν − uµ uν ) µ0

B σ Bσ + ε0 E σ Eσ . (1.4.9) µ0

Combining (1.4.3) and (1.4.9), the total energy-momentum tensor for the system of fluid and electromagnetic field is B σ Bσ µν 2 σ T = ηm c + E + P − − ε0 E Eσ uµ uν µ0 B σ Bσ BµBν σ 1 − P− − 2 ε0 E Eσ g µν − − ε0 E µ E ν . (1.4.10) 2µ0 µ0

22

1 Covariant fluid models for magnetized plasmas

1.4.4 Derivation from kinetic theory Consider a species a, with rest mass ma , charge qa , and with distribution function Fa (x, p). Fluid equations are obtained by considering moments of the distribution function. For simplicity in writing, the x dependences of all quantities are suppressed in the following equations. The zeroth order moment defines the proper number density, napr , for species s and the corresponding proper mass density is Z d4 p (ηm )a = ma napr , napr = Fa (p). (1.4.11) (2π)4 The first moment defines the fluid 4-velocity, uµa : Z d4 p pµ Fa (p) . napr uµa = 4 (2π) ma

(1.4.12)

Each species satisfies a continuity equation of the form (1.2.1), specifically ∂µ (napr uµa ) = 0.

(1.4.13)

The fluid is described by its proper mass density, ηm , and a fluid 4-velocity, uµ , given by X X ηm = ma napr , uµ = (1.4.14) ma napr uµa /ηm , a

a

respectively. The continuity equation (1.4.1) for the fluid is then satisfied as a consequence of (1.4.13) with (1.4.14). The second moment of the distribution defines the energy-momentum tensor for species s. This includes a contribution, (ηm )a uµa uνa , that corresponds to the rest mass energy in the rest frame of the fluid, It is convenient to separate the energy-momentum tensor into a part corresponding to its rest energy and a part due to internal motions in the fluid: µν µν Taµν = Tarm + Tath ,

µν Tarm = (ηm )a uµa uνa ,

(1.4.15)

where ‘rm’ denotes rest mass and ‘th’ denotes thermal motions. One has Z d4 p µν Fa (p) ma (uµ − uµa )(uν − uνa ) = (Ea + Pa )uµa uνa − Pa g µν , Tath = (2π)4 (1.4.16) where Ea is the internal energy density and Pa is the partial pressure for species a. Assuming that the only force is electromagnetic, the equation of fluid motion for species a is ∂µ Taµν = F ν ρ Jaρ ,

(1.4.17)

1.4 Relativistic magnetohydrodynamics

23

with Jaµ = qa napr uµa . On summing (1.4.17) over all species a, the net 4-force, F ν ρ J ρ , on the right hand side may be written as a 4-gradient and included in the left hand side. From Maxwell’s equations one has J ρ = ∂µ F µρ /µ0 , and hence this 4-force becomes F ν ρ ∂µ F µρ /µ0 , which may be rewritten as µν −∂µ TEM , in terms of the energy-momentum tensor (1.4.6). The equation of motion then reduces to (1.4.2). 1.4.5 Generalized Ohm’s law and infinite conductivity A characteristic difference between MHD and kinetic theory is the appeal in MHD to some form of Ohm’s law to place a restriction on the electromagnetic field. Two examples of Ohm’s law are discussed briefly here: that for a nonrelativistic, collisional, electron-ion plasma, and that for a relativistic, collisionless pair plasma. A two-fluid model is assumed in both cases, with the fluids being electrons and ions, and electrons and positrons, respectively. In an electron-ion plasma, the ratio, me /mi , of the mass of the electron to the mass of the ion is a small parameter in which one can expand. The fluid velocity is equal to the velocity of the ions to lowest order in me /mi , and the current density is determined by the flow of the electrons relative to the ions. In the presence of collisions, there is a drag on the electrons that may be represented by a frictional force equal to −νe times the momentum of the electrons, where νe is the electron collision frequency. In an isotropic plasma, the effect of the collisions is described by a conductivity σ0 = ωp2 /ε0 νe . In a quasi-neutral plasma, when the charge density is assumed negligible, the static response may be written in the 4-tensor form J µ = σµ ν E ν ,

σ µν = σ0 (g µν − uµ uν ).

(1.4.18)

The limit of infinite conductivity corresponds to σ0 → 0, νe → 0, that is, to the collisionless limit. In this limit, for the current to remain finite one requires E µ = 0, which is the formal justification for assuming E µ = 0 in MHD. In the presence of a magnetic field the conductivity is anisotropic. The conductivity tensor may be obtained from the response tensor for a cold, magnetized electron gas by replacing the frequency, ω, in the rest frame by ω + iνe to take account of the collisions. This corresponds to identifying σ µν as iΠ µν /ku with Π µν given by the cold plasma form (1.3.7). In (1.3.7) one makes the replacement k˜ u → ku + iνe , and projects onto the 3-dimensional hyperplane orthogonal to uµ . This gives σ µν =

iωp2 [τ µν (ku + iνe ) − uµ uν ] , ε0 (ku + iνe )

with τ µν (ω) given by (1.2.21). In the static limit, ku → 0, one has µν ωp2 g⊥ + (Ωe /νe )f µν µν µ ν . σ = −b b + ε0 νe 1 + Ωe2 /νe2

(1.4.19)

(1.4.20)

24

1 Covariant fluid models for magnetized plasmas

In the limit νe → 0 only the component along bµ bν becomes infinite, and this requires the condition Eb = −Ez = 0; the Hall term (∝ f µν ) remains finite, µν and the Pedersen terms (∝ g⊥ ) tends to zero in this limit. 1.4.6 Two-fluid model for a pair plasma A two-fluid model for a relativistic electron-positron plasma with no thermal motions enables one to calculate E µ and to discuss the assumption that it is zero in relativistic MHD. In this case there is no obvious small parameter, such as the mass ratio, that allows one to justify a simple approximation to Ohm’s law. One may rearrange the fluid equations (1.4.13) and (1.4.17) for a = ±, m± = me , E± = 0 = P± into equations for the variables n = n+ + n− ,

uµ =

n+ uµ+

+ n

n+ uµ−

,

ρ = e(n+ − n− ), J µ = e(n+ uµ+ − n− uµ− ).

(1.4.21)

The first two of the variables (1.4.21) are the proper number density and the proper charge density, respectively. The equations of continuity (1.4.13) for s = ± imply continuity equations ∂µ (nuµ ) = 0 for mass and ∂µ J µ = 0 for charge. The equations of motion (1.4.17) for s = ± imply an equation of motion for the fluid of the form (1.4.2) and an equation for the current. The equation of motion for the fluid is µν ∂µ TM = F νβJβ,

µν TM = nmuµ uν +

m (J µ − ρuµ )(J ν − ρuν ) . (1.4.22) ne2 1 − ρ2 /n2 e2

The term nmuµ uν is the conventional energy-momentum tensor for a cold fluid, and conventional MHD is justified only if the additional term in (1.4.22) can be neglected. The generalized Ohm’s law is identified by calculating E ν = F ν β uβ with u given by (1.4.21). Using the equations of motion (1.4.17) together with (1.4.21) gives µ ν 1 u J + uν J µ − ρ(uµ uν + J µ J ν /n2 e2 ) , (1.4.23) Eν = ∂µ ne 1 − ρ2 /n2 e2 which gives the electric field in the rest frame of the fluid. The assumption E µ = 0 in conventional MHD applies to a pair plasma only if one can justify neglecting the right hand side of (1.4.23). If the assumption cannot be justified, relativistic MHD is not valid and should not be used. Relativistic MHD can break down for a variety of reasons (Melatos and Melrose 1996), including that there are too few charges to carry the required current density.

1.4 Relativistic magnetohydrodynamics

25

1.4.7 Lagrangian density for relativistic MHD Relativistic MHD is amenable to a Lagrangian formulation (Dewar 1977; Achterberg 1983). The action principle is Z δ d4 x Λ(x) = 0, Λ(x) = −ηm c2 − E + B α Bα /2µ0 , (1.4.24) where ηm is the proper mass density and E is the thermal energy density of the fluid. One may regard the form (1.4.24) as a postulate that defines relativistic MHD. The final term in (1.4.24) arises from the Lagrangian for the electromagnetic field, LEM = −F αβ Fαβ /4µ0 , with F µν = B µ uν − B ν uµ for E µ = 0. The derivation of the equation of motion in the form (1.4.2) follows from the fact that the Lagrangian (1.4.24) may be regarded as a functional of ηm , uµ and B µ : the dependence on x is implicit in this functional dependence. The energy-momentum tensor is implied by (1.4.24) reproduces (1.4.10) for E µ = 0: |B|2 |B|2 µ ν |B|2 b b , uµ uν − P + hµν − T µν = ηm c2 + E + 2µ0 2µ0 µ0 hµν = g µν − uµ uν ,

(1.4.25)

with |B|2 = −B σ Bσ and where hµν projects onto the 3-dimensional hypersurface orthogonal to the fluid 4-velocity. The equation of motion in the form (1.4.2) corresponds to the conservation law ∂µ T µν = 0. 1.4.8 MHD wave modes The properties of (small amplitude) waves in relativistic MHD are determined by a wave equation that may be derived from the Lagrangian. Suppose that there is an oscillating part of the fluid displacement, such that one has xµ → xµ0 + δxµ ,

δxµ = ξ µ e−iΦ + c.c.,

(1.4.26)

where xµ0 is the fluid displacement in the absence of the fluctuations. (The subscript 0 is omitted below after making the expansion.) The phase, or eikonal, Φ, satisfies kµ = ∂µ Φ. The 4-velocity has a small perturbation, given by the derivative of (1.4.26) with respect to proper time. This gives uµ → uµ0 + δuµ , with δuµ = −iωξ to first order in the perturbation. The normalization u2 = 1 must be preserved, and to lowest order, this requires ξu = 0. In the following it is assumed that ξ µ , like B µ , has no component along uµ . On averaging the action (1.4.24) over the phase, only terms of even power in ξ remain. The first two terms implied by the phase average are (Achterberg 1983)

26

1 Covariant fluid models for magnetized plasmas

L(0) (x) = −ηm c2 − E + B α Bα /2µ0 , (1.4.27) (2) 2 2 ∗ ∗ L (x) = − ηm c + P + E (ku) ξξ − Γ P kξ kξ 1 − [Bξ Bξ ∗ − B α Bα ξξ ∗ ](ku)2 − Ω α Ωα , µ0 Ω µ = kξ B µ − kB ξ µ ,

(1.4.28)

respectively, with A˜µ = hµν Aν for any 4-vector Aµ . The wave equation for MHD waves follows from ∂L(2) /∂ξµ∗ = 0 with L(2) given by (1.4.28). The resulting equation is of the form Γ˜ µν ξν = 0,

Γ

µν

Γ˜ µν = hµα hνβ Γαβ (k, u),

(1.4.29)

(kB)2 µν (ku)2 µ ν B σ Bσ 2 2 (ku) + g − B B = − ηm c + P + E − µ0 µ0 µ0 B σ Bσ kB µ ν − ΓP − (k B + k ν B µ ), (1.4.30) kµ kν − µ0 µ0

with B σ Bσ = −|B|2 . There are only three independent (orthogonal to u) components of ξ µ . One way to proceed is to choose a set of basis vectors that span the 3-dimensional space orthogonal to u. One choice consists of the direction of the magnetic field, bµ , the component of the wave 4-vector orthogonal to both b and u, κµ⊥ ∝ k µ − ku uµ + kb bµ , and the direction orthogonal to these, aµ = ǫµ νρσ uν bρ κσ⊥ = f µ ν κν⊥ . The perpendicular and parallel component of k are introduced by writing k⊥ = −kκ⊥ , kz = −kb, so that one has k µ = ku uµ + k⊥ κµ⊥ + kz bµ , and g µν = uµ uν − κµ⊥ κν⊥ − aµ aν − bµ bν . Equation (1.4.29) may be written as three simultaneous equations for the components ξ⊥ = −ξκ⊥ , ξa = −ξa, ξb = −ξb. The matrix form of these equations is    A 0 B ξ⊥  0 D 0   ξa  = 0, (1.4.31) B 0 C ξb

with the matrix components given by

D = (ηm c2 + E + P + |B|2 /µ0 )(ku)2 − kz2 |B|2 /µ0 ,

2 A = (ηm c2 + E + P + |B|2 /µ0 )(ku)2 − Γ P k⊥ − |k|2 |B|2 /µ0 ),

B = −Γ P k⊥ kz ,

C = (ηm c2 + E + P )(ku)2 − Γ P kz2 ,

(1.4.32)

with Γ = ∂(ln P )/∂(ln ηm ) for adiabatic changes. The result dispersion equation for relativistic MHD is D(AC − B 2 ) = 0.

(1.4.33)

References

27

The dispersion equation (1.4.33) factorizes into D = 0, which gives the dispersion relation for the Alfv´en mode, and AC − B 2 = 0, which has two solutions corresponding to the fast and slow magnetoacoustic modes. It is convenient to define the Alfv´en speed and the sound speed by 2 vA =

|B|2 , µ0 (ηm + E/c2 + P/c2 )

c2s =

ΓP . ηm + E/c2 + P/c2

(1.4.34)

The dispersion relation for the Alfv´en mode becomes ωA =

|kz |vA , 2 /c2 )1/2 (1 + vA

vφ2 =

2 vA cos2 θ 2 /c2 , 1 + vA

(1.4.35)

where vφ = ω/|k| is the phase speed. The fluid displacement in Alfv´en waves is along aµ . The dispersion equation for the fast and slow modes is 2 v2 vA 4 2 2 2 2 2 2 2 1+ A ω − 1 + c k − c k − v |k| ω 2 + c2s vA |k|2 kz2 = 0. s z s ⊥ A c2 c2 (1.4.36) Solving for the phase speed, the dispersion relations for the two modes are of 2 the form vφ2 = v± , with 1 v 2 c2 2 2 v± = vA + c2s + A2 s cos2 θ 2 2 2(1 + vA /c ) c #1/2 " 2 2 2 v c 2 2 sin2 θ . (1.4.37) ± vA − c2s − A2 s cos2 θ + 4c2s vA c The solution for the fluid displacement in the two modes is of the fom µ ξ± ∝ sin ψ± κµ⊥ + cos ψ± bµ ,

tan ψ± =

2 v± − c2s cos2 θ c2s sin θ cos θ

=

(1.4.38)

c2s sin θ cos θ 2 /c2 )v 2 − v 2 − c2 vA ± s A

. (1.4.39) (1 + sin2 θ For either a very low density or a very strong magnetic field, satisfying |B|2 /µ0 ≫ ηm c2 + E + P , the conventional Alfv´en speed can exceed the speed of light, and for vA ≫ c. In this case, when the sound speed is negligible, 2 the MHD speed becomes vA /(1 + vA /c2 )1/2 . At sufficiently high (relativistic) temperature, the adiabatic index is Γ = 4/3, the pressure satisfies P√= E/3 ≫ ηm c2 , implying that the sound speed approaches the limit cs → c/ 3.

References 1. 2. 3. 4. 5.

A.E. Shabad (1975) Ann, Phys. 90, 166 Lichnerowicz 1967; Anile 1989; Uchida 1997 Hartree 1931 Appleton 1932 Melrose, D.B., Gedalin, M.E., Kennett, M.P., Fletcher, C.S. 1999 J. Plasma Phys. 62, 233

2 Response tensors for magnetized plasmas

The generalization of the covariant classical kinetic theory of plasmas from an unmagnetized to a magnetized plasma involves a considerable incsrease in algebraic complexity. As in the unmagnetized case, the forward-scattering and Vlasov methods give alternative expressions for the linear response tensor, and these are related by a partial integration. A third method used n the unmagnetized case, based on combining a fluid (cold plasma) approach with a Lorentz transformation, does not generalize to the magnetized case: this method cannot be used to include the effect of the gyration of particles. For a relativistic thermal (J¨ uttner) distribution, the linear response tensor can be evaluated using a procedure due to Trubnikov. For most purposes, it is convenient to use forms for the response tensor obtained by expanding in Bessel functions. The response may then be interpreted in terms of dispersive contributions due to gyroresonant interactions. After expanding in Bessel functions, the forward-scattering and Vlasov methods give alternative expressions related by a partial integration and sum rules for the Bessel functions. The quadratic and cubic nonlinear response tensors can be calculated exactly using the same methods.

30

2 Response tensors for magnetized plasmas

2.1 Perturbation theory for a spiraling charge In a uniform magnetostatic field, the unperturbed orbit of a charged particle is a spiraling motion about the magnetic field lines. By solving the covariant form of Newton’s equation of motion, one identifies a tensor that determines the 4velocity at the particles proper time τ in terms of the 4-velocity at an arbitrary initial proper time τ = 0, whose Fourier transform is the tensor τ µν (ω) that appears in the cold plasma response tensor (1.3.7). The generalization to an arbitrary distribution of particles requires calculation of the 4-current due to an individual charge whose orbit is perturbed by a fluctuating electromagnetic field leads to a perturbation in the orbit. This perturbation is treated using the weak turbulence expansion, which involves expanding in the amplitude of the fluctuating field, A(k). The first order terms is needed to derive the response tensor using the forward-scattering method, and higher order terms are needed to derive the nonlinear response tensor. The terms in the expansion of the 4-current associated with an individual charge in A(k) are evaluated in this section. 2.1.1 Orbit of a spiraling charge In the absence of a fluctuating field, A(k), the classical equation of motion for a particle with charge q and mass m in a static electromagnetic field F0µν is q duµ (τ ) = F0µν uν (τ ), dτ m

(2.1.1)

where τ is the proper time of the particle. For a magnetostatic field, F0µν = Bf µν , (2.1.1) becomes duµ (τ ) = ηΩ0 f µν uν (τ ), dτ

η=

q , |q|

Ω0 =

|q|B . m

(2.1.2)

where η is the sign of the charge and Ω0 is the nonrelativistic gyrofrequency also called the cyclotron frequency. In solving (2.1.2) one is free to choose a frame in which the magnetic field is along the 3-axis and such that the initial 4-velocity, uµ (0) = uµ0 , corresponds to uµ0 = (γ, γv⊥ cos φ0 , −ηγv⊥ sin φ0 , γvz ), (2.1.3) where φ0 is an initial gyrophase. For a single particle one is free to choose the initial conditions such that φ0 = 0, and the 3-velocity in the 1–3 plane. One finds that all of γ, v⊥ , vz and p⊥ = γmv⊥ , pz = γmvz are constants of the motion. Let the solution for the orbit be written in the form X µ (τ ) = xµ0 + tµν (τ )u0ν ,

(2.1.4)

2.1 Perturbation theory for a spiraling charge

31

where x0 is a constant 4-vector. On separating k-components in the 0-3 plane and ⊥-components in the 1-2 plane, the parallel motion is constant rectilinear motion, which corresponds to Xkµ (τ ) = xµ0k + uµ0 τ . Hence, on writing tµν (τ ) = µν µν µν tµν k (τ ) + t⊥ (τ ), one identifies tk (τ ) = gk τ . The equation of motion (2.1.1) projected onto the 1-2 plane leads to a differential equation for tµν ⊥ (τ ): t¨µν (τ ) = ηΩ0 f µ ρ t˙ρν (τ ),

(2.1.5)

where a dot denotes differentiation with respect to τ . The solution that satisµν µν fies the initial conditions is t˙µν sin Ω0 τ . The derivative ⊥ (τ ) = g⊥ cos Ω0 τ +ηf of (2.1.4) with respect to τ then has the explicit form µν t˙µν (τ ) = gkµν + g⊥ cos Ω0 τ + ηf µν sin Ω0 τ. (2.1.6)

uµ (τ ) = t˙µν (τ )u0ν , Integration gives

µν tµν (τ ) = gkµν τ + g⊥

cos Ω0 τ sin Ω0 τ − ηf µν . Ω0 Ω0

(2.1.7)

The solution for 4-velocity at proper time τ is uµ (τ ) = (γ, γv⊥ cos(φ0 + Ω0 τ ), −ηγv⊥ sin(φ0 + Ω0 τ ), γvz ),

(2.1.8)

and the solution for the orbit is X µ (τ ) = xµ0 + (γτ, R sin(φ0 + Ω0 τ ), ηR cos(φ0 + Ω0 τ ), γvz τ ),

(2.1.9)

where R = γv⊥ /Ω0 is the gyroradius. 2.1.2 Properties of tµν (τ ), t˙µν (τ ) The tensors tµν (τ ) and t˙µν (τ ) characterize the spiraling motion of a charge in a magnetic field. The tensor t˙µν (τ ) satisfies the differential equation (2.1.5). Integrating (2.1.5) gives t˙µν (τ ) = gkµν + ηΩ0 f µ ρ tρν (τ ),

(2.1.10)

where t˙µν (0) = g µν , ˙µν

tνµ (0) = −ηf µν /Ω0

(2.1.11)

are used. The tensor t (τ ) also satisfies t˙µν (−τ ) = t˙νµ (τ ),

t˙µ ν (τ1 )t˙νρ (τ2 ) = t˙µρ (τ1 + τ2 ).

(2.1.12)

The contravariant components of tensor tµν (τ ) are   Ω0 τ 0 0 0 1  0  ,  0 − sin Ω0 τ η cos Ω0 τ tµν (τ ) =   0 −η cos Ω τ − sin Ω τ 0 Ω0 0 0 0 0 0 −Ω0 τ

(2.1.13)

32

2 Response tensors for magnetized plasmas

and the contravariant components of t˙µν (τ ) are   1 0 0 0  0 − cos Ω0 τ −η sin Ω0 τ 0   t˙µν (τ ) =   0 η sin Ω0 τ − cos Ω0 τ 0  , 0 0 0 −1

(2.1.14)

which follows by differentiation of (2.1.13).

2.1.3 Characteristic response due to a spiraling charge The response of a spiraling charges is characterized by the Fourier transform of t˙µν (τ ). Specifically, one may regard t˙µν (τ ) as a causal function, which vanishes for τ < 0, so that its Fourier transform is defined by writing Z ∞ i (2.1.15) dτ eiωτ t˙µν (τ ) = τ µν (ω), ω 0 the integral reproduces the tensor τ µν (ω), given by (1.2.21), specifically τ µν (ω) = gkµν +

ω2

ω µν ωg⊥ + iηΩ0 f µν . 2 − Ω0

(2.1.16)

A matrix representation of τ µν (ω) is given by (1.2.22). Being a causal function, one is to interpret the poles, at ω = 0, ±Ω0 , in terms of the Landau prescription: give ω and infinitesimal imaginary part, ω → ω + i0, and use the Plemelj formula 1 1 − iπ δ(ω − ω0 ), =℘ ω − ω0 + i0 ω − ω0

(2.1.17)

where ℘ denotes the Cauchy principal value. µ The 4-current density, Jsp (x) say, for a charge q spiraling in a magnetic field is Z µ Jsp (x) = q dτ uµ (τ ) (2π)4 δ 4 x − X(τ ) . (2.1.18)

The Fourier transform of (2.1.18) is Z Z µ µ Jsp (k) = d4 x Jsp (x) eikx = q dτ uµ (τ ) eikX(τ ) .

(2.1.19)

The explicit forms (2.1.6) for uµ (τ ) and (2.1.4) for X µ (τ ) are to be inserted into (2.1.19). On inserting the expression (2.1.4) for X µ (τ ) into the exponent in (2.1.19) becomes (2.1.20) kX(τ ) = kx0 + (ku0 )k τ + kµ tµν (τ )u0ν ⊥ ,

where the notation (ab)k = gkµν aµ bν is used. Let the wave 3-vector be written in cylindrical polar coordinates, so that the 4-vector has components

2.1 Perturbation theory for a spiraling charge

k µ = (ω, k⊥ cos ψ, k⊥ sin ψ, kz ). Then (2.1.20) contains the term µν kµ t (τ )u0ν ⊥ = −k⊥ R sin(φ0 + Ω0 τ + ηψ),

33

(2.1.21)

(2.1.22)

with R = γv⊥ /Ω0 = p⊥ /|q|B the radius of gyration of the particle. To perform the τ -integral in (2.1.19), one first expands in Bessel functions in such a way that all the dependence on τ is in exponents. 2.1.4 Expansion in Bessel functions The expansion in Bessel functions is based on the generating function eiz sin φ =

∞ X

einφ Jn (z).

(2.1.23)

n=−∞

The expansions needed here are (2.1.23) and ∞ X cos φ (n/z)Jn (z) eiz sin φ = einφ , i sin φ Jn′ (z) n=−∞

(2.1.24)

which follow from (2.1.23) by differentiating with respect to φ, z, respectively. One may also obtain (2.1.24) from (2.1.23) by using the recursion formulas n (2.1.25) Jn−1 (z) + Jn+1 (z) = 2 Jn (z), z Jn−1 (z) − Jn+1 (z) = 2Jn′ (z).

(2.1.26)

The actual expansion required is for the integrand in (2.1.19): uµ (τ )eikX(τ ) = eikx0

∞ X

e−isηψ ei[(ku0 )k −sΩ0 ]τ U µ (s, k),

(2.1.27)

s=−∞

which involves Fourier-Bessel components of the 4-velocity U µ (s, k) = γJs (k⊥ R), γV (s, k) ,

with the 3-velocity components given by V (s, k) = 12 v⊥ eiηψ Js−1 (k⊥ R) + e−iηψ Js+1 (k⊥ R) , − 21 iηv⊥ eiηψ Js−1 (k⊥ R) − e−iηψ Js+1 (k⊥ R) , vz Js (k⊥ R) .

(2.1.28)

(2.1.29)

The final result for the single particle 4-current (2.1.19) is µ Jsp (k) = qeikx0

∞ X

s=−∞

e−isηψ U µ (s, k) 2πδ (ku)k − sΩ0 .

(2.1.30)

The identity kµ U µ (s, k) = 0 ensures that the charge continuity condition, µ kµ Jsp (k) = 0 is satisfied.

34

2 Response tensors for magnetized plasmas

2.1.5 Gyroresonance condition The gyroresonance condition, expressed by the δ-function in (2.1.30) (ku)k − sΩ0 = 0,

εω − s|q|B − pz kz = 0,

(2.1.31)

with pµ = muµ , p0 = ε, p3 = pz . The condition (4.1.8) sometimes referred to as the Doppler condition. The gyroresonance condition in (4.1.8) may be rationalized by multiplying by εω + s|q|B + pz kz . This gives (ω 2 − kz2 )p2z − 2s|q|Bkz pz + (m2 + p2⊥ )ω 2 − s2 q 2 B 2 = 0.

(2.1.32)

In p⊥ -pz space, (2.1.32) is a conic section, being an ellipse with minor axis along the p⊥ -axis for ω 2 − kz2 > 0, and a hyperbola for ω 2 − kz2 < 0. Only one arm of the hyperbola is physical for ω 2 − kz2 > 0, with the spurious solution being introduce by the rationalization process. One may also regard (2.1.32) as a quadratic equation for pz , and solve it for the two solutions pz = pz± . One finds pz± = kz fs ± ωgs ,

fs =

s|q|B , ω 2 − kz2

gs2 = fs2 −

ω2

ε2⊥ , − kz2

(2.1.33)

with ε2⊥ = m2 + p2⊥ . The corresponding resonant values of the energy, ε± = (ε2⊥ + p2z± )1/2 are ε± = ωfs ± kz gs , (2.1.34) where the sign of the square root is chosen such that one has ε± ω − s|q|B − pz± kz = 0. The physically allowed region for the resonance corresponds to gs2 > 0. Solutions that do not satisfy ε± ≥ m are spurious. An alternative way of factoring the resonance condition involves replacing the momentum components p⊥ , pz by ε⊥ , t, defined by ε2⊥ = m2 + p2⊥ ,

pz = ε ⊥

2t , 1 − t2

ε = ε⊥

1 + t2 , 1 − t2

(2.1.35)

with −1 < t < 1. One has εω − pz kz − s|q|B = (ε⊥ ω + s|q|B)

(t − t+ )(t − t− ) , 1 − t2

(2.1.36)

with the resonant values given by 1 + t2± . 1 − t2± (2.1.37) The physically allowed solutions are those with t± real and in the range with −1 < t± < 1. t± =

ε⊥ kz ± (ω 2 − kz2 )gs , ε⊥ ω + s|q|B

pz± = ε⊥

2t± , 1 − t2±

ε± = ε⊥

2.1 Perturbation theory for a spiraling charge

35

2.1.6 Perturbation expansion of the 4-current On including the fluctuating field, A(k), the orbit may be expanded in powers of A(k). The equation of motion becomes duµ (τ ) q µν = F uν (τ ) + S µ (τ ), dτ m 0 where the perturbing electromagnetic field, A(k), is included in Z 4 d k1 −ik1 X(τ ) iq µ e k1 u(τ ) Gµν k1 , u(τ ) Aν (k1 ), S (τ ) = m (2π)4

(2.1.38)

(2.1.39)

with Gµν (k, u) = g µν − k µ uν /ku. The tensor t˙µν (τ ) plays the role of an integrating factor in the sense that it allows one to integrate (2.1.38) to find Z τ µ µν ˙ dτ ′ t˙µν (τ − τ ′ )Sν (τ ′ ), u (τ ) = t (τ )u0ν + 0

µ

X (τ ) =

xµ0

µν

+ t (τ )u0ν +

Z

τ

dτ

′′

Z

τ ′′

dτ ′ t˙µν (τ ′′ − τ ′ )Sν (τ ′ ),

(2.1.40)

which follow by integrating once and twice, respectively. A perturbation expansion of the orbit in powers of A(k) may be written X X u(n)µ (τ ), (2.1.41) X (n)µ (τ ), uµ (τ ) = X µ (τ ) = n=0

n=0

where u(n)µ (τ ) = dX (n)µ (τ )/dτ is of nth order in A(k). The expansion may be made by first inserting the expansions (2.1.41) into the expression (2.1.39) for S µ (τ ), and expanding it is the same form as (2.1.41): X S (n)µ (τ ), (2.1.42) S µ (τ ) = n=1

which has no zeroth order term. The perturbation expansion is straightforward: given the solution to order n − 1, the expression (2.1.41) is used to find S (n) (τ ), and the nth order equation of motion is solved to find Z τ u(n)µ (τ ) = dτ ′ t˙µν (τ ′′ − τ ′ )Sν(n−1) (τ ′ ), X (n)µ (τ ) =

Z

τ

dτ ′′

Z

τ ′′

dτ ′ t˙µν (τ ′′ − τ ′ )Sν(n−1) (τ ′ ).

(2.1.43)

2.1.7 Expansion of the 4-current The expansion of the 4-current leads to an nth order term of the form

36

2 Response tensors for magnetized plasmas (n)µ Jsp (k)

=q

Z

dτ j (n)µ (τ ) eikX

(0)

(τ )

,

(2.1.44)

with X (0)µ (τ ) identified with the zeroth order orbit (2.1.4). The zeroth order current is given by (2.1.19), and corresponds to j (0)µ (τ ) = t˙µν (τ )u0ν ,

(2.1.45)

where t˙µν (τ ) is given by (2.1.6). The first and second order terms are j (1)µ (τ ) = u(1)µ (τ ) + ikX (1) (τ ) u(0)µ (τ ), j

(2)µ

(τ ) = u

(2)µ

(1)

(2.1.46)

(1)µ

(τ ) + ikX (τ ) u (τ ) (1) 2 (0)µ (1) 1 (τ ), + ikX (τ ) − 2 kX (τ ) u

(2.1.47)

and so on for the higher order currents. 2.1.8 First order current

The first order current follows from (2.1.44) with (2.1.46). Writing u(1)µ (τ ) = dX (1)µ (τ )/dτ and partially integrating, the result reduces to Z (0) (1)µ (2.1.48) Jsp (k) = iq dτ ku(0) (τ ) Gαµ k, u(0) (τ ) Xα(1) (τ ) eikX (τ ) ,

with

ku(τ ) Gµν k, u(τ ) = ku(τ ) g µν − k µ uν (τ ).

(2.1.49)

On inserting the first order perturbation in the expression (2.1.48) for the orbit, one finds Z τ ′′ Z Z τ q2 (1)µ ′′ Jsp (k) = − dτ ′ dτ dτ m Z 4 d k1 i[kX (0) (τ )−k1 X (0) (τ ′ )] ˙ e tαβ (τ ′′ − τ ′ ) × (2π)4 (2.1.50) ×ku(0) (τ ) Gαµ k, u(0) (τ ) k1 u(0) (τ ′ ) Gβν k1 , u(0) (τ ′ ) Aν (k1 ). To carry out the integrals over τ , τ ′ and τ ′′ in (2.1.50) one first expands the integrand in Bessel functions, using (2.1.27), so that all the dependences on proper time are in exponents. The resulting expression is Z 4 ∞ X d k1 −i(k−k1 )x0 −iη(sψ−s1 ψ1 ) (1)µ Jsp (k) = e e 4 (2π) s,s1 =−∞ ×Gαµ (s, k, u)ταβ (ku)k − sΩ0 G∗βν (s1 , k1 , u) (2.1.51) ×2πδ (ku)k − (k1 u)k − (s − s1 )Ω0 ,

where τ µν (ω) is given by (2.1.15), and where the azimuthal angles ψ, ψ1 are defined by (2.1.21). The tensor Gµν (s, k, u) is given by Gµν (s, k, u) = g µν Js (k⊥ R) −

k µ U ν (s, k) . (ku)k − sΩ0

(2.1.52)

2.1 Perturbation theory for a spiraling charge

37

2.1.9 Small gyroradius limit An important limiting case in which the foregoing formulae simplify considerably is the limit of small gyroradii. The argument of the Bessel functions is k⊥ R, and in the limit R → 0 one has Js (0) = 1 for s = 0 and Js (0) = 0 for s 6= 0. Applying this approximation to U µ (s, k), as given by (2.1.28), one finds that V (s, k) has nonzero contributions only for s = 0, ±1: V (±1, k) = 12 v⊥ e±iηψ (1, ∓iη, 0).

V (0, k) = (0, 0, vz ),

(2.1.53)

In this approximation (2.1.28) gives U µ (0, k) = uµk ,

U µ (±1, k) = 21 u⊥ e±iηψ (0, 1, ∓iη, 0),

(2.1.54)

with uµk = γ(1, 0, 0, vz ), u⊥ = γv⊥ . In the small gyroradius limit, the current (2.1.30) reduces to µ Jsp (k) = qeikx0 uµk 2πδ[(ku)k , (2.1.55)

which is equivalent to the current for a charge in constant rectilinear motion. in the small-gyroradius limit, the function Gµν (s, k, u), defined by (2.1.52) is nonzero only for s = 0, when it simplifies to Gµν (0, k, u) = Gµν (k, uk ) = g µν − k µ uνk /kuk . The first-order current (2.1.51) then simplifies to (1)µ Jsp (k)

=

Z

d4 k1 −i(k−k1 )x0 e (2π)4

One has

×Gαµ (k, uk )ταβ (kuk )Gβν (k1 , uk ) 2πδ (k − k1 )uk .

Gαµ (k, uk )ταβ (kuk )Gβν (k ′ , uk ) = τ µν (kuk ) −

uµk kα τ αν (kuk ) + τ µβ (kuk )kβ′ uνk

with k ′ uk = kuk .

kuk

+

kα kβ′ τ αβ (kuk )uµk uνk (kuk )2

(2.1.56)

, (2.1.57)

38

2 Response tensors for magnetized plasmas

2.2 General forms for the linear response 4-tensor Two general methods for calculating response tensors are discussed in Volume I: the forward-scattering method and the Vlasov method. The two methods are used here to derive general expressions for the response tensor for a magnetized plasma. Both expressions are useful in different contexts. 2.2.1 Forward-scattering method for a magnetized plasma The derivation of the linear response tensor using the forward-scattering method involves averaging the first-order single-particle current over the distribution of particles. For a magnetized particle the first order, single-particle current is given by (2.1.50), with the unperturbed orbit given by (2.1.4), viz. X µ (τ ) = xµ0 +tµν (τ )u0ν , where x0 describes the initial conditions and u0 is the initial 4-velocity. The average over a distribution of particles follows by noting that the distribution F (x0 , p0 ) represents the number of world lines (one per particle) threading the 7-dimensional surface d4 x0 d4 p0 /(2π)4 dτ . Hence, the appropriate average follows by replacing the integral over dτ in (2.1.50) by the integral over d4 x0 d4 p0 /(2π)4 times F (x0 , p0 ). Assuming a uniform distribution in space and time implies that F (p0 ) does not depend on x0 ; x0 appears only in an exponential factor exp[i(k − k1 )x0 ], and the x0 -integral gives (2π)4 δ 4 (k − k1 ). The k1 -integral in (2.1.50) is performed over the resulting δ-function, with the implied identity k1µ = k µ being the forward-scattering condition. This leads to an expression of the form J (1)µ (k) = Π µν (k)Aν (k), from which one identifies Z τ Z τ ′′ Z 4 q2 d p(τ ) ′′ dτ Π µν (k) = − F (p) dτ ′ exp ik X(τ ) − X(τ ′ ) 4 m (2π) ′′ ˙ (2.2.1) ×tαβ (τ − τ ′ ) ku(τ ) Gαµ k, u(τ ) ku(τ ′ ) Gβν k, u(τ ′ ) , where the superscript (0) on u(τ ) is now omitted, with u(τ ) given by (2.1.6), viz. uµ (τ ) = t˙µν (τ )u0ν , with tµν (ξ) given by (2.1.7) and where the dot denotes the derivative with respect to τ . Also in (2.2.1) it is noted that the initial value of τ is related to the initial gyrophase, φ0 say, and one is free to choose φ0 = Ω0 τ and to write the integral over d4 p0 as an integral over d4 p(τ ). The distribution F (p) is assumed independent of gyrophase, and hence of τ . The dependence on proper times in (2.2.1) simplifies after a partial integration: Z ∞ Z 4 q2 d p(τ ) Π µν (k) = F (p) dξ exp ik X(τ ) − X(τ − ξ) 4 m (2π) 0 αµ k, u(τ ) ku(τ − ξ) Gβν k, u(τ − ξ) , (2.2.2) ×Tαβ (ξ) ku(τ ) G where it is convenient to introduce the tensor

T µν (ξ) = tµν (ξ) − tµν (0).

(2.2.3)

2.2 General forms for the linear response 4-tensor

39

The tensor T µν (ξ) plays an important role in the following; its matrix representation follows from (2.1.13):   Ω0 τ 0 0 0 1  − sin Ω0 τ −η(1 − cos Ω0 τ ) 0   0 . T µν (τ ) = (2.2.4) − sin Ω0 τ 0  Ω0  0 η(1 − cos Ω0 τ ) 0 0 0 −Ω0 τ The property T µν (τ ) = −T νµ (−τ ) allows one to separate the integrand in (2.2.2) into parts that are even and odd functions of ξ. These correspond to the hermitian and antihermitian parts, respectively, of the response tensor. The antihermitian part of (2.2.2) is Z ∞ Z 4 d p(τ ) q2 Aµν dξ exp ik X(τ ) − X(τ − ξ) Tαβ (ξ) F (p) Π (k) = 4 2m (2π) −∞ (2.2.5) ×ku(τ ) Gαµ k, u(τ ) ku(τ − ξ) Gβν k, u(τ − ξ) .

The hermitian part is implicit in (2.2.2) and it is not particularly helpful to identify it explicitly. 2.2.2 Forward-scattering form summed over gyroharmonics

Further evaluation of the response tensor in the forward-scattering form (2.2.1) involves expanding in Bessel functions. This enables one to perform the integrals over proper time (equivalent to gyrophase here) explicitly, and interpret the result in terms of a sum over gyroresonant contributions. To perform the τ ′ and τ ′′ integrals in (2.2.1) and the ξ-integral in (2.2.2) one first expands in Bessel functions using (2.1.27). An alternative procedure is to start from the first order current in the form (2.1.51) in which the expansion in Bessel functions is already made. One replaces the δ-function in (2.1.51) by writing Z 2πδ (ku)k − (k1 u)k − (s − s1 )Ω0 = dτ ei[(ku)k −(k1 u)k −(s−s1 )Ω0 ] . (2.2.6)

The integral over dτ is replaced by one over d4 x0 d4 p0 times F (x0 , p0 ), and the calculation proceeds as in the derivation of (2.2.1). The resulting expression for the linear response tensor is Z ∞ X q2 d4 p Π µν (k) = − F (p) Gαµ (s, k, u)ταβ ((ku)k − sΩ0 )G∗βν (s, k, u), m (2π)4 s=−∞ (2.2.7) where the subscript 0 on u and p = mu is now redundant. In (2.2.7) the function Gµν (s, k, u) = g µν Js (k⊥ R) −

k µ U ν (s, k) (ku)k − sΩ0

(2.2.8)

40

2 Response tensors for magnetized plasmas

is the appropriate generalization of Gµν (k, u) = g µν − k µ uν /ku that appears in the unmagnetized case. The identity kν Gµν (s, k, u) = 0 ensures that the charge-continuity and gauge-invariance conditions, kµ Π µν (k) = 0, kν Π µν (k) = 0, are manifestly satisfied. The 4-vector U µ (s, k) is given by (2.1.28) with (2.1.29), and τ µν (ω) is given by (2.1.15). The form (2.2.7) for the linear response tensor is in a concise notation, and one needs to write it either in component form or in some other more convenient form in order to evaluate it for specific distributions. One procedure which retains the manifestly covariant form involves introducing four basis 4vectors that span the 4-dimensional space. One such choice consists of the µ µ µ four 4-tensors, kkµ , k⊥ , kG , kD , defined by (1.1.15). One has µ ν kD , (k 2 )k gkµν = kkµ kkν − kD

µν µ ν µ ν (k 2 )⊥ g⊥ = k⊥ k⊥ + kG kG ,

(2.2.9)

2 with (k 2 )⊥ = −k⊥ . It is convenient to separate U µ into parallel and perpendicular components by writing (2.1.28) with (2.1.29) in the form µ U µ (s, k) = Ukµ (s, k) + U⊥ (s, k),

Js (k⊥ R) µ , (ku)k kkµ − (kD u)k kD 2 (k )k s γv⊥ µ µ µ U⊥ (s, k) = Js (k⊥ R) k⊥ − iη Js′ (k⊥ R) kG . k⊥ k⊥ R Ukµ (s, k) =

(2.2.10)

One way of writing (2.2.7) in terms of such tensor components follows by substituting the explicit forms (2.2.8) for Gµν to obtain Z ∞ X q2 d4 p µν Π (k) = − F (p) τ µν Js2 m (2π)4 s=−∞ U µ kα τ αν + τ µβ kβ U ∗ν kα τ αβ kβ U µ U ∗ν , (2.2.11) − Js + (ku)k − sΩ0 [(ku)k − sΩ0 ]2 where the argument ((ku)k − sΩ0 ) and (s, k) of τ µν and U µ , respectively, are omitted for simplicity in writing. Substituting the explicit expression (2.1.15) for τ µν gives Z ∞ X q2 d4 p Π µν (k) = − F (p) m (2π)4 s=−∞ gkµν

Js2

−

kkµ U ∗ν + kkν U µ

Js +

(k 2 )k U µ U ∗ν [(ku)k − sΩ0 ]2

(ku)k − sΩ0 µ ∗ν ν µ [(ku)k − sΩ0 ]2 k⊥ U + k⊥ U (k 2 )⊥ U µ U ∗ν µν 2 g J − + J + s [(ku)k − sΩ0 ]2 − Ω02 ⊥ s (ku)k − sΩ0 [(ku)k − sΩ0 ]2 µ ∗ν ν µ iηΩ0 [(ku)k − sΩ0 ] kG U − kG U µν 2 f Js + Js . (2.2.12) + [(ku)k − sΩ0 ]2 − Ω02 (ku)k − sΩ0

2.2 General forms for the linear response 4-tensor

41

Further separation into components in the parallel subspace, the perpendicular subspace and the cross-terms between these two subspaces follows by inserting (2.2.9) into (2.2.12). 2.2.3 Vlasov method for a magnetized plasma The Vlasov method leads to a form for the response tensor that is equivalent to but superficially different from that obtained using the forward-scattering method. The method is based on the covariant Vlasov equation, which is duµ ∂ ∂ F (x, p) = 0, (2.2.13) uµ µ + m ∂x dτ ∂pµ with duµ /dτ determined by the equation of motion (2.1.38). On Fourier transforming (2.2.13) with (2.1.38) and using (2.1.39), one obtains Z ∂ ∂F (k2 , p) , −iku + q F0µν uν µ F (k, p) = −iq dλ(2) k1 u Gµν (k1 , u)Aν (k1 ) ∂p ∂pµ (2.2.14) where the right hand side is the convolution of the Lorentz force due to the perturbing electromagnetic field and the perturbed distribution function. One may interpret the operator q F0µν uν ∂/∂pµ on the left hand side as d/dτ , that is, as the derivative with respect to proper time along the orbit. One may integrate (2.2.14) once before making an expansion in powers of the amplitude of the fluctuating field. With u interpreted as the unperturbed orbit u(τ ), given by (2.1.6)–(2.1.7), integrating (2.2.14) once gives Z Z τ ′ −ikX(τ ′ ) ikX(τ ) dλ(2) k1 u(τ ′ ) dτ e F k, p(τ ) = −iq e ∂F (k2 , p(τ ′ )) ×Gαν k1 , u(τ ′ ) Aν (k1 ) . ∂pα (τ ′ )

(2.2.15)

It is convenient to omit the subscript 0 on the initial values in (2.1.6)–(2.1.7), so that one has uµ (τ ) = t˙µν (τ )uν ,

uµ = uµ (0) = (γ, u⊥ , 0, uz ),

(2.2.16)

with u⊥ = γv⊥ , uz = γvz and with pµ = muµ = (ε, p⊥ , 0, pz ). The explicit form for the tensor t˙µν (τ ) is given by (2.1.6). One also has kX(τ ) = kx0 + (ku)k τ − k⊥ R sin(ηψ + Ω0 τ ), with R = u⊥ /Ω0 the gyroradius.

(2.2.17)

42

2 Response tensors for magnetized plasmas

2.2.4 Linearized Vlasov equation In the Vlasov method, the weak turbulence expansion involves an expansion of F (k, p) in powers of A(k): F (k, p) = F (p) (2π)4 δ 4 (k) +

∞ X

F (n) (k, p).

(2.2.18)

n=1

On inserting (2.2.18) into (2.2.14) one obtains a hierarchy of equations, starting with the linearized Vlasov equation, ∂ ∂F (p) µν −iku + q F0 uν µ F (1) (k, p) = −iq ku Gαν (k, u)Aν (k) , (2.2.19) ∂p ∂pα with the nth term in the expansion determined by the (n − 1)th term by ∂ −iku + q F0µν uν µ F (n) (k, p) ∂p Z ∂F (n−1) (k2 , p) . (2.2.20) = −iq dλ(2) k1 u Gαν (k1 , u)Aν (k1 ) ∂pα The solution of this hierarchy of equations is obtained by substituting (2.2.18) into (2.2.15). The solution of the linearized Vlasov equation is the first order term in the resulting hierarchy of equations. The explicit form for the first order perturbation to the distribution function is Z ∞ (1) k, p(τ ) = −iq Aν (k) F dξ eik[X(τ )−X(τ −ξ)] 0

×ku(τ − ξ) Gαν k, u(τ − ξ)

∂F (p) , ∂pα (τ − ξ)

(2.2.21)

where the variable τ ′ in (2.2.15) is replaced by ξ = τ − τ ′ in (2.2.21), and where t˙µν (τ ) = t˙νµ (−τ ) is used, cf. (2.1.7). The linear response tensor is found by writing the induced current in the form Z 4 d p(τ ) µ u (τ ) F (1) k, p(τ ) , (2.2.22) J (1)µ (k) = q 4 (2π) and equating the right hand side to Π µν (k)Aν (k) to identify Π µν (k).

2.2.5 Vlasov form for the linear response tensor The resulting expression for the Vlasov form for the linear response tensor, for any specific distribution of particles, is

2.2 General forms for the linear response 4-tensor

Π µν (k) = −iq 2

Z

d4 p(τ ) (2π)4

Z

∞

43

dξ uµ (τ ) eik[X(τ )−X(τ −ξ)]

0

×ku(τ − ξ) Gαν k, u(τ − ξ)

∂F (p) . ∂pα (τ − ξ)

(2.2.23)

The Vlasov form of the response tensor for the plasma is found by summing the contributions (2.2.23) for each species of particle. The Vlasov form (2.2.23) is equivalent to the forward-scattering form (2.2.2), as may be shown by partially integrating. The relatively lengthy calculation is facilitated somewhat by noting the identity ∂ ∂pα (τ )

ku(τ ) Gαν k, u(τ ) = 0.

(2.2.24)

In evaluating the derivative ∂F (p)/∂pα (τ − ξ) in (2.2.26), one needs to take account of the fact that F (p) is independent of the gyrophase, and hence of τ or ξ. The derivative in (2.2.26) may be written in the form # " u⊥α (τ − ξ) ∂ ∂ ∂F (p) + α F (p). (2.2.25) =− ∂pα (τ − ξ) u⊥ ∂p⊥ ∂pk Terms proportional to ku(τ − ξ) in the integrand in (2.2.26) are perfect differentials, ku(τ − ξ) e−ikX(τ −ξ) = −id[e−ikX(τ −ξ) ]/dξ, and may be integrated trivially. The integrand of the integrated term depends on τ only through uµ (τ ), which may be replaced by uµk . With these changes (2.2.23) becomes Π

µν

ν Z 4 d4 p µ uk ∂ ∂ d p(τ ) µ 2 u + u (τ ) (k) = q F (p) − iq (2π)4 k u⊥ ∂p⊥ ∂pkν (2π)4 # " Z ∞ (ku)k ∂ ∂ + k α α F (p). × dξ eik[X(τ )−X(τ −ξ)] uν (τ − ξ) u ∂p ∂p ⊥ ⊥ 0 k (2.2.26) 2

Z

The term in (2.2.26) that does not involve the ξ-integral is symmetric in µ, ν, as may be seen by partially integrating the derivative with respect to pkν . Hence, in (2.2.26) one may replace uµk ∂/∂pkν by uνk ∂/∂pkµ , or by half the sum of the two to give an obviously symmetric form. The form (2.2.23) includes both hermitian and antihermitian parts, with the latter given by Z 4 Z iq 2 d p(τ ) ∞ Π Aµν (k) = − dξ uµ (τ ) uν (τ − ξ) eik[X(τ )−X(τ −ξ)] 2 (2π)4 −∞ # " (ku)k ∂ α ∂ F (p). (2.2.27) +k × u⊥ ∂p⊥ ∂pα k

44

2 Response tensors for magnetized plasmas

2.2.6 Vlasov form summed over gyroharmonics Further reduction of (2.2.26) involves expanding in Bessel functions, using (2.1.27), and performing the ξ-integral. The expression for the linear response tensor obtained from (2.2.26) is CHECK SIGN IN ANISOTROPIC TERM ! Z uµk uνk ∂ d4 p µ ∂ µν 2 Π (k) = q − uk (2π)4 u⊥ ∂p⊥ ∂pkν ! ∞ X U µ (s, k)U ∗ν (s, k) (ku)k ∂ α ∂ + kk α F (p), (2.2.28) − (ku)k − sΩ0 u⊥ ∂p⊥ ∂pk s=−∞ with U µ (s, k) given by (2.1.28) with (2.1.29). The antihermitian part follows either by applying the Landau prescription to (2.2.28) or by making the expansion in Bessel functions and performing the ξ-integral in (2.2.27): Π Aµν (k) = iπq 2

Z

∞ d4 p X µ U (s, k)U ∗ν (s, k) (2π)4 s=−∞

×δ (ku)k − sΩ0

(ku)k ∂ ∂ + kkα α u⊥ ∂p⊥ ∂pk

!

F (p).

(2.2.29)

The equivalence of the Vlasov form (2.2.28) and the forward-scattering form (2.2.7) follows from a (tedious) partial integration. 2.2.7 Explicit forms for the components of Π µν (k) In practice it is convenient to have explicit expressions for the components of the response tensor. Explicit expressions are given here for both the forwardscattering and Vlasov forms. The components of the response 4-tensor in the forward-scattering form (2.2.12) may be written in the form Π µν (k) = −

q2 m

Z

∞ X d3 p f (p) Aµν (s, k, u), (2π)3 γ s=−∞

Aµν given explicitly in table 2.1, and where the relation Rwith 3 d p f (p)/(2π)3 γ is used.

(2.2.30) R

d4 p F (p)/(2π)4 =

2.2.8 Response 3-tensor

In non-covariant notation the response may be described by the dielectric tensor, K i j (ω, k). The relation between the notations implies

2.2 General forms for the linear response 4-tensor

45

Table 2.1. The components of Aµν = Aµν (s, k, u) in (2.2.30), with the argument of the Bessel functions, Js = Js (k⊥ R), omitted.

A00 = A01 =

2 (γkz vz + sΩ0 )2 − γ 2 kz2 γ 2 k⊥ − 2 [(ku)k − sΩ0 ] [(ku)k − sΩ0 ]2 − Ω02

A

11

=

(ku)2k (k2 )k s2 Ω02 Js2 − 2 2 [(ku)k − sΩ0 ] k⊥ [(ku)k − sΩ0 ]2 − Ω02

A33 = − A12 = iη

(k2 )k (Ω0 Js + γk⊥ v⊥ Js′ )2 (γv⊥ Js′ )2 − 2 [(ku)k − sΩ0 ] [(ku)k − sΩ0 ]2 − Ω02

(γω − sΩ0 )2 − (ωγvz )2 (k⊥ γvz )2 − Js2 2 [(ku)k − sΩ0 ] [(ku)k − sΩ0 ]2 − Ω02 ω 2 − kz2 sΩ0 Js γv⊥ Js′ [(ku)k − sΩ0 ]2 k⊥ c −

γk⊥ c (Ω0 Js + γk⊥ v⊥ Js′ )Js [(ku)k − sΩ0 ]2 − Ω02

2 γ 2 vz k⊥ c γ(ωvz + kz c)sΩ0 − γ 2 ωkz (1 − vz2 ) − J2 2 [(ku)k − sΩ0 ] [(ku)k − sΩ0 ]2 − Ω02 s

A22 = −Js2 +

A13 =

γkz (ωvz − kz ) + ωsΩ0 γv⊥ Js′ Js [(ku)k − sΩ0 ]2 −

Js2

γ 2 k⊥ c(ω − kz vz ) ω(γkz vz + sΩ0 ) − γkz2 sΩ0 − Js2 2 [(ku)k − sΩ0 ] k⊥ c [(ku)k − sΩ0 ]2 − Ω02

A02 = iη

A03 =

(ku)k (Ω0 Js + γk⊥ v⊥ Js′ )Js [(ku)k − sΩ0 ]2 − Ω02

γk⊥ vz (ku)k γω(ωvz − kz ) + kz csΩ0 sΩ0 − Js2 [(ku)k − sΩ0 ]2 k⊥ c [(ku)k − sΩ0 ]2 − Ω02

A23 = −iη

γω(ωvz − kz c) + kz csΩ0 γv⊥ Js′ [(ku)k − sΩ0 ]2

A

−A

k⊥ γvz − (Ω0 Js + γk⊥ v⊥ Js′ ) Js [(ku)k − sΩ0 ]2 − Ω02

A10 = A01 21

12

A20 = −A02

A

23

= −A

32

A30 = A03 A31 = A13

K i j (ω, k) = δ i j +

Π i j (k) , ε0 ω 2

(2.2.31)

with the component Π i j (k), with k µ = [ω, k], numerically equal to minus the µ = i, ν = j component of Π µν (k). The terms in (2.2.30) that depend on the sign, η, of the charge are associated with gyrotropy, which leads to elliptical polarization of waves in a magnetized plasma. In a pure pair plasma, in which the distributions of elec-

46

2 Response tensors for magnetized plasmas

trons and positrons are identical, the electron and positron contributions to the gyrotropic terms cancel, whereas they add for the non-gyrotropic terms. The 3-tensor components of (2.2.28), (2.2.29) follow from the µ = i, ν = j components of U µ (s, k)U ∗ν (s, k). The explicit expression, (2.1.28) with (2.1.29) for U µ (s, k) implies

U i U ∗j

 2 sΩ0 sΩ0 2 sΩ0 2 ′ J iηγv J J γv J ⊥ s s z s  k⊥ k⊥ k⊥ s      sΩ0 ′ 2 ′2 2 ′ , = Js Js (γv⊥ ) Js −iηγ v⊥ vz Js Js   −iηγv⊥ k⊥     sΩ0 2 2 2 2 2 γvz Js iηγ v⊥ vz Js (γvz ) Js k⊥ 

(2.2.32)

where the arguments of U µ (s, k), Js (k⊥ R) are omitted. On expressing F (p) in 8-dimensional phase space in terms of the distribution function f (p) in 6dimensional phase space, using d4 p F (p)/(2π)4 = d3 p f (p)/(2π)3 γ, the derivative ∂F (p)/∂p0 in (2.2.26) is omitted. such that the term proportional to kkα ∂F (p)/∂pα k is replaced by a term proportional to k · ∂f (p)/∂p for example. The space-components of (2.2.28) become Z ∂ ∂ vz d3 p i j − v v b b γ Π ij (k) = q 2 z ⊥ (2π)3 γ v⊥ ∂pz ∂p⊥ ∞ i ∗j X ω − kz vz ∂ ∂ U (s, k)U (s, k) + kz + f (p). (2.2.33) γ(ω − kz vz ) − sΩ0 v⊥ ∂p⊥ ∂pz s=−∞

2.3 Response of a relativistic thermal plasma

47

2.3 Response of a relativistic thermal plasma The linear response tensor for a relativistic, thermal, magnetized plasma was first calculated by [3]. A covariant generalization of Trubnikov’s calculation is presented in this section, starting with the Vlasov form for the response tensor. The evaluation of the forward-scattering form leads to a superficially different result; the two forms are related by identities satisfied by the Trubnikov functions that appear. Trubnikov’s method is also applied to strictly-parallel and strictly-perpendicular J¨ uttner distributions. 2.3.1 Trubnikov’s response tensor for a magnetized plasma A relativistic thermal distribution is the J¨ uttner distribution 2π 2 nρ e−ργ , m3 K2 (ρ) (2.3.1) where ρ = m/T is the inverse temperature in units of the rest energy of the particle. Note that all integrals over momentum space are associated with factors 2π in the denominator. The normalization corresponds to Z Z Z d4 p d3 p f (p) d3 p npr = F (p) = f (p), (2.3.2) , n = (2π)4 (2π)3 γ (2π)3 F (p) =

(2π)3 nρ δ(p2 − m2 ) exp[−ρ(p˜ u/m)], m2 K2 (ρ)

f (p) =

where npr is the proper number density, and n is the number density in the rest frame. On inserting the J¨ uttner distribution (2.3.1) into (2.2.23), one needs to evaluate the derivative of F (p) ∝ δ(p2 − m2 ) exp(−ρ u˜ u). The relevant derivative gives ∂F (p) ρ ku(τ ′ ) Gαν k, u(τ ′ ) t˙α β (τ ′ ) =− ku(τ ′ ) u ˜ν − k˜ u uν (τ ′ ) F (p), β ∂p m (2.3.3) with τ ′ = τ − ξ. The first term on the right hand side of (2.3.3) leads to a trivial ξ-integral: Z ∞ dξ ku(τ − ξ) eik[X(τ )−X(τ −ξ)] = i. (2.3.4) 0

In this way (2.2.26) reduces to Z Z ∞ d4 p q2 ρ µ ν µ ˙νσ iR(ξ)u , F (p) u ˜ u ˜ + i k˜ u dξ u t (−ξ)u e Π µν (k) = − σ m (2π)4 0 (2.3.5) where the 4-vector (2.3.6) Rµ (ξ) = ckα tαµ (−ξ) − tαµ (0)

48

2 Response tensors for magnetized plasmas

is introduced. Using the definition (2.2.3) of T µν (ξ), an alternative form for (2.3.6) is ˜ ν (ξ) = ckβ T βν (ξ), R

Rµ (ξ) = ckα T µα (ξ),

(2.3.7)

˜ ν (ξ) is defined for a later purpose. where R It is convenient to define the function Z ′ d4 p ′ F (p) eiR(ξ)u+(s+s )u I(ρ, ξ, s + s ) = (2π)4 Z ′ d4 p nρ δ(p2 − m2 ) e−[ρ˜u−iR(ξ)−(s+s )]u . = 2π(mc)2 K2 (ρ) (2π)4 (2.3.8) The integral over d4 p reduces to a standard integral for the MacDonald function K1 , with a complex argument: nρ K1 r(ξ) ′ I(ρ, ξ, s + s ) = , (2.3.9) K2 (ρ) r(ξ) with the complex function r(ξ) defined by r(ξ) =

2 1/2 ρ˜ uµ − iRµ (ξ) − c(sµ + s′µ ) 2

→ (ρ − iωξ) +

kz2 ξ 2

1/2 2 2k⊥ , + 2 (1 − cos Ω0 ξ) Ω0

(2.3.10)

where the final expression applies in the rest frame of the plasma for sµ = 0, s′µ = 0. The response tensor (2.3.5) may be re-expressed in terms of the function I(ρ, ξ): Z ∞ q2 ρ µ ν ν µ σ µν ˙ n˜ u u ˜ + i k˜ u dξ t σ (−ξ)ˆ u uˆ I(ρ, ξ) , (2.3.11) Π (k) = − m 0 where u ˆµ denotes differentiating with respect to sµ and setting sµ = 0. Evaluating the derivatives of the Macdonald functions using the identity (A.1.16), one has K2 r(ξ) µ K1 r(ξ) µ u ˆ = a (ξ) 2 , r(ξ) r (ξ) K3 r(ξ) µ ν K1 r(ξ) µν K2 r(ξ) µ ν u ˆ u ˆ = −g + a (ξ)a (ξ) , r(ξ) r2 (ξ) r3 (ξ) aµ (ξ) = ρ˜ uµ − icRµ (ξ),

˜ µ (ξ). a ˜µ (ξ) = ρ˜ uµ − icR

The resulting expression for the response tensor is

(2.3.12)

2.3 Response of a relativistic thermal plasma

Π µν (k) = −

Z

49

∞

q 2 nρ µ ν k˜ uρ u ˜ u ˜ −i dξ m K2 (ρ) 0 K3 r(ξ) K2 r(ξ) (2)µν (1)µν −t (ξ) × t (ξ) , r2 (ξ) r3 (ξ)

(2.3.13)

with t˙µν (ξ) given by (2.1.14) and with t(1)µν (ξ) = t˙νµ (−ξ) = T˙ µν (ξ),

t(2)µν (ξ) = aµ (ξ)˜ aν (ξ),

(2.3.14)

with aµ (ξ), a ˜µ (ξ) defined by (2.3.12). The form (2.3.13) with (2.3.14) is a covariant version of the response tensor derived by [3]. For some purposes it is convenient to introduce a matrix representation of the two tensorial quantities inside the integrand in (2.3.13). The following matrix representations apply in the rest frame of the plasma, u ˜µ = [1, 0], with the axes oriented to give k = (k⊥ , 0, kz ), b = (0, 0, 1). One has   1 0 0 0  0 − cos Ω0 ξ −η sin Ω0 ξ 0   T˙ µν (ξ) =  (2.3.15)  0 η sin Ω0 ξ − cos Ω0 ξ 0  , 0 0 0 −1   ξ 0 0 0 1 − cos Ω0 ξ sin Ω0 ξ   0 − −η 0   µν Ω Ω 0 0 (2.3.16) T (ξ) =  . sin Ω0 ξ   0 η 1 − cos Ω0 ξ − 0   Ω0 Ω0 0 0 0 −ξ Explicit forms for the 4-vectors (2.3.7) are sin Ω0 ξ 1 − cos Ω0 ξ µ R (ξ) = ωξ, k⊥ , −ηk⊥ , kz ξ , Ω0 Ω0 1 − cos Ω0 ξ sin Ω0 ξ ν ˜ , ηk⊥ , kz ξ . R (ξ) = ωξ, k⊥ Ω0 Ω0

(2.3.17)

2.3.2 Manifestly gauge-invariant form The response tensor in the form (2.3.13) must satisfy the charge-continuity and gauge-invariance conditions, kµ Π µν (k) = 0, kν Π µν (k) = 0. On writing down the relevant conditions one finds that they are not trivially satisfied, despite the fact that (2.3.13) is derived from an initial expression that does satisfy the relevant conditions trivially. Thus, in order for these relations to be satisfied one infers that certain identities must be satisfied. These identities are of the form Z ∞ Kν+1 r(ξ) df (ξ) Kν r(ξ) Kν (ρ) dξ f (0) ν + + if (ξ) ka(ξ) = 0, ρ dξ rν (ξ) rν+1 (ξ) 0 (2.3.18)

50

2 Response tensors for magnetized plasmas

with arbitrary f (ξ) and ν. The identity is confirmed by a partial integration, noting that r2 (ξ) = aµ (ξ)aµ (ξ) with (2.3.12) and (2.3.7) implies dr(ξ)/dξ = −ika(ξ)/r(ξ). The specific identities required to confirm that (2.3.13) satisfies the charge-continuity and gauge-invariance conditions correspond to ν = 2 and f (ξ) = 1, f (ξ) = ξ, f (ξ) = sin Ω0 ξ and f (ξ) = 1 − cos Ω0 ξ in (2.3.18). To rewrite (2.3.13) in a form that manifestly satisfies the charge-continuity and gauge-invariance conditions, the integral is re-expressed in a form that involves only K3 r(ξ) /r3 (ξ). This involves using dr(ξ)/dξ = −ika(ξ)/r(ξ) with ν = 2 twice, with f (ξ) = T µν (ξ), df (ξ)/dξ = t(1)µν (ξ), and then with f (ξ) = u ˜µ u ˜ν . The result is Z q 2 nρ2 k˜ u ∞ dξ kα T µα (ξ) kβ T βν (ξ) Π µν (k) = i mK2 (ρ) 0 K3 r(ξ) −kα kβ T αβ (ξ) T µν (ξ) − iρ k˜ u T˜µν (ξ) , (2.3.19) r3 (ξ) which involves the tensor T˜ µν (ξ) = Tαβ (ξ) Gαµ (k, u˜)Gβν (k, u˜).

(2.3.20)

Contracting (2.3.19) with either kµ or kν gives zero as required. In the rest frame, explicit expressions for the tensor quantities in (2.3.20) follow from 2 (2.3.17). In particular one has kα kβ Tαβ (ξ) = (ω 2 − kz2 )ξ − (k⊥ /Ω0 ) sin Ω0 ξ. 2.3.3 Forward-scattering form of Trubnikov’s tensor An alternative procedure for deriving Trubnikov’s response tensor is to start from the expression for the response tensor derived using the forwardscattering method, cf. (2.2.2). The steps involved are as follows: insert the form (2.3.1) for the J¨ uttner distribution into (2.2.2), write uµ (τ − ξ) = ˙tµ σ(−ξ)uσ (τ ), include the dependences on u(τ ) in exponential form using uµ = ∂esu /∂sµ with sµ → 0, evaluate the integral over d4 p(τ ) using (2.3.8), and carry out the differentiations using (2.3.12). After writing ku Gαµ (k, u) = (k σ g αµ − k α g σµ )uσ , ku Gβν (k, u) = (k τ g βν − k β g τ ν )uτ , this procedure gives Z ∞ q 2 nρ dξ Tαβ (ξ) (k σ g αµ − k α g σµ )(k τ g βν − k β g τ ν ) Π µν (k) = K2 (ρ)m 0 K2 r(ξ) K3 r(ξ) η ˙ ×tτ (−ξ) − gση . (2.3.21) + aσ (ξ)aη (ξ) r2 (ξ) r3 (ξ) The coefficients of the K2 r(ξ) /r2 (ξ) and K3 r(ξ) /r3 (ξ) terms are Tαβ (ξ) (k σ g αµ − k α g σµ )(k τ g βν − k β g τ ν ) t˙τ η (−ξ)gση

= (d/dξ)[T µν (ξ)kα kβ T αβ (ξ) − kβ T µβ (ξ)kα T αν (ξ)],

(k σ g αµ − k α g σµ )(k τ g βν − k β g τ ν ) t˙τ η (−ξ)aσ (ξ)aη (ξ) = ρ2 (k˜ u)2 T˜µν (ξ) ˜ ν (ξ)], −[2iρ k˜ u + kσ kτ T στ (ξ)][T µν (ξ)kα kβ T αβ (ξ) − Rµ (ξ)R (2.3.22)

2.3 Response of a relativistic thermal plasma

51

respectively, where (2.3.17) and (2.3.20) are used. The fact that the coefficient of K2 r(ξ) /r2 (ξ) in (2.3.21) is a perfect derivative allows one to use the identity (2.3.18) to rewrite (2.3.21) in the form (2.3.19), in which only Kν r(ξ) /rν (ξ) with ν = 3 appears. One finds that (2.3.21) reproduces (2.3.19), which is equivalent to (2.3.13). This establishes that (2.2.2) and (2.2.26) are equivalent starting points for the calculation based on Trubnikov’s method. 2.3.4 Relativistic plasma dispersion functions (RPDFs) For an arbitrary distribution, the general forms (2.2.12), (2.2.28) for the response tensor summed over gyroharmonics can be expressed as integral over p⊥ , pz involving a resonant denominator (ku)k − sΩ0 . The forward-scattering form, as in (2.2.12), involves terms with this resonant denominator, with this denominator squares, and with analogous resonant denominators with s → s ± 1. The Vlasov form (2.2.28) involves only a single resonant denominator. The forward-scattering form does not simplify significantly for a J¨ uttner distribution. The Vlasov form (2.2.28) simplifies, for a J¨ uttner distribution, to Z ∞ Z ∞ ∞ X U µ (s, k)U ∗ν (s, k) −ρu˜u q 2 nρ2 k˜ u Π µν (k) = dpz dp p e . ⊥ ⊥ 4 2m K2 (ρ) 0 (ku)k − sΩ0 −∞ s=−∞ (2.3.23) There are important simplifying features of dispersion in a nonrelativistic thermal plasma that do not apply to a relativistic thermal plasma. In the nonrelativistic limit, the resonant denominator reduces to ω − sΩ0 − kz vz , 2 which does not involve v⊥ , and a Maxwellian distribution, ∝ exp[−(v⊥ + 2 2 vz )/2V ], factorizes into a function of v⊥ and a function of vz . This allows the integral over momentum space to factorize into two separate integrals. The integral over v⊥ involves only the Bessel functions, whose argument is proportional to v⊥ , and the integral over vz can be evaluated in terms of the conventional nonrelativistic plasma dispersion function, as discussed in §2.4.1. In contrast, in the relativistic case, the resonant denominators depends on both γ and pz , and the distribution function depends on the Lorentz factor, γ = (m2 + p2⊥ + p2z )1/2 /m, and does not factor into functions of p⊥ and pz . As a consequence, there is no obvious counterpart of the nonrelativistic result. Some progress can be made towards evaluating the relativistic result by rationalizing the denominator, as in (2.1.32), to remove the square root. The integral over pz can then be rewritten in terms of a sum of terms with resonant denominators pz − pz± , with pz± given by (2.1.33). The pz -integrals can then be reduced to two forms: Z Z dpz e−ργ e−ργ , . (2.3.24) dpz pz − p± γ pz − p±

52

2 Response tensors for magnetized plasmas

These integral may be evaluated in terms of the relativistic plasma dispersion function (RPDF) used to evaluate the response tensor for the J¨ uttner distribution in the unmagnetized case. The definition of this relativistic dispersion function is [34] Z 1 e−ργ T (v0 , ρ) = , (2.3.25) dv v − v0 −1 with v0 = ω/|k| in the unmagnetized case, such that the Cerenkov resonance at v = v0 . In the magnetized case, the resonances occur at v = vz± = pz± /ε± , determined by (2.1.33), (2.1.34). An intermediate step in the evaluation is to write pz in terms of the parameter t introduced in (2.1.35): this involves writing pz /ε⊥ = 2t/(1−t2 ), so that one has γ = ε/m = (ε⊥ /m)(1+t2 )/(1−t2 ). It is convenient to define the plasma dispersion function J(t0 , ρ⊥ ) =

Z

2

1

dt −1

2

e−ρ⊥ (1+t )/(1−t ) , t − t0

(2.3.26)

with t0 = t± and ρ⊥ = ρε⊥ /m. The dispersion integral (2.3.26) can be expressed in terms of (2.3.25): 1 (1 − v02 ) ′ (1 − v02 )1/2 J(t0 , ρ⊥ ) = 2K1 (ρ⊥ ) + T (v0 , ρ⊥ ) + T (v0 , ρ⊥ ) , − 2 v0 ρ⊥ (2.3.27) with v0 = 2t0 /(1 + t20 ). 2.3.5 Alternative form of Π µν (k) for a J¨ uttner distribution Carrying out the procedure outlined above leads to an expression for Π µν (k) for a J¨ uttner distribution that involves an integral over p⊥ with the integrand depending on the plasma dispersion function (2.3.27). This form of the response tensor is of interest as the nonquantum limit of a form derived using relativistic quantum theory. The following explicit steps are involved in the evaluation of Π µν (k) in the form (2.3.23) using this procedure. The resonant denominator, (ku)k − sΩ0 = (εω − s|q|B − pz kz )/m, is rationalized by multiplying by εω + s|q|B + pz kz . This leads to the resonant denominator being replaced by the resonance condition in the form (2.1.32), that is by a quadratic function pz . The integrand in (2.3.23) can then be rewritten using U µ (s, k)U ∗ν (s, k) εω + s|q|B + pz kz U µ (s, k)U ∗ν (s, k) X ±m = , (2.3.28) (ku)k − sΩ0 2ωgs (ω 2 − kz2 ) pz − pz± ± with gs given by (2.1.33). It is convenient to write µ U µ (s, k) = Ukµ (s, k) + U⊥ (s, k),

Ukµ (s, k) = uµk Js (k⊥ R),

(2.3.29)

2.3 Response of a relativistic thermal plasma

53

with muµk = pµk = (ε, 0, 0, pz ) and the other functions in (2.3.29) independent of pz . After some manipulations, the sum of (2.3.28) over s reduces to ∞ X 1 U µ (s, k)U ∗ν (s, k) = 2 [uµk kkν + uνk kkµ − (ku)k gkµν ] (ku) − sΩ (k )k 0 k s=−∞

+

∞ X X

s=−∞ ±

±m

µ ∗ν U± (s, k)U± (s, k) ε + ε± , 2 2gs (k )k pz − pz±

(2.3.30)

with (k 2 )k = ω 2 − kz2 , with ε± given by (2.1.34), and with µ Uk± (s, k) = uµk± Js (k⊥ R),

and

µ U⊥± (s, k)

=

µ U⊥ (s, k)

uµk± = (ε± , 0, 0, pz± )/m,

(2.3.31)

independent of pz .

2.3.6 Strictly-parallel distribution Trubnikov’s method may be used to evaluate the response tensor the response tensor for strictly-parallel and strictly-perpendicular J¨ uttner-like distributions. The steps involved in the derivations closely follow those for an isotropic distribution. For a strictly-parallel distribution, the distribution function, F (p) of f (p) is proportional to δ(p2⊥ ), implying that all particles have p⊥ = 0. It is convenient to introduce a 1D distribution function, g(pz ), such that Z Z Z g(pz ) d3 p f (p⊥ , pz ) d4 p npr = dpz F (p), = = 3 γ (2π) γ (2π)4 Z Z Z d4 p d3 p f (p , p ) = γ F (p), (2.3.32) n = dpz g(pz ) = ⊥ z (2π)3 (2π)4 are the proper number density and the actual number density, respectively. A strictly-parallel J¨ uttner distribution corresponds to g(pz ) =

n e−ργ npr , e−ργ = . 2mK1 (ρ) 2mK0 (ρ)

(2.3.33)

The evaluation of the counterpart of the result (2.3.21) for the response tensor is modified in two ways: the orders of the Macdonald functions are reduced by unity from the isotropic case, and the tensors inside the square brackets in (2.3.21) are projected onto the (two-dimensional) parallel subspace. Specifically, (2.3.21) is replaced by Z ∞ q 2 nρ dξ − (k 2 )k T µν (ξ) − kβ T µβ (ξ)k ν − k ν kα T αν (ξ) Π µν (k) = K1 (ρ)m 0 2 µν K1 rk (ξ) αβ + kak (ξ) T µν (ξ) − kak (ξ)aµk (ξ)kα T αν (ξ) +kα kβ T (ξ)gk rk (ξ) K2 rk (ξ) µ αβ ν µβ ν −kak (ξ)kβ T (ξ)ak (ξ) + kα kβ T (ξ)ak (ξ)ak (ξ) , (2.3.34) rk2 (ξ)

54

2 Response tensors for magnetized plasmas

with aµk (ξ) = gkµν aν (ξ), rk2 (ξ) = aµk (ξ)akµ (ξ). Various alternative forms of (2.3.34) follows by appealing to the identity (2.3.18), in this case with ka(ξ) reinterpreted as kak (ξ). For some purposes it is convenient to separate into the perpendicular and parallel subspaces by writing T αβ (ξ) = T⊥αβ (ξ) + gkαβ ξ,

(2.3.35)

where T⊥αβ (ξ) contains only the central four components in (2.2.4) that involve trignometric functions. 2.3.7 Strictly-perpendicular thermal distribution Trubnikov’s method may also be used to evaluate the response tensor for the strictly-perpendicular relativistic thermal distribution F (p) =

nρ1/2 (2π)1/2 mcK

3/2 (ρ)

δ(p2 − m2 ) δ(pz ) exp[−ρ(p¯ u)/m].

(2.3.36)

When (2.3.36) is inserted into the form (2.2.23) for the response tensor, derived using the Vlasov approach, some terms involve the derivative of δ(pz ). Such terms are to be evaluated only after a partial integration is performed. An alternative procedure, adopted here, is to start from the forward-scattering form (2.2.2) rather than (2.2.23). On inserting (2.3.36) into (2.2.2), the calculation is closely analogous to that leading to (2.3.21) for the isotropic thermal distribution. There are three notable changes: (i) the orders of all the functions Kν (z)/z ν are reduced by 1/2; (ii) the argument, r(ξ), of these functions is replaced by r⊥ (ξ), which is equal to r(ξ) evaluated at kz = 0; and (iii) in the counterpart of (2.3.21) the expression inside the square brackets is replaced by the corresponding expression with the components along the magnetic field set to zero. In place of (2.3.9) one has nρ1/2 K1/2 rP (ξ) , (2.3.37) I(ρ, ξ) = 1/2 K3/2 (ρ) r (ξ) P

and in place of (2.3.10) and (2.3.12) one has 1/2 2 2k⊥ 2 rP (ξ) = = (ρ − iωξ) + 2 (1 − cos Ω0 ξ) , Ω0 K3/2 rP (ξ) µ µ K1/2 rP (ξ) = aP (ξ) , u ˆ 1/2 3/2 rP (ξ) rP (ξ) K5/2 rP (ξ) µ µ ν K1/2 rP (ξ) µν K3/2 rP (ξ) ν u ˆ u ˆ = −˜ g + aP (ξ)aP (ξ) , 1/2 3/2 5/2 rP (ξ) rP (ξ) rP (ξ) (2.3.38)

1/2 aµP (ξ)aPµ (ξ)

2.3 Response of a relativistic thermal plasma

55

respectively, and with aµP (ξ) = aµ (ξ) − bµ ba(ξ). The subscript P denotes a projection onto the 3-dimensional subspace orthogonal to the magnetic field, µν in particular with gP = g µν + bµ bν . The resulting expression for the response tensor that replaces (2.3.21) for the strictly-perpendicular distribution (2.3.36) is q 2 nρ1/2 mK3/2 (ρ)

Z

∞

− T µν (ξ)kσ kτ T˙Pστ (ξ) − kβ T µβ (ξ)kσ T˙Pσν (ξ) 0 µτ µν K3/2 r⊥ (ξ) αβ αν ˙ ˙ −kα T (ξ)kτ TP (ξ) + kα kβ T (ξ)TP 3/2 r⊥ (ξ) 2 µν + kaP (ξ) T (ξ) − kaP (ξ)aµP (ξ)kα T αν (ξ) K5/2 r⊥ (ξ) µ µβ ν αβ ν −kaP (ξ)kβ T (ξ)˜ aP (ξ) + kα kβ T (ξ)aP (ξ)˜ aP (ξ) , 5/2 r⊥ (ξ) (2.3.39)

Π µν (k) =

dξ

with aµP (ξ) = ρ˜ uµP − ikβ TPµβ (ξ), a ˜νP (ξ) = ρ˜ uνP − ikα TPαν (ξ), and TPµν (ξ) = µν µ ν T (ξ) − ξb b . Various alternative forms are obtained by using the identity (2.3.18), with ka(ξ) reinterpreted as kaP (ξ) In particular, the µ = ν = 3 term in (2.3.39) simplifies considerably, reducing to Z ∞ K3/2 r⊥ (ξ) q 2 nρ1/2 dξ ξ kσ kτ T˙Pστ (ξ) Π 33 (k) = 3/2 mK3/2 (ρ) 0 r⊥ (ξ) 2 K5/2 r⊥ (ξ) q 2 npr = − kaP (ξ) , (2.3.40) 5/2 m r (ξ) ⊥

where (2.3.18) is used. One has npr = nK1/2 (ρ)/K3/2 (ρ), for the distributions (2.3.36). The form (2.3.39) is a covariant generalization of the result derived by [4]. Note also that the Macdonald functions of half-integer order in (2.3.39) can be expressed in terms of a rational function times an exponential function, π 1/2 e−z π 1/2 e−z , K (z) = (1 + z), 3/2 2 2 z 1/2 z 3/2 π 1/2 e−z (3 + 3z + z 2 ). (2.3.41) K5/2 (z) = 2 z 5/2

K1/2 (z) =

56

2 Response tensors for magnetized plasmas

2.4 Weakly relativistic thermal plasma Trubnikov’s response tensor contains all relativistic effects for a magnetized thermal plasma, but it is too cumbersome for most practical applications, and approximations to it need to be made. An important application of the general result is in deriving relativistic correction to the response tensor of a nonrelativistic magnetized thermal plasma, referred to here as the weakly relativistic approximation. This involves assuming ρ ≫ 1 (T ≪ 5 × 109 K for electrons or positrons) and expanding in powers of 1/ρ. When this expansion is made, weakly relativistic effects are described in terms of appropriate RPDFs, referred to here as Shkarofsky functions, or Dnestrovskii functions in the case of perpendicular propagation. In this section the response tensor is first evaluated directly for a nonrelativistic thermal distribution, and this nonrelativistic approximation is rederived starting from Trubnikov’s response tensor. The corrections and other changes associated with weakly relativistic effects are then discussed. 2.4.1 Nonrelativistic magnetized thermal plasma In the nonrelativistic limit, the J¨ uttner distribution reduces to a Maxwellian distribution. The nonrelativistic approximation corresponds formally to c → ∞, and it is helpful to use ordinary units, rather than natural units, by including c explicitly in discussing this limit. The nonrelativistic limit corresponds to γ → 1, p⊥ → mv⊥ , pz → mvz . In the J¨ uttner distribution (2.3.1) one has ρ = mc2 /T → ∞, when the asymptotic approximation to the Macdonald functions apply. For z ≫ 1 one has Kν (z) ≈ (π/2z)1/2 e−z .

(2.4.1)

To obtain a nontrivial result, one needs to retain the next order term in the 2 expansion of γ = 1 + (v⊥ + vz2 )/2c2 + · · ·, so that the J¨ uttner distribution (2.3.1) gives f (p) =

2 2 2 2π 2 ¯ h3 nρ e−ργ → (2π)3/2 ¯h3 ne−(v⊥ +vz )/2V , (mc)3 K2 (ρ)

(2.4.2)

which is a Maxwellian distribution. Note that the normalization is the conventional one used in quantum statistical mechanics, Z ∞ Z ∞ Z f (p) d3 p dpz f (p) = 2π dp p , (2.4.3) n= ⊥ ⊥ (2π¯ h)3 (2π¯ h)3 −∞ 0 rather than the one used conventionally in classical kinetic theory, where the factor (2π¯ h)3 is omitted in (2.4.3), so that the factor (2π)3/2 ¯h3 in (2.4.2) is replaced by 1/(2π)3/2 . A major simplification occurs in the evaluation of the response tensor in the nonrelativistic limit for a Maxwellian distribution: the integrals over p⊥

2.4 Weakly relativistic thermal plasma

57

and pz factorize into two independent integrals. The resonant denominator in the response tensor (2.3.23) reduces to ω − sΩ0 − kz vz , which does not involve v⊥ . The integral over v⊥ involves only the Bessel functions, whose argument is proportional to v⊥ ; the integral can be evaluated in terms of modified Bessel function, Is (λ), using   Z ∞ Js2 (z) 2 2 dv⊥ v⊥  v⊥ Js′ (z)Js (z)  e−v⊥ /2V 2 V 2 ′2 0 v⊥ Js (z)   Is (λ)  , (2.4.4) (Ω0 /k⊥ )[Is (λ) − Is′ (λ)] = e−λ  (Ω0 /k⊥ )2 {s2 Is (λ) − 2λ2 [Is (λ) − Is′ (λ)]} 2 2 with z = k⊥ v⊥ /Ω0 , λ = k⊥ V /Ω02 , and where the latter two identities follow from the first using the properties (A.1.2)–(A.1.3) of Js and (A.1.6)–(A.1.7) of Is . The vz -integral involves the factor exp(−vz2 /2V 2 ) in the Maxwellian distribution and the resonant denominator. This integral can be evaluated in terms of the nonrelativistic plasma dispersion function. A conventional definition of this function is the function Z(y), which is often written in terms of a related function φ(y) = −yZ(y). The definition is [?]

Z(y) =

−φ(y) = π −1/2 y

Z

∞

2

dt

−∞

e−t , t−y

(2.4.5)

which is a complex error function. Z(y) satisfies the differential equation dZ(y) = −2 1 + yZ(y) , dy

and for real y has real and imaginary parts Z y √ 2 2 2 dt et , Im Z(y) = i πe−y . Re Z(y) = −2 e−y

(2.4.6)

(2.4.7)

0

Expansions for large and small y give   1 + 1 + 3 + · · · for y ≫ 1, 1 2y 2 4y 4 Re Z(y) = − y  2y 2 + 4 y 4 + · · · for y ≪ 1. 3

(2.4.8)

The analytic properties of Z(y) for complex y are well known [10]. The specific the response function corresponds √ integral that appears in √ to t → vz / 2V , y → ys = (ω − sΩ0 )/ 2kz V . The argument ys may be interpreted in terms of the line profile for gyromagnetic absorption at the sth harmonic. Absorption is described by the imaginary part of Z(ys ), and

58

2 Response tensors for magnetized plasmas

according to (2.4.7) this √ corresponds to a gaussian profile, centered on ω = sΩ0 , with a line width 2|kz |V determined by the Doppler effect due to the thermal spread in vz . The real part of Z(ys ) describes the dispersion corresponding to this absorption. The real and imaginary parts of any causal function, f (ω), being related by the Kramers-Kronig relation Z ∞ dω ′ i Im f (ω ′ ), (2.4.9) Re f (ω) = − ℘ ′ π −∞ ω − ω where

℘ denotes the principal value, with f (ω) = Z(ys ) here.

2.4.2 Response tensor for a Maxwellian distribution The resulting expression for the dielectric tensor (2.2.31), with the contribution of only one (unlabeled) species retained explicitly, corresponds to the 3-tensor components of Π µ ν (k) in the rest frame of the plasma. These components are # " X ys Z(ys ) ω q 2 nω i i i b bj + N j (s, ω, k) , (2.4.10) Π j (ω, k) = m kz2 V 2 ω − sΩ0 s √ with ys = (ω − sΩ0 )/ 2kz V , and with s2 Is λ

 s k⊥ ∆s Is   kz λ   k⊥ s2  i −λ  ′ ′ ′ N j (s, ω, k) = e  −iηs(Is − Is ) Is − 2λ(Is − Is ) −iη ∆s (Is − Is )  ,   λ kz  k  2 s k⊥ k⊥ ∆2s ⊥ ′ ∆s Is iη ∆s (Is − Is ) I s kz λ kz kz2 λ (2.4.11) with ∆s = (ω − sΩ0 )/Ω0 , and where the argument λ of the modified Bessel functions is omitted. 

iηs(Is′ − Is )

2.4.3 Cold plasma limit The result (2.4.10) with (2.4.11) reproduces the cold-plasma response tensor (3.1.13) with (3.1.14). The cold limit is V → 0, which implies λ → 0, ys → ∞, with ys Z(ys ) → −1 for ys → ∞. The power series expansion of the modified Bessel functions, s+2k ∞ X 1 λ Is (λ) = , (2.4.12) k!(s + k)! 2 k=0

implies that in the limit λ → 0, only s = 0, ±1 contribute with I0 (λ) → 0, I± (λ) → 12 . These leading terms suffice to reproduce the cold plasma limit for the component with i, j = 1, 2. For the 13- and 31-components, in this limit

2.4 Weakly relativistic thermal plasma

59

the contributions from s = ±1 sum to zero for each species. For the 23- and 32-components, one obtains zero in the cold plasma limit only after summing over species and assuming that the plasma is charge-neutral. Separating the sum over species, a, into sums over the positively charges, a+, and negatives;y charged, a−. charge neutrality requires which corresponds to X ωpa+ X X X ωpa− |qa− |na− , qa+ na+ = = , (2.4.13) Ωa+ Ωa− a+ a− a+ a− where ωpa± and Ωa± are the plasma frequencies and cyclotrons frequencies, respectively. For the 33-component, for s = 0 one needs to retain the next order term in the expansion (2.4.8) of Z(y0 ) for y0 ≫ 1 to reproduce the cold plasma result. Thermal corrections to the cold plasma limit follow from result (2.4.11) follow by expanding in λ ∝ V 2 and in 1/ys ∝ V . When only the terms of zeroth and first order in V 2 are retained, there are contributions from s = 0, ±1, ±2. The general form of the response tensor to this order is rather cumbersome, and it is usually appropriate to make other simplifying assumptions before making this expansion. 2.4.4 Thermal corrections to longitudinal response tensor A particular simplifying assumption involves considering the longitudinal response. The contribution of a single species to the longitudinal part of the dielectric tensor (2.2.31), K L (ω, k) = 1 + Π L (ω, k)/ε0 ω 2 , follows from the longitudinal part of (2.4.10) with (2.4.11), which give " # 2 2 X ω ω q n ys Z(ys ) 1+ Π L (ω, k) = e−λ Is (λ) . (2.4.14) m |k|2 V 2 ω − sΩ0 s Expanding in powers of V 2 gives ω 2 sin2 θ cos2 θ q 2 n ω 2 − Ω02 cos2 θ |k|2 V 2 + 3 cos4 θ − Π L (ω, k) = − 2 2 2 m ω − Ω0 ω Ω02 ω 4 sin4 θ ω 4 sin4 θ ω 6 (ω 2 + 3Ω02 ) sin2 θ cos2 θ − 2 2 . (2.4.15) + 2 2 + Ω0 (ω − Ω02 ) Ω02 (ω 2 − Ω02 )3 Ω0 (ω − 4Ω02 ) For example, the dispersion relation for longitudinal waves in a magnetized thermal electron gas follows from ω 2 +µ0 Π L (ω, k) = 0 with (2.4.15) evaluated for electrons. 2.4.5 MHD-like limit Another relevant particular simplifying assumption applies to MHD-like waves in an electron-ion plasma. Adding a label a for species a = e, i, with ys → yas ,

60

2 Response tensors for magnetized plasmas

this limit corresponds to ye0 ≪ 1, with yes ≫ 1 for s 6= 0 and yis ≫ 1 for all s. For the 23- and 32-components, the cancelation between electron and ion contributions, due to the charge-neutrality condition (2.4.13), no longer occurs, so that these components are nonzero. The resulting approximate expression for the dielectric tensor is 2 2 X ωpi X ωpi ω ω2 1 1 1+ K 1 3 = 0, , K = i K 1 =1+ 2 2 2 2 Ω , 2) − ω Ω (Ω Ω i i i i i i K 2 3 = −i

2 X ωpi Ωi k⊥ 2 ω k , Ω z i i

K 33 = 1 −

2 X ωpi i

ω2

+

ωp2 2 kz Ve2

(2.4.16)

with K 1 1 = K 2 2 , K 1 2 = −K 2 1 , K 1 3 = K 3 1 , K 2 3 = −K 3 2 , and with P 2 2 2 2 i ωpi /Ωi = c /vA in ordinary units.

2.4.6 Nonrelativistic limit of Trubnikov’s tensor

It is of interest to rederive the foregoing results starting from Trubnikov’s response tensor. The nonrelativistic limit of Trubnikov’s response is found by taking the limit ρ = mc2 /T ≫ 1 in the response tensor in either the forms (2.3.13) or (2.3.21) with (2.3.22). On expanding r(ξ), defined by (2.3.10), in powers of 1/ρ, one finds r(ξ) = ρ − iωξ +

k2 kz2 ξ 2 + ⊥2 (1 − cos Ω0 ξ) + O(1/ρ). 2ρ ρΩ0

(2.4.17)

With this approximation, ρ ≫ 1 implies |r(ξ)| ≫ 1, and the asymptotic limit (2.4.1) of the Macdonald functions is then justified. This gives 2 Kν r(ξ) π 1/2 e−ρ iωξ kz2 ξ 2 k⊥ (1 − cos Ω ξ) . ≈ exp − − 0 rν (ξ) 2 ρ 2ρ ρΩ02 ρν+1/2 (2.4.18) The final term involving cos Ω0 ξ can be expanded, effectively in gyroharmonics, using the generating function for modified Bessel functions, (A.1.5). The modified Bessel functions, Is (λ), satisfy the differential equation (A.1.6) and the recursion relations (A.1.7). Using these relations, one finds     1 Is (λ) ′ cos Ω0 ξ Is (λ)     ∞     X sin Ω0 ξ i(s/λ)Is (λ)   λ cos Ω0 ξ   −isΩ0 ξ = ,  e  e 2 ′′ cos Ω0 ξ Is (λ)     s=−∞     2 2 2 ′ sin Ω0 ξ −(1/λ )[s Is (λ) − λIs (λ)] sin Ω0 ξ cos Ω0 ξ −i(s/λ2 )[Is (λ) − λIs′ (λ)] (2.4.19) Using (2.4.19) in the form (2.3.13) of Trubnikov’s response tensor allows one to write all the ξ-dependence in the integrand in exponential form, allowing the ξ-integral to be performed.

2.4 Weakly relativistic thermal plasma

61

The resulting integral over proper time ξ in either of the forms (2.3.13) or (2.3.21) of Trubnikov’s response tensor can be evaluated in terms of the plasma dispersion function (2.4.7). Three integrals appear:     Z ∞ 1 1 2 2 y Z(y ) k ξ s s  ∆s Ω0 /kz2 V 2  , dξ  ξ  exp i(ω − sΩ0 )ξ − z = −i 2ρ ω − sΩ0 2 0 ∆2s Ω02 /kz4 V 4 ξ (2.4.20) √ with ∆s = (ω − sΩ0 )/Ω0 = ys 2kz V /Ω0 . For the µ, ν = 1, 2 components the integral with unity in the square brackets appears, for the components with either µ = 1, 2, ν = 0, 3 or µ = 0, 3, ν = 1, 2 the additional factor ξ appears, and for the µ, ν = 0, 3 components the factor ξ 2 appears. A derivation of the result (2.4.20), with unity in the square brackets, involves√writing ρ = 1/V 2 , replacing the variable of integration by ξ by t = kz V ξ/ 2, completing the square in the exponent, which becomes −(t − iys )2 − ys2 , regarding the integral over 0 ≤ t < ∞ as a contour integral in complex-t space, and deforming the contour such that it along the imaginary-t axis from the origin to Im t = ys , and then parallel to the real-t axis to infinity, with the first of these two integrals giving (−i times) the real and imaginary parts of the function Z(ys ) in (2.4.7). To evaluate the integrals with ξ or ξ 2 , one includes the additional factors of t or t2 in the integral representation (2.4.5) for Z(y) to derive the results given in (2.4.20). It is then straightforward to show that the space components of the form (2.3.13) of Trubnikov’s response tensor reproduce the nonrelativistic result (2.4.10) with (2.4.11). 2.4.7 Shkarofsky’s response 3-tensor Trubnikov’s response tensor includes all relativistic effects in a thermal magnetized plasma, and the foregoing discussion shows how it reproduces the known nonrelativistic limit to it is reproduced in the limit of very large ρ. To treat weakly relativistic effects one needs to make somewhat different approximations to the exact expression. The initial approximations made in deriving the weakly relativistic limit are the same as in the derivation of the nonrelativistic limit. First, the asymptotic approximation (2.4.1) to the Macdonald functions is made, using (2.4.18). Second, the generating function for modified Bessel functions is used, effectively to introduce a sum over gyroharmonics, as in (2.4.19). To achieve this one may write r(ξ) ≈ r0 (ξ)+ Λ(1 − cos Ω0 ξ),

r0 (ξ) = [(ρ− iωξ)2 + kz2 ξ 2 ]1/2 ,

Λ=

2 k⊥ , 2 Ω0 r0 (ξ)

(2.4.21) where Λ is implicitly a function of ξ. The results (2.4.19) apply, with λ → Λ. With these approximations,Trubnikov’s response leads to Shkarofsky’s [5] approximation to the response 3-tensor. Specifically, the 3-tensor components of (2.3.13) become

62

2 Response tensors for magnetized plasmas

q 2 nω Π (k) = −i m ij



Z

∞

0

s2 Is Λ

∞ X eρ−Λ ˆ ij (k) e−r0 (ξ)−isΩ0 ξ , H dξ [r0 (ξ)/ρ]5/2 s=−∞ s

iηs(Is′ − Is )

  s2 Is  ij ˆ Hs (k) =  −iηs(Is′ − Is ) − 2Λ(Is′ − Is )  Λ  k ∂ ∂ kz z sIs Λ(Is′ − Is ) iη k⊥ ∂s k⊥ ∂s

∂ kz sIs k⊥ ∂s



  ∂  kz ′ Λ(Is − Is ) −iη , k⊥ ∂s   ∂ Is 1 + kz ∂kz (2.4.22)

with r0 (ξ) given by (2.4.21), and where the argument Λ of the modified Bessel functions is omitted for simplicity in writing. The derivatives ∂/∂s and ∂/∂kz are understood to operate on the exponential functions exp(−isΩ0 ξ) and exp[−r0 (ξ)], respectively. 2.4.8 Shkarofsky functions The next approximation allows one to express the integral over ξ in (2.4.22) in terms of functions defined by Shkarofsky. This requires Λ ≪ 1, so that the modified Bessel functions may be approximated by the leading term in their expansion, Is (Λ) ≈ Λs /2s s!. The dependence of the integrand in (2.4.22) on 2 ξ is made explicit by writing Λ = λρ/r0 (ξ), λ = k⊥ /ρΩ02 . The integral over ξ can be written in terms of the generalized Shkarofsky functions [22] Z ∞ at2 (it)r exp izt − dt , (2.4.23) Fq,r (z, a) = −i (1 − it)q (1 − it) 0 where the dimensionless arguments, z = ρ(ω − sΩ0 )/ω, a = kz /ω, are introduced by writing ξ = t/ω, with t dimensionless. The Shkarofsky functions [5] correspond to the special case r = 0: Fq (z, a) = Fq,0 (z, a).

(2.4.24)

The resulting approximation to the space components of the response 4tensor, expanded to low orders in λ, was written down by [5], and generalized by [7]. The latter form includes contributions from harmonics s = 0, ±1, ±2 and is of the form (ordinary units) Π µ ν (k) = −

ωp2 ρ µ X ν, ε0

with the components given by 1+ 1+ 2+ 0+ 1+ X 1 1 = F5/2 − λ(F7/2 − F7/2 ), X 2 2 = X 1 1 + 2λ(F7/2 − F7/2 ), 1− 1− 2− X 1 2 = iη F5/2 − λ(2F7/2 − F7/2 ) ,

(2.4.25)

2.4 Weakly relativistic thermal plasma

63

0+ 0+ 1+ X 3 3 = F5/2 − λ(F7/2 − F7/2 ) i 2 2 h k c ρ 0+ 1+ 2+ 0+ 0+ 1+ − 4F11/2,2 + F11/2,2 ) , − λ(F9/2.2 − F9/2,2 ) + 14 λ2 (3F11/2,2 + z 2 F7/2,2 ω k⊥ kz 1− 1− 2− 1 F7/2,1 − 21 λ(2F9/2,1 X 3= − F9/2,1 ) , ωΩ0 k⊥ kz 0+ 1+ 0+ 1+ 2+ F − F7/2,1 − 41 (3F9/2,1 X 2 3 = −iη − 4F9/2,1 + F9/2,1 ) , ωΩ0 7/2,1 (2.4.26)

where the following notation is introduced: ρ(ω − sΩ0 ) . ω (2.4.27) Further physical approximations need to be made to the plasma dispersion functions to derive results that are useful for application. The special case of perpendicular propagation is of particular relevance in identifying the effects that distinguish the weakly relativistic approximation from the nonrelativistic approximation. (s±) Fq,r =

1 2

Fq,r (zs , kz2 c2 ρ/2ω 2 )±Fq,r (z−s , kz2 c2 ρ/2ω 2) ,

z±s =

2.4.9 Dnestrovskii functions For perpendicular propagation, kz = 0, the function (2.4.21) simplifies to r0 (ξ) = ρ − iωξ, and the response tensor (2.4.22) simplifies considerably. The relativistic plasma dispersion functions that appear can be written in terms of [?] Z ∞ eizt−Λ dt Is (Λ), (2.4.28) Rl (z, λ, s) = −i (1 − it)l 0

2 with Λ = λ/(1 − it) and λ = ρk⊥ /Ω02 . An approximate form obtained by expanding in λ < ∼ 1, each term in the expansion can be written in terms of a Dnestrovskii function [15]. A generalized Dnestrovskii function is defined as the counterpart for perpendicular propagation of the generalized Shkarofsky function (2.4.23) [24]: Z ∞ (it)r eizt Fq,r (z) = −i dt , Fq (z) = Fq,0 (z). (2.4.29) (1 − it)q 0

The special case r = 0 corresponds to the Dnestrovskii function [15]: Z ∞ eizt . (2.4.30) Fq (z) = −i dt (1 − it)q 0 An alternative definition is [19, 20] 1 Fq (z) = Γ (q)

Z

0

∞

dx

xq−1 e−x . x+z

(2.4.31)

64

2 Response tensors for magnetized plasmas q 1/2

-q Fq (z)

0 q

- q 1/2

0

z

Fig. 2.1. A plot of the Dnestrovskii function Fq (z) for real z, solid line for the real part and dotted line for the imaginary part; the function 1/z is indicated by the dashed curve [24, 25].

Properties of these functions are summarized in Appendix A.2.3. The Dnestrovskii function Fq (z) is plotted in Figure 2.1. For comparison, the nonrelativistic counterpart, 1/z, for kz → 0 is shown. Features to note are (a) the resonance at z = 0 is removed, (b) there is a skewing so that the maximum value occurs below z = 0, and (c) damping (which is absent in the nonrelativistic case) isis nonzero in a region below z = 0. An approximation to the Dnestrovskii function that is useful in understanding dispersion in the weakly relativistic case can be derived from an expansion of Fq (z) in derives of the nonrelativistic plasma dispersion function: Fq (z) = −

Z ′′′ (ψ) Z(ψ) − ···, − 1/2 12q (2q)

(2.4.32)

with ψ = (z + q)/(2q)1/2 . Retaining only the leading term in (2.4.32) would imply Fq (z) = 0 at z = −q, which is close to actual zero in Figure 2.1. For large h the leading term in (2.4.32) reproduces the leading term in an expansion that applies for real z ≥ 0 and q > 0: q(q2 z) 1 q + + ··· . (2.4.33) Fq (z) = 1+ z+q (z + q)2 (z + q)4 The leading term, 1/(z + q) provides an approximation to Fq (z) for large z. 2.4.10 Weakly relativistic dispersion for kz 6= 0 For kz 6= 0, the Shkarofsky functions and the Dnestrovskii functions are related by an expansion in modified Bessel functions: Fq (z, a) =

∞ X

s=−∞

e−2a Is (2a) Fq−s (z),

(2.4.34)

2.4 Weakly relativistic thermal plasma

65

with q = s + 5/2, z = ρΩ0 ∆s /ω, a = ρkz2 /2ω 2 . An approximation analogous to (2.4.32) is [28] Fq (z, a) = −

Z(ψ) (3a + q)Z ′′′ (ψ) + ···, − 3(4a + 2q)2 (4a + 2q)1/2

(2.4.35)

with ψ = (z + q)/(4a + 2q)1/2 . Comparison of (2.4.32) and (2.4.35) suggests an approximation Fq (z, a) ≈ Fq+2a (z) [28]. The Shkarofsky functions include both the nonrelativistic dispersion that exists for kz 6= 0 and the weakly relativistic dispersion described by the Dnestrovskii functions for kz = 0. The center of the line, which is at ω = sΩ0 in the nonrelativistic case, is downshifted to z + q = 0 in the approximation (2.4.33) for kz = 0, and this is unchanged for kz 6= 0, so that the center of the line is determined approximately by (ordinary units with ρ → c2 /V 2 ω − sΩ0 V2 (2.4.36) ≈ −(s + 5/2) 2 ω c √ for q = s + 5/2. The line width, which is δωs = 2|kz |V in the nonrelativistic case, is broadened to (ordinary units with ρ → c2 /V 2 ) 2 2 1/2 kz V ω2 V 2 δωs + (s + 5/2) 2 2 = , ω ω2 Ω0 c

(2.4.37)

with ω ≈ sΩ0 . A notable feature of the inclusion of relativistic effect occurs at sufficiently high harmonics, s2 > ∼ ρ. According to (2.4.37), for sufficiently large s the line width is dominated by the relativistic contribution, which increases ∝ s3/2 . In comparison, the separation between neighboring harmonics implied by (2.4.37) decreases more slowly, ∝ s. Hence, for sufficiently large s, the relativistic broadening exceeds the separation between harmonics. The resulting smoothing out of the harmonics into a continuum is a characteristic relativistic effect. 2.4.11 Comparison of approximate and exact results [29]

66

2 Response tensors for magnetized plasmas

2.5 Response tensor for a synchrotron-emitting gas Highly relativistic electrons in a magnetic field emit and absorb synchrotron radiation. Dispersion is related to dissipation, and the dispersion due to highly relativistic electrons is related to synchrotron absorption. For a thermal distribution, such dispersion can be treated using Trubnikov’s response tensor in the limit ρ ≪ 1 [30]. However, synchrotron-emitting particles typically have power-law energy spectra, and not thermal spectra. The response tensor for an arbitrary nonthermal distribution of synchrotron-emitting particles can be evaluated by making the highly relativistic approximation [31, 32], which involves Airy functions. 2.5.1 Method of stationary phase The synchrotron approximation involves expanding in inverse powers of the Lorentz factor of the radiating electron. The emission is strongly beamed into the forward direction, implying that the difference between the angle of emission and the pitch angle, |θ − α|, is of first order in 1/γ, cf. figure 2.2. An observer at θ ≈ α receives a pulse of radiation each time the particle’s motion is directly towards the observer, who can see the particle only for a fraction ∼ 1/γ of its orbit each gyroperiod. Consider the integral over proper time, ξ, in the expression (2.2.2) or (2.2.23) for the response tensor. The proper time parameterizes the gyrophase, and the foregoing argument implies that the dominant contribution to the ξ-integral is from a small range of ξ corresponding to emission in the direction of the observer. It is appropriate to approximate such an integral using the method of stationary phase. The method of stationary phase applies to an integral, over z say, in which the integrand contains a phase factor, exp[if (z)] say, such that the integral is dominated by points of stationary phase, where f ′ (z) = df (z)/dz is zero. Assuming only one stationary phase point, at z = z0 , the stationary phase approximation corresponds to 1/2 Z iπ dz G(z) eif (z) ≈ G(z0 ) eif (z0 ) , (2.5.1) f ′′ (z0 ) where G(z) is a slowly varying function. The derivation of (2.5.1) involves expanding in a Taylor series about z = z0 , with f ′ (z0 ) = 0. Only terms up to second order are retained in the phase f (z) = f (z0 ) + 12 (z − z0 )2 f ′′ (z0 ), and G(z) ≈ G(z0 ) is regarded as a constant. The resulting integral gives (2.5.1). In the application considered here, integrals appear in which G(z) has a zero near z = z0 . To evaluate such integrals one appeals either to Hermite integration or, more simply, to a partial integration to find 1/2 Z (z − z0 )2 i/f ′′ (z0 ) iπ if (z) dz e ≈ eif (z0 ) , (2.5.2) f ′′ (z0 ) (z − z0 )4 −3/[f ′′ (z0 )]2 and so on, with integrals of odd powers of z − z0 vanishing.

2.5 Response tensor for a synchrotron-emitting gas

67

1/γ

Fig. 2.2. Radiation from a highly relativistic particle moving in a circle is seen as a sequence of pulses, one per gyroperiod, by observers whose line of sight is within an angle ∼ 1/γ of the plane of the circle.

2.5.2 Synchrotron approximation to the response tensor The response tensor in the synchrotron limit is obtained by starting either from (2.5.10), viz. Z ∞ Z 4 d p(τ ) q2 F (p) dξ exp ik X(τ ) − X(τ − ξ) Π µν (k) = 4 m (2π) 0 ×Tαβ (ξ) ku(τ ) Gαµ k, u(τ ) ku(τ − ξ) Gβν k, u(τ − ξ) , (2.5.3)

or from the Vlasov form (2.2.23), viz. Z Z 4 d p(τ ) ∞ Π µν (k) = −iq 2 dξ uµ (τ ) eik[X(τ )−X(τ −ξ)] (2π)4 0 ∂F (p) . ×ku(τ − ξ) Gαν k, u(τ − ξ) t˙α β (τ − ξ) ∂pβ

(2.5.4)

The phase factor in the integrand is k[X(τ ) − X(τ − ξ)], with the orbit given by X µ (τ ) = xµ0 + tµν (τ )u0ν . The initial 4-velocity, uµ0 includes an arbitrary initial gyrophase, φ, and X µ (τ ) involves trignometric functions of φ − Ω0 τ . Hence, the phase factor involves trignometric functions of both φ − Ω0 τ and φ − Ω0 (τ − ξ) The integral over d4 p(τ ) contains an integral over gyrophase, which can be written either as an integral over φ or as an integral over Ω0 τ . Using (2.1.21) and (2.1.22) one finds k[X(τ ) − X(τ − ξ)] = (ku0 )k ξ − k⊥ R[sin(φ + ηψ) − sin(φ + ηψ − Ω0 ξ)], (2.5.5) with φ = φ0 + Ω0 τ . The points of stationary phase with respect to φ occur at cos(φ + ηψ) = cos(φ + ηψ − Ω0 ξ).

(2.5.6)

One solution is at φ + ηψ = 21 Ω0 ξ. There are two solutions each period, and the two solutions contribute equally. Thus the integral over φ is approximated by making the stationary phase approximation for one of these solution, and multiplying the result by 2 to take account of the other. After applying the method of stationary phase to the φ-integral, the phase factor (2.5.5) is approximated by

68

2 Response tensors for magnetized plasmas

k[X(τ ) − X(τ − ξ)] = (ku0 )k ξ − 2k⊥ R sin( 21 Ω0 ξ).

(2.5.7)

Writing k⊥ = |k| sin θ, v⊥ = v sin α, (2.5.7) gives k[X(τ )−X(τ −ξ)] = γξ[(ω −|k|v cos(α−θ)]−γ|k|v sin α sin θ[ξ −2 sin( 12 Ω0 ξ)]. (2.5.8) The integral over pitch angle can also be approximated using the method of stationary phase. The points of stationary phase in the α-integral occur at 2 sin( 21 Ω0 ξ) sin(α − θ) = cos α sin θ 1 − . (2.5.9) Ω0 ξ Thus the condition for stationary phase for the α-integral can be approximated by (Ω0 ξ)2 sin(α − θ) = π1 , π1 = sin θ cos θ. (2.5.10) 24 As for the φ-integral, there are two stationary phase points that contribute equally, and only one need be retained with the result multiplied by 2. In evaluating the integrals by the method of stationary phase, it is convenient to write y = Ω0 ξ,

a=

γ(ω − |k|v) , Ω0

b=

γ|k|v sin2 θ , 8Ω0

(2.5.11)

and to change variables to δφ = φ − ξ/2, δα = α − θ − π1 . The phase factor in (2.5.4) reduces to k[X(τ ) − X(τ − ξ)] = ay +

by 3 sin α (δα)2 + 4by + 4by (δφ)2 , 3 sin θ sin2 θ

(2.5.12)

and in the integrand one makes the replacement Ω0 τ = δφ + 21 y, α = δα + θ + π1 . The variables y = Ω0 ξ, δα and δφ are all small quantities of O(γ −1 ). The correction π1 is O(γ −2 ). 2.5.3 Expansion about a point of stationary phase On making the foregoing approximations, one needs to evaluate the slowly varying functions in the integrand in (2.5.3) or (2.5.4) before performing the integral over ξ or y = Ω0 ξ. It is conventional to describe a distribution of highly relativistic particles in terms of the energy spectrum: in terms of Lorentz factors, let N (γ)dγ be the number density of particles in the range γ to γ + dγ. This corresponds to F (p) =

N (γ)φ(α) n δ(p2 ) , 2 2 2πm c γ(γ 2 − 1)1/2

(2.5.13)

where φ(α) is the pitch angle distribution. The normalization conditions are

2.5 Response tensor for a synchrotron-emitting gas

Z

dγ N (γ) = 1,

Z

d cos α φ(α) = 2.

69

(2.5.14)

The integral over pitch angle is evaluated by the method of stationary phase, and φ(α) is evaluated at α = θ + δα + π1 , giving cos θ cos θ , sin α = sin θ 1 + δα , (2.5.15) φ(α) = φ(θ) 1 + δα g(θ) sin θ sin θ with g(θ) = tan θ φ′ (θ)/φ(θ). The contribution to dispersion from the highly relativistic particles is usually of interest only for relatively high frequency waves, where the waves are transverse to an excellent approximation. It is convenient to project the response tensor onto the two transverse 4-vectors tµ = [0, t],

t = (cos θ, 0, − sin θ),

aµ = [0, a],

a = (0, 1, 0).

(2.5.16)

The components along these 4-vectors are denoted as the 1- and 2-components, respectively. For the response tensor in the forward-scattering form (2.2.2) the following expansions are required: 1 |k|2 sin2 θ y 3 −y − 12 η y 2 T µν (ξ) = , kα kβ T αβ (ξ) = , 1 2 −y Ω0 2 η y Ω0 6 |k| sin θ − cos θ 61 y 3 , − 21 η y 2 , Ω0 |k| sin θ − cos θ 61 y 3 , 21 η y 2 . kα T αν (ξ) = Ω0

kβ T µβ (ξ) =

(2.5.17)

The matrix representation in (2.5.17) represents the two components tµ , aµ , defined by (2.5.16). The Vlasov form involves a derivative of the distribution function; this gives ∂ 1 ∂ ku(τ ) Gαν k, u(τ ) t˙α β (τ ) β = [ku(τ ) u˜ν − k˜ u uν (τ )] ∂p mγc2 ∂γ ∂ 1 , −[ku(τ ) aν (τ ) − ka(τ ) uν (τ )] 2 2 mγ v ∂α aµ (τ ) =

d µ u (τ ), dα

ka(τ ) = γ|k|v δα,

(2.5.18)

where only the leading term in the approximate expression for ka(τ ) is retained. 2.5.4 Transverse components of the response tensor There are only three independent transverse components for the response tensor, and it is convenient to write them in terms of tµν = µ0 Π µν , with the

70

2 Response tensors for magnetized plasmas

2-dimensional transverse subspace spanned by the 4-vectors tµ , aµ , defined by (2.5.16). The three independent components contain a class of integrals that need to be evaluated by the method of stationary phase. This class is defined by Z ∞

I (n) (a, b) =

0

dy y n exp[iay + i 31 by 3 ],

(2.5.19)

with n an integer. The forward-scattering form for the response tensor gives tµν (k) = −

2 ωp0 Ω0 φ(θ) ω

Z

dγ

N (γ) µν J (a, b), γ2v

(2.5.20)

with the three independent components J tt (a, b) = 43 bI (1) (a, b) − a, J aa (a, b) = 4bI (1) (a, b) + 38 ia2 I (0) (a, b) + 53 a, 2 (1) (0) (a, b) − 20 (a, b) − 2iI (−1) (a, b) + 2i J ta (a, b) = − 12 η cos θ 16 9 ia I 9 aI (2.5.21) +g(θ) − 34 aI (0) (a, b) − 2iI (−1) (a, b) + i 13 .

The Vlasov form (2.5.4) leads to

µν tµν (k) = tµν (γ) (k) + t(α) (k), Z 2 ωp0 Ω0 d N (γ) tµν (k) = − φ(θ) dγ γv hµν , γ ω dγ γ 2 v Z 2 h i ωp0 Ω0 N (γ) 1 tta (k) = φ(θ) dγ ( 2 η cos θ) −g(θ) iI (−1) (a, b) , α ω γ

(2.5.22)

with hµν given by     htt −aI (−1) (a, b) − bI (1) (a, b)      aa    −aI (−1) (a, b) − 3bI (1) (a, b) h  =   . (2.5.23)     (−1) 1 4 4 ta (0) h η cos θ 2 + g(θ) iI (a, b) + aI (a, b) − i ] 2 3 3 The forms (2.5.20), (2.5.22) are related to each other by a partial integration. However, the two expressions (2.5.20), (2.5.22) are not equivalent because the constant terms in the diagonal components in (2.5.20) are not reproduced by the partial integration of (2.5.22). This inconsistency reflects a more fundamental problem in taking the extreme relativistic limit. 2.5.5 Extreme relativistic limit In the extreme relativistic limit the gyroradii of the particles become arbitrarily large and their gyroperiods become arbitrarily long. The particle motions

2.5 Response tensor for a synchrotron-emitting gas

71

are better approximated as constant rectilinear motion than as gyromagnetic motion in this extreme limit. The inconsistencies that arise in taking the extreme relativistic limit are related to this breakdown in the assumption that the motions are gyromagnetic. One procedure that avoids the inconsistencies is to note that in the extreme limit the system is effectively unmagnetized, and to use this fact to remove the terms that cause the inconsistencies. Specifically, in taking the extreme relativistic limit one first subtracts the constant terms in (2.5.20), (2.5.22) that remain in this limit, ensuring that the remaining terms are well behaved. One then adds the known result for the response tensor for an unmagnetized plasma to replace the subtracted terms. This procedure involves separating the response into two parts: the unmagnetized part and the magnetic correction to it. This separation corresponds to µν tµν (k) = tµν 0 (k) + tmag (k),

(2.5.24)

where µ, ν = 1, 2 span only the 2-dimensional transverse plane. The unmag2 µν netized part corresponds to tµν at high frequencies, where ωp0 0 (k) = ωp0 δ is the proper plasma frequency. The magnetized part, tµν mag (k), in (2.5.24) describes the dispersion associated with the synchrotron emitting particles. The method developed here gives tµν mag (k) when terms independent of the magnetic field are discarded. 2.5.6 Airy integral approximation The integrals I (n) (a, b) that appear in (2.5.20) and (2.5.22) may be evaluated in terms of the Airy functions Ai (z) and Gi (z), cf. (A.1.23). Identifying z = a/b1/3 , one finds I (0) (a, b) = πb−1/3 Ai (z) + iGi (z) , I (1) (a, b) = −iπb−1/3 Ai′ (z) + iGi′ (z) , Z z (2.5.25) I (−1) (a, b) = iπ dz ′ Ai (z ′ ) + iGi (z ′ ) . 0

The Airy function Ai (z) can be represented as a Bessel function of order 1/3, and in the case of relevance to synchrotron emission these are Macdonald functions. The relevant representations are 1 a 1/2 1 a Re I (0) (a, b) = √ K1/3 (R), Im I (1) (a, b) = √ K2/3 (R), 3 b 3 b Z ∞ 2a3/2 1 dt K1/3 (t), R = 1/2 . Im I (−1) (a, b) = − √ (2.5.26) 3b 3 R

These functions appear in the discussion of synchrotron emission and absorption in §4.5. The approximations available for Gi (z) are for large and small z. The asymptotic expansion for z ≫ 1 is given by (A.1.24), and the expansion for z ≪ 1 by (A.1.25).

72

2 Response tensors for magnetized plasmas

2.5.7 Response at high frequencies Approximations available for Gi (z) are relevant to the evaluation of the hermitian part. Some known results are summarized in Appendix B. The case of most interest is high frequencies, corresponding to a ≫ 1. In this case the asymptotic expansions of the plasma dispersion functions give 1 i 2b I (1) (a, b) = − 2 , I (0) (a, b) = 1+ 3 , I (−1) (a, b) = − ln a, a a a (2.5.27) with a = ω/2Ω0 γ. The leading terms in the hermitian part of the response tensor in the form (2.5.20) become 2 Ω0 3 2 γ sin θ, 3 ω

J aa (a, b) = 7J tt (a, b), ω J ta (a, b) = 21 iη cos θ[1 + g(θ)] ln , 2Ω0 γ

J tt (a, b) = −

(2.5.28)

where the logarithmic term is assumed to dominate in tta (k). A result similar to (2.5.28) with (2.5.20) was derived in a different way by [4]. 2.5.8 Power-law distribution The case of most interest in astrophysical plasmas is a power-law distribution. For present purposes a power-law distribution is defined by N0 γ −β for γ1 < γ < γ2 , N (γ) = 0 otherwise, ( (β − 1)/(γ11−β − γ21−β )−1 for β 6= 1, N0 = (2.5.29) ln(γ2 /γ1 ) for β = 1, Inserting (2.5.29) into (2.5.20) with (2.5.28) gives tµν (k) =

2 ωp0 δ µν + ∆tµν (k), c2

(2.5.30)

with δ µν the unit tensor in the 2-dimensional subspace, and with tt 2 ωp0 ∆t (k) Ω02 2ζφ(θ) sin2 θ 1 = , c2 ω 2 3 ∆taa (k) 7 2 ω Ω0 β −1 1 i ωp0 ln cos θφ(θ)[1 + g(θ)] ∆tta (k) = − η 2 , 2 c ω β − 2 γ1 Ω0 γ1 sin θ (2.5.31)

2.5 Response tensor for a synchrotron-emitting gas

ζ=

 β−1     β − 2 γ1

73

for β > 2,

 β − 1 1−β   (γ − γ21−β )−1  2−β 1

ω Ω0 sin θ

(2.5.32)

(2−β)/2

for β < 2,

with the proviso that the latter approximation applies only for frequencies (ω/Ω0 sin θ)1/2 < ∼ γ2 . 2.5.9 Highly relativistic thermal distribution Application of the foregoing method to a J¨ uttner distribution with a highly relativistic temperature, ρ ≪ 1, is of formal interest in comparing the method with Trubnikov’s method. For a highly relativistic thermal distribution, (2.5.29) is replaced by N (γ) =

γ 2 ρ e−ργ ≈ 21 γ 2 ρ3 e−ργ , K2 (ρ)

(2.5.33)

with K2 (ρ) ≈ 2/ρ2 for ρ ≪ 1. The resulting expression for the transverse components are of the form (2.5.24), with the magnetized part, tµν mag (k), having diagonal components t11 mag (k) =

2 2ωp0 Ω02 sin2 θ , ρ2 ω 2

11 t22 mag (k) = 7tmag (k),

(2.5.34)

and off-diagonal components 21 t12 mag (k) = −tmag (k) = −iη cos θ

2 ωp0 Ω0 ln ω

ωρ 2Ω0

,

(2.5.35)

Alternatively, the result (2.5.34), (2.5.35) may be derived by making the ultrarelativistic approximation in Trubnikov’s response tensor. A relevant approximation to the function r(ξ), defined by (2.3.10), is ω 2 sin2 θ (Ω0 ξ)4 , Ω02 12 (2.5.36) where the term ρ2 is neglected, and vacuum dispersion, kz2 → ω 2 cos2 θ, 2 k⊥ → ω 2 sin2 θ is assumed in evaluating δr2 (ξ). In evaluating tµν mag (k) one is to subtract the contribution that is nonzero for B → 0. Inspection of the response tensor (2.3.13) shows that the term proportional to K2 r(ξ) /r2 (ξ) has a nonzero contribution for B → 0, and subtracting it using (2.5.36) gives K2 r0 (ξ) K2 r(ξ) δr2 (ξ) K3 r0 (ξ) − =− . (2.5.37) r2 (ξ) r02 (ξ) 2 r03 (ξ) r2 (ξ) = r02 (ξ) + δr2 (ξ),

r02 (ξ) = −2iωρξ,

δr2 (ξ) = −

The integral over ξ in (2.3.13) can then be evaluated, either using the identity

74

2 Response tensors for magnetized plasmas

Z

0

∞

dx xµ Kν (ax) = 2µ−1 a−µ−1 Γ

1+µ+ν 2

Γ

1+µ−ν 2

,

(2.5.38)

or by first making the approximation Kn (r0 ) ≈ 2n−1 (n − 1)!/r0n for |r0 | ≪ 1. The result (2.5.34) for the diagonal terms is reproduced. For the off-diagonal R term, and integral of the form dξ ξ K2 (r0 )/r02 appears, and with r02 ∝ ξ and K2 (r0 ) ≈ 2/r02 , this integral is logarithmically divergent. The result (2.5.35) is reproduced, except for the argument of the logarithm. This argument is not well determined: as in (2.5.31) it effectively depends on a lower energy cutoff, which is ill-defined for a relativistic thermal distribution. This uncertainty in the argument of the logarithm in the following discussion of the dispersion.

2.6 Response tensor for a pulsar plasma

75

2.6 Response tensor for a pulsar plasma TBA 2.6.1 Properties of pulsar plasma The pulsar plasma of interest in connection with the radio emission is dominated by pairs created in the pulsar plasma itself, as required by the electrodynamics § ??. However, the detailed properties of the pulsar plasma are not well determined, with major uncertainties in the details of the pair creation and in the screening of the parallel electric field, Ez . One constraint imposed by the electrodynamics is that the number density of the pairs exceed nGJ , given by (??), so that the difference between the number densities of electrons and positrons is equal to nGJ . The number density of pairs is usually expressed as a multiplicity factor, M , times nGJ at the height of their creation, with M ≫ 1 required for screening of Ez about this height. Another requirement of the electrodynamics is that there be a current, I, of order enGJ c times the area of the polar cap, in order to explain the power loss in terms of IΦ. This suggests that the pairs must be streaming outward relativistically. Due to the superstrong magnetic field, all electrons and positrons quickly radiated away any perpendicular energy, relaxing to their ground state, p⊥ = 0. This leads to an important simplification: the pulsar plasma is strictly one dimensional (1D). The plasma parameters needed to describe the pulsar plasma include the number densities n± , streaming 4-speeds u± , and the spread, ∆u± say, in 4-speeds about the mean values. The wave dispersion depends on the specific form of the unknown distribution functions, g ± (u), but is relatively insensitive to the choice of distribution functions which give the same values of n± , u± and ∆u± in the highly relativistic case u± ≈ γ ± , ∆u± ≈ ∆γ ± . Due to the electrons and positrons being created together, one expects their distribution functions to be similar, implying ∆u+ = ∆u− , but Ez 6= 0 causes the bulk streaming velocities to be different, u+ 6= u− ; this difference is large in an oscillating model. Models for the pair creation provide an estimate of the bulk 4-speed u ¯ ≈ γ¯ when this effect of Ez is ignored. Detailed models for the pair creation are based on curvature emission and inverse Compton emission by primary particles accelerated in a polar gap [?, ?, ?]. Subject to considerable uncertainty, arguably plausible values are M ∼ 10–103, ∆γ ∼ 1–10, γ¯ ∼ 10–103. The height of the emission region is also poorly constrained, and this leads to a further uncertainty in the plasma parameters in the emission region. There is evidence for the emission region being located at heights of several tens to hundreds of stellar radii on field lines at angles 0.5–0.7 times the angle subtended by the last closed field line [?, ?]. The magnetic field, B, and the number density of pair (in the absence of further pair creation) decreases ∝ (R∗ /r)3 , which decreases by ∼ 103 between r ∼ 10 R∗ and r ∼ 100 R∗.

76

2 Response tensors for magnetized plasmas

The presence of a parallel electric field, Ez , implies that the motion of the electrons and positrons is different, and the reflection of some particles of one sign leads to their number densities not being the same, and allowing the difference to equal nGJ . Quite generally, acceleration by Ez implies γ + 6= γ − . In the frame in which the bulk streaming is zero, this implies that the electrons and positrons have oppositely directed bulk motions. Oscillations of Ez in a LAEW imply that this counter-streaming is itself highly relativistic. A general treatment that includes all these effects in the wave dispersion is needed. It is convenient to consider simpler models that allow the effects of these various different features of the distribution functions to be considered separately. 2.6.2 Cold pair-plasma model The simplest useful model for wave dispersion in a pulsar plasma is based on the assumptions that the highly relativistic streaming motion is the same for both electrons and positrons, and that the intrinsic spread in there Lorentz factors is zero. One can then eliminate the streaming motion by making a Lorentz transformation to the rest frame of the plasma. In this frame the wave properties are those of a cold electron-positron plasma. The theory of dispersion in a cold plasma is discussed in §3.2, and applied to the specific case of an electron gas (the ‘magnetoionic theory’) in § 3.2.1. The effect of an admixture of positrons is discussed in §3.2.7. The cold-plamsa approximation for the modes of a pulsar plasma correeponds to this last case in the limit Ωe ≫ ωp . There are only two natural modes at any given frequency in a cold plasma, and these are often referred to as the ordinary and extraordinary modes. For ω ≪ Ωe in a pulsar plasma these are often denoted th O and X modes, respectively, corresponding to the magnetoionic o and z modes in §3.2.1. The the dielectric tensor, K i l (ω) = δji + µ0 Π i j (ω)/ω 2 , for a cold plasma is given by (3.1.15) in terms of three functions S, D, P , with K 1 1 = K 2 2 = S, K 1 2 = −K 2 1 = −iD, K 3 3 = P and all other components equal to zero. For a cold electron-positron gas, S, D, P are given by (3.2.2), specifically, S = 1 − X/(1 − Y 2 ), D = ηXY /(1 − Y 2 ), P = 1 − X, where the magnetoionic parameters are X = ωp2 /ω 2 , Y = Ωe /ω and with η = (n+ − n− )/(n+ + n− ) the average charge per particle, with η = −1 for a pure electron gas, η = 0 for a pure pair plasma. The cold plasma dielectric tensor does not depend on k, implyinh that the dispersion equation is a qudratic equation in |k|2 , and that at any frequency there are only two modes of a cold plasma. For an electron gas, both modes (‘ordinary’ and ‘extraordinary’) are separated into two propagating bands separated by a stop band, which is a range of frequencies where the waves are evanescent, with a resonance, |k| → ∞, on the lower-frequency side, and a cutoff, |k| → 0, on the upper-frequency side. The presence of positrons introduces as additional cutoff frequency in the ex-

2.6 Response tensor for a pulsar plasma

77

traodrinary mode near the cyclotron frequency, Y ≈ 1, the cutoof frequencies satisfy (3.2.21). O and X modes At frequencies ω ≪ Ωe the two modes are referred to as the O and X modes in the cold-plasma model for a pulsar. The condition ω ≪ Ωe applies in the radio emission region, and for a range of heights between the emission region and the cyclotron-resonance region. Throughout the magnetosphere, one has 2 X/Y 2 = ωp2 /Ωe2 ≪ 1, and this parameter may be interpreted as 1/βA , with βA ≫ 1 is the Alfv´en speed in the pair plasma. For βA ≫ 1, the characteristic 2 1/2 MHD speed, βA /(1 + βA ) , is close to the speed of light. The properties of the two cold-plasma wave modes are derived in §3.2.1 by first solving for the axial ratio, T , of the polarization ellipse. T , satisfies the quadratic equation (3.2.4), which may be written as T 2 − RT − 1 = 0 with R=

Y sin2 θ (P S − S 2 + D2 ) sin2 θ = (1 − E), P D cos θ (1 − X)η cos θ

E=

X(1 − η 2 ) . 1−Y2 (2.6.1)

The solutions for for T are T± = 21 R ± 12 (R2 + 4)1/2 ,

R2 + 4 =

F2 , P 2 D2 cos2 θ

(2.6.2)

with F 2 given in terms of S, D, P by (3.1.28). The term E in (2.6.1) is nonzero due to the charge imbalance, η 2 6= 1. It appears from S 2 − D2 =

(1 − X)2 − Y 2 + XY 2 E . 1−Y2

(2.6.3)

E is small at all frequencies with the exception of a small range near the 2 cyclotron frequency, |1 − Y | < ∼ 1/βA . Near the cyclotron resonance, E 6= 0 leads to an additional cutoff in the extraordinary mode. However, near the cyclotron resonance the neglect of the spread in Lorentz factors is not well justified, and the cold plasma model is not a valid approximation. The region near the cyclotron resonance is discussed separately below. For ω ≪ Ωe , the approximation E → 0 applies. The two modes have relatively simple properties in three ranges: low frequencies, ω ≪ ωp , intermediate frequencies, ωp ≪ ω ≪ Ωe , and high frequencies, ω ≫ Ωe . Low-frequency modes 2 At low and intermediate frequencies, one has S ≈ 1/β02 , D ≈ −ηY /βA ,P = 2 1/2 1 − X, where β0 = βA /(1 + βA) is the MHD speed, with β0 ≈ 1 for βA ≫ 1. Except for a small range of angles about sin θ = 0, one has R2 ≫ 4, implying

78

2 Response tensors for magnetized plasmas

that the two modes are approximately linearly polarized. The identification of the ± modes depends on the sign of R, which is determined by the sign σ = (1 − X)η cos θ/|(1 − X)η cos θ|. For σ = 1, T+ ≈ R corresponds to the O mode, and T− ≈ −1/R corresponds to the X mode; for σ = −1, T− ≈ −R = |R| corresponds to the O mode, and T+ ≈ 1/R = −1/|R| corresponds to the X mode. At low frequencies, ω ≪ ωp , the two modes are MHD-like. In the limit Y → ∞, using the first and last of (3.1.29) with T+ → ∞ and T− → 0, respectively, one obtains the approximations n2O ≈

1−X PS ≈ , A 1 − X cos2 θ

n2X ≈ S ≈ 1.

(2.6.4)

The polarization vectors in the same approximation are eO =

LO κ + TO t , (L2O + TO2 )1/2

X sin θ cos θ LO =− , TO 1 − X cos2 θ

eX = ia.

(2.6.5)

The O mode dispersion relation (2.6.4) separates into a lower-frequency range, ω 2 < ωp2 cos2 θ, and a higher-frequency range ω 2 > ωp2 . At low frequencies, ω ≪ ωp , the dispersion relation becomes n2 = 1/ cos2 θ and the polarization is along the 1-axis. These properties correspond to the Alfv´en mode in the limit βA → ∞. The upper limit of the low-frequency branch is at the resonance at ω 2 = ωp2 cos2 θ, where the polarization becomes longitudinal. There is a stop band, ωp2 cos2 θ < ω 2 < ωp2 , between the lower and upper branches. The upper branch starts at a cutoff at ω 2 = ωp2 . At higher frequencies, still satisfying ω ≪ Ωe , the dispersion relation approaches n2O ≈ 1−(ωp2 /ω 2 ) sin2 θ, which is equivalent to ω 2 − |k|2 ≈ ωp2 sin2 θ. The X mode has vacuum-like properties, n2X ≈ 1/β02 ≈ 1, and its polarization is strictly transverse in this approximation. Cold plasma modes at small angles The foregoing properties of the two modes are derived under the assumption that the modes are nearly linearly polarized, corresponding to R ≫ 1 in (2.6.1). One has R = 0 for sin θ = 0 and the opposite condition, R ≪ 4, is satisfied at sufficient;y small angles, where the two modes are oppositely circularly polarized. The magnetoionic dispersion equation may be written in the form (1 − X)(n2 − n2+ )(n2 − n2− ) − sin2 θ

XY 2 2 2 n (n − 1 + E) = 0, 1−Y2

(2.6.6)

with E defined by (2.6.1), and with n2± defined by σ = ±1 in n2σ = 1 −

X(1 − σ|η|Y ) . 1−Y2

Tσ = σ

η cos θ , |η cos θ|

(2.6.7)

2.6 Response tensor for a pulsar plasma

79

For sin θ = 0 there are three solutions, X = 1, corresponding to longitudinal oscillations at ω = ωp , and two oppositely circularly polarized modes with 2 n2 = n2σ , T = Tσ . In the strong-B limit, X/Y 2 = 1/βA ≪ 1 and Y ≫ 1 2 2 otherwise, the refractive indices approach n = 1 ± ηX/Y = 1 ± ηY /βA , where 2 Y ≪ βA (ω ≫ ωp /βA ) is assumed. The handedness of the circular polarization is determined by the sign σ = (1−X)η cos θ/|(1−X)η cos θ|, which reverses at ω = ωp . As sin θ increases, |R| increases, and for |R| > ∼ 4 these modes become the O and X modes. depending on the sign σ: Tσ corresponds to the O mode, and T−σ corresponds to the X mode. Cold-plasma modes near the cyclotron resonance Escaping pulsar radiation necessarily passes through a region where the wave frequency is equal to the cyclotron frequency. The cyclotron resonance is smoothed out when a spread in Lorentz factors is included. Although this invalidates the model, nevertheless the cold plasma assumption provides a useful guide to identifying how the polarization varies from well below to well above the cyclotron resonance. As Y = Ωe /ω varies from Y ≫ 1 to Y ≪ 1, the parameter R, defined by (2.6.1) with E = 0, varies from |R| ≫ 1 to |R| ≪ 1. The axial ratios vary from T± ≈ R, −1/R for |R| ≫ 1 to T± = ±1 + 12 R, corresponding to a change from nearly linear to nearly circular polarizations. It follows that the nearly linear polarizations of the wave modes at intermediate frequencies change to the nearly circular polarizations characteristic of the wave modes of any magnetized plasma at frequencies high frequencies, ω ≫ ωp , Ωe . The rapid change in the shape of the polarization ellipse implies that the gradient in B causes wave in one mode to couple with the other mode, such that it becomes a mixture of the two modes. Mode coupling is said to be weak when the effect of the inhomogeneity is unimportant, and waves in a given mode remain in that mode; this implies that the polarization changes from linear to circular as the cyclotron resonance is crossed. In the opposite limit, when mode coupling is strong, the initial linear polarization is preserved. The interpretation of the observed pulsar polarization, involving jumps between orthogonal polarizations that can be significantly elliptical [?], seems to require that mode coupling be relatively strong at the cyclotron resonance [?], allowing a partial conversion of linear into circular polarization [?]. 2.6.3 Cold counter-streaming electrons and positrons The presence of a parallel electric field, Ez 6= 0, accelerates the electrons and positrons, causing their bulk velocities to be different. Unlike the case where these is a single bulk velocity, the effect of relative streaming motions cannot be removed by a Lorentz transformation. One can always transform to a frame in which the average streaming velocity is zero, and in this frame the electrons and positrons are streaming in opposite directions. Another possible choice of

80

2 Response tensors for magnetized plasmas

frame in which these counter-streaming motions are equal, at ±βc. These two frames are the same only if the number densities n± are equal. An implication of the presence of relative streaming motions is that the dielectric tensor, K i j (ω, k), is a function of k in any frame. For a cold plasma, the resulting dispersion equation is a polynomial equation in |k|2 of order higher than two. It follows that there can be more than two solutions. Hence, the presence of relative streaming motions not only modifies the properties of the existing natural wave modes but also introduces intrinsically new wave modes. These new modes are called beam modes in an appropriate limit. The beam modes may lead to intrinsically growing modes when, as a function of some parameter, a beam mode and another real solution become a double solution and then a complex conjugate pair of solutions. Response tensor for counter-streaming pairs Consider the frame in which the bulk velocities, ±β, of electrons and positrons are equal and opposite. Let their the number densities be written n± = n(1 ± η), n = n+ + n− , η = (n+ − n− )/(n+ + n− ). One can identify two sources of gyrotropy is counter-streaming pair plasma. One is due to a charge imbalance, η 6= 0. The other is due to a nonzero current: the current density is J = −e(n+ β+ − n− β− )c, with β± = ±βb here. The sign of the current-induced gyrotropy is determined by the sign of β. One can write the response tensor in the form [?] ¯ µν (k) + η Π ˆ µν (k), Π µν (k) = Π

(2.6.8)

with K i j (k) = δji + µ0 Π i j (k)/ω 2 . It is convenient to introduce the notation ωp2 =

e2 n , ε0 m

ω0 = γω,

ωk = γkz βc,

ω⊥ = γk⊥ βc.

(2.6.9)

The components of the response tensor that do not depend on η are ¯ 2 2 = ωp2 ¯ 1 1 = µ0 Π µ0 Π ¯3

µ0 Π

3

=

ωp2

¯ 13 = ω2 µ0 Π p

"

(ω02 − ωk2 )2 − Ω 2 (ω02 + ωk2 ) (ω02 + ωk2 − Ω 2 )2 − 4ω02 ωk2

ω 2 (ω02 + ωk2 ) (ω02 − ωk2 )2

+

2 ω⊥ (ω02 + ωk2 − Ω 2 )

(ω02 + ωk2 − Ω 2 )2 − 4ω02 ωk2

ωk ω⊥ (ω02 − ωk2 + Ω 2 )

(ω02 + ωk2 − Ω 2 )2 − 4ω02 ωk2

¯ 1 2 = −iω 2 µ0 Π p ¯ 2 3 = iω 2 µ0 Π p

,

,

,

Ωe ωk (ω02 − ωk2 + Ωe2 )

(ω02 + ωk2 − Ωe2 )2 − 4ω02 ωk2 Ωe ω⊥ (ω02 + ωk2 − Ωe2 )

#

(ω02 + ωk2 − Ωe2 )2 − 4ω02 ωk2

.

,

(2.6.10)

2.6 Response tensor for a pulsar plasma

81

50 50 40 40 30 30

ω 20 20

3 2

4

10 10

1 0

5 20

0

40 40

k

60 60

80 80

100 100

Fig. 2.3. Dispersion curves, ω vs. k for nearly parallel propagation (θ = 0.1 rad) in a cold counter-streaming plasma with β = 0.3, where ωp = 10 and Ωe = 30 (arbitrary units). Dashed lines are imaginary parts and solid lines are real parts. Numbers indicate regions expanded in figure 2.4.

The response of a counter-streaming pair plasma is gyrotropic even when the number densities are equal (η = 0). This is due to the current from the oppositely directed flows of oppositely charged particles; the sign of the current is determined by the sign of β, which is included in ωk , ω⊥ in (2.6.10). The components proportional to η are ˆ 22 = ω2 ˆ 1 1 = µ0 Π µ0 Π p

2ω0 ωk Ωe2 , (ω02 + ωk2 − Ωe2 )2 − 4ω02 ωk2

# " 2 2ω⊥ ω0 ωk2 2ω 2 ω0 ωk 3 2 ˆ − 2 , µ0 Π 3 = −ωp − 2 (ω0 − ωk2 )2 (ω0 + ωk2 − Ωe2 )2 − 4ω02 ωk2 ˆ 1 3 = −ωp2 µ0 Π

(ω02 + ωk2 − Ωe2 )2 − 4ω02 ωk2

ˆ 1 2 = −iωp2 µ0 Π ˆ 2 3 = iω 2 µ0 Π p

ω0 ω⊥ (ω02 − ωk2 − Ωe2 )

Ωe ω0 (ω02 − ωk2 − Ωe2 )

,

(ω02 + ωk2 − Ωe2 )2 − 4ω02 ωk2

(ω02

,

2Ωe ω0 ω⊥ ωk . + ωk2 − Ωe2 )2 − 4ω02 ωk2

(2.6.11)

The specific forms (2.6.10), (2.6.11) apply to counter-streaming electrons and positrons; some of the signs are different for counter-streaming electron beams [?]. The absence of streaming corresponds to ωk → 0, ω⊥ → 0, ω0 → ω. Wave modes of a counter-streaming pair plasma The inclusion of counter-streaming leads to a rich variety of dispersive effects, even in the cold approximation, as illustrated in figure 2.3. In order to illustrate these effects, the parameters chosen in figure 2.3 correspond to

82

2 Response tensors for magnetized plasmas 1

1

12

2

3 17.4

11.75 0.8

17.2

11.5 0.6

11.25

0.4

10.75

ω

ω

17

ω

11

16.8

10.5 0.2

16.6 10.25 0.5

k

4

1

1.5

11

2 0.5

11

13

14

16.4

33

k

34

35

36

0.4

10.8

0.3

ω

ω 10.6

0.2

10.4

0.1

10.2

k

5

12

57.5

58

k

58.5

59

59.5

60

94.5

95

k

95.5

96

96.5

97

Fig. 2.4. Dispersion curves, ω vs. k for propagation at a small angle (θ = 0.1 rad) in a cold counter-streaming plasma with β = 0.3, where ωp = 10 and Ω = 30. These subfigures show close-up views of the numbered regions in figure 2.3, and display the behavior in the interaction regions.

Ωe /ωp = 3, β = 0.3; the topological structure of the dispersion curves is insensitive to the choice of these parameters. The cutoffs, |k| = 0, in figure 2.3 are near ω = 0, ωp , Ωe . As β is reduced to zero, the modes with cutoffs at ω = 0, ωp , Ωe reduce to the cutoffs in the Alfv´en, O and in X modes, respectively. The number of different modes reduces in this limit due to modes becoming conincident, which occurs with the two that cut off near Ωe , or disappearing, which occurs for the beam modes that intersect ω = 0 in figure 2.3. An important feature of dispersion in a cold plasma with counter-streaming is that there can be intrinsically growing modes: a complex conjugate pair of solutions arises two real modes merge (as some parameter is varied) to become a pair of modes with one growing and the other decaying. Complex modes appear in figure 2.3 at the points laebled 1–5. New modes appear in qualitatively different ways at nonzero frequency, as illustrated in cases 2–5 in figure 2.4. In some cases two real modes reconnect to create two different real modes, as in cases 2 and 3. One class of unstable mode is purely growing, with zero real frequency, as illustrated by the dashed curve in figure 2.3 and by cases 1 and 5 in figure 2.4. In other cases two real modes become a complex conjugate pair of modes, one of which is necessarily a growing mode, as in case 4 in figure 2.4. Consider, for example, the specific dispersion curve in figure 2.4 that begins at ω = 0 for k = 0. This branch is initially imaginary, becoming real and propagating at k ≃ 1. As the intersection labeled 2 is approached, it deflects away from the other modes and remains a single real mode, and again at intersection 3. At intersection 4, it joins another mode and forms a pair of complex conjugate solutions, which then become real again for higher k.

2.6 Response tensor for a pulsar plasma

83

2.6.4 Dispersion in intrinsically relativistic pulsar plasmas The assumption that the pulsar plasma is cold in its rest frame is unrealistic. The distributions of electrons and positrons are expected to have a relativistic spread in energies. With the 1D assumption, the dispersion may be defined in terms of averages, denoted by angular brackets, over the 1D distributions functions g± (u). Response tensor for a 1D pair plasma The response tensor for an arbitrary 1D distribution of electrons and positrons follows from (1.3.9). Assuming that the distributions of electrons and positrons are the same, apart from their number densities, the response tensor reduces to X e 2 n± 1 k µ k ν 1 µν (ku)2 g⊥ − Π µν (k) = − D D2 + 2 2 m γ (ku) (ku) − Ω e ± µ ν µ ν ν ν 2 µ ν , u − uµ kG −ku (k⊥ u + uµ k⊥ ) − k⊥ u u ∓ i Ωe ku f µν + kG

(2.6.12)

where the angular brackets denote the average XZ XZ hKi = du g± (u) K, du g± (u) = 1, ±

(2.6.13)

±

for any function K, where g± (u) are the 1D distribution functions, with u = γβ, γ = (1 − β 2 )−1/2 . Relativistic plasma dispersion functions (RPDFs With ku = γ(ω − kz β), the denominators that appear in (2.6.12) can be rewritten using (ku)2 = kz2 γ 2 (z−β)2 ,

(ku)2 −Ωe2 = kz2 γ 2 (1+y 2 )(β−z+ )(β−z− ), (2.6.14)

with z = ω/kz , y = Ωe /kz , z± =

z ± y(1 + y 2 − z 2 )1/2 . 1 + y2

(2.6.15)

The result may be expressed in terms of three relativistic plasma dispersion functions (RPDFs), defined by 1 1 1 , R(z) = , S(z) = . W (z) = γ 3 (z − β)2 γ(z − β) γ 2 (z − β) (2.6.16)

84

2 Response tensors for magnetized plasmas

Dielectric tensor in terms of RPDFs The reulting expression for the dielectric tensor is [?] K

1

1

K 33 K 12 K 13 K 23

ωp2 1 (z − z+ )2 R(z+ ) − (z − z− )2 R(z− ) 1 + = K 2 =1− 2 , ω 1 + y2 γ z+ − z− 2 2 ωp2 z+ R(z+ ) − z− R(z− ) tan2 θ 1 2 , + = 1 − 2 z W (z) + ω 1 + y2 γ z+ − z− ωp2 y (z − z+ )S(z+ ) − (z − z− )S(z− ) = −iη 2 , ω 1 + y2 z+ − z− ωp2 1 (z − z+ )z+ R(z+ ) − (z − z− )z− R(z− ) =− 2 , ω 1 + y2 z+ − z− ωp2 y tan θ z+ S(z+ ) − z− S(z− ) , (2.6.17) = iη 2 ω 1 + y2 z+ − z− 2

with K 2 1 = −K 1 2 , K 3 1 = K 1 3 , K 2 3 = −K 3 2 . Low-frequency limit The general form of the response tensor (2.6.17) simplifies at frequencies well below the cyclotron frequency. This limit corresponds to y → ∞, z± → ±1. With hβi = 0, the RPDFs associated with the cyclotron resonance simplify according to R(±1) = ∓hγi, S(±1) = ∓1. (2.6.18) In this limit the dielectric tensor (2.6.17) may be approximated by

ωp2 kz2 2 z hγi − 2z hγβi + γβ 2 , ω 2 Ωe2 2

ωp2 2 k⊥ 2 = 1 − 2 z W (z) − 2 γβ , ω Ωe

ωp2 k⊥ kz z hγβi − γβ 2 , =− 2 2 ω Ωe ωp2 kz c ωp2 k⊥ c = iη 2 hβi , z − hβi , K 2 3 = −iη 2 ω Ωe ω Ωe

K 11 = 1 + K 33 K 13 K 12

(2.6.19)

Differences between electrons and positrons are included in two different ways in (2.6.19). One is through η, and the other is through the averages hβi, hγβi. These averages can be nonzero due to a net streaming velocity, in which case they can be chosen to be zero by transforming to the rest frame of the plasma. However, these averages are also nonzero if there is a difference in the bulk flow velocities of electrons and positrons, and then these averages are

2.6 Response tensor for a pulsar plasma

85

4 3

z2 W(z)

2 1 0

z2 R(z)

-1 -2 0.6

0.8

1

1.2

z

Fig. 2.5. Plots of z 2 W (z) and z 2 R(z) for a J¨ uttner distribution with ρ = 1 (dotted) and ρ = 2.5 (solid) [?].

nonzero in any frame. The effect of relative streaming motions is discussed further below in connection with counter-streaming. When relative motions of electrons and positrons is neglected, so that these averages are zero in the rest frame, it is convenient to write the remaining parameters in (2.6.19) in terms of Ωe hγβ 2 i = hγi δβ 2 . (2.6.20) βA = hγi−1/2 , ωp This definition of the Alfv´en speed is the generalization for hγi > 1 of the definition βA = Ωe /ωp for a cold plasma. The RPDFs (2.6.16) have been evaluated for two classes of distribution: a relativistic thermal (J¨ uttner) distribution, and water-bag and related distributions. In the highly relativistic approximation the properties of the RPDFs are dominated by relativistic effects that are not particularly sensitive to the specific form of the distribution function. 1D J¨ uttner distribution A 1D J¨ uttner distribution is e−ργ K0 (ρ) 1 g(u) = = , , 2K1 (ρ) γ K1 (ρ)

hγi =

K0 (ρ) + K2 (ρ) , 2K1 (ρ)

(2.6.21)

where two relevant examples of averages are given. The RPDF associated with dispersion in the parallel direction is Z 1 T ′ (z, ρ) e−ργ W (z) = , T (z, ρ) = dβ . (2.6.22) 2K1(ρ) β−z −1 The two RPDFs associated with the cyclotron resonance are 2K0 (ρ)T (z, ρ) 1 T ′ (z, ρ) , R(z) = , S(z) = − (1 − z 2 )2K1 (ρ) z 2K1 (ρ)

(2.6.23)

86

2 Response tensors for magnetized plasmas 10 8

z2 W(z)

6 4 2 0

z2 R(z) -2 0.6

0.8

1

1.2

z

Fig. 2.6. As for figure 2.5 but for a soft bell distribution with um = 0.977098 (dotted) and um = 0.9 (solid) [?].

The properties of the RPDF T (z, ρ) are discussed in §4.? of volume 1. The RPDFs z 2 W (z) and z 2 R(z) are illustrated in figure 2.5 for two mildly relativistic J¨ uttner distributions. Water-bag and bell distribution A water-bag distribution is one in which the distribution function is a constant between two limits and zero otherwise, so that the distribution function is discontinuous at these limits. In a bell distribution the step functions at the two limits are replaces by a power-law variation. These distribution functions are of the form g(u) = (u2m − u2 )n H(u2m − u2 ), (2.6.24) where H(x) is the step function, and where the two limits are at u = ±um . The water-bag distribution corresponds to n = 0, and n = 1, 2, 3 have been called hard bell, soft bell and squishy bell, respectively. These distributions have the advantage that the RPDFs can be evaluated in terms of powers and logarithmic functions [?, ?]. The argument of the logarithms is (z − βm )/(z + βM ), with um = γm βm , and this has an imaginary part for −βm < z < βm , where resonance is possible. The RPDFs z 2 W (z) and z 2 R(z) are illustrated in figure 2.6 for two soft bell distributions with the same values of hγi as the two J¨ uttner distributions in figure 2.5.

2.7 Nonlinear response tensors

87

2.7 Nonlinear response tensors General covariant forms for the quadratic and cubic response tensors may be derived by extending the calculation of the first-order current in §2.1 to calculate the second and third order currents, and applying the forward-method to find the nonlinear response tensors. 2.7.1 Higher order currents Extending the derivation of the first-order current (2.1.51) to higher orders gives the higher order currents. Here a general form for the nth order current is noted, and the explicit expression for n = 1, 2 are given. The counterpart of (2.1.51) for the nth order current is (n)µ Jsp (k) =

∞ X

s,s1 ,...,sn =−∞

Z

d4 k1 ··· (2π)4

Z

d4 kn −i(k−k1 −···−kn )x0 e (2π)4

β (n)µν1 ...νn (s, k; s1 , k1 ; . . . ; sn , kn ) ×2πδ (ku)k − (k1 u)k − · · · (kn u)k − (s − s1 − · · · − sn )Ω0 .

×e

−iη(sψ−s1 ψ1 −···−sn ψn )

(2.7.1)

The explicit expressions for n = 2 is

β (2)µν1 ν2 (s, k; s1 , k1 ; s2 , k2 ) = Gαµ (s, k, u)G∗βν (s1 , k1 , u)G∗γρ (s2 , k2 , u)fαβγ (s, k; s1 , k1 ; s2 , k2 ; u), 1 k1σ τ ασ (˜ ω2 ) ω )τ βγ (˜ ω2 ) kσ τ σβ (˜ ω1 )τ αγ (˜ f αβγ (s, k; s1 , k1 ; s2 , k2 ; u) = + 2 ω ˜ ω ˜1 k1σ τ σγ (˜ ω2 )τ αβ (˜ ω) + (ν, s1 k1 ) ↔ (ρ, s2 , k2 ) , (2.7.2) + ω ˜2 with ω ˜ = (ku)k − sΩ0 , ω ˜ n = (kn u)k − sn Ω0 , and where the final entry in (2.7.2) implies three additional terms obtained from those written by making the interchange indicated. 2.7.2 Nonlinear response tensor for a magnetized plasma The forward-scattering method leads to general expressions for the quadratic and cubic nonlinear response tensors by averaging the currents (2.7.1) for n = 2 and n = 3, respectively, over the distribution of particles. The result for the quadratic response tensor is, with k0 = −k, Z X q3 d4 p Π (2)µνρ (k0 , k1 , k2 ) = − eiη(s0 ψ0 +s1 ψ1 +s2 ψ2 ) F (p) 2 4 2m (2π) s +s +s =0 0

∗αµ

×G

∗βν

(s0 , k0 , u)G

∗γρ

(s1 , k1 , u)G

1

2

(s2 , k2 , u)fαβγ (s0 , k0 ; s1 , k1 ; s2 , k2 ; u), (2.7.3)

88

2 Response tensors for magnetized plasmas

f αβγ (s0 , k0 ; s1 , k1 ; s2 , k2 ; u) = − +

k1σ τ σα (˜ ω0 )τ βγ (˜ ω2 ) ω ˜0

k2σ τ σα (˜ ω0 )τ γβ (˜ ω1 ) k0σ τ σβ (˜ ω1 )τ αγ (˜ ω2 ) k0σ τ σβ (˜ ω2 )τ αγ (˜ ω1 ) + + ω ˜0 ω ˜1 ω ˜2 ω1 )τ γα (˜ ω0 ) k1σ τ σγ (˜ ω2 )τ βα (˜ ω0 ) k2σ τ σβ (˜ + , (2.7.4) + ω ˜2 ω ˜1

with ω ˜ n = (kn u)k − sn Ω0 . The result for the cubic response tensor in an arbitrary magnetized plasma is Π (3)µνρσ (k0 , k1 , k2 , k3 ) Z q4 d4 p F (p) =− 3 6m (2π)4 s

X

eiη(s0 ψ0 +s1 ψ1 +s2 ψ2 +s3 ψ3 )

0 +s1 +s2 +s3 =0

∗αµ

×G

∗βν

(s0 , k0 , u)G

(s1 , k1 , u)G∗γρ (s2 , k2 , u)G∗δσ (s3 , k3 , u)

×fαβγδ (s0 , k0 ; s1 , k1 ; s2 , k2 ; s3 , k3 ; u),

(2.7.5)

f αβγδ (s0 , k0 ; s1 , k1 ; s2 , k2 ; s3 , k3 ; u) k1φ τ βα (˜ ω0 ) + k0φ τ αβ (˜ ω1 ) k1η τ ηα (˜ ω0 )g β φ − = ω ˜0 ω ˜0 + ω ˜1

k0η τ ηβ (˜ ω1 )g α φ θφ + τ (˜ ω0 + ω ˜1) ω ˜1 k2θ τ γδ (˜ ω3 ) + k3θ τ δγ (˜ ω2 ) k2η τ ηδ (˜ ω3 )g γ θ k3η τ ηγ (˜ ω2 )g δ θ − + × ω ˜2 ω ˜2 + ω ˜3 ω ˜3 +(1, ν) ↔ (2, ρ) + (1, ν) ↔ (3, σ) k0θ k0η τ αβ (˜ ω1 )τ θγ (˜ ω2 )τ ηδ (˜ ω3 ) + 11 other terms, (2.7.6) + ω ˜2ω ˜3

where +(1, ν) ↔ (2, ρ) + (1, ν) ↔ (3, σ) refers to two other terms obtained from that written by making the interchanges indicated, and where “11 other terms” refers to those obtained by completely symmetrizing the term written over (0, µ), (1, ν), (2, ρ), (3, σ). 2.7.3 Cold plasma limit The cold plasma limit for the quadratic and cubic response tensor follows from (2.2.7) and (2.7.5), respectively, by setting F (p) = n(2π)4 δ 4 (p − m˜ u), with u˜ = [1, 0] in the rest frame of the plasma. This corresponds to neglecting all motion of the particles, including the spiraling motion. In particular, it corresponds to setting the gyroradius of the particle and hence the argument of the Bessel functions to zero, which is the strictly-parallel limit. Then the only term that contributes in the sum has s0 = s1 = s2 = 0. The contribution

References

89

of a single species of cold particles to the quadratic response tensor may be written Π µνρ (k0 , k1 , k2 ) = −

q 3 n αµ G (k0 , u ˜)Gβν (k1 , u ˜)Gγρ (k2 , u ˜) fαβγ (k0 , k1 , k2 , u ˜), 2m2 (2.7.7)

with

k1σ ασ k2σ ασ τ (k0 u)τ βγ (k2 u) + τ (k0 u)τ γβ (k1 u) k0 u k0 u k0σ σβ k0σ σγ + τ (k1 u)τ αγ (k0 u) + τ (k2 u)τ αβ (k0 u) k1 u k2 u k2σ σβ k1σ τσγ (k2 u)ταβ (k0 u) − τ (k1 u)τ αγ (k0 u), (2.7.8) − k2 u k1 u

f αβγ (k0 , k1 , k2 , u) = −

The analogous expression for the cubic response tensor follows from (2.7.5) and (2.7.6) by the same procedure, with only the terms with s0 = s1 = s2 = s3 = 0 contributing. 2.7.4 Quadratic response with one slow disturbance TBA 2.7.5 Cubic response with one slow disturbance TBA 2.7.6 Effective cubic response TBA

References 1. F. J¨ uttner: Ann. der Phys. 34, 856 (1911) 2. J.L Synge: The Relativistic Gas, (North-Holland, Amsterdam 1957) 3. B.A.Trubnikov: Magnetic emission of high temperature plasma, doctoral dissertation, Moscow Institute of Engineering and Physics (1958); English translation in AEC-tr-4073, US Atomic Energy Commission, Oak Ridge, Tennessee (1960) 4. Trubnikov and Yakubov (1963) 5. I.P. Shkarofsky (1966a) PF 9, 561 6. I.P. Shkarofsky (1986) PF 35, 319 7. Robinson (1986) 8. P.A. Robinson (1988) Aust. J. Phys. 39, 57–70 9. Robinson P A 1986 J. Math. Phys. 27 1206 10. Fried, B D and Conte S D 1961 The Plasma Dispersion Function, Academic Press, New York

90 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

2 Response tensors for magnetized plasmas Gross (1951) Bernstein (1958) Dnestrovskii and Kostomarov (1961) Dnestrovskii and Kostomarov (1962) Dnestrovskii V N, Kostomorov, D P and Skrydlov N V 1964 Sov. Phys. Tech. Phys. 8 691 S. Puri, F. Leuterer, M. Tutter (1973) Journal of Plasma Physics, 9, 89 S. Puri, F. Leuterer, M. Tutter (1975) Journal of Plasma Physics, 14, 169 M. Bornatici et al.(1983) M. Bornatici, U. Raffina (1988) PPCF 30, 115–132 M.J. Bruggen-Kerkhof, L.P.J. Kamp, F.W. Sluijter (1993) JPA 26, 5505–5521 Godfrey B B, Newberger B S, Taggart K A 1975 IEEE Trans. Plasma Sci. PS-3 60 V. Krivenski, A. Orefice (1983) JPP 30, 125 P.A. Robinson (1986) AJP 39, 57 P.A. Robinson (1987a) JMP 28, 1203 P.A. Robinson (1987b) JPP 37, 435 P.A. Robinson (1987c) JPP 37, 449 C. Morali, V. Petrillo (1981) Phys. Scr. 24, 955 P.A. Robinson (1989) JMP 30, 2484 D.G. Swanson (2002) PPCF 44, 1329 D.B. Melrose (1997) JPP 58, 735 V.N. Sazonov, (1969) JETP 29, 587 D.B. Melrose (1997) Phys. Rev. E 56, 3527

3 Wave dispersion in relativistic magnetized plasma

TBA

92

3 Wave dispersion in relativistic magnetized plasma

3.1 Dispersion in cold magnetized plasma The theory of wave dispersion is applied to determines the properties of wave modes of a cold plasma. These wave properties are derived in this section by reducing the covariant formalism to a conventional 3-tensor approach in the rest frame of the plasma. 3.1.1 Invariant dispersion equation The covariant theory for wave dispersion is rather cumbersome, and it is often simpler to use a non-covariant formalism. The covariant theory is based on the wave equation, derived from the Fourier the covariant form of Maxwell’s equation. Following the notation used in volume 1, the wave equation is µ (k), Λµν (k)Aν (k) = −µ0 Jext

(3.1.1)

where the current is separated into an induced part and and extraneous part; the linear induced current is included on the left hand side of (3.1.1), with Λµ ν (k) = k 2 δνµ − k µ kν + tµ ν (k),

tµν (k) = µ0 Π µν (k),

(3.1.2)

and the extraneous part remains as a source term in (3.1.1). The source term is set to zero to obtain the homogeneous wave equation, which is to be solved for the wave properties. Without the source term, (3.1.1) may be interpreted as four simultaneous equations for the four components of Aµ (k). The usual condition for a solution to exist is that the determinant of the coefficients vanish, which corresponds to det [Λµ ν (k)] = 0. However, the charge-continuity and gauge-invariance conditions imply that this determinant is identically zero, and the Aµ (k) ∝ k µ is a trivial solution of the homogeneous wave equation. Non-trivial solutions are determined by the condition that the matrix of cofactors vanish. Let λµν (k) be the matrix of cofactors of Λµν (k); it is necessarily of the form λµν (k) = λ(k) k µ k ν , (3.1.3) where λ(k) is an invariant. The invariant form of the dispersion equation is λ(k) = 0. An explicit form for λ(k) may be obtained in terms of the traces of powers of tµν (k), defined by (3.1.2). Let the trace of the nth power be denoted t(n) (k), so that one has t(1) (k) = tµ µ (k), t(2) (k) = tµ ν (k)tν µ (k), and so on. The explicit form is 2 λ(k) = k 4 + k 2 t(1) (k) + 12 t(1) (k) − t(2) (k) 3 1 + 2 t(1) (k) − 3t(1) (k)t(2) (k) + 2t(3) (k) . (3.1.4) 6k

However, the evaluation of the traces is cumbersome, even for the relatively simple case of a cold plasma.

3.1 Dispersion in cold magnetized plasma

93

Use of the invariant form (3.1.4) of the dispersion equation can be illustrated for the case of a cold unmagnetized plasma, which corresponds to ταβ → gαβ in (1.3.7). This is the leading term in an expansion of (1.3.7) in powers of Ω0 /k¯ u → 0, referred to here as the weak-anisotropy limit. To zeroth order in Ω0 /k¯ u → 0 (1.3.7) gives k2 u ¯µ u ¯ν kµ u ¯ν + k ν u¯µ µν µν 2 + , (3.1.5) t (k) = −ωp g − k¯ u (k¯ u)2 with ωp2 = q 2 n/ε0 m. The traces are t(n) (k) = (−ωp2 )n

k2 (k¯ u)2

n

+2 ,

(3.1.6)

with n = 1, 2, 3. Then (3.1.4) gives λ(k) =

ωp2 1− (k¯ u)2

!

(k 2 − ωp2 )2 .

(3.1.7)

The solutions of λ(k) = 0 for an unmagnetized cold plasma are (k¯ u)2 = ωp2 , corresponding to longitudinal oscillations at the plasma frequency, and k 2 = ωp2 , corresponding to two degenerate transverse modes. The inclusion of the term of order Ω0 /k¯ u breaks this degeneracy, leading to the weak-anisotropy approximation, which is discussed further in §??. 3.1.2 3-tensor form of the wave equation A 3-tensor form for the wave equation is obtained by choosing a specific gauge and frame. A convenient choice is the temporal gauge, A0 (k) = 0. Only the space components of the wave equation (3.1.1) are then relevant: i Λi j (k)Aj (k) = −µ0 Jext (k).

(3.1.8)

In the specific frame, which is usually the rest frame of the plasma, it is convenient to divide both sides of (3.1.8) by ω 2 , so that it becomes µ0 i |k|2 (k). (3.1.9) − 2 (δji + κi κj ) + K i j (k) Aj (k) = − 2 Jext ω ω with κ = k/|k, and where the equivalent dielectric tensor is defined by K i j (k) = δji +

1 Π i j (k), ε0 ω 2

(3.1.10)

with µ0 = 1/ε0 in natural units. The homogeneous wave equation follows from (3.1.9) by setting the source term to zero. The condition for a solution to exist is that the determinant

94

3 Wave dispersion in relativistic magnetized plasma

of the 3 × 3 matrix be set to zero. This determinant is equal to λ(k)/ω 6 . Introducing the refractive index, which is n = |k|c/ω in ordinary units, and choosing κ = (sin θ, 0, cos θ) to be in the 1-3 plane at an angle θ to b along the 3-axis, the determinant becomes −n2 cos2 θ + K 1 1 (k) K 1 2 (k) n2 sin θ cos θ + K 1 3 (k) . K 2 1 (k) −n2 + K 2 2 (k) K 2 3 (k) λ(k) = ω 6 2 3 3 2 n sin θ cos θ + K 1 (k) K 2 (k) −n sin2 θ + K 3 3 (k) (3.1.11) The determinant may be evaluated directly, and this is the simplest procedure in the case of a cold plasma. More generally, as shown in § 2.2.4 of volume I, (3.1.11) is equivalent to λ(k) = ω 6 n4 K L (k) − n2 [K L (k)K1 (k) − K2L (k)] + det [K i j (k)] , (3.1.12)

with K1 (k) = K s s (k), K L (k) = −κi κj K i j (k), K2L (k) = −κi κj K i s (k) K s j (k). 3.1.3 Dielectric tensor in the rest frame of a cold plasma The covariant form (1.3.7) for the linear response tensor due to a single species of cold particles has space components given by (1.2.25) in the rest frame of the particles. On inserting a label a to denote the species, inserting (1.2.25) into the expression (3.1.13) for the dielectric tensor, and summing over the species, one obtains K i j (k) = δji −

2 X ωpa a

ω2

ταi j (ω),

(3.1.13)

2 with ωpa = qa2 na /ε0 ma . The 3-tensor components of (1.2.21) are

ω2 iηa Ωa ω  ω2 − Ω2 ω2 − Ω2 a a  ω2 ταi j (ω) =   − iηa Ωa ω  ω2 − Ω2 ω2 − Ω2 a a 0 0 

 0  , 0  1

(3.1.14)

where Ωa = |qa |B/ma and ηa = qa /|qa | are the cyclotron frequency and sign of the charge of species a. The dielectric tensor (3.1.13) with (3.1.14) may be written in a standard form (Stix 1962)   S(ω) −iD(ω) 0 K i j (ω) =  iD(ω) S(ω) 0  , (3.1.15) 0 0 P (ω)

with

3.1 Dispersion in cold magnetized plasma

S(ω) = 12 [R+ (ω) + R− (ω)], R± (ω) = 1 −

2 X ωpa a

95

D(ω) = 12 [R+ (ω) − R− (ω)],

ω , ω 2 ω ± ηa Ωa

P (ω) = 1 −

2 X ωpa a

ω2

.

(3.1.16)

The dispersion equation, λ(k) = 0, for a cold plasma follows by inserting (3.1.15) into (3.1.11), which gives S(ω) − n2 cos2 θ −iD(ω) n2 sin θ cos θ = 0. iD(ω) S(ω) − n2 0 (3.1.17) 2 2 n2 sin θ cos θ 0 P (ω) − n sin θ

A particular solution of the dispersion equation is interpreted as the dispersion relation for a particular wave mode. An arbitrary wave mode is labeled M . 3.1.4 Polarization vector

The dispersion equation is the condition for a solution of the homogeneous wave equation to exist. Given the solution of the dispersion equation for a mode M , one may solve for the 4-potential for the mode M A covariant µ covariant form for the dispersion relation is k µ = kM , with AµM (k) = Aµ (kM ) denoting the solution for the 4-potential. A practical complication is that the amplitude, phase and gauge of AµM (k) are arbitrary, and it is not possible to define a unique normalization for an arbitrary choice of gauge. It is convenient to specify that the solution for AµM (k) be in the temporal gauge, and then AµM (k)AMµ (k) is necessarily negative, and may be normalized by setting it to −1. The quantity defined in this way is referred to as the polarization vector for the mode M , denoted eµM (k) = [0, eM (k)], with eM (k) · e∗M (k) = 1 the polarization 3-vector. The Onsager relations, for k, b in the 1-3 plane, imply that the 2-component is out of phase with the 1- and 3-components, allowing one to choose the phase such that these are imaginary and real, respectively. The polarization vector, chosen in this way, can be written in the form eM =

LM κ + TM t + ia , 2 + 1)1/2 (L2M + TM

(3.1.18)

with t, a defined by (1.1.22). For the specific choice of coordinate axes indicated, one has κ = (sin θ, 0, cos θ),

t = (cos θ, 0 − sin θ),

a = (0, 1, 0).

(3.1.19)

3.1.5 Solution of the homogeneous wave equation Solution of the homogeneous wave equation is equivalent to solving a set of simultaneous linear algebraic equation: the condition for a solution to exist is that the determinant of the matrix of coefficients vanish, and the solution is

96

3 Wave dispersion in relativistic magnetized plasma

found by constructing the matrix of cofactors, choosing any column of it, and identifying the solution as the resulting column matrix when the determinant vanishes. As already noted, in the covariant, gauge-independent formulation this is complicated by the determinant vanishing identically. The condition for a non-trivial solution to exist is then that the matrix of cofactors (3.1.3) vanish, and the solution involves the second order matrix of cofactors. Denoting this by λµναβ (k), it satisfies (3.1.20) Λµ ρ (k)λρναβ (k) = λ(k) g µα k ν k β − g µβ k ν k α ,

A solution satisfying a specific gauge condition Gµ Aµ (k) = 0 is found by projecting onto Gµ Gα , and identifying the solution in this gauge with any column of the matrix representation of Gµ Gα λµναβ (k). In the temporal gauge, Gµ = [1, 0], this projection gives the 3-tensor λ0i 0j (k). For a specific soluµ tion k µ = kM of λ(k) = 0, and column of λ0i 0j (kM ) is proportional to the contravariant component, eiM (k), of the polarization vector in the temporal gauge for the mode M . A specific form derived in volume I is λ0i 0j = n4 κi κj − n2 (κi κj K1 + δji K L − κi κr K r j − κj κs K i s ) + 21 δji [(K1 )2 − K2 ] + K i s K s j − K1 K i j .

(3.1.21)

On inserting the dispersion relation for mode M into (3.1.21), the result is proportional to the outer product eiM e∗Mj . For the cold plasma dielectric tensor in the form (3.1.15), the matrix of form of the 3-tensor (3.1.21) is  (S − n2 )(P − n2 sin2 θ) 0i λ 0j =  −iD(P − n2 sin2 θ) −(S − n2 )n2 sin θ cos θ iD(P − n2 sin2 θ) P S − n2 (S sin2 θ + P cos2 θ) −iDn2 sin θ cos θ

(3.1.22)

 −(S − n2 )n2 sin θ cos θ , iDn2 sin θ cos θ S 2 − D2 − n2 S(1 + cos2 θ) + n4 cos2 θ

where arguments ω are omitted in S, D, P . The polarization 3-vector for a cold plasma mode is found by inserting the dispersion relation into any of the columns in (3.1.22), and choosing the phase and normalization appropriately. 3.1.6 Cold plasma wave modes The dispersion equation (3.1.17) reduces to (Stix 1962) An4 − Bn2 + C = 0,

(3.1.23)

3.1 Dispersion in cold magnetized plasma

97

with the coefficients given by A = S sin2 θ + P cos2 θ,

B = (S 2 − D2 ) sin2 θ + P S(1 + cos2 θ),

C = P (S 2 − D2 ).

(3.1.24)

The dispersion equation (3.1.22) is a quadratic equation for n2 , and the two solutions are n2 = n2± =

B±F , 2A

F = B 2 − 4AC

1/2

,

(3.1.25)

where the ± labeling is arbitrary. The parameters LM , TM in the polarization vector in the form (3.1.18) are found by setting n2 = n2M , with M = ±, in the middle column of (3.1.22): LM =

(P − n2M )D sin θ , An2M − P S

TM =

DP cos θ . An2M − P S

(3.1.26)

Rather than solve the dispersion equation for n2 , it is sometimes convenient to use (3.1.26) to choose T as the independent variable in place of n2 . One solves (P S − S 2 + D2 ) sin2 θ T2 − T − 1 = 0, (3.1.27) P D cos θ for T = T± , finding T± =

(P S − S 2 + D2 ) sin2 θ ± F −2P D cos θ , = 2P D cos θ (P S − S 2 + D2 ) sin2 θ ∓ F F 2 = (P S − S 2 + D2 )2 sin4 θ + 4P 2 D2 cos2 θ.

Inverting (3.1.26) gives n2M and LM in terms of TM , with M = ±, P D cos θ S 2 − D2 P [S cos θ + DTM ] n2M = S+ = = , A TM S − DTM cos θ P cos θ + DTM sin2 θ LM =

(3.1.28)

(3.1.29)

sin θ sin θ (P S − S 2 + D2 )TM cos θ − P D [(P − S)TM cos θ − D] = . A P S − DTM cos θ (3.1.30)

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3 Wave dispersion in relativistic magnetized plasma

3.2 Waves in cold plasmas The simplest model for wave dispersion in a magnetized plasma is one in which the thermal motions of particles are neglected, called a cold plasma. A further simplification applies at frequencies well above the ion plasma and cyclotron frequencies, when on the dispersion due to the electrons need be considered. Dispersion in a cold electron gas is referred to here as the magnetoionic theory, and the natural wave modes are referred to as the magnetoionic modes. 3.2.1 Magnetoionic waves For historical reasons, a cold plasma in which only the contribution of the electrons is retained is referred to as a magnetoionic medium, and the natural wave modes of a cold electron gas are referred to as the magnetoionic modes. The magnetoionic theory was developed in the early 1930s in connection with radio wave propagation in the ionosphere [3, 4], and is also called AppletonHartree theory. The magnetoionic theory was developed before the presentday meaning of “ion” became accepted. Ions, in the modern-day sense, play no role in the magnetoionic theory. In summarizing the theory of wave dispersion in a magnetoionic medium here one minor generalization is made here: it is assumed that there may be (cold) positrons and well as electrons present. This is achieved by allowing the parameter η to be variable, being equal to η = −(n+ − n− )/(n+ + n− ), where n± are the number densities of electrons and positrons, respectively. (The ± labeling corresponds to the sign ǫ used in QED, with ǫ = +1 for the particle and ǫ = −1 for the antiparticle, which are electron and positron, respectively, here.) A pure electron gas corresponds to η = −1, and a pure pair plasma (equal numbers of electrons and positrons) corresponds to η = 0. It is convenient to define the magnetoionic parameters X=

ωp2 , ω2

Y =

Ωe . ω

(3.2.1)

In terms of these parameters, the components in the dielectric tensor (3.1.15) become ηXY X , D= , P = 1 − X, (3.2.2) S =1− 2 1−Y 1−Y2 where the arguments, ω, are omitted. The coefficient (3.1.24) in the dispersion equation (3.1.23) become X (1 − Y 2 cos2 θ), 1−Y2 X2 2 2 + (1 − η Y ) sin2 θ (1 − Y 2 )2 X +(1 − X) 1 − (1 + cos2 θ), 1−Y2

A=1−

B = 1−

2X 1−Y2

3.2 Waves in cold plasmas

C = (1 − X) 1 −

2X 1−Y2

X2 2 2 + (1 − η Y ) . (1 − Y 2 )2

99

(3.2.3)

It is straightforward to write down the solutions of the quadratic equation (3.1.23), and these define the two magnetoionic modes. In the alternative treatment in which one first solves for the axial ratio, T , equation (3.1.27) becomes T2 −

(1 − E)Y sin2 θ T − 1 = 0, η(1 − X) cos θ

E=

(1 − η 2 )X . 1−Y2

(3.2.4)

The term E is nonzero only when both positrons and electrons are present. The solutions of (3.2.4) are T = Tσ =

ηY (1 − X) cos θ = 2 1 2 2 (1 − E)Y sin θ − σ∆

1 2 (1

− E)Y 2 sin2 θ − σ∆ , ηY (1 − X) cos θ

∆2 = 41 (1 − E)2 Y 4 sin4 θ + η 2 (1 − X)2 Y 2 cos2 θ.

(3.2.5)

In place of (3.1.29) one has n2σ = 1 − Lσ =

XTσ X(1 − X)(1 + Y Tσ cos θ) =1− , Tσ − Y cos θ 1 − X − Y 2 + XY 2 cos2 θ

Tσ XY sin θ(1 + Y Tσ cos θ) XY sin θ = , 1 − X Tσ − Y cos θ 1 − X − Y 2 + XY 2 cos2 θ

(3.2.6)

where the alternative forms are related by the quadratic equation (3.2.4). 3.2.2 Parallel and perpendicular propagation In the limit of parallel propagation, sin θ = 0, the wave properties simplify, but with a subtle complications. The dispersion equation (3.1.23) with (3.1.24) reduces to P [(n2 −S)2 −D2 ] = 0. The solution P = 0 corresponds to longitudinal oscillations at ω = ωp , and the other two solutions correspond to oppositely circularly polarized modes. This is a general feature of parallel propagation in an arbitrary magnetized plasma: the polarization of the modes for parallel propagation are either longitudinal or (transverse and) circular. The solutions of (n2 − S)2 − D2 = 0 may be written n2 = R± , with R± defined by (3.1.16). However, the ± labeling of these solutions is not consistent with the ± labeling in (3.1.28), which corresponds to T± = ± cos θP D/| cos θP D| for parallel propagation. The dependence on the sign of cos θ reflects two different conventions for the handedness of the polarization. The axial ratio T , defined by T = TM in (3.1.18), corresponds to a screw sense relative to the wave-normal direction κ. The other handedness is that of a charge spiraling about a magnetic field line, which is right hand for negative charges and left hand for positive charges, with the handedness corresponding to a screw sense relative to b. These two definitions of handedness are

100

3 Wave dispersion in relativistic magnetized plasma

n2 20

w

10

x 0 1 z

2

ω/ωp

o

-10 Fig. 3.1. The dispersion curves for the o, x and z banches of the magnetoioinic modes are indicated schematically for ωp < Ωe . The whistler branch is off the diagram in the upper left hand corner.

equivalent for cos θ > 0 and opposite for cos θ < 0. The ± sign is R± depends on the sign of the charge, ηa , in (3.1.16). Physically one expects the mode polarized with the same handedness as particles spiral to have a resonance at the cyclotron frequency for the particles. Thus one expects the mode with n2 = R+ , which has a resonance for negatively charged particles, to have a right handed screw sense relative to b, corresponding to T = cos θ/| cos θ, and for the mode with n2 = R− to have the opposite polarization. As expected, on setting T = cos θ/| cos θ| in the second of (3.1.29) for | cos θ| = 1, one finds n2M = S + D = R+ . However, the solution for T± also involves the sign of D, and inserting T± = ± cos θD/| cos θD| in (3.1.29) gives n2M = S ± |P D|/P , so that the labeling of the modes reverses relative to their handedness whenever P D changes sign. This seems inconsistent with the physical argument relating to the handedness. The subtle point mentioned above is that the solutions for the wave properties in the limit of parallel propagation are not the same of the limit for parallel propagation of the solutions for oblique propagation. If one plots the solutions for parallel propagation, the labeling of the modes at points where the two solutions cross. One such crossing occurs at the plasma frequency, where P changes sign. In a cold plasma with multiple ion species, R− has resonances at each of the ion cyclotron frequencies, where is changes sign by passing through infinity; there are solutions of R+ = R− , implying D = 0 in between the cyclotron frequencies. The actual solutions for the modes, for an arbitrarily small angle, deviate away from the crossing point, as illustrated schematically in figure 3.2.

3.2 Waves in cold plasmas ω

ω

1

1

k

0 -

1 (a)

i

0

0

2

1 +

+ 0

101

2 (b)

k

Fig. 3.2. Schematic illustration of the dispersion curves for nearly parallel propagating modes. (a) For two modes labeled 0 and 1, near the cross-over point: the solid curves are for θ = 0 and light dashed curves show how the curves reconnect for θ 6= 0 into two modes labeled + and −. (b) For three modes, labeled 0, 1 and 2, with the dashed curves defining three reconnected modes labeled +, i and −.

For perpendicular propagation, cos θ = 0, the solutions of the dispersion equation (3.1.23) reduce to n2 = P , n2 = (S 2 − D2 )/S. The fact that the solution n2 = P does not depend on the magnetic field led to the mode being called ‘ordinary’, with the other mode being called ‘extraordinary’ in the context of a cold electron gas. These names continue to be used in connection with the magnetoionic modes, as discussed in §3.2.1. 3.2.3 Cutoffs and resonances A cutoff is defined as a zero of refractive index, and a resonance is defined as an infinity in refractive index. The dispersion equation in the form (3.1.23) implies that cutoff frequencies are determined by C = 0, and resonances are determined by A = 0. (Formally, these conditions should be C/A = 0, and A/C = 0, respectively, because there are infinities is A and C at the cyclotron resonances; however, these infinities cancel in the ratios A/C, C/A except for parallel propagation.) One implication of cutoffs and resonances is that they separate different branches of the cold plasma modes. The refractive index squared is positive just above a cutoff frequency and negative just below it. Regions with n2 < 0 correspond to evanescence: the disturbance oscillating at frequency ω decays spatially, at a rate |n|ω/c in ordinary units. The refractive index squared is negative just above a resonance and positive just below it. Thus propagating branches of a wave mode are separated from each other by regions of evanescence. In the presence of inhomogeneity, waves are refracted towards the direction of increasing refractive index, so that they refract away from a cutoff and towards a resonance. The cold plasma assumption is well satisfied near a cutoff, but breaks down near a resonance, where thermal motions are important, and lead to gyromagnetic absorption. The cutoff frequencies in a cold plasma are determined by P (S 2 −D2 ) = 0, and hence are independent of angle of propagation. At frequencies well above ion cyclotron frequencies, only the contribution from the electrons is important

102

3 Wave dispersion in relativistic magnetized plasma

and the wave modes are the magnetoionic modes (§3.2.1). The cutoff frequency at P = 0 corresponds to ω = ωp , and is in the o mode. There are two solutions (with ω > 0) of S 2 − D2 = 0, and these are the cutoff frequencies in the x mode and the z mode. There are two resonant frequencies, and these are in the z mode and the whistler mode. The resonances at perpendicular propagation define so-called hybrid frequencies. For cos θ = 0, the condition S(ω) = 0 reduces to 1−

2 X ωpi ωp2 − =0 ω 2 − Ωe2 ω 2 − Ωi2 i

(3.2.7)

where it is assumed that the only negatively charged particles are electrons, and where the sum is over all ionic species, with the ith species having plasma frequency ωpi and cyclotron frequency Ωi . The upper hybrid frequency is the solution of (3.2.7) found by neglecting the ions, and the lower hybrid frequency is the solution found by assuming Ωi2 ≪ ω 2 ≪ Ωe2 . These are given by 2 ωUH = ωp2 + Ωe2 ,

2 ωLH =

X i

2 ωpi

Ωe2 2 , ωUH

(3.2.8)

respectively. The magnetoionic waves in a pure electron gas (η = −1) have four branches: two high frequency branches, referred to as the o mode and the x mode, and two lower frequency branches, referred to as the z mode and the whistler mode. The o mode and the x mode exist above cutoff frequencies, ωp , ωx , respectively, the z mode exists above a cutoff frequency, ωz , and below a resonant frequency, ω+ (θ), and the whistler mode exists below a resonant frequency, ω− (θ). These branches are illustrated in figure 3.1. The condition, C = 0, for a cutoff has three (positive frequency) solutions in the magnetoinic theory. The solution corresponding to 1− X = 0 is ω = ωp , which is the cutoff for the o mode. For an electron gas (η = −1), there are two positive frequency solutions corresponding to the other factor in square brackets in (3.2.3). Writing these as ω = ωx and ω = ωz , one finds ωx = 12 Ωe + 21 (4ωp2 + Ωe2 )1/2 ,

ωz = − 21 Ωe + 21 (4ωp2 + Ωe2 )1/2 .

(3.2.9)

These are the cutoff frequencies for the x and z modes, respectively. The condition A = 0 for a resonance has two solutions in the magnetoionic theory. Writing these two resonant frequencies as ω± (θ), one has 2 ω± (θ) = 12 (ωp2 + Ωe2 ) ±

1 2

(ωp2 + Ωe2 )2 − 4ωp2 Ωe2 cos2 θ

1/2

.

(3.2.10)

For ωp > Ωe the resonance at ω = ω+ (θ) is in the z mode and the resonance at ω = ω− (θ) is in the whistler mode; for ωp < Ωe these roles are reversed. For perpendicular propagation, the higher resonant frequency, ω+ (θ), reduces to the upper hybrid frequency, defined by (3.2.8). For the lower resonant

3.2 Waves in cold plasmas n

w

n

(a)

103

(b)

z c

o

z o 0 Ωe

ωp

x x

ω

0

ωp

ω

Fig. 3.3. a) A schematic illustration of the refractive index curves for ωp ≫ Ωe in a cold electron gas (solid line) and the modifications introduced by a small admixture of positrons (dashed lines). (b) As for (a) but for ωp ≪ Ωe and omitting the whistler mode.

frequency, the neglect of the ions is not justified when ω− (θ) is comparable with or less than the lower hybrid frequency. As discussed in §3.2.2, the solutions of the dispersion equation in the limit sin θ = 0 are different from the limit sin θ → 0 of the solutions for sin θ 6= 0. In the magnetoionic theory, this effect occurs at the plasma frequency. The dispersion relations for sin θ = 0 are n2 = 1 − X/(1 ± Y ), together with X = 1 for longitudinal oscillations. The solution n2 = 1 − X/(1 + Y ) crosses with the solution X = 1 at n2 = Y /(1 + Y ). For arbitrarily small angles, these solutions reconnect, as illustrated schematically in figure 3.2(a). In the limit sin θ → 0, the solution n2 = 1 − X/(1 + Y ) corresponds to the o mode for ω > ωp and to the z mode for ω < ωp . 3.2.4 Dispersion equation for streaming cold plasma The direct method for solving for the wave properties in the presence of streaming motions involves starting from the wave equation with the streaming motions included in the response tensor. The dispersion equation becomes Using the same notation for the parameters S, D, P as in the cold plasma case, cf. (3.1.28)–(3.2.6), and adding additional terms in the relativistic case (2.6.19), the dispersion equation is S − n2 cos2 θ −iD Q + n2 sin θ cos θ = 0. iD S − n2 −iR (3.2.11) 2 2 Q + n2 sin θ cos θ iR P − n sin θ Evaluating the determinant gives a dispersion equation that can be written in the form An4 − Bn2 + C = 0, with A = P cos2 θ + S sin2 θ + 2Q sin θ cos θ, B = AS + P S − Q2 − (D sin θ − R cos θ)2 , C = P (S 2 − D2 ) − S(Q2 + R2 ) − 2DQR.

(3.2.12)

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3 Wave dispersion in relativistic magnetized plasma

Unlike the non-streaming case, where the coefficients A, B, C are functions of ω, θ, in the streaming case, the coefficients (3.2.12) are implicit functions of |k| = nω. Approximations need to be made to make progress in solving for the wave modes of interest. A specific case considered below is the weak-anisotropy approximation, which allows the properties of the two high-frequency modes to be determined. The dispersion equation can be written as a polynomial equation in n. This is achieved by rationalizing, by multiplying by factors (ω − kz va )2 and (ω − kz va )2 − Ωa2 /γa2 that cancel the resonant denominators associated with each streaming species a. A polynomial equation of order N has N solutions. There are analytic solutions of a polynomial equation for N ≤ 4, and elementary numerical methods may be used to solve polynomial equations of higher order. Although there are N solutions of a polynomial equation of order N , it does not follow that there are N distinct modes. Some of the solutions may be spurious, being introduced by the rationalization procedure. To illustrate this point, consider a streaming electron gas, when only the contribution of the electrons is retained. In this case, the wave properties are known to be given by the magnetoionic theory in the absence of streaming. Introducing a streaming motion is equivalent to interpreting the wave properties from a moving frame. The rationalization of the dispersion equation involves multiplying by (ω − kz ve )2 and (ω − kz ve )2 − Ωe2 /γe2 , so that it becomes a polynomial equation of order N = 8 in n. The existence of N = 8 modes can be interpreted by considering solving for kz rather than n. Then there a four magnetoionic modes, which are pairwise degenerate in the absence of streaming, with the degeneracy being broken by the introduction of streaming. The other four solutions can be interpreted in terms of modified forms of the solutions of (ω − kz ve )2 = 0 and (ω − kz ve )2 − Ωe2 /γe2 = 0. These are sometimes referred to as ‘beam modes’. Beam modes can become intrinsically growing modes in the presence of relative streaming motions between different components in a plasma. However, when all the components in the plasma are streaming with the same velocity, the beam modes must be spurious solutions of the dispersion equation. 3.2.5 Magnetoionic waves in a streaming plasma In the case where all species are streaming with the same velocity, one may treat wave dispersion in two alternative ways, one direct and one indirect. The direct way is to start from the wave equation and the dispersion equation with the streaming motion included in the response tensor, and solve for the wave properties directly. The foregoing discussion illustrates the difficulties with the direct method: it requires solving a high-order polynomial equation, and introduces spurious solutions associated with beam modes. The other method is to solve for the wave properties in the rest frame, and Lorentz transform these properties to the frame in which the plasma is streaming. While the latter procedure is the simpler, this is not necessarily the case.

3.2 Waves in cold plasmas

105

The transformed wave properties are implicit functions of the transformed frequency and parallel wave number: solving for explicit expressions involves the similar difficulties to the direct method. To illustrate the foregoing point, consider the magnetoionic waves in a frame in which the (cold) electron gas is streaming. Let the rest frame of the plasma be the unprimed frame and let the primed frame be the laboratory frame in which the plasma is streaming. A Lorentz transformation between the wave 4-vectors in the two frames is       ′  γ(ω ′ − kz′ β) ω γ(ω + kz β) ω ′ ′  .  k⊥  =  ,  k⊥ (3.2.13) k⊥ k⊥ = ′ ′ γ(kz − ω ′ β) kz γ(kz + ωβ) kz

The dispersion equation for the magnetoionic waves in the form (3.1.22), viz., An4 − Bn2 + C = 0, with coefficients, A, B, C, given by (3.1.24). These coefficients depend only on n2 , θ and ω, with ω appearing only in the magnetoionic parameters X = ωp2 /ω 2 , Y = Ωe /ω. The invariant forms of these quantities follow from the identifications n2 →

(k˜ u)2 − k 2 , (k˜ u)2

sin2 θ →

2 k⊥ , 2 (k˜ u) − k 2

ω → k˜ u,

(3.2.14)

with u ˜µ = [1, 0] in the rest frame. This allows one to rewrite the dispersion equation in an arbitrary frame by making the replacements k → k ′ , u ˜ → u, ′ with uµ = [γ, γβ], and with k⊥ = k⊥ . However, the dispersion equation An4 −Bn2 +C = 0 is no longer a quadratic equation in the new variables, e.g., in n′2 , θ′ , ω ′ , and solving it for the dispersion relations is not straightforward. If one starts with the solutions n2 = n2± = (B ±F )/2A, cf. (3.1.25), and makes the substitutions (3.2.14), this leads to implicit equations for the dispersion relations in terms of the transformed variables. Approximations need to be made to find explicit solutions. 3.2.6 Transformation of the polarization vector Proceeding with the example of the transformation of the magnetoionic waves to a streaming frame, consider the transformation of the polarization 3-vector in the general form (L± κ + T± t + ia) , (3.2.15) e± = (L2± + T±2 + 1)1/2

with L± , T± given by (3.2.4)–(3.2.6) for the magnetoionic waves. One can transform L± , T± to the new frame by re-expressing them in terms of the primed variables. One also needs to transform the vectors κ, t, a. Let the transformed vectors be denoted κt , tt , at . In the primed frame the polarization vector are then given by e′± =

(L± κt + T± tt + iat ) , (L2± + T±2 + 1)1/2

(3.2.16)

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3 Wave dispersion in relativistic magnetized plasma

with L± , T± rewritten as functions of the primed variables. In the case X ≪ 1, (3.2.6) implies that the longitudinal part, L± , is of order X, and (3.2.4) is approximated by T 2 + (Y sin2 θ/η cos θ) T − 1 = 0, which has solutions T± =

−Y sin2 θ ∓ (Y 2 sin4 θ + 4η 2 cos2 θ)1/2 . η cos θ

(3.2.17)

To lowest order in an expansion in X, the polarization vectors (3.2.16) in the primed frame reduce to e′± = (T± t′ + ia′ )/(T±2 + 1)1/2 , with T± given by expressing the right hand side of (3.2.17) in terms of the variables in the primed frame. In evaluating the transformed vectors κt , tt , at , one needs to take the gauge condition into account. The form (3.2.15) applies in the temporal gauge, A0 (k) = 0, and this gauge condition is not preserved by a Lorentz transformation. One needs to make a gauge transformation to the temporal gauge in the new frame. This transformation is discussed in §2.6.4 of volume 1. The transformed vectors are γω κt = ′ (|k|′ sin θ′ , 0, |k|′ cos θ′ − ω ′ β), (3.2.18) ω |k| β(1 − n′2 ) 1 ′ ′ , 0, −n sin θ , at = a. (3.2.19) n′ cos θ′ − tt = n 1 − n′ β cos θ′ One may rewrite (3.2.19) as 1 (1 − n′2 )β cos θ′ ′ 1 (1 − n′2 )β sin θ′ ′ ′ tt = n − κ, t − n 1 − n′ β cos θ′ n 1 − n′ β cos θ′

at = a′ ,

(3.2.20) with κ′ = (sin θ′ , 0 cos θ′ ), t′ = (cos θ′ , 0, − sin θ′ ), a′ = (0, 1, 0). Using (3.2.18), (3.2.20) one may rewrite (3.2.16) in the form e′± = (L′± κ′ + T±′ t′ + ′2 1/2 ia′ )/(L′2 , with L′± , T±′ identified in terms of L± , T± by relatively ± + T± + 1) cumbersome expressions. To solve for wave properties explicitly in the presence of streaming it is usually necessary to make some simplifying assumption. An approximation that applies at high frequencies is based on the assumption of weak anisotropy, discussed in §3.5. 3.2.7 Pair plasma The inclusion of an admixture of positrons (η 6= −1) modifies the magnetoionic wave properties by introducing an additional branch of solutions. This branch exists between an additional cutoff frequency, and an additional resonance (at Ωe ), as shown in figure 3.3. There are four cutoff frequencies for η 6= ±1: ωp , and the three positive-frequency solutions of 1−

2X X 2 (1 − η 2 Y 2 ) + = 0. 1−Y2 (1 − Y 2 )2

(3.2.21)

3.2 Waves in cold plasmas

107

The resonant frequency are determined by A/C = 0, and although resonances at the frequencies (3.2.10) are independent of η, there is an additional resonance at Ωe for |η| = 6 1. This resonance arises from C, given by (3.2.3), diverging quadratically for Y → 1 for η 2 6= 1, whereas it diverges only linearly for η 2 = 1. The wave properties simplify considerably for a pure pair plasma, η = 0. For η = 0 one has D(ω) = 0 in (3.2.2), and the equivalent dielectric tensor (3.1.15) reduces to the same form as for a uniaxial crystal. The dispersion equation becomes (n2 − S)(An2 − P S) = 0. For θ 6= 0, the ordinary mode (as defined in magnetoionic theory) has dispersion relation and polarization vector (ω 2 − ωp2 )(ω 2 − ωp2 − Ωe2 ) 2 (θ)] [ω 2 − ω 2 (θ)] , [ω 2 − ω− +

(P cos θ, 0, −S sin θ) , (P 2 cos2 θ + S 2 sin2 θ)1/2 (3.2.22) and the extraordinary mode (as defined in magnetoionic theory) has n2o =

n2x =

ω 2 − ωp2 − Ωe2 , ω 2 − Ωe2

eo =

ex = (0, i, 0),

(3.2.23)

where the coordinates axes have B along the 3-axis and k in the 1-3 plane, 2 and where ω± (θ) are given by (3.2.10). There is only one cutoff frequency, at 2 2 1/2 (ωp + Ωe ) , which is common to both modes. There is an inconsistency in the labeling of the modes: the conventional labeling of the modes from magnetoionic theory is opposite to that for the labeling implied by the analogy with a uniaxial crystal. For a uniaxial crystal the ‘ordinary’ mode is the one that does not depend on angle, and the ‘extraordinary’ mode is the one that does depend on angle. 3.2.8 High-frequency limit At high frequency the magnetoionic waves may be approximated by expanding in powers of X ≪ 1, Y ≪ 1. At sufficiently high frequency X ∝ 1/ω 2 becomes much smaller than Y ≪ 1/ω, and the wave properties may be approximation by taking the limit X → 0 in (3.2.4). The solutions (3.2.5) for X = 0 become Tσ =

1 2Y

−η cos θ = sin2 θ − σδ

1 2Y

sin2 θ + σδ , η cos θ

δ=

1 2 4Y

sin4 θ + η 2 cos2 θ

1/2

.

(3.2.24) An expansion in Y is valid except for a small range of angles about perpen1 dicular propagation, specifically, except for | cos θ| < ∼ 2 Y , and (3.2.24) reduces to Tσ = ησ cos θ/| cos θ|, corresponding to circular polarizations. In this approximation, the refractive indices (3.2.6) become n2σ = 1 − X(1 − σY ).

(3.2.25)

108

3 Wave dispersion in relativistic magnetized plasma n2

w (a)

c z Ω e

0

o

x

ωp

ω

n2

(b) z- c

o

c- z

x 0

ωp

Ω e

ω

Fig. 3.4. (a) A schematic illustration of the refractive index curves for ωp ≫ Ωe in a cold electron gas (solid line) and the modifications introduced by a small admixture of positrons (dashed lines). (b) As for (a) but for ωp ≪ Ωe and omitting the whistler mode.

The longitudinal part of the polarization, Lσ , is very small, being proportional to XY . The neglect of the longitudinal part of the polarization is the basis for the weak-anisotropy approximation, discussed below. The approximation (3.2.25) suffices for a treatment of Faraday rotation, which is the rotation of the plane of linear polarization for radio waves propagating through a magnetized plasma. On separating linearly polarized radiation into oppositely circularly polarized modes, the difference between the refractive indices imply that a phase difference develops between the two modes. The plane of linear polarization rotates at a rate per unit length ω(nR −nL )/2c per unit length, where nR and nL are the refractive indices for right- and lefthand polarized modes, respectively. In an electron plasma (η = −1), the rate implied by (3.2.24), (3.2.24) is (ωp2 Ωe /2ω 2 c) cos θ/| cos θ|. 3.2.9 Low-frequency waves in cold plasma The contribution of the ions to wave dispersion is negligible at high frequencies but it becomes important a low frequencies, comparable with or below the cyclotron frequency of ions. In the low-frequency limit one has ω ≪ ωp , corresponding to P large and negative. The appropriate limit of cold plasma theory is found by expanding in powers of 1/P . To lowest order (3.1.28) gives

3.2 Waves in cold plasmas

S sin2 θ ± [S 2 sin4 θ + 4D2 cos2 θ]1/2 2D cos θ

T± =

where the ± labeling is arbitrary. To the same order (3.1.29) gives D cos θ S 2 − D2 1 2 S + = n = , cos2 θ T S − DT cos θ

109

(3.2.26)

(3.2.27)

and L = T tan θ. Relevant sums over the ionic species give 2 X ωpi i

Ωi2

=

2 X ωpi

c2 2 , vA

i

Ωi

=

ωp2 , Ωe

(3.2.28)

where the inertia of the electrons is neglected in the Alfv´en speed, vA , and where the latter condition follows from charge neutrality. One has (in ordinary units) S =1+

2 2 X ωpi ω c2 + 2 − ω2) , 2 2 (Ω vA Ω i i i

D=−

X i

2 ωpi ω . 2 Ωi (Ωi − ω 2 )

(3.2.29)

At frequencies ω ≪ Ωi one has D ≪ S, and a further approximation involves expanding in powers of D/S. To lowest order in D/S, the solutions of (3.2.26) reduce to T+ = ∞, T− = 0. This limit corresponds to linearly polarized modes. To next order the solutions become T+ = S sin2 θ/D cos θ = −1/T−, implying that the modes are elliptically polarized when ω/Ωi is assumed non-negligible. The two modes are cold plasma counterparts of two MHD wave modes, with the + mode corresponding to the Alfv´en mode and the − mode to the fast magnetoacoustic mode (with sound speed zero in a cold plasma). The solution for the Alfv´en (A) mode corresponds to n2A =

S D2 cos2 θ , + 2 cos θ S sin2 θ

TA =

S sin2 θ , D cos θ

LA = TA tan θ.

(3.2.30)

The solution for the magnetoacoustic (m) mode corresponds to n2m = S −

D2 , S sin2 θ

Tm = −

D cos θ , S sin2 θ

Lm = Tm tan θ.

(3.2.31)

These reduce to the MHD modes for zero sound speed in the limit ω → 0, corresponding to D = 0. The modification to the MHD dispersion relations for ω/Ωi 6= 0 becomes substantial as the ion cyclotron frequency is approached. A nonzero parallel component of the polarization vector appears when one includes the next order term in an expansion is 1/P , or more specifically in ω 2 /ωp2 . It is convenient to introduce the skin depth, λe = c/ωp , one has P ≈ −1/ω 2λ2e . The resulting modification is important for Alfv´en waves propagating at large angles. Setting D = 0 and retaining 1/P 6= 0, it is convenient

110

3 Wave dispersion in relativistic magnetized plasma

to write the resulting dispersion relation for Alfv´en waves in terms of the frequency, ω = ωA (k). Approximating the polarization vector by eA ≈ (1, 0, ez ), with |ez | ≪ 1, one finds 2 ωA ≈

kz2 2 λ2 ) , S(1 + k⊥ e

ez ≈ −

S sin θ ≈ |k|2 λ2e sin θ. A

(3.2.32)

2 In the approximation S → 1/vA ≫ 1 waves described by (3.2.32) are referred to as inertial Alfv´en waves.

3.3 Waves in weakly relativistic thermal plasmas

111

3.3 Waves in weakly relativistic thermal plasmas There is a rich variety of natural wave modes in a relativistic magnetized thermal plasma. These modes include modified forms of the modes of a cold magnetized plasma, and additional modes that depend intrinsically on thermal effects. One class of additional modes have frequencies close to harmonics of the electron cyclotron frequency for nearly perpendicular propagation, θ ≈ π/2. Weakly relativistic effects modify these cyclotron harmonic modes. In this section the properties of waves in weakly relativistic, magnetized, thermal, electron and electron-positron plasmas are discussed, emphasizing the cyclotron harmonic modes. 3.3.1 Wave modes for perpendicular propagation The cyclotron harmonic wave modes are often called Bernstein modes. However, this name is appropriate only for a specific class of longitudinal waves discussed [12] and considered earlier by [11]. There are two further classes of cyclotron harmonic wave modes, discussed by [13, 14], that are related to the transverse-like ordinary and extraordinary modes in a cold plasma. The properties of all three classes of cyclotron harmonic modes in the nonrelativistic case were described by [16, 17], cf. also the review by [18]. Here we follow [24, 25] in referring to these as the Gross-Bernstein (GB) modes, the ordinary modes, and the Dnestrovskii-Kostomarov (DK) modes. Cyclotron harmonic modes were first identified for strictly perpendicular propagation in a nonrelativistic plasma, and it is appropriate to start by considering this case. Setting kk = 0 in (2.4.22), the 13-, 23-, 31- and 32components of the response 3-tensor are zero. The dispersion equation then becomes 1 K 1 K 12 0 2 = (K 3 3 − n2 )[K 1 1 K 2 2 − n2 ) − K 1 2 K 2 1 ] = 0, K 1 K 2 2 − n2 0 3 2 0 0 K 3−n (3.3.33) where it is noted that K 1 2 = −K 2 1 is imaginary. There are two relevant solutions n2 = K 3 3 , K 1 1 (K 2 2 − n2 ) − |K 1 2 |2 = 0, (3.3.34) corresponding to the ordinary and extraordinary modes, respectively. The extraordinary mode is usually factorized according to n2 = K 2 2 −

|K 1 2 |2 , K 11

K 1 1 = 0,

(3.3.35)

corresponding to transverse and longitudinal modes. The solutions of the two dispersion equations (3.3.35) are identified as the Dnestrovskii-Kostomarov (DK) modes and the Gross-Bernstein (GB) modes, respectively. However,

112

3 Wave dispersion in relativistic magnetized plasma

there is no guarantee that the factorization (3.3.35) is valid, and one can only justify a posteriori that a solution of the general dispersion relation (3.3.34) for the extraordinary mode is indeed a solution of K 1 1 = 0, and corresponds to a longitudinal mode for perpendicular propagation. 3.3.2 Cyclotron harmonic modes in a nonrelativistic plasma The cyclotron harmonic modes have their simplest form for perpendicular propagation in a nonrelativistic thermal electron gas. The relevant dielectric tensor follows from (2.4.22). For perpendicular propagation in the nonrelativistic limit, in (2.4.22) one sets kz = 0, r0 (ξ) = ρ − iωξ in the exponential function and r0 (ξ) = ρ elsewhere, corresponding to the nonrelativistic limit. Also the quantity Λ introduced in (2.4.21), reduces in the nonrelativistic limit to λ, defined by (3.3.36), which is the square of the perpendicular wavenum2 ber, k⊥ , times the square V 2 Ω02 , V 2 = 1/ρ of the gyroradius of a particle with the thermal speed V . Then (2.4.22) gives K i j = δji −

∞ hij (λ)e−λ ωp2 X , ω 2 s=−∞ ∆s

h1 1 (λ) = s2 Is (λ)/λ,

λ=

2 k⊥ n2 ω 2 = , ρΩ02 ρΩ02

∆s =

ω − sΩ0 , ω

h1 2 = iηs[Is (λ) − Is′ (λ)],

h2 2 = h1 1 − 2λ[Is (λ) − Is′ (λ)],

h3 3 = Is (λ),

(3.3.36)

with η = −1 for a pure electron gas. Ordinary modes On inserting (3.3.36) in the dispersion relation (3.3.34) for ordinary mode waves, one obtains n2 = 1 −

∞ ωp2 X s2 Is (λ)e−λ . ω 2 s=−∞ λ∆s

(3.3.37)

Qualitatively, the factor ∆s in (3.3.37) becomes arbitrarily small in the limit ω → sΩe , and no matter how small the numerator there is always a solution of (3.3.37). Hence, there is one ordinary mode per harmonic s ≥ 1. The numerator contains the factor s s λ /2 s! for λ → 0, −λ (3.3.38) Is (λ)e ≈ 1/(2πλ)1/2 for λ → ∞. The qualitative form of the dispersion relations is illustrated in Figure 3.5. Each mode has both a cutoff (n2 → 0, λ → 0) and a resonance (n2 → ∞, λ → ∞). The properties of these modes can be summarized as follows: (i) As the cutoff at the sth harmonic is approached, the dispersion relation (3.3.37) is approximated by

3.3 Waves in weakly relativistic thermal plasmas ω / Ωe

ω / Ωe

(a)

8

113

(b)

+1 s

7 6 5 4 3 s

2 1 0

1

λ / ω2

λ

Fig. 3.5. The nonrelativistic ordinary modes are illustrated schematically for perpendicular propagation in a warm plasma: (a) the undamped modes for 6 < ωp /Ωe < 7, (b) including a complex mode, with dotted line indicating the imaginary part, between two harmonics with s + 1 < ωp /Ωe . [After [17] and [24, 25].]

λs = −2s s!∆s (ωp2 − s2 Ωe2 )/ωp2 .

(3.3.39)

This implies that for λ → 0 the dispersion curve approaches the harmonic from below for sΩe < ωp and from above for sΩe > ωp , as illustrated in Figure 3.5. (ii) As the resonance at the sth harmonic is approached, the dispersion relation (3.3.37) is approximated by λ3/2 = −ωp2 /Ωe2 (2π)1/2 ∆s ρ.

(3.3.40)

This implies that for λ → ∞ the dispersion curve approaches the harmonic from below for sΩe < ωp and from above for sΩe > ωp , as illustrated in Figure 3.5. (iii) For ω ≥ ωp each dispersion curve has nearly horizontal portions in which ω varies slowly with small λ at small and large λ, plus a portion in which the dispersion curve from just above one harmonic to just below the next harmonic. The latter portions of the curves for each harmonic form an envelope approximately along n2 = 1 − ωp2 /ω 2 . (iv) Close to the sth harmonic the sum in (3.3.37) is approximated by retaining only the terms labeled 0 and s. This gives ∆s =

ωp2 Is (λ)e−λ , s2 Ωe2 − ωp2 I0 (λ)e−λ − λρΩe2

|∆s |max ≈

ωp2 , 2ρs3

(3.3.41)

which is valid for |s∆s | ≪ 1. The maximum value, |∆s |max determines the maximum deviation of the dispersion curve from the harmonic line, and it occurs for λ ≈ s2 /3.

114

3 Wave dispersion in relativistic magnetized plasma

Extraordinary modes The extraordinary mode includes both the DK modes and the GB modes; [17] referred to these as the extraordinary magnetic induction mode and the extraordinary electrostatic mode, respectively. Their properties can be summarized as follows: (i) In the cold plasma limit, λ → 0, the DK mode corresponds to the extraordinary mode of magnetoionic theory, with dispersion relation given n2 =

(ω 2 − ωx2 )(ω 2 − ωz2 ) 2 ) ω 2 (ω 2 − ωUH

(3.3.42)

where ωUH = (ωp2 + Ωe2 )1/2 is the upper hybrid frequency (3.2.8). The xmode is the branch above the cutoff at ωx , and the z-mode is the branch in the range ωz < ω < ωUH where the cutoff frequencies are given by (3.2.9). (ii) The thermally modified z-mode joins onto a GB mode at ωUH . The relevant GB mode is the one that has its cutoff at the harmonic sΩe immediately below ωUH . (iii) Close to the sth harmonic, for s|∆s | ≪ 1, the sum in (3.3.37) is approximately by retaining only the terms labeled ±1 and s. This gives (A − y)2 − ξ 2 (B + y)2 + Ey 2 = 0, A=1−

ωp2 I1 (λ)e−λ ωλ

y=

ωp2 λs−1 e−λ , Ωe2 2s s! ∆s

1 1 (s + 2)2 2 + , E= 2 λ , ω − Ωe ω + Ωe s (s + 1) ωp2 [I1′ (λ) − I1 (λ)]e−λ 1 1 − B= . (3.3.43) ω ω − Ωe ω + Ωe

For s = 1 the terms with ω − Ωe in the denominator are absent. (iv) Near the cutoff at the sth harmonic, the dispersion curves for the DK and GB modes approach each other (K 2 2 → K 1 1 , |K 1 2 |2 /K 1 1 → 0), and the two solutions of (3.3.43) give the cutoffs for the DK and GB modes. The two modes that emerge from cutoffs at the sth harmonic have dispersion relations, for λ → 0, Ωe2 A + ξ 2 B ± [(A + ξ 2 B)2 − (1 − ξ 2 + E)(A2 − ξ 2 B 2 )]1/2 , ωp2 1 − ξ2 + E (3.3.44) where A, B are evaluated at λ = 0. For sΩe < ωz both modes emerge below the harmonic, for ωz < sΩe < ωx one emerges below the harmonic and the other above the harmonic, and for sΩe > ωx both emerge above the harmonic. (v) For λ → ∞ the resonance correspond to λs−1 = 2s s!∆s

3.3 Waves in weakly relativistic thermal plasmas

115

ω / Ωe 8 7 6 ωUH 5 4 3 2 1 0 1/ ρ

1/3

λ / ω2

Fig. 3.6. As in Figure 3.5 but for the extraordinary mode. The DK and GB modes may be distinguished by whether they approach the harmonic at large λ from below or above respectively. The two cutoffs that do not occur at harmonic are ωz and ωx , and the upper hybrid frequency, ωUH , is indicated.

3/2

λ

=

ωp2 /Ωe2 (2π)1/2 ∆s ρ −ωp2 /Ωe2 (2π)1/2 ∆s ρ

for GB modes, for DK modes.

(3.3.45)

Thus the GB mode approaches the harmonic from above and the DK approaches the harmonic from below. The qualitative form of the dispersion curves for the extraordinary mode are illustrated in Figure 3.6. 3.3.3 Weakly relativistic modifications The foregoing properties cyclotron wave modes in a nonrelativistic plasma are modified when either weakly relativistic effects [24, 25] or non-perpendicular propagation [8] are included. Weakly relativistic effects, which are included through the Shkarofsky functions, include a frequency downshift, a broadening of each resonance, and an associated broadening of the frequency range in which damping occurs. For convenience let φ = π/2 − θ 6= 0 be the angular deviation from perpendicular propagation. A semiquantitative prescription for generalizing the nonrelativistic modes to include these effects is as follows [8]. (1) Solve the dispersion equation in the nonrelativistic limit, ignoring the weakly relativistic effects, to find the “nonrelativistic” modes, as above. (2) Downshift the nonrelativistic modes by a fractional amount of order 1/ρ. (3) Determine the frequency range of the core of the cyclotron absorption band below each harmonic. Any downshifted, large-λ mode lying in this range is heavily damped.

116

3 Wave dispersion in relativistic magnetized plasma

(4) Introduce resonance broadening over a fractional frequency range of order Ωe /ρ around the downshifted cyclotron harmonic frequency. (5) To allow for φ 6= 0 include damping over a frequency range of order kz /ρ1/2 , with kz = |k| cos θ ≈ |k|φ. As a result, for kz > ωρ1/2 damping extends above the harmonic. The following discussion of the specific wave modes amplifies these points. 3.3.4 Ordinary modes Weakly relativistic effects are included in the dispersion relation (3.3.34) for the ordinary mode by evaluating K 3 3 using the approximate form (2.4.26). A somewhat improved approximation [8] gives ωp2 Is (λ)e−λ ρ n =1− ω2 2

∂ 1 + 2a Fq (zs , a), ∂a

(3.3.46)

λ , + λ2 )

(3.3.47)

q = 5/2 + (s2 + λ2 )1/2 − λ −

2(s2

with a = kz2 ρ/2ω 2 and with the Shkarofsky function approximated by the form (2.4.35). The properties of Fq (z) illustrated in Figure 2.1 underlie the points (1)–(5) above in describing weak relativistic effects on the dispersion. Using (3.3.46), these weakly relativistic effects are confined to the range |(zs + q)/(4a + 2q)1/2 | < ∼ 3.

(3.3.48)

As a result the large-λ ordinary modes cannot exist outside the range s < ∼ smax , φ< φ , with max ∼ ωp2 (2ρ|∆s |max − 7)2 1 3 2 3 −7 . , φmax = 2 smax ≈ 2 Ωe 7 + 6(7 + 23 ρ2 tan2 φ 2ρ 36 (3.3.49) It follows that weakly damped ordinary mode waves exist only for ωp2 ∼ > 23Ωe2 , and then only for ω ≪ ωp . Dnestrovskii-Kostomarov modes The DK modes are modified in a similar way to ordinary mode waves by weakly relativistic effects. [25] estimated that weakly damped DK modes exist only for 3/2 1 ωp . (3.3.50) s< ∼ 8 Ωe In the range where they exist, the DK mode approximate their nonrelativistic counterparts closely.

3.3 Waves in weakly relativistic thermal plasmas

117

Gross-Bernstein modes The weakly relativistic dispersion relation for the GB modes is 0=1−

ωp2 ρIs (λ)e−λ X 2 s Fq−1 (zs , a), ω2λ s

(3.3.51)

with q given by (3.3.47), and with the Shkarofsky function approximated by (2.4.34). The modifications from the nonrelativistic GB modes are similar to those for the ordinary modes. The maximum harmonic for which GB modes exist for perpendicular propagation, and the maximum value, φmax , for which off-angle propagation is possible for given s are [8]: ! 3ωp2 ωp2 ρ 1 3 2 2 3 φ < s < (3.3.52) ∼ φmax = 2ρ2 Ω 2 s3 − 5 . ∼ ρΩ 2 15 + ρ2 tan2 φ , e e In summary, weakly relativistic effects are severely limiting on the range of existence of the cyclotron harmonic wave modes. This is due primarily to damping just below the cyclotron harmonic associated with the relativistic downshift in the cyclotron frequency, Ωe /γ, compared with the nonrelativistic case, Ωe . The nonrelativistic limit for perpendicular propagation implies no damping except exactly at ω = sΩe , and weakly relativistic effects imply damping in a range below sΩe , as illustrated in Figure 2.1. The inclusion of a small angular deviation, φ = π/2 − θ, from perpendicular propagation allows damping at ω 6= sΩe , which is qualitatively similar to the weakly relativistic effect. The damping allowed for φ 6= 0 (that is, kz 6= 0) is symmetric in frequency about the cyclotron line, whereas the damping due to the relativistic effect, Ωe → Ωe /γ, is always below the nonrelativistic resonance. When an admixture of positrons is present, the electrons and positrons contribute with the same sign to the non-gyrotropic components of the response tensor, and with the opposite sign to the gyrotropic components. Assuming that the electrons and positrons are thermal with the same temperature, this property allows one to include an admixture of positrons trivially in the response tensor. An admixture of positrons affects only the K 1 2 = −K 2 1 . It follows that the presence of positrons affects only the DK modes. 3.3.5 Smoothing of the cyclotron resonance The cyclotron resonance, which leads to the DK and GB modes in a warm plasma, cf. §??, is smoothed out in the highly relativistic limit. For a onedimensional distribution, the cyclotron resonance for a specific particle occurs at ω = Ωe /γ(1−nβ cos θ), and for a distribution of highly relativistic particles this implies that cyclotron resonance occurs over a broad range of frequency around ω ∼ Ωe /hγi, corresponding to z/y ∼ 1/hγi. The arguments z± of R(z± ), S(z± ) in (??) change from z± ≈ (z ± y)/(1 + y 2) in the nonrelativistic case to z± ≈ ±y/|y| in the highly relativistic case. Although all the plasma

118

3 Wave dispersion in relativistic magnetized plasma

dispersion functions have sharp peaks at 1−z ∼ 1/hγi2 , in (??) the coefficients of the terms involving R(z), S(z) are of order 1/y 2 ∼ 1/ < γi2 , 1/y ∼ 1/ < γi, respectively, and these factors counteract the peaks, smoothing out the cyclotron resonance in the response tensor. As a consequence, the dispersion curves for the two modes do not exhibit a cyclotron resonance in a highly relativistic plasma.

3.4 Waves in pulsar plasma

119

3.4 Waves in pulsar plasma TBA 3.4.1 Low-frequency waves in pulsar plasmas At frequencies well below the cyclotron resonance, the wave properties in a pulsar plasma are modified from those of a cold plasma through the RDPF z 2 W (z). As illustrated in figures 2.5 and 2.6. this function is dominated by a peak just below z = 1. As in the cold plasma case, at low and intermediate frequencies, the effect of the gyrotropic terms is small, and a useful first approximation is to neglect them. Wave dispersion in non-gyrotropic approximation In the non-gyrotropic case. the dispersion equation reduces to K 1 1 − n2 cos2 θ 0 K 1 3 + n2 sin θ cos θ = 0. 0 K 2 2 − n2 0 3 2 K 1 + n sin θ cos θ 0 K 3 3 − n2 sin2 θ

(3.4.53)

with components of the dielectric tensor are given by (2.6.19) with η = 0, hβi = 0, hγβi = 0. The dispersion equation (3.4.53) factors into two equations K 2 2 − n2 = 0,

which describes the X mode, and K 1 1 K 3 3 − K 1 3 K 3 1 − n2 K 1 1 sin2 θ + K 3 3 cos2 θ = 0,

(3.4.54)

(3.4.55)

which has a high-frequency branch identified as the O mode, and a lowfrequency branch identified as the Alfv´en mode. The X mode

On inserting the expression (2.6.19) for K i j , the dispersion relation for the X mode becomes 2 1 + 1/βA n2X = , (3.4.56) 2 2 1 − (δβ /βA ) cos2 θ

where δβ 2 characterizes the spread in velocities, with δβ 2 → 1 when the spread is highly relativistic. The X mode is strictly transverse, polarized orthogonal 2 2 2 to B, k. For βA ≫ δβ 2 < ∼ 1, (3.4.56) may be approximated by nX = 1/β0 , 2 1/2 β0 = βA /(1 + βA ) , which is the dispersion relation for magnetoacoustic 2 waves in a cold plasma. For δβ 2 > βA the mode is unstable, corresponding to a firehose instability associated with the extremely anisotropic (strictly parallel) distribution of particles [12].

120

3 Wave dispersion in relativistic magnetized plasma ω

ω

=

kc

O

X

ωc

L

ω max A

k

Fig. 3.7. The dispersion curves for the X, Alfv´en and O modes are shown schematically for a pulsar plasma for a small value of hγi and a relatively small θ. The resonance in the Alfv´en mode is denoted by ωmax and the cutoff in the longitudinal (L) portion of the O mode by ωc . The O mode crosses the light line (dashed diagonal line) for very small θ, but not otherwise. For hγi ≫ 1 the qualitative shape of the dispersion curves remains the same, but drawn out along the light line.

Equation (3.4.55) describes the Alfv´en and O modes. For parallel propagation, sin θ → 0, (3.4.55) factors into K 3 3 = 0 and n2 = K 1 1 . The mode defined by K 3 3 = 0 is longitudinal. The mode defined by n2 = K 1 1 is the Alfv´en mode, which is degenerate with the X mode for sin θ = 0; the degeneracy is broken and the modes become oppositely circularly polarized when gyrotropic effects are included. The dispersion curves for the parallel longitudinal mode intersects both the other modes for sin θ = 0, and for small sin θ 6= 0 they reconnect to form the O mode and the Alfv´en mode on the high and low frequency sides of a stop band. The X mode passes continuously through this band, reversing its handedness across it. Longitudinal mode The dispersion relation for the parallel longitudinal (L) mode is [?] ω = ωL (z),

2 ωL (z) = ωp2 W (z).

The L mode has a cutoff at ω = ωc given by

2 ωc2 = ωL (∞) = ωp2 γ −3 .

(3.4.57)

(3.4.58)

The dispersion curve crosses the light line at ω = ω1 , given by 2 ω12 = ωL (1) = ωp2 hγi (1 + δβ 2 ).

(3.4.59)

As for Langmuir waves in a nonrelativistic plasma, the parallel L mode has a maximum frequency at a phase speed of order the mean speed of the particles

3.4 Waves in pulsar plasma

121

z -2

1.5

A 1

L-O

0.5

-2

-1

0

1

2

/ p logÖωÖω

Fig. 3.8. The dispersion relations for the Alfv´en (A) mode and the O modes are shown for three values of θ: the faint curve corresponds to θ = 0, where the resonance (turnover at high z −2 ) is in the L mode and the Alfv´en mode is a horizontal line that crosses the faint curve at ω = ω1 ; for a very small but non-zero value of θ these two dispersion curves reconnect and separate, as shown by the inner pair of solid curves, with the O mode extending slightly into the region z −2 > 1; the outer pair of solid curves are for a much larger value of θ.

(∼ (δβ 2 )1/2 ), which is very close to the speed of light in a highly relativistic plasma. Landau damping results from resonance at ω = kz v, and is strong for phase speeds near and below the mean speed of the particles. As a consequence, the L mode effectively ceases to exist for phases speeds near and below this maximum. Parallel Alfv´ en mode The dispersion relation for the parallel Alfv´en mode may be written in terms of the refractive index as n2 = n2A with n2A =

2 1 + βA + δβ 2 . 2 βA

(3.4.60)

The parallel L mode and Alfv´en modes intersect at the cross-over frequency ω = ωco = ωL (1/nA ), which may be expressed in terms of the frequencies, ωc given by (3.4.58), at which the parallel L mode crosses the light line, and ω1 given by (3.4.59), at which the parallel L mode has its cutoff. One has 2 ω1 for nA − 1 ≪ δn, 2 ωco = ωp W (1/nA ) ≈ (3.4.61) 2 −2 ωc (nA − 1) for nA − 1 ≫ δn, with nA given by (3.4.60) and with δn = ωc /ω1 ∼ 1/hγi1/2 . For nA − 1 ≫ 1/hγi1/2 , the Landau damping by the bulk pair plasma is strong, and the waves

122

3 Wave dispersion in relativistic magnetized plasma

are strongly damped except at very small angles. For nA − 1 ≪ 1/hγi1/2 one has ωco ≈ ω1 . Obliquely propagating modes For slightly oblique propagation, as illustrated in figure 3.7, the two parallel modes reconnect, and may be regarded as a higher-z mode and a lower-z mode, where z = ω/kz is the phase speed. The higher phase-speed mode is longitudinal near the cutoff frequency and becomes nearly transverse at frequencies ω ∼ > ωco . This branch is labeled the O mode in figure 3.7. The lower phase-speed mode corresponds to the oblique Alfv´en mode, with 1 δβ 2 ω2 (3.4.62) z 2 = 2 1 − 2 1 − 2 tan2 θ , nA ωco βA at low frequencies ω ≪ ωco . At higher frequencies the mode is limited by the maximum frequency for the oblique Alfv´en mode, as illustrated in figure 3.7. The minimum frequency for the O mode is the cutoff frequency, as illustrated in figure 3.7. At higher frequencies the phase speed of the O mode increases, with the mode being superluminal (z > 1) except for a small range of angles θ ≈ 0. Analytic approximations to the dispersion relation of the O mode at ω > ∼ ωco are derived in the weak-anisotropy limit, cf. (3.5.95), (3.5.96) below. 3.4.2 Relativistic plasma emission The most widely favored pulsar radio emission mechanism is one based on a beam instability. This has some analogy with plasma emission in solar radio bursts, and is referred to here as ‘relativistic plasma emission’. A model for this mechanism involves a beam with a small velocity spread propagating at a relativistic speed through a denser background pair plasma. The instability can be either maser or reactive. Beam-driven maser growth Consider a beam with number density nb ≪ ne , mean Lorentz factor γb and spread ∆γb about the mean propagating through the secondary pair plasma with number density ne and mean Lorentz factor hγi ≪ γb in its rest frame. A distribution function for an idealized beam with γb ≫ 1 is 2

2

nb e−(γ−γb ) /2(∆γb ) , (3.4.63) (2π)1/2 ∆γb R where the normalization is nb = dγ g(γ). The absorption coefficient for waves in a mode M for the distribution (3.4.63) is g(γ) =

3.4 Waves in pulsar plasma

γM (k) = 2i

RM (k) Π A (kM ), ε0 ωM (k) M

123

(3.4.64)

A where ΠM (kM ) fis the anithermitian part of the response tensor for the distributions (3.4.63) projected onto the polarization vector for the mode M and µ evaluate at the dispersion relation k µ = kM for the mode M , and . This gives Z 2RM (k)|eMk |2 dγ dg(γ) γM (k) = nb δ(kM u) , (3.4.65) ε0 mωM (k) γ dγ

with eMk (k) = b · eM (k). On inserting (3.4.63) into (3.4.65), and carrying out the integral over the δ-function, it is convenient to change notation to ω, θ as the independent variables, omitting the label M for the mode and making the approximation RM (k) → 12 . One finds γ(ω, θ) =

2 ωpb |ek |2 (γ0 − γb )zγ0 −(γ0 −γb )2 /2(∆γb )2 e , ω 2(2π)1/2 (∆γb )3

(3.4.66)

2 with γ0 = (1 − z 2 )−1/2 , ωpb = q 2 nb /ε0 m. Wave growth occurs for γ0 < γb , which corresponds to z < βb . The maximum growth rate for this maser instability is near γ0 = γb − ∆γb , and is given approximately by

γmax (ω, θ) = −

2 ωpb |ek |2 γb , ω 2(2πe)1/2 (∆γb )2

(3.4.67)

where ∆γb ≪ γb is assumed. The bandwidth of the growing waves is ∆ω ≈ ω∆γb /γb3 . The ratio of the maximum growth rate to the bandwidth of the growing waves is γmax (ω, θ) nb |ek |2 γb4 ≈ (3.4.68) ne hγi(∆γb )3 2(2πe)1/2 , ∆ω 1/2

where ω ≈ ωp γb

is assumed.

Reactive beam instability When the velocity spread of the beam is neglected, the beam is cold, and the only possible instability is reactive. For a cold beam with number density nb , velocity βb and Lorentz factor γb propagating through a cold background plasma, the dispersion equation for parallel, longitudinal waves is 1−

ωp2 ωp2 nb − = 0. ω2 ne γb3 (ω − kz βb )2

(3.4.69)

The dispersion equation is a quartic equation for ω. We are interested in the case where the beam is weak, in the sense nb /γb3 ≪ ne .

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3 Wave dispersion in relativistic magnetized plasma

In the limit of arbitrarily large kz βb , the four solutions of (3.4.69) approach ω = ±ωp , kz βb ± ωp (nb /ne γb3 )1/2 . The solution near ω = −ωp is of no interest here, and it is removed by approximating the quartic equation by the cubic equation nb (ω − ωp )(ω − kz βb )2 − ωp ω 2 = 0. (3.4.70) 2ne γb3 The solutions of the cubic equation simplify in two cases: the ‘resonant’ case kz βb ≈ ωp , and the ‘nonresonant’ case ω ≪ ωp . The approximate solutions for the growth rate in these two cases are  √ 1/3  ω p 3 nb   , resonant,   ωp + i γb 2 2ne (3.4.71) ω≈ 1/2  nb ωp   , nonresonant.  kz βb + i 3/2  2ne γb Growth of the beam modes at low frequency corresponds to a nonresonant version of the beam instability, which applies only at frequencies below the resonant frequency, ω = kz βb . As the resonant frequency is approached the nonresonant instability transforms into the faster-growing resonant instability. Counter-streaming instability An oscillating model for the pulsar magnetosphere involves counter-streaming electrons and positrons, This differes from the weak-beam case in that there is no background plasma at rest. Various instabilities arise in this case, as illustrated in figure 2.4. A simple analytic model involves equal number densities and oppositely directed velocities, ±βb . In this case one has W (z) = (1/2γb3 )[1/(z − βb )2 + 1/(z + βb )2 ]. The dispersion equation for parallel longitudinal waves, 1 − (ωp2 /ω 2 )z 2 W (z) = 0, gives (ω 2 − kz2 βb2 )2 −

ωp2 2 (ω + kz2 βb2 ) = 0. γb3

2 The two solutions, ω 2 = ω± say, are ( 1/2 ) ωp2 8kz2 βb2 γb3 2 2 2 ω± = kz βb + 3 1 ± 1 + . 2γb ωp2

(3.4.72)

(3.4.73)

The higher frequency branch has a cutoff (kz = 0) at ω = ωp /γb3 , with the frequency increasing monotonically with increasing kz , given approximately by ( [ωp2 /γb3 + 3kz2 βb2 ]1/2 for 8kz2 βb2 ≪ ωp2 /γb3 , (3.4.74) ω≈ kz βb for 8kz2 βb2 ≫ ωp2 /γb3 .

3.4 Waves in pulsar plasma

125

There is no reactive instability associated with this branch. The lower frequency branch of (3.4.73) has a frequency that is purely 3/2 imaginary for kz2 βb2 < ωp2 /γb . The frequency is real for kz2 βb2 > ωp2 /γb3 , and imaginary for kz2 βb2 < ωp2 /γb3 : ω≈

(

ikz βb kz βb −

ωp /21/2 γb3

for 8kz2 βb2 ≪ ωp2 /γb3 ,

for 8kz2 βb2 ≫ ωp2 /γb3 .

(3.4.75)

Thus the lower frequency branch implies a beam-type instability for kz2 βb2 < ωp2 /γb3 . The maximum growth rate is for kz2 βb2 = 3ωp2 /8γb3 , and is Γmax

1 ωp = √ 3/2 , 2 2 γb

√ 3 ωp kz = √ 3/2 . 2 2 γb βb

(3.4.76)

It follows that the maximum growth rate is smaller the higher the Lorentz factor of the flow, so that maximum growth is when the flow is nonrelativistic or mildly relativistic, γb ≈ 1. The growth rate for a counter-streaming instability decreases with increasing Lorentz factor and is a maximum when the counter-streaming velocity is nonrelativistic. The assumption that the beams are cold is vlaid only if the counter-streaming speed is large compared with the intrinsic velocity spread in each beam. The maximum growth rate then occurs when the Lorentz factor of the counter-streaming is comparable with the intrinsic spread in Lorentz factors. Growth of Alfv´ en waves A beam instability can generate Alfv´en waves. The resonance at z = βb occurs in the O mode for βb > 1/nA , with nA given by (3.4.60), and in the Alfv´en mode for βb < 1/nA . For Alfv´en waves near the cross-over frequency ω ∼ ω1 , with ω1 given by (3.4.59), the maser instability may be treated using (8.1.29). For Alfv´en waves at low frequency the conditions for beam-driven growth are special. The Cerenkov resonance requires z = β, that is, ω = kz βb , and for low frequency Alfv´en waves this implies βb = 1/nA . In a pulsar magnetosphere, the quantity nA varies with position in the magnetosphere, and in particular with radial distance, r, along a given magnetic field line. A particular beam satisfies the condition βb = 1/nA only at one point along a given field line [?, ?, ?]. Moreover, if this condition is satisfied, Alfv´en waves over a wide frequency range resonate simultaneously with the beam. In the case of a weak beam, this suggest that beam-driven Alfv´en turbulence develops only in localized regions where the resonance conidition is satisfied. Alfv´en waves cannot escape directly from the plasma, and some nonlinear process is required to convert the energy in the Alfv´en turbulence into escaping radiation in either the O or X-modes.

126

3 Wave dispersion in relativistic magnetized plasma

3.5 Weak-anisotropy approximation TBA 3.5.1 Projection onto the transverse plane The weak-ansiotropy approximation is discussed in §2.5.7 of volume 1. The important assumption is that the waves can be approximated as transverse. In the magnetoionic theory, this is justified at high frequencies due to the longitudinal part of the polarization Lσ ∝ XY , as given by (3.2.6), approaching zero ∝ 1/ω 3 for ω ≫ ωp , Ωe . More generally, the assumption that the waves are transverse is a hypothesis on which the weak-anisotropy approximation is based, and its validity needs to be checked by comparison with exact results. The relevant choice of gauge is then the radiation gauge, where there are components only in the transverse plane, with zero time-like and longitudinal components. One may choose the two 4-vectors tµ , aµ , defined by (1.1.23). The wave equation (3.1.2) reduces to the 2-dimensional equation [k 2 δνµ + tµν (k)]Aν (k) = 0,

(3.5.77)

where µ, ν run over only the transverse components t, a. The axial ratio, T , is defined in terms of the ratio of the components t- and a-components of Aµ , with At : Aa = T : i. Hence, (3.5.77) implies (k 2 + tt t )T + itt a = 0, ta t T + i(k 2 + ta a ) = 0. Eliminating T between these two equation gives the dispersion equation k 4 + (tt t + ta a )k 2 + tt t ta a − tt a ta t = 0.

(3.5.78)

Equation (3.5.78) reproduces the dispersion equation (2.5.24) of volume 1, viz. 2 k 4 + k 2 t(1) + 12 t(1) − t(2) = 0, (3.5.79) with the traces given explicitly by t(1) = tt t + ta a ,

t(2) = (tt t )2 + (ta a )2 + 2tt a ta t .

(3.5.80)

Eliminating k 2 between the two equations leads to a quadratic equation for the axial ratio: tt t − ta a T + 1 = 0, (3.5.81) T2 − itt a where itt a = −ita t is real. The solutions of (3.5.79) with (3.5.80) are 2 k 2 = k± = − 21 (tt t + ta a ) ±

The solutions of (3.5.81) are

1 2

1/2 t . (t t − ta a )2 + 4tt a ta t

(3.5.82)

3.5 Weak-anisotropy approximation

T± =

−(tt t − ta a ) ± (tt t − ta a )2 + 4tt a ta t −2itta

1/2

,

127

(3.5.83)

with T+ T− = −1. The two components of the equations imply 2 k± = −ta a − itt a T± = −tt t +

itt a . T±

(3.5.84)

The accuracy of this approximation may be checked by applying it to a cold non-streaming electron gas, and comparing the result with the exact results of the magnetoionic theory. In the rest frame of a cold plasma, one has tt a ta t = − = iD cos θ. ω2 ω2 (3.5.85) The accuracy can be checked by considering the form of the quadratic equation for T and the expressions for kσ2 = ω 2 (1 − n2σ ). The coefficient of the term proportional to −T in (3.5.81) becomes tt t = −(S cos2 θ + P sin2 θ − 1), ω2

ta a = −(S − 1), ω2

(P − S) sin2 θ Y sin2 θ tt t − ta a = = , t it a D cos θ η cos θ

(3.5.86)

where the final expression follows from (3.2.2). The weak-anisotropy approximation gives the axial ratio of the polarization ellipse accurately for X ≪ 1. The expressions for Tσ and n2σ reproduce the results (3.2.24), (3.2.25) obtained in the high-frequency approximation to the magnetoionic waves. 3.5.2 Modes of a synchrotron emitting gas In the absence of cold particles, the dispersion in a synchrotron emitting gas would be determined by the dispersion due to the synchrotron emitting particles themselves. Assuming that the dispersion is dominated by the relativistic particles, the medium is birefringent and the dispersion relations can be written in the form 1/2 2 . (3.5.87) k 2 = k± = − 12 t(1) ± 12 2t(2) − (t(1) )2 In terms of the refractive indices, (3.5.87) corresponds to k 2 = ω 2 (1 − n2 ), so 2 that the dispersion relations become n2 = n2± , with n2± = 1 − k± /ω 2 . The two parameters that appear in (3.5.87) are t(1) = −(ttt + taa ),

t(1) = (ttt )2 + (taa )2 + 2|tat |2 ,

(3.5.88)

where arguments (k) are omitted, and where the two transverse vectors (??) are chosen. Inserting (2.5.32) in (3.5.87) gives 2 k± =

2 1/2 ωp0 . + ∆ttt − ∆taa ± (∆ttt − ∆taa )2 + 4|∆tat |2 2 c

(3.5.89)

128

3 Wave dispersion in relativistic magnetized plasma

The polarization vectors of the two modes are eµ± =

T± tµ + iaµ 1/2 , T±2 + 1

with axial ratios determined by 1/2 ∆ttt − ∆taa ± (∆ttt − ∆taa )2 + 4|∆tat |2 , T± = 2i∆tta

(3.5.90)

(3.5.91)

with the orthogonality of the two modes corresponding to T+ T− = −1. One has |∆tta | ≪ |∆ttt − ∆taa |, except for very small sin θ, so that the natural modes of a synchrotron-emitting gas are nearly linearly polarized except for nearly parallel propagation. The modes of a relativistic plasma are nearly linearly polarized in contrast with the modes of a cold plasma, which are nearly circularly polarized at high frequency. The difference between the refractive indices of the cold plasma modes causes Faraday rotation, in which the plane of linear polarization to rotate, at a rate per unit length ∝ ω −2 along the ray path. The difference between the refractive indices of a relativistic plasma modes causes a generalized form of Faraday rotation. For radiation polarized at angle to the polarization vector either the natural modes, this causes the polarization to become elliptical, with the shape of the ellipse changing periodically at a rate ∝ ω −3 along the ray path. In the particular case of radiation polarized at π/4 to either natural mode, this generalized Faraday rotation causes the polarization ellipse to pass through circular, linear at π/2 to the initial polarization, and circular with the opposite handedness before returning to the initial polarization, and repeating the cycle. 3.5.3 Weak-anisotropy approximation for a pulsar plasma The weak-anisotropy approximation, discussed in §3.5, is valid provided that the longitudinal part of the polarization is negligible. One can then solve the wave equation by first projecting it onto the (2D) transverse plane. The polarizations of the wave modes are described by a single parameter, chosen to be the axial ratio, T ± = −1/T∓, of the polarization ellipse of one of the modes, and the dispersion relation, written in terms of the refractive index, involves an isotropic part, that is the same for both modes, and a part that has opposite sign for the two modes and depends on T± . Projection of response tensor The treatment of the weak-anisotropy approximation in § 3.5 is based on projection the response tensor tµ ν = µ0 Π µν onto the transverse 4-vectors tµ , aµ , defined by (1.1.23), and denoted here by components µ = 1, 2, respectively. For the response tensor (2.6.17) this gives

3.5 Weak-anisotropy approximation

t1 1

t2 2

129

ωp2 = − 2 z 2 W (z) sin2 θ c X 1 1 1 2 2 α(z cos θ − z ) R(z ) , + + α α (1 + y 2 ) cos2 θ γ z+ − z− α=± # " X ωp2 1 1 1 2 =− 2 α(z − zα ) R(zα ) , + c 1 + y2 γ z+ − z− α=±

t1 2 = iη

X ωp2 y 1 α(z cos2 θ − zα )S(zα ), 2 2 c (1 + y ) cos θ z+ − z− α=±

(3.5.92)

with z = ω/kz , y = Ωe /kz , and z± defined by (2.6.15). The properties of the two modes are given by 1/2 2 , k 2 = k± = − 12 (t1 1 + t2 2 ) ± 12 (t1 1 − t2 2 )2 + 4(t1 2 )2 1/2 t1 1 − t2 2 ± (t1 1 − t2 2 )2 + 4(t1 2 )2 T± = , 2it1 2

(3.5.93)

2 with k± = (1 − n2± )ω 2 /c2 . Another form of the dispersion relation (3.5.93) is 2 k± = −t1 1 + it1 2 T± = −t2 2 + it1 2 /T± ,

(3.5.94)

2 allowing one to make approximations to k± in terms of approximations to T± .

Non-gyrotropic case At low frequencies the O and X modes in a pulsar plasma are nearly linearly polarized. This case may be treated by neglecting the gyrotropic term t1 2 . 2 2 The dispersion relations become kO = −t1 1 , and kX = −t2 2 , which give  2 2   θ /θ1 for θ <  ∼ θ0  2 sin θ   2 2 kO = ω 2 1 − n2A + + θ02 , sin(2θ) 2  βA for θ > θ  2 0 ∼ 2 θ1 4 sin (θ/2) 1 2 2 2 2 2 (3.5.95) kX = ω 1 − nA cos θ − 1 − 2 sin θ , βA respectively, with n2A given by (3.4.60) and with θ0 = ωc /ω1 ∼ 1/hγi1/2 . 2 The refractive indices are given by kM = (ω 2 /c2 )(1 − n2M ), with M = O, X. For θ → 0, the approximation (3.5.95) for the O mode reduces to the parallel Alfv´en mode, and this is not correct for frequencies below the crossover frequency ωco ∼ ω1 . Thus (3.5.95) is valid for ω > ∼ ω1 , cf. figure 3.8.

130

3 Wave dispersion in relativistic magnetized plasma

The O mode at ω > ∼ ω1 is subluminal for sufficiently small θ and superluminal for larger θ. Let θ = θ1 be the value of θ below which subluminal waves exist. From (3.5.95) one has θ1 ≈

(n2A − 1)1/2 ω = ω1

1 + δβ 2 2 βA

1/2

ω , ω1

(3.5.96)

where (3.4.60) is used and θ1 < θ0 is assumed. With βA ∼ Ωe /ω1 , the angle θ1 ∼ ω/Ωe is very small for ω ∼ ω1 . Inclusion of gyrotropy due to charge imbalance The generic effect of gyrotropy is to cause the wave modes to be elliptically polarized, which is described here in terms of finite axial ratios T+ = −1/T− for the polarization ellipses. The axial ratios of the two modes are given by (3.5.93). These may be written T± = 12 R ± 21 (R2 + 4)1/2 , where the parameter R is given by (2.6.1) in the cold plasma limit, and by R = (t1 1 − t2 2 )/it1 2

(3.5.97)

in the weak-anisotropy approximation. Relevant approximations to t1 1 , t2 2 are used in (3.5.95) and an analogous approximation to t1 2 is required. The gyrotropy is due to both charge and current densities, but only the charge density contributes for θ → 0. One simplifying approximation is to neglect the current density. Then (??) implies t1 2 = −iη(ωp2 ω/Ωe c2 ) cos θ, and (3.5.97) reduces to sin2 θ Ωe 2 ω 2 1 − δβ 2 sin2 θ Ωe ω12 (θ) R= z W (z) − 2 ≈ . (3.5.98) 2 η cos θ ω ω p βA η cos θ ω ωp2 In the approximate form in (3.5.98), the second term inside the square brackets 2 is neglected, due to βA ≫ 1 in a pulsar plasma and δβ 2 → 1 for a highly relativistic velocity spread, and z 2 W (z) is approximately by (??). The modes are nearly circularly polarized for R ≪ 1 and nearly linearly polarized for R ≫ 1. The change from circular to linear occurs over a small range of angles, as illustrated in figure ??. Let θc be the angle around which this change from circular to linear occurs. Setting R = 1 in (3.5.98) gives ω ω 2 1/2 η ω 1/2 p . ≈ (3.5.99) θc ≈ η Ωe ω12 Ωe hγi The angle θ ∼ θc at which the modes change from circular to linear, may be compared with the angle θ ∼ θ1 at which the o mode changes from subluminal to superluminal. Using (3.5.96) and (3.5.99) the ratio of the angles is η Ωe 1/2 θc . ≈ (3.5.100) θ1 ωhγi

3.5 Weak-anisotropy approximation

The axial ratios of the two modes are given by " 4 1/2 # θ θ2 η . ± +1 T± = |η| 2θc2 4θc4

131

(3.5.101)

The corresponding approximations for the dispersion relations for the two modes follows from (3.5.94). For small angles, θ < ∼ θ1 one finds 2 k±

ω2 = 2 c

"

1−

n2A

η θc2 θ2 + 2± 2θ1 |η| θ12

θ4 +1 4θc4

1/2 #

.

(3.5.102)

For θ ≫ θc the two solutions (3.5.102) reproduce the small-angle limit of the modes (3.5.95) in the nongyrotropic limit. For θ ≪ θc (3.5.102) implies η θc2 ω2 2 (3.5.103) ≈ 2 1 − n2A ± k± c |η| θ12 for the two circularly polarized modes, with θc /θ1 given by (3.5.100). Inclusion of gyrotropy due to relative streaming In the absence of streaming motions, a pair plasma with equal numbers of electrons and positrons is non-gyrotropic, so that the wave modes have no circularly polarized component. In the present of counter-streaming this is not the case, even when the number densities of electrons and positrons are equal. This is evident from the cold-plasma model: the response tensor (2.6.10) has gyrotropic components even for η = 0. TBC 3.5.4 Mode coupling In a homogeneous medium, radiation in different modes propagate independently of each other along ray paths that are straight lines. In an inhomogeneous medium, the ray paths are different, and mode coupling causes radiation initially in one mode (of a locally homogeneous medium) to become a mixture of the two modes. Even in the absence of coupling between the modes, the anisotropy causes the ray paths to be different: radiation at given ω, θ in the two modes propagates along ray paths that are not parallel (the group velocity is not parallel to k in general). This implies that radiation in the two modes seen by a distant observer, traced back along the ray path to the source, originates from different points in the source. Mode coupling results from gradients in the plasma parameters causing gradients in the properties of the natural modes. In the presence of weak gradients, one identifies the wave modes at each point as those of a locally

132

3 Wave dispersion in relativistic magnetized plasma

homogeneous plasma. The gradients in the plasma parameters imply gradients in the wave properties (notably in the polarizations vectors) that are into account by introducing the concept of mode coupling. The concept of a mode coupling is somewhat counter-intuitive, in that it suggests that a physical process, called ‘mode coupling’ exists. This is not the case. The relevant physical process in an anisotropic medium is the independent propagation of two orthogonal modes in a birefringence medium,. This actively changes the polarization of radiation passing through the medium. This effect causes Faraday rotation of the plane of linear polarization in a medium in which the modes are circularly polarized, and it is called generalized Faraday rotation in the general case where the modes are elliptically polarized. Gradients in the wave properties reduce the effectiveness of the medium in causing the two modes to propagate independently. The concept of mode coupling is particularly confusing when it is ‘strong’ and negates the effect of the anisotropy, so that the initial polarization is preserved as in an isotropic medium. Mode coupling is not a physical effect, but rather the partial negation of the physical effects associated independent propagation of the two modes in a homogeneous, birefringent medium. Mode coupling also occurs at sharp gradients. Waves in one mode incident on a sharp boundary gives rise to reflected and transmitted waves in both modes in general. Whether or not sharp gradients exist in association with local structures in a pulsar magnetosphere is not known. It is possible that the secondary plasma, generated by a pair cascade, is highly inhomogeneous both along and across field lines, with sharp gradients in both the pair density and the bulk Lorentz factor. In an oscillating model for the magnetosphere, there are also temporal gradients in the Lorentz factors. Mode coupling due to temporal variations in plasma parameters must occur, but there is no detailed theory for this effect. The following discussion of mode coupling is restricted to propagation in a weakly anisotropic plasma. Mode coupling in a weakly anisotropic plasma The important gradient in a weakly anisotropic plasma is in the polarization of the natural modes. The polarization vectors for the two modes may be written T± , (1 + T±2 )1/2

1 , (1 + T±2 )1/2 (3.5.104) with T± determined by (2.6.2) in a cold plasma and by (3.5.93) in the weak anisotropy approximation. Without loss of generalization one can choose T+ > 0, with T− = 1/T+ < 0. Then one has χ+ − χ− = π/2. The derivative e′± , of e± with respect to distance along the direction of the gradient, depends on the derivative χ′± = ±χ′+ , and on the derivatives θ′ , φ′ of the polar angles of κ relative to the magnetic field direction b. One has e± = sin χ± t + i cos χ± a,

sin χ± =

cos χ± =

References

∂t = −κ, ∂θ

∂t = cos θ a, ∂φ

∂a = − cos θ t − sin θ κ. ∂φ

133

(3.5.105)

The derivative of (3.5.104) gives e′± = χ′± (cos χ± t − i sin χ± a) + φ′ cos θ(sin χ± a − i cos χ± t) −(θ′ sin χ± + iφ′ cos χ± sin θ)κ,

(3.5.106)

Ignoring the component along κ, (3.5.106) gives ′ e+ −iφ′ cos θ sin 2χ+ χ′+ − iφ′ cos θ cos 2χ+ e+ . = e− −χ′+ + iφ′ cos θ cos 2χ+ −iφ′ cos θ sin 2χ+ e′− (3.5.107) It follows fom the off-diagaonal terms in (3.5.107) that coupling rate per unit length from one mode to the other due to the inhomogeneity is proportional to the rate of change of the shape of the polarization ellipse, described by χ′+ , or to the rate of twisting of the magnetic field, described by φ′ . These changes are opposed by the components in the two modes getting out of phase as the rate per unit length ∆k = ω(n+ − n− )/c. Mode coupling is strong or weak depending on which of these two rates, respectively, is the larger. At frequencies well below the cyclotron resonance, the modes are nearly linearly polarized (T± = 0, ∞) and only changes in φ are important in causing mode coupling. The shape of the polarization ellipse changes rapidly as the cyclotron resonance is crossed, where the changes in χ+ become important. Lorentz transform to the pulsar frame The foregoing results apply in the rest frame of the plasma. On Lorentz transforming to the pulsar frame in which the plasma is streaming with a bulk Lorentz factor γp , the polarization vectors remain transverse, with the axial ratio, T ′ = half R′ ± 21 (R′2 + 4)1/2 , with R′ ≈

γp2 θ′2 8Ωe γs 1 . ′ 2 ′2 2 ηhγiω (1 + γp θ ) 1 − γp2 θ′2

(3.5.108)

The condition for R′ = 1 to occur at small angles, θ′ ≪ 1, corresponds to ω′ ≫

References 1. TBA

8Ωe . |η|hγiγp3

(3.5.109)

4 Gyromagnetic processes

Gyromagnetic emission is the generic name for emission due to the spiraling motion of a particle in a magnetic field. Gyromagnetic emission by nonrelativistic particles is referred to as cyclotron emission, which is dominated by the fundamental and first few harmonics of the cyclotron frequency. Gyromagenetic emission by highly relativistic particles is referred to as synchrotron emission, which is dominated by very high harmonics which overlap and form a continuum. Gyromagenetic emission by mildly relativistic particles is referred to as gyro-synchrotron emission. These emission processes are treated in this chapter. The generalization of Thomson scattering to the scattering of waves in a magnetized plasma is also discussed. The 4-current due to a single spiraling particle is evaluated in §2.1 by expanding in Bessel functions. General formulae for gyromagnetic emission are written down in §4.1. The special case of emission in vacuo is treated in §4.2. Cyclotron emission is discussed in §4.3 with emphasis on a relativistic effect that is important for electron cyclotron maser emission. Approximations that apply in the gyro-synchrotron and synchrotron limits are discussed in §4.4. Synchrotron emission is treated in §4.5. Scattering of waves by electrons in a magnetic field is discussed in §4.6.

136

4 Gyromagnetic processes

4.1 Gyromagnetic emission Gyromagnetic emission is treated in this section by constructing the current associated with a charge executing gyromagnetic motion, and inserting this in the general emission formula. The presence of a magnetic field also modifies other processes, including the scattering of waves by particles. A formal theory for these processes is also presented in this section. Kinetic equations for these processes are written down using semi-classical arguments. 4.1.1 Probability of emission for periodic motion A particle spiraling in a magnetic field is an example of emission by a particle whose motion is periodic. It is convenient to consider the general case of a charge in periodic motion emitting radiation and to apply the result to the particular case of gyromagnetic emission. A general formula is derived in § 5.1 of volume I for emission of radiation in an arbitrary wave mode M by an arbitrary spontaneous current due to an arbitrary extraneous current Jext . The emission formula may be written in terms of the probability per unit time that a wave quantum in the mode M in the range d3 k/(2π)3 . An explicit expression for this quantity is wM (k) =

µ0 RM (k) ∗ µ |e (k)Jext (kM )|2 , T |ωM (k)| Mµ

(4.1.1)

where T is an arbitrarily long normalization time. For a particle whose orbit is x = X(τ ), implying the 4-velocity uµ (τ ) = dX µ (τ )/dτ , the current is Z µ J (k) = q dτ uµ (τ ) eikX(τ ) . (4.1.2) For periodic motion at a frequency ω0 , the orbit is of the form ˜ µ (χ), X µ (τ ) = xµ0 + u¯µ τ + X

˜ µ (χ + 2π) = X ˜ µ (χ), X

χ = ω0 τ,

(4.1.3)

with the 4-velocity given by uµ (τ ) = u¯ + u˜µ (χ),

˜ µ (χ)/dχ. u ˜µ (χ) = ω0 dX

(4.1.4)

One may expand the periodic motion in Fourier series by writing ˜

uµ (τ ) eikX (ω0 τ ) =

∞ X

U µ (s, k) eisω0 χ ,

s=−∞

U µ (s, k) = The 4-current becomes

ω0 2π

Z

0

2π/ω0

˜

dτ uµ (τ ) e−isω0 τ eikX(ω0 τ ) .

(4.1.5)

4.1 Gyromagnetic emission

J µ (k) = qeikx0

∞ X

s=−∞

U µ (s, k) 2πδ(k¯ u − sω0 ),

137

(4.1.6)

µ which is identified with Jext (k) in (4.1.2). On inserting (4.1.6) into (4.1.1), the square of the δ-function appears, and this is rewritten using [2πδ(k¯ u − sω0 )]2 = (T /¯ γ )2πδ(k¯ u − sω0 ), where ¯ ] implies k¯ u ¯µ = [¯ γ , γ¯ v u = γ¯ (ω − k · ¯c) in the δ-function. The probability of emission becomes

wM (k) =

µ0 q 2 RM (k) X ∗ |e (k)U µ (s, kM )|2 2πδ(kM u¯ − sω0 ). γ¯ |ωM (k)| s Mµ

(4.1.7)

The probability (4.1.7) describes emission by a particle with an arbitrary periodic motion, at frequency ω0 about an average 4-velocity u¯. The emission separates into contributions from harmonics of the oscillation frequency, with the harmonic number, s, taking on all integer values. However, depending on the wave properties, emission is possible at a specific harmonic only if the resonance condition, described by the δ-function in (4.1.7), can be satisfied. For example, for waves with phase speed greater than the speed of light, ω/|k| > 1, k¯ u is positive for |v| < 1, and emission at s ≤ 0 is forbidden. 4.1.2 Probability of gyromagnetic emission Gyromagnetic motion consists of such a periodic motion, specifically circular motion, in the direction perpendicular to B and an average 4-velocity parallel to B. The circular motion about the magnetic field, and the periodicity is in the gyrophase, φ, which evolves a φ = Ω0 τ = Ωt, with Ω = Ω0 /γ, where Ω0 = |q|B/m is the cyclotron frequency. The average motion corresponds to u ¯µ → uµk . The current (2.1.30) due to a spiraling charge is of the form (4.1.6), specifically, J µ (k) = qeikx0

∞ X

s=−∞

e−isηψ U µ (s, k) 2πδ (ku)k − sΩ0

(4.1.8)

corresponds to ω0 → Ω0 , u ¯µ → uµk , and to a specific form for U µ (s, k) in (4.1.6). The probability of gyromagnetic emission in the mode M at the sth gyroharmonic follows from (4.1.7): wM (s, k, p) =

2 q 2 RM (k) ∗ eMµ (k)U µ (s, kM ) 2πδ (kM u)k − sΩ0 . (4.1.9) hε0 γωM (k) ¯

4.1.3 Gyroresonance condition

The gyroresonance condition, also called the Doppler condition, implied by the δ-function in (4.1.9) is (ku)k −sΩ0 = 0 with k = kM for waves in the mode

138

4 Gyromagnetic processes

M . Classically, the resonance condition is interpreted by noting that (ku)k is the frequency of the wave in the frame in which the gyrocenter of the particle is at rest. Hence the resonance corresponds to the wave frequency being an integral multiple, s = 0, ±1, ±2, . . ., of the gyrofrequency of the particle in the inertial frame in which the motion of the gyrocenter is at rest. Resonance at s > 0 is referred to as the normal Doppler effect, and resonance at s < 0 as the anomalous Doppler effect. (The resonance at s = 0 is sometimes referred to as the Cerenkov effect, but this can lead to confusion with the resonance condition for an unmagnetized particle.) From a semiclassical viewpoint, in which the particles are treated quantum mechanically and the waves are treated classically, the resonance condition follows from conservation of energy and momentum during the emission of a wave quantum. A relativistic quantum treatment of a particle in a magnetic field leads to energy eigenvalues (ordinary units are used to discuss the quantum recoil) εn (pz ) = (m2 c4 + p2z c2 + 2n¯hΩ0 mc2 )1/2 , (4.1.10) where n = 0, 1, 2, . . . is the principal quantum number. In (4.1.10) the spin of the particle is ignored. For spin- 12 particles one has 2n = 2l + 1 + σ, where l = 0, 1, 2, . . . describes the orbital motion and σ = ±1 is the spin quantum number. The classical limit corresponds to h ¯ → 0, n → ∞, h ¯ n → p2⊥ /Ω0 m. ′ As a result of emission, pz changes to pz = pz − ¯hkz and the energy of the particle decreases by ¯hω. Suppose that the initial energy is given by (4.1.10) and that the final principal quantum number is n′ = n − s, where s is an integer. The final energy satisfies (in ordinary units) εn′ (p′z ) = εn−s (pz − ¯hkz ) = εn (pz ) − ¯hω.

(4.1.11)

On squaring (4.1.11) and using (4.1.10) one obtains (in ordinary units) εn (pz )ω − ¯ hpz kz − sΩ0 mc2 − ¯h(ω 2 − kz2 c2 )/2 = 0. An expansion in h ¯ gives (in ordinary units) (ku)k − sΩ0 − ¯h(ω 2 − kz2 c2 )/2mc2 + · · · = 0,

(4.1.12)

where the neglected terms, denoted + · · ·, are of higher order in h ¯ . The lowest order term, (ku)k − sΩ0 = 0, in (4.1.12) reproduces the classical Doppler condition. The next order term in (4.1.12) is interpreted in terms of the quantum recoil due to the emission. The resonance conditions for absorption differs from that for emission in that the quantum recoil has the opposite sign. Quantum mechanically, the normal and anomalous Doppler effects correspond to the particle jumping to lower and higher n, respectively, on emission of a wave quantum. That is, with n → n − s in (4.1.10), the contribution 2n¯ hΩ0 mc2 of the motion perpendicular to the field lines decreases for s > 0 and increases for s < 0. The total energy of the particle must decrease, and in the anomalous Doppler effect this occurs due to the decrease in p2z exceeding the increase in p2⊥ = 2n¯ hΩ0 m.

4.1 Gyromagnetic emission

139

v⊥

(c) (b) (a) vk Fig. 4.1. Examples of resonance ellipses: (a) a semicircle centered on the origin, (b) an ellipse inside v = 1, (c) an ellipse touching v = 1.

4.1.4 Resonance ellipses The gyroresonance condition is amenable to a graphical interpretation. If one plots the resonance curve in v⊥ –vz space for given ω and kz , the resonance condition for each harmonic defines a resonance ellipse. The resonance ellipse corresponds to all the values of v⊥ and vz for which resonance with a wave at given ω, kz and s is possible. That is, a given wave resonates with all particles that lie on the resonance ellipse that it defines. Similarly, a given particle resonates with all waves that define resonance ellipses that pass through the representative point of the particle in v⊥ –vz space. The resonance ellipse is centered on the vz axis at vz = vc , with semi-major axis a perpendicular to the vz axis, and with eccentricity e = (a2 − b2 )1/2 /a. This ellipse is described by v2 (vz − vc )2 + ⊥2 = 1, 2 b a 2 2 s Ω0 + kz2 − ω 2 a2 = , s2 Ω02 + kz2

ωkz , s2 Ω02 + kz2 k2 e2 = 2 2 z 2 . s Ω0 + kz

vc =

(4.1.13)

Some examples of resonance ellipses in v⊥ –vz space are illustrated schematically in figure 4.1. Note that for ω 2 < kz2 the resonance ellipse touches the circle, v = 1, and that the outer segment of the curve in nonphysical, as shown by the dashed segment in figure 4.1. The resonance ellipse in vz -v⊥ space is convenient when considering nonrelativistic and mildly relativistic particles, but not for relativistic particles, which are near the unit circle, v ≈ 1. Alternatively, one may plot the resonance condition in pz -p⊥ space. In this case the resonance condition becomes p2⊥ (pz − pc )2 + = 1, m2 a ˜2 m2˜b2 ω 2 (s2 Ω02 − ω 2 + kz2 ) a ˜2 = , (ω 2 − kz2 )2

sΩ0 mkz , ω 2 − kz2 2 2 2 2 ˜b2 = s Ω0 − ω + kz . ω 2 − kz2 pc =

(4.1.14)

140

4 Gyromagnetic processes

For ω 2 > kz2 , (4.1.14) defines an ellipse with major axis along the pz -axis. For ω 2 < kz2 , ˜b2 in (4.1.14) is negative and the curve is an hyperbola. The dashed segment of curve (c) in figure 4.1 correspond to a nonphysical arm of the hyperbola in p⊥ -pz space. For emission in vacuo, with kz = ω cos θ, (4.1.14) reduces to (|p| cos α − pc )2 |p|2 sin2 α = 1, + m2 a ˜2 m2˜b2 a ˜2 =

s2 Ω02 − ω 2 sin2 θ , ω 2 sin4 θ

pc =

sΩ0 m cos θ , ω sin2 θ

˜b2 = a ˜2 sin2 θ,

(4.1.15)

where the cylindrical coordinates p⊥ , pz are rewritten in terms of polar coordinates |p|, α. It follows that emission at given ω and θ is possible in vacuo only at harmonics, s, which satisfy ω < sΩ0 / sin θ. 4.1.5 Differential changes In semiclassical theory, changes due to emission and absorption may be treated in terms of a differential operator. In the unmagnetized case, the change due to p → p−k is treated using the differential operator k α ∂/∂pα . In the magnetized case the changes for emission are pµk → pµk − kkµ and n → n − s, with the inverse changes corresponding for true absorption. The differential operator corresponds to ∂ ∂ ∂ k α α → kkα α + s . ∂p ∂pk ∂n 2 In the classical limit, 2nΩ0 m is interpreted as p2⊥ = m2 γ 2 v⊥ . Hence, the n-derivative reduces to its classical counterpart

s

(kp)k ∂ sΩ ∂ ∂ = , = ∂n v⊥ ∂p⊥ p⊥ ∂p⊥

(4.1.16)

where in the final form the Doppler condition is used. The transfer equation for the waves due to gyromagnetic emission and absorption by a distribution of particles is of the same form in the unmagnetized case. In terms of the occupation number, the transfer equation is µν ν ν ∂µ TM (k) = SM (k) − γM (k) PM (k),

(4.1.17)

µν µ ν where TM (k) = vgM (k)kM NM (k) is the energy-momentum tensor for the ν ν waves, and PM (k) = kM NM (k) is the 4-momentum in the waves. In (6.4.38) spontaneous emission is described by the term Z d4 p ν ν k γwM (k, p) F (p). (4.1.18) SM (k) = (2π)4 M

The final term in (6.4.38) describes absorption, with the absorption coefficient given by

4.1 Gyromagnetic emission

γM (k) = −

∞ X

s=−∞

Z

141

d4 p ˆ (p), γ wM (s, k, p) DF (2π)4

ˆ = sΩ ∂ + k α ∂ . D k v⊥ ∂p⊥ ∂pα k

(4.1.19)

4.1.6 Quasilinear equations The covariant version of the quasilinear equation that describes the effect of gyromagnetic emission and absorption on a distribution of magnetized particles is derived in an analogous manner to the quasilinear equation for the unmagnetized case, cf. § 5.2 of Volume I. The quasilinear equation is #) ( " dF (p) ∂F (p) ∂F (p) 1 ∂ ν + D⊥k (p) = p⊥ −A⊥ (p) F (p) + D⊥⊥ (p) dτ p⊥ ∂p⊥ ∂p⊥ ∂pνk ∂F (p) ∂F (p) ∂ µ µν (p) + Dkk (p) , (4.1.20) + µ −Aµk (p) F (p) + Dk⊥ ∂pk ∂p⊥ ∂pν where the subscript M denoting the wave mode is omitted. The coefficients ! ! ∞ Z X A⊥ (p) sΩ/v⊥ d3 k (4.1.21) =− γwM (s, k, p) Aµk (p) kkµ (2π)3 s=−∞ describe the effect of spontaneous emission. The diffusion coefficients in (4.1.20) are     (sΩ/v⊥ )2 D⊥⊥ (p) Z ∞ X d3 k µ   Dµ (p)  γwM (s, k, p) NM (k)  (sΩ/v⊥ )kk  ,  k⊥ = 3 (2π) µν s=−∞ kkµ kkν Dkk (p) (4.1.22) µ µ with D⊥k (p) = Dk⊥ (p).

142

4 Gyromagnetic processes

4.2 Gyromagnetic emission in vacuo The classical theory of gyromagnetic emission in vacuo is amenable to an exact analysis, in the sense that the power radiated can be calculated in closed form. Such an analysis can be carried out in a variety of ways starting from the probability (4.1.9). Besides being of interest in themselves, some of the exact results are useful as a basis for various approximations to gyroemission, including the synchrotron limit. Of formal interest is the fact for gyroemission in vacuo one can compare two different descriptions of the effect of the back reaction of the emission on the particles: one description is in terms of the quasilinear equations and the other is in terms of the radiation reaction force. 4.2.1 Gyromagnetic emission of transverse waves In the case of transverse waves with two degenerate states of polarization, all relevant information on the polarization is retained by modifying the probability (4.1.9) so that it becomes a polarization tensor: wαβ (s, k, p) =

˜s∗α U ˜sβ q2 U 2πδ (ku)k − sΩ0 , 2ε0 γω

(4.2.1)

˜sµ is U µ (s, k), as given by (2.1.27) with (2.1.28), projected onto the where U transverse plane. The projection onto the transverse plane is carried out by (a) setting both the time-component and the longitudinal component of U µ (s, k) to zero, and (b) choosing two basis vectors to span the remaining two-dimensional transverse plane. Let the two transverse directions be those introduced in (1.1.22), specifically, e1 = t = (cos θ, 0, − sin θ),

e2 = a = (0, 1, 0),

(4.2.2)

with k = ω(sin θ, 0, cos θ). This corresponds to ψ = 0 in (2.1.28), which gives cos θ − vz v⊥ sin θ ′ α ˜ Js (sx), −iηv⊥ Js (sx) , x= , (4.2.3) Us = sin θ 1 − vz cos θ where the resonance condition, (ku)k − sΩ0 = 0 is used. Emission at the sth harmonic is completely polarized. The polarization vector corresponds to an elliptical polarization with axial ratio T = −η

cos θ − vz Js (sx) , v⊥ sin θ Js′ (sx)

(4.2.4)

relative to the t-direction. For nonrelativistic particles, the Bessel functions are approximated by the leading term in their power series expansion, and (4.2.4) gives T ≈ −η(cos θ − vz )/(1 − vz cos θ) ≈ −η cos θ. Emission by an electron (η = −1) has a right hand (T > 0) circular component for cos θ > 0.

4.2 Gyromagnetic emission in vacuo

143

Emission by a highly relativistic particle is dominated by a broad range of high harmonics, and although the emission at each harmonic is completely polarized, the resulting sum over harmonics leads to partial polarization. The polarization in this case needs to be determined in other ways, rather than using (4.2.4). 4.2.2 Power emitted in gyromagnetic emission in vacuo Starting from the probability (4.2.1) for emission in vacuo it is possible to evaluated several quantities explicitly by performing the sum over s and integral over k-space. Consider the power radiated, written as a polarization tensor: ∞ Z X d3 k αβ ω wαβ (s, k, p). (4.2.5) P = 3 (2π) s=1

The total power radiated is P = P 11 + P 22 , and the power radiated in either linear polarization is P 11 , P 22 . One may evaluate the integral in (4.2.5) by introducing polar coordinates: the integral over azimuthal angle, φ, is trivial, and the integral over |k| = ω/c is performed over the δ-function. One finds P

αβ

=

Z ∞ X q 2 s2 Ω 2 s=1

4πε0

0 γ2

1

d cos θ −1

˜s∗α U ˜sβ U . (1 − vz cos θ)3

(4.2.6)

The P 11 , P 22 terms can be evaluated exactly, and only these components are considered in the remainder of the calculation. It is possible to perform the calculation either by integrating over cos θ and then performing the sum over s, or by carrying out these steps in the opposite order. Both procedures are summarized below. The cos θ-integral in (4.2.6) is performed after making a transformation to the frame in which the gyrocenter is at rest. The power radiated is an invariant, and so is unchanged by this transformation. (The power is the ratio of the time-components of two 4-vectors, the energy-momentum in the emitted radiation and xµ , and this ratio is unchanged by a Lorentz transformation.) Let the particle have speed v in the laboratory frame and speed v ′ in this rest frame. Let the emitted wave be at an angle θ′ in this rest frame. One has cos θ′ =

cos θ − vz , 1 − vz cos θ

v ′ sin θ′ =

v⊥ sin θ , 1 − vz cos θ

v′ =

v⊥ . (1 − vz2 )1/2

(4.2.7)

One has d cos θ/(1 − vz cos θ)2 = d cos θ′ /(1 − vz2 ), and 1/(1 − vz cos θ) = (1 + vz cos θ′ )/(1 − vz2 ). The remainder of the integrands for P 11 , P 22 are even functions of cos θ′ , so that the term vz cos θ′ does not contribute to them. One requires the following integral identities: Z 1 Z v′ Jt2 (sv ′ sin θ′ ) J2t (2sy)/v ′ ′ d cos θ =2 dy . (4.2.8) Jt2 (sv ′ sin θ′ )/ sin2 θ′ J2t (2sy)/y −1 0

144

4 Gyromagnetic processes

Using the recursion relations (2.1.25), (2.1.26) for the Bessel functions, one obtains P 11,22 =

X 2q 2 s2 Ω 2 0

s=1

2

2

v ′ (1 − v ′ ) Gs11,22 ,

Z v′ 1 J2s (2sy) − ′3 dy dy J2s (2sy), y v 0 0 Z ′ Z v′ 1 1 ′ dy 1 v ′ = ′ J2s (2sv ) − ′ 2 J2s (2sy) + ′ dy J2s (2sy). sv y v 0 v 0 G11 s

G22 s

4πε0

1 = ′2 v

Z

v′

(4.2.9)

The next step is to perform the sum over s. The relevant sums are Kapteyn series, and the following series were first evaluated explicitly for the present purpose by [1]: X s=1

s2 J2s (2sy) =

y 2 (1 + y 2 ) , 2(1 − y 2 )4

X s=1

′ sJ2s (2sy) =

y , 2(1 − y 2 )2

(4.2.10)

which apply for 0 ≤ y < 1. The remaining integral is elementary, and the final result for the power radiated is P = P 11 + P 22 =

2 q 2 Ω02 p2⊥ q 2 Ω02 γ 2 v⊥ = , 2 6πε0 m 6πε0

2 2 + v⊥ − 2vz2 2m2 + 3p2⊥ P 11 − P 22 = − = − . P 11 + P 22 4(1 − vz2 ) 4(m2 + p2⊥ )

(4.2.11) (4.2.12)

The total power P is well known, and can be derived more simply, as discussed below. Note that the total power radiated, P , depends only on the perpendicular component, p⊥ , of momentum, and is independent of pz . Formula (4.2.12) shows that the power in gyromagnetic emission is preferentially polarized along e2 rather than along e1 . In the nonrelativistic limit (4.2.12) implies a degree of polarization ≈ −1/2, but this is not particularly meaningful; it refers to an average over all angles and according to (4.2.4) the polarization depends relatively strongly on angle in the nonrelativistic limit. In the highly relativistic limit, the degree of polarization implied by (4.2.12) is ≈ −3/4. This is a characteristic value for the polarization of synchrotron emission; the actual polarization of synchrotron emission is a function of frequency. 4.2.3 Angular distribution of gyromagnetic emission An alternative procedure involves performing the sum over s in (4.2.6) before carrying out the integral over θ. The required summation formulae were also evaluated by [1] for this purpose:

4.2 Gyromagnetic emission in vacuo

X

s2 Js2 (sx) =

s=1

x2 (4 + x2 ) , 16(1 − x2 )7/2

X

s2 Js′2 (sx) =

s=1

145

4 + 3x2 , (4.2.13) 16(1 − x2 )5/2

for 0 ≤ x < 1. One finds 2 Z 1 2 q 2 Ω02 v⊥ (cos θ − vz )2 [4(1 − vz cos θ)2 + v⊥ sin2 θ] P = d cos θ 2 2 2 sin θ]7/2 64πε0 γ −1 [(1 − vz cos θ)2 − v⊥ 2 2 4(1 − vz cos θ) + 3v⊥ sin2 θ + . (4.2.14) 2 sin2 θ]5/2 [(1 − vz cos θ)2 − v⊥ The integral over cos θ is performed using standard integral identities. The result (4.2.11), (4.2.12) is reproduced. 4.2.4 The quasilinear coefficients Using the techniques outlined above, it is possible to evaluate the quasilinear coefficients A⊥ (p), Aµk (p), as given by (4.1.21), for gyromagnetic emission in vacuo. The coefficients (4.6.5) are     A⊥ (p) sΩ /γv Z 0 ⊥ ∞ X d3 k   αβ  A0 (p)  (4.2.15) ω =−  γw (s, k, p).  k 3  (2π) s=1 3 ω cos θ Ak (p)

The evaluation of these coefficients closely parallels the evaluation of P . One finds A⊥ (p) = −

1 − vz2 γP, 2 v⊥

A0k (p) = −γP,

A3k (p) = −γvz P,

(4.2.16)

with P given by (4.2.11). The mean rate of change of the 4-momentum of the particle per unit time follows from (4.2.16), which implies dpz dp⊥ dε 1 − vz2 =− = −P, = −γvz P. (4.2.17) P, dt v⊥ dt dt For a nonrelativistic particle, v⊥ , |vz | ≪ 1, (4.2.17) implies that the rate of change of pz is small in comparison with the rate of change of p⊥ , with v⊥ hdp⊥ /dti ≈ hdε/dti. However, for a highly relativistic particle, v → 1, (4.2.17) implies hdp⊥ /dti ≈ − sin α P , hdpz /dti ≈ − cos α P , where α is the pitch angle. In this case the rate of change in the pitch angle is small, such that the particle loses energy essentially at constant α. This is seen directly by comparing the rates of change of p and α: P P dp dα =− , =− cot α, (4.2.18) dt v dt mγ 3 v 2

146

4 Gyromagnetic processes

with P ∝ sin2 α given by (4.2.14). Comparing the rates of change of p and α, (4.2.18) implies that they are in the ratio p−1 hdp/dti : hdα/dti = tan α : 1/γ 2 v, so that the change in α is small for γ 2 ≫ 1. A distribution of particles that is initially isotropic becomes anisotropic as a result of gyromagnetic losses. For highly relativistic particles, each particle moves along a nearly radial line in p-space, that is, with |p| decreasing at nearly constant α. The anisotropy arises because P ∝ sin2 α implies that particles with larger α move to smaller |p| faster than those with smaller α. Particles that lie initially on a circle in p⊥ -pz space, with p2⊥ +p2z = p20 initially say, define an ellipse with semi-major axis along the pz -axis fixed at p0 , and with the axial ratio of the ellipse decreasing with time. For nonrelativistic particles the trajectory is along a line that is radial in the cylindrical sense: decreasing p⊥ at constant pz . The diffusive coefficients in the quasilinear equation (4.1.20) cannot be evaluated in the same way even for emission in vacuo. The reason is that these terms depend on the distribution of waves, e.g., described by the occupation number N (ω, θ). After performing the integral over ω using the δ-function, as in the derivation of (4.2.6), one has ω → sΩ0 /γ(1 − vz cos θ), so that the occupation number becomes an implicit function of both s and θ, precluding the sum over s being performed exactly and precluding the integral over cos θ being performed exactly even for an isotropic distribution of radiation. 4.2.5 Emission by accelerated particles The power radiated in vacuo by an accelerated charge can be treated by combining the Larmor formula with a Lorentz transformation. This approach leads to a general formula for the power radiated in vacuo, but it does not give any information on how the radiation is distributed in frequency and angle, nor on its polarization. The Larmor formula gives the power radiated in vacuo by an accelerated charge in its instantaneous rest frame: P (t) =

q 2 |a(t)|2 , 6πε0

(4.2.19)

where a(t) is the instantaneous acceleration. The acceleration in (4.2.19) is related to the force acting on the particle by a(t) = F (t)/m. In the instan˙ where the dot implies taneous rest frame one has v = 0, γ = 1, and a = v, differentiation with respect to t. In terms of 4-vectors, in the rest frame one ˙ with dτ = dt in this frame. Using has uµ = [1, 0], aµ = duµ /dτ = [0, v], aµ aµ = −v˙ 2 , (4.2.19) becomes P =−

q 2 aµ aµ , 6πε0

(4.2.20)

in the rest frame. The left hand side of (4.2.20) is the ratio of the time components of two 4-vectors, and hence is an invariant, and the right hand side is

4.2 Gyromagnetic emission in vacuo

147

already in invariant form. Hence the special theory of relativity implies that (4.2.20) applies in all inertial frames. It is therefore the desired generalization of the Larmor formula. In the laboratory frame, in which the particle has instantaneous velocity v, (4.2.20 is rewritten in 3-vector notation by noting the relations ˙ 2 . (4.2.21) ˙ v · v˙ v + (1 − v 2 )v], ˙ aµ = γ 4 [v · v, aµ aµ = −γ 6 v˙ 2 − |v × v|

The generalization of the Larmor formula is P =

˙ 2 q 2 v˙ 2 − |v × v| . 6πε0 (1 − v 2 )3

(4.2.22)

The acceleration is a somewhat artificial quantity for a relativistic particle and it is more relevant to write (4.2.22) in terms of the 4-force. dpµ /dτ , acting on the particle. One has m2 |a|2 = −(dpµ /dτ )(dpµ /dτ ), and hence the Larmor formula has the covariant generalization P =−

dpµ dpµ q2 , 6πε0 m2 dτ dτ

(4.2.23)

which applies in an arbitrary frame. The power in gyromagnetic radiation follows by substituting the equation of motion, dpµ /dτ = qF0µν uν , into (4.2.23). Writing F0µν = Bf µν , the right hand side of (4.2.23) contains a factor αβ f µα uα fµ β uβ = g⊥ uα uβ = −u2⊥ = −p2⊥ /m2 .

Then (4.2.23) reproduces the result (4.2.11) for the power radiated, P . 4.2.6 The radiation reaction 4-force The radiation reaction force in conventional classical electrodynamics is F react (t) =

¨ (t) q2 v , 6πε0

(4.2.24)

where a dot denotes differentiation with respect to t. The covariant generalization to the radiation reaction 4-force was written down by Dirac: 2 µ q2 dpν dpν pµ d p µ . (4.2.25) Freact = + 6πε0 m dτ 2 dτ dτ m2 On inserting the equation of motion dpµ /dτ = qF0µν uν , and proceeding as in the derivation of (4.2.23), one finds µ Freact =−

m2 pµ⊥ + p2⊥ pµ q 4 B 2 m2 pµ⊥ + p2⊥ pµ = −P , 6πε0 m3 m2 mp2⊥

(4.2.26)

148

4 Gyromagnetic processes

which is the radiation reaction 4-force for gyromagnetic emission in vacuo. Comparison of (4.2.26) and (4.2.16) shows that the parallel components of µ Freact are equal to Aµk ; the perpendicular components are equivalent provided that one averages (4.2.26) over gyrophase, and notes the identity 2 (m2 + p2⊥ )/p2⊥ = (1 − vz2 )/v⊥ . This confirms that in the case of gyromagnetic emission the terms in the quasilinear equation (4.1.20) that describe the effects of spontaneous emission are equivalent to radiation reaction 4-force in conventional classical electrodynamics.

4.3 Relativistic effects in cyclotron emission

149

4.3 Relativistic effects in cyclotron emission Cyclotron emission is gyromagnetic emission from nonrelativistic particles. In this section, cyclotron emission and absorption in a nonrelativistic plasma is discussed emphasizing intrinsically relativistic effects. It seems counterintuitive that intrinsically relativistic effects can be important for nonrelativistic particles, but this is the case. Why this should be so can be understood by noting that the formal nonrelativistic limit corresponds c → ∞, and this is inconsistent with Maxwell’s equations, which require c to be finite. The most important context where this inconsistency arises is in the resonance condition for nearly perpendicular propagation where the “semi-relativistic” approximation is needed even for nonrelativistic particles. 4.3.1 Emissivity in the magnetoionic modes The only radiation that can escape from a nonrelativistic magnetized plasma is in the magnetoionic modes, specifically, either in the x-mode or the o-mode. The relevant waves properties are summarized in §3.2.1. The polarization vector, (3.1.18), and the ratio of electric energy are eµσ =

Lσ κµ + Tσ tµ + iaµ , (L2σ + Tσ2 + 1)1/2

Rσ =

1 + L2σ + Tσ2 , 2(1 + Tσ2 )nσ (∂[ωnσ ]/∂ω)

(4.3.1)

with σ = ±1 labeling the magnetoionic waves, the high frequency branches of which are labeled o, x. It is conventional to introduce the emissivity, ησ (s, ω, θ), to describe the emission by a single particle at the sth harmonic in the mode σ. The emissivity is the power radiated per unit frequency and per unit solid angle and hence it is related to the probability for gyromagnetic emission by Z ∞ Z 1 Z d3 k ω wσ (s, k, p), (4.3.2) 2π dω d cos θ ησ (s, ω, θ) = (2π)3 0 −1 with wσ (s, k, p) given by (4.1.9). A factor (∂[ωnσ ]/∂ω) from the change of variable of integration from |k| = nσ ω to ω in (4.3.2) cancels with this factor in (4.3.1). The emissivity at the sth harmonic reduces to q 2 nσ ω 2 v 2 sin2 α Lσ sin θ + Tσ (cos θ − nσ v cos α) ησ (s, ω, θ) = Js 8π 2 ε0 c(1 + Tσ2 ) nσ v sin α sin θ 2 −ηJs′ δ[ω(1 − nσ v cos θ cos α) − sΩ0 ], (4.3.3) where the arguments (ω, θ) of nσ , Tσ and Lσ are omitted, and similarly the arguments (ω/Ω0 )nσ v sin α sin θ of Js and Js′ are omitted. The sign η of the charge is equal to −1 for electrons, Ω0 → Ωe . A form of the transfer equation in terms of these variables is

150

4 Gyromagnetic processes

vgσ

dIσ (ω, θ) = Jσ (ω, θ) − γσ (ω, θ)Iσ (ω, θ), dℓ

(4.3.4)

where Iσ is the specific intensity (power per unit solid angle per unit frequency per unit area) in the mode M , and where ℓ denotes distance along the ray path. The coefficient Jσ (ω, θ) is the volume emissivity: Z ∞ Z 1 X 2 Jσ (ω, θ) = 2π d|p| |p| ησ (s, ω, θ) f (|p|, cos α). (4.3.5) d cos α 0

−1

s

The absorption coefficient (4.1.19) has the following form in terms of the emissivity: Z ∞ Z 1 X (2πc)3 ησ (s, ω, θ) γσ (ω, θ) = −2π d|p| |p|2 d cos α vn2σ ω 2 ∂(ωnσ )/∂ω 0 −1 s ∂ cos α − nσ v cos θ ∂ + × f (|p|, cos α). (4.3.6) ∂|p| |p| sin α ∂α In applying these formulae here, first the strictly nonrelativistic limit is considered, and then “semirelativistic” effects are taken into account. 4.3.2 Gyromagnetic emission by thermal particles In the strictly nonrelativistic limit, cyclotron emission by nonthermal electrons occurs at lines centered on the cyclotron frequency and its harmonics, with the intensity of the lines decreasingly rapidly with increasing harmonic number s. These lines have a finite width as a result of Doppler broadening due to the random thermal motions of the electrons along B. A thermal distribution function of particles in the strictly nonrelativistic limit (γ = 1, p = mv) is the Maxwellian distribution f (p) =

2 nρ3/2 e−ρv /2 , 3/2 3 (2π) m

(4.3.7)

with ρ = m/T , where T is the temperature in energy units. For a thermal distribution the volume emissivity and the absorption coefficient are proportional to each other (Kirchhoff’s law), with the specific relation being γσ (ω, θ) =

(2π)3 ρ mω 2 n2σ ∂(ωnσ )∂ω

Jσ (ω, θ).

(4.3.8)

In the following the volume emissivity at the sth harmonic, Jσ (s, ω, θ), is calculated, and the absorption coefficient follows from (4.3.8). The integrals in (4.3.5) over velocity space for the distribution (4.3.7) separate in cylindrical coordinates. This follows by writing (4.3.7) in the form 2 exp(−ρv 2 /2) = exp(−ρv⊥ /2) exp(−ρvz2 /2) and performing the v⊥ -integral over the ordinary Bessel functions using

4.3 Relativistic effects in cyclotron emission

151

Js2 (z) Is (λ) , dv⊥ v⊥ e zJs (z)Js′ (z)  = e−λ  λ[Is′ (λ) − Is (λ)] 2 ′2 2 2 ′ 0 z Js (z) s Is (λ)−2λ [Is (λ)−Is (λ)] (4.3.9) with z = k⊥ v⊥ /Ω0 and where the argument of the modified Bessel functions, Is , is λσ = (ωnσ sin θV /Ω0 )2 , with V = 1/ρ1/2 the thermal speed. The resulting expression for the emission coefficient is Z

∞



2 − 12 ρv⊥ 







1/2 2 ωp m nσ Aσ (s, ω, θ) −ρ(ω−sΩ0 )2 /2ω2 n2 cos2 θ, 2 σ e π ωc ρ1/2 | cos θ| (4.3.10) 2 −λσ ω e Is (Lσ cos θ − Tσ sin θ) tan θ + sTσ sec θ Aσ (s, ω, θ) = 2 1 + Tσ Ω0 λσ ω (Lσ cos θ − Tσ sin θ) tan θ + sTσ sec θ (Is′ − Is ) +2 Ω0 2 s −η + 2λσ Is − 2λM Is′ . (4.3.11) λσ Jσ (s, ω, θ) =

The general form (4.3.11) is unnecessarily cumbersome for most purposes and approximations need to be made. One simplifying assumption is the small gyroradius limit, λσ ≪ 1. This corresponds to the argument of the modified Bessel functions being small, when only the leading terms in the power series expansion of the modified Bessel functions are retained. In the small-gyroradius limit, (4.3.11) reduces to 2 ( 21 λσ )s−1 ω (Lσ cos θ − Tσ sin θ) tan θ + sTσ sec θ − sη . Aσ (s, ω, θ) ≈ 2s!(1 + Tσ2 ) Ωe (4.3.12) Further simplification occurs when (i) the longitudinal component can be neglected (Lσ → 0), and (ii) by setting ω = sΩ0 , in which case the quantity in square brackets reduces to s(Tσ cos θ − η), with η = −1 for electrons. The exponential function in (4.3.10) determines the line profile. The line emission at the sth harmonic is centered on ω = sΩ0 and it is Doppler broadened with a characteristic width (∆ω)s =

nσ | cos θ| sΩ0 . ρ1/2

(4.3.13)

The Doppler broadening is due to the component of the random thermal motions along the magnetic field lines. In the strictly nonrelativistic limit the perpendicular motions do not contribute to the line width, whereas relativistic effects imply a nonzero broadening due to the so-called transverse Doppler effect.

152

4 Gyromagnetic processes

4.3.3 Semirelativistic approximation The strictly nonrelativistic limit breaks down near perpendicular propagation even for v ≪ 1. This may be seen by noting that in the nonrelativistic approximation, γ = 1, the resonance condition (4.1.11) reduces to a vertical line in velocity space, and for perpendicular propagation the resonance condition reduces to a semicircle centered on the origin, cf. figure 4.1. For nearly perpendicular propagation the resonant ellipse is well approximated by a semicircle with its center on the vz -axis away from the origin. Relativistic effects must be taken into account to treat the emission and absorption correctly for θ ≈ π/2, even for seemingly nonrelativistic particles (corresponding to ρ ≫ 1). The simplest useful approximation to the resonance condition that allows one to treat perpendicular propagation is the “semirelativistic” approximation γ = 1 + 21 v 2 . This leads to the Doppler condition being approximated by 2 ω(1 − nσ vz cos θ) − sΩe (1 − 21 v⊥ − 21 vz2 ) = 0.

(4.3.14)

which corresponds to a resonance ellipse that is a circle with its center at vz = vc . v⊥ = 0, and radius a, given by a2 = n2σ cos2 θ + 2(sΩe − ω)/sΩe .

vc = nσ cos θ,

(4.3.15)

4.3.4 Integration around the resonant semicircle The integral over velocity space in the evaluation of the emission or absorption coefficient corresponds to an integral around the resonant semicircle, which has center at vz = vc and radius a, given by (4.3.15). The integral is performed by changing variables from v⊥ , vz to a and φ, with vz = vc − a cos φ,

v⊥ = a sin φ,

−1 ≤ cos φ ≤ 1.

(4.3.16)

The integrals that appear in evaluating the emission and absorption coefficients are of the form Z ∞ Z ∞ dvz g(v⊥ , vz ) δ[ω − sΩe (1 − 12 v 2 ) − ωlnσ vz cos θ] dv⊥ v⊥ −∞

0

=

a 2ω

Z

1

−1

d cos φ g(a sin φ, nσ cos θ − a cos φ),

(4.3.17)

with g(v⊥ , vz ) is an arbitrary function. Specifically, the volume emissivity (4.3.5) with (4.3.3) and (4.3.7) becomes Z 1 ωp2 mρ3/2 a3 nσ ω −ρ(a2 +vc2 )/2 e d cos φ sin2 φ eρavc cos φ 8π(2π)3/2 c(1 + Tσ2 ) −1 2 Lσ sin θ + Tσ [cos θ − nσ (vc − a cos φ)] ′ × Js − ηJs , (4.3.18) nσ a sin θ sin φ

Jσ (s, ω, θ) =

4.3 Relativistic effects in cyclotron emission

153

with vc = nσ cos θ, and where the arguments of the Bessel functions is (ω/Ω0 )nσ a sin θ sin φ. The remaining integral in (4.3.18) simplifies for a ≪ 1 and s ≪ 1/a. The Bessel function is replaced by the leading term in its power series expansion, Js (x) = (x)s /2s s!, and (4.3.18) gives Jσ (s, ω, θ) =

ωp2 mρ3/2 a3 nσ ω −ρ(a2 +v2 )/2 s2s 2s−2 c (nσ a sin θ) e (2s s!)2 4π(2π)3/2 (1 + Tσ2 ) 2 ∂ × Lσ sin θ + Tσ cos θ − nσ vc − a − ηs ∂A 2s sinh A ∂2 × 1− , (4.3.19) ∂A2 A

with A = ρavc . In contrast with (4.3.10), which implies zero emission and absorption at θ = π/2 except exactly at ω = sΩe , (4.3.19) implies emission and absorption over a nonzero frequency range for all θ ≈ π/2. The integral in (4.3.18) may be performed in closed form for perpendicular propagation, θ = π/2. The specific integrals involved are Is(1) (v)

=

Z

1

1 2

Z

1

1 2

−1

d cos αJs2 (v sin α)

1 = v

Z

v

dy J2s (2sy),

0

1 χs (v) − 2 3 = 4sv 2γ v

Z

v

1 3χs (v) − 2 3 = d cos α sin sin α) = 4sv 2γ v −1 Z v J2s (2sv) 1 ′ χs (v) = J2s (2sv) − + dy J2s (2sy). 2sv 2sv 2 0

Z

v

Is(2) (v) Is(3) (v)

=

1 2

2

d cos α cos

−1

Z

1

2

αJs2 (v sin α)

αJs′2 (v

dy J2s (2sy),

0

dy J2s (2sy),

0

(4.3.20)

For perpendicular propagation (4.3.18) simplifies not only through cos θ → 0, sin θ → 1, but also because the wave properties simplify. With M = x, o for the two modes, one has Tx → 0, To → −∞. For simplicity the refractive index is also approximated by unity. Then (4.3.18) reduces to Jx,o (s, ω, θ) =

ωp2 mρ3/2 ωa3 −ρa2 /2 (3) Is (a), Is(2) (a) . e 4π(2π)3/2

(4.3.21)

For small a and s ≪ 1/a the Bessel functions in (4.3.21) with (4.3.20) are approximated by the leading term in the power series expansion, giving Z x J2s (2sx) xJ2s (2sx) (sx)2s ′ J2s (2sx) = , J2s (2sx) = , dy J2s (2sy) = . (2s)! x 2s 0 (4.3.22) With these approximations, one has

154

4 Gyromagnetic processes

Is(1) (a) = Is(2) (a) =

J2s (2sa) , 2s

Is(3) (a) =

J2s (2sa) . 2sa2

(4.3.23)

It follows that perpendicular emission is stronger for x-mode than for the o-mode by a factor a2 = 2(sΩe − ω)/ω. 4.3.5 Relativistic frequency downshift and line broadening Two intrinsically relativistic effects on cyclotron emission are (i) a frequency downshift and (ii) a line broadening due to the transverse Doppler effect. Both of these are treated here using (4.3.21) with (4.3.22) assuming perpendicular propagation. The line width may be characterized by the spread in frequency around the mean frequency. This involves moments of the frequency distribution, with the mth moment defined by Z ∞ Z ∞ hω m (s, θ)iσ = dω ω m−1 Jσ (s, ω, θ) dω ω −1 Jσ (s, ω, θ). (4.3.24) 0

0

The mean frequency is given by the moment m = 1, and it determines the line center. The second moment, m = 2, determines the variance in the frequency: the bandwidth may be characterized by the square root of the variance, ∆ω = (hω 2 i − hωi2 )1/2 . The moments (4.3.24) may be evaluated approximately for emission at θ = π/2 using the method of steepest descent. The method of steepest descent applies to an integral of the form Z I = dy G(y) exp[−F (y)], (4.3.25) where G(y) is slowly varying and exp[−F (y)] is sharply peaked about some value y = y0 . The range of integration is assumed to cover the important contribution from this peak. The value of y0 is determined by the solution of F ′ (y0 ) = 0, where the prime denotes differentiation. The integral is approximated by 1/2 2π G(y0 ) exp[−F (y0 )]. (4.3.26) I= F ′′ (y0 ) The specific integrals of interest here are written as integrals over a2 = 2(sΩe − ω)/sΩe , with (4.3.21) and (4.3.22) implying F (a2 ) = ρa2 /2 − s ln(a2 ) for the x mode and F (a2 ) = ρa2 /2 − (s + 1) ln(a2 ) for the o mode. One finds (s + 12 ± 21 ) s + 1 ± 12 hω(s, π/2)i = sΩe 1 − , [∆ω(s, π/2)]2 = (sΩe )2 , ρ ρ2 (4.3.27) for these two modes, respectively. Thus there is a frequency downshift of the center of the cyclotron line by ∼ s/ρ, and a relativistic broadening of ∆ω/sΩe ∼ s1/2 /ρ. The total line broadening for θ 6= π/2 is

4.3 Relativistic effects in cyclotron emission

155

v sin α

v sin α (a)

(b)

α = αc

v cos α

v cos α

Fig. 4.2. (a) A nonthermal loss cone distribution, with loss cone angle α0 , is assumed to be confined to the lightly shaded region, with the lower-energy thermal distribution filling the darkly shaded region. The dotted semicircle indicates a resonance ellipse that samples only regions where ∂f /∂p⊥ is positive. (b) A shell distribution with a resonance ellipse inside the shell in the region where f is an increasing function of v.

2

2

[∆ω(s, θ)] = (sΩe )

cos2 θ s + 32 ± + ρ ρ2

1 2

.

(4.3.28)

In a nonrelativistic plasma, ρ ≫ 1, the relativistic broadening is significant 1/2 only for angles |θ − π/2| < ∼ s /ρ. 4.3.6 Electron cyclotron maser emission Electron cyclotron absorption can be negative under relatively mild conditions in a nonrelativistic plasma. This results in electron cyclotron maser emission (ECME) near the fundamental or low harmonics of the cyclotron frequency. An important type of ECME is due to the intrinsically relativistic effect (Twiss 1958; Wu and Lee 1979; Melrose 1986; [refs]) and this is treated using the semirelativistic approximation (4.3.14). Consider the absorption coefficient (4.1.19) in the semirelativistic approximation. The integral over momentum space in (4.1.19) reduces to an integral around the resonance ellipse as in (4.3.17). The sign of the absorption is deˆ around the resonance termined by a weighted average of the value of −Df ˆ given by (4.1.19). Specifically, one has ellipse, with D ˆ (p⊥ , pz ) = − sΩe ∂ + kz ∂ f (p⊥ , pz ). (4.3.29) −Df v⊥ ∂p⊥ ∂pz For |kz | ≪ ω the contribution from the pz -derivative is small compared with the contribution from the p⊥ -derivative in (4.3.29). It follows that if the resonance ellipse for a wave with given kz , ω, s is such that ∂f /∂p⊥ is positive everywhere around the ellipse, then the absorption coefficient is necessarily negative. Two examples of such an ellipse are illustrated schematically in figure 4.2, (a) for a loss-cone distribution, that is, a distribution in which the number of particles decreases inside a loss cone α < αc , and (b) for a shell distribution, where the particles are confined to a shell in velocity space. The

156

4 Gyromagnetic processes

resonance ellipse in figure 4.2(a) is chosen not to intersect the region where the thermal electrons are located; waves corresponding to resonance ellipses that intersect this region suffer strong thermal gyromagnetic absorption. The resonance ellipse in figure 4.2(b) is for perpendicular propagation; in a magnetoionic medium such waves are below the cutoff frequency for the x mode and so cannot escape. ECME due to a shell distribution leads to escaping radiation effectively only in the absence of thermal plasma, and this proviso seems to be satisfied in two important applications of ECME: to the Earth’s auroral kilometric radiation (AKR) and to Jupiter’s decametric radio emission (DAM) [refs]. This form of ECME depends intrinsically on a relativistic effect, as is evident from the fact that the resonance ellipse is approximated by a circle rather than a vertical straight line. The following quantum mechanical argument also emphasizes the essentially relativistic nature of this form of ECME. Maser emission is understood in terms of a quantum state of higher energy being overpopulated relative to a state of lower energy. The perpendicular energy is quantized with the Landau quantum number n introduced in (4.1.10). Let the occupation number be N (n). Stimulated emission n → n − s is proportional to N (n) and true absorption n − s → n is proportional to N (n − s), so that the net absorption is proportion to N (n) − N (n − s). (In the classical limit one has N (n) − N (n − s) = s∂N (n)/∂n with ∂/∂n → (Ω0 /v⊥ )∂/∂p⊥ .) This suggests that for N (n) > N (n − s) the absorption is negative. This conclusion is correct in a relativistically correct treatment, but not in a strictly nonrelativistic theory. This follows from the nonrelativistic counterpart of (4.1.10), εn = m + p2z /2m + nΩ0 , which implies that the harmonics are equally spaced in the strictly nonrelativistic case. The emission at ω = sΩ0 comes from all transitions n → n − s, that is, from all values of n. Hence, one needs to sum over n, to include the effect of transitions n ↔ n − s for all n. This sum gives ∞ X

n=s

N (n) − N (n − s) = N (∞) − N (0).

(4.3.30)

One necessarily has N (∞) = 0 and N (0) ≥ 0, so that the absorption is nonnegative and there can be no maser action. However, when the correct relativistic formula (4.1.10) for the energy εn is used, it implies that each transition n ↔ n − s occurs at a slightly different frequency, which is referred to an anharmonicity. Hence maser action between two specific quantum states can be considered in isolation from transitions for all other n. Then N (n) > N (n − s) suffices for the absorption to be negative for the relevant frequency of transition. Thus, quantum mechanically, this form of ECME is attributed to the anharmonicity in cyclotron transitions implied by the relativistically correct form of the energy (4.1.10).

4.4 Gyrosynchrotron emission

157

4.4 Gyrosynchrotron emission Gyromagnetic emission in the cyclotron (nonrelativistic) limit consists of a sequence of narrow well-separated lines at ω ≈ sΩ0 that decrease in intensity rapidly with increasing s. Gyromagnetic emission in the synchrotron (highly relativistic) limit becomes a continuum consisting of an extremely large number of overlapping, highly broadened harmonics. The transition between these two cases is provided by the mildly relativistic regime, where gyrosynchrotron emission consists of a modest number (a few to ∼ 100) of overlapping harmonics. The synchrotron limit is amenable to a well-known analytic treatment, cf. §4.5. The analytic treatment in the mildly relativistic case is less familiar and more cumbersome. In this section, synchrotron and gyrosynchrotron emission and absorption are treated in a unified way, with the main difference being the approximations made to Bessel functions. 4.4.1 Synchrotron approximation There are three simple approximations that apply in the synchrotron limit. First, one has γ ≫ 1, so that an expansion in powers of 1/γ converges rapidly. Second, simple arguments imply that emission from any highly relativistic particle is strongly concentrated in a forward cone with half-angle ∼ 1/γ. Hence, emission by a particle with pitch angle α (p⊥ = p sin α, pz = p cos α) is strongly concentrated around θ = α. This may be inferred from (4.2.14), where the integral is dominated by contributions from the minimum value of the denominators, which occurs very near α = θ for γ ≫ 1. It follows that α − θ is of order 1/γ and that the expansion in powers of 1/γ also involves an expansion in α − θ. Third, synchrotron emission is dominated by high harmonics, s ∼ (γ sin θ)3 . An excellent approximation is to treat s as a continuous variable and to replace the sum over s by an integral. Some of these characteristics of synchrotron emission may also be inferred from the resonance condition in the form (4.1.15). The synchrotron limit corresponds to |u|/c ≈ γ, sΩ0 ≫ ω sin θ. The resonance ellipses for a range of s are centered on γ cos α ≈ sΩ0 cos θ/ sin2 θ, with major axes along the parallel axis, eccentricity e = | cos θ| and semi-minor axes b = sΩ0 /ω sin θ. The maximum emission at a given s is for particles with the maximum allowed value of sin α, and this corresponds to tan α = b/uc, γ = (b2 + u2c )1/2 (tan α = bc/uc , γ = (b2 + u2c /c2 )1/2 in ordinary units). This gives maximum emission at a given s for particles with tan α = tan θ, γ = sΩ0 /ω sin2 θ. The mildly relativistic regime has similar characteristics to the synchrotron limit. One needs to retain exact expressions for the speed and energy of the emitting particle, rather than expanding in powers of 1/γ. There is a systematic set of approximations in the mildly relativistic case that parallels those made in the highly relativistic case (Petrosian 1981, Robinson and Melrose 1984). The important qualitative point is that the emission from a particle with given v and α is concentrated around a specific angle θ = α0 . Retaining

158

4 Gyromagnetic processes

the refractive index, n ≈ (1 − ωp2 /ω 2 )1/2 for ω ≫ ωp , the relevant angle is determined by cos α0 = nv cos θ. This reduces to α0 = θ in the synchrotron limit, that is, for n → 1 and v → 1. Three steps are involved in evaluating the emission and absorption coefficients for a distribution of particles. First, one carries out the integral over the pitch angle distribution of the particles, expanding in powers of the parameter (cos α − cos α0 ) before carrying out the integral. For this first step the harmonic number, s, is assumed to be an integer, and relevant integrals are performed exactly in terms of Bessel functions. Second, the sum over s is replaced by an integral, which is performed over the δ-function in (4.1.9), so that s is reinterpreted as a continuous function of ω, θ and v. Third, the Bessel functions are replaced by appropriate approximate forms. Only at this stage does one distinguish between the gyrosynchrotron and synchrotron limits, with these being described by the Carlini and Airy integral approximations to the Bessel functions, respectively. (At this third stage the theory may also be applied to treat cyclotron emission, by making the approximation (4.3.22) to the Bessel functions.) The final step is to integrate over the energy spectrum of the particles. 4.4.2 Average over pitch-angle distribution The emissivity (4.3.3) for gyromagnetic emission of magnetoionic waves is used to define an average emissivity for gyromagnetic emission by averaging over the pitch angle distribution. It convenient (but not necessary) to assume that the distribution factorizes into an energy spectrum, expressed as a function of the Lorentz factor, N (γ), and pitch-angle distribution, φ(α): d4 p 1 F (p) = dγ N (γ) d cos α φ(α), (2π)4 γ Z 1 Z ∞ 1 d cos α φ(α) = 1, dγ N (γ) = n, 2

(4.4.1)

−1

1

where n is the number density in the rest frame. The average emissivity for gyromagnetic emission is defined by averaging (4.3.3) over φ(α) and integrating over harmonic number s. This gives η¯σ (ω, θ, γ) =

1 2

Z

1

d cos α φ(α)

−1

Z

∞

ds ησ (s, ω, θ).

(4.4.2)

0

The first step is to carry out the integral over cos α, which is facilitated by changing variables to v′ =

nv sin θ , sin α0

cos α′ =

cos α − cos α0 , 1 − cos α cos α0

cos α0 = nv cos θ.

(4.4.3)

4.4 Gyrosynchrotron emission

159

The primed variables have a physical interpretion in terms of a Lorentz transformation in which a particle with pitch angle α0 in the laboratory frame has pitch angle α′0 = π/2 in the new (primed) frame. There are two additional types of term that need to be included. One is from the derivative of the pitch-angle distribution, which is nonzero for an anisotropic distribution, and φ(α) is expanded about the pitch angle at which the emission is maximized, giving φ(α) ≈ φ(α0 ) −

cos α − cos α0 ′ φ (α0 ). sin α0

(4.4.4)

The other type of term arises when replacing the sum over harmonic number, s, by an integral. Integrals of the following form appear: Z ∞ d 1 1−s cos α′ cos α0 f (s), ds f (s) δ [γω(1−cos α cos α0 )−sΩ0 ] = Ω0 ds 0 (4.4.5) where the right hand side is to be evaluated at the value of s determined by the δ-function in (4.4.5). The average over pitch angle involves the three integrals that appear in (4.3.20) with α → α′ , a → v ′ . In addition, two other integrals appear: Z 1 1 d (1) (4) 1 d cos α 2 sin α Js (v sin α)Js′ (v sin α) = I (v), Is (v) = 2 s dv s −1 Z 1 1 d (2) (5) 1 Is (v) = 2 d cos α 2 sin α cos2 α Js (v sin α)Js′ (v sin α) = I (v), s dv s −1 (4.4.6) (1)

(2)

with Is and Is given by (4.3.20). The average emissivity for a specific (unlabeled) mode, described by parameters n, L and T , cf. (4.3.3), reduces to q 2 ω 2 sin2 α0 γ sin2 α0 H(ω, θ, γ), A= 2 , (4.4.7) Ω0 8π ε0 cn sin2 θ h i 1 2 (1) ′ 2 ˆ 1 c2 I (2) (v ′ ) c I (v ) + c + 2 dc H(ω, θ, γ) = 2 s 1 + T2 1 s 2 (3) ′ (4) ′ +c3 Is (v ) − ηc1 c3 Is (v ) − η dˆc2 c3 Is(5) (v ′ ) , η¯(ω, θ, γ) = A

c1 = L sin θ + (1 − n2 v 2 ) T cos θ,

c3 = v ′ sin2 α0 ,

c2 = nv sin θ (L cos θ − T sin θ), φ′ (α0 ) d +s dˆ = −nv cos θ 4 + tan α0 , (4.4.8) φ(α0 ) ds

with the harmonic number determined by s=

γω(1 − cos α0 cos θ) . Ω0

(4.4.9)

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4 Gyromagnetic processes

4.4.3 Gyromagnetic absorption coefficient The absorption coefficient (4.1.19) is evaluated in a manner that closely parallels the evaluation of the emissivity. Compared with the calculation of the emissivity, an extra term arises in the absorption coefficient, due to a derivaˆ in (4.1.19) is written in tive with respect to pitch angle when the operator D terms of derivatives with respect to |p| and α, cf. (4.3.6). The calculation of the absorption coefficient gives γ(ω, θ) = γγ (ω, θ) + γα (ω, θ),

(4.4.10)

with the two terms arising from the |p|- and α-derivatives, respectively, in (4.3.6): Z Z 1 (2π)4 c ∞ d ln[N (γ)/γ 2 v] , dγ d cos α φ(α) η(ω, θ, γ) γγ (ω, θ) = − 2 ω m 1 dγ −1 Z (2π)4 c ∞ γα (ω, θ) = − 2 dγ N (γ) ω m 1 Z 1 cos α − cos α0 d ln[φ(α)] 1 d cos α φ(α) η(ω, θ, γ) , (4.4.11) 2 γv 2 sin α dα −1 where the refractive index is set equal to unity where it appears as a multiplicative factor. The evaluation of the term γγ (ω, θ) follows the same steps as the derivation of (4.4.7). The evaluation of the term γα (ω, θ) gives 3 Z ∞ A N (γ) γ 2πc 2 sin α φ(α ) γα (ω, θ) = 4π dγ 0 0 ω γv 2 Ω0 1 + T2 1 ˆ 1 c3 I (5) (v ′ ) + dc ˆ 2 I (6) (v ′ ) , ˆ 2 I (2) (v ′ ) + c2 c3 + dc 2c1 c2 + dc s 3 s 2 s Is(6) (v) =

1 2

Z

1

−1

d cos α cos2 α sin2 α Js′2 (v sin α),

(4.4.12)

with A given by (4.4.7), and with dˆ = −nv cos θ(5 + sd/dt). The integral (6) 2 2 + Js+2 ) − 21 Js2 , Is (v) is evaluated in closed form by writing Js′2 = 14 (Js−2 and using the integrals (4.3.20) together with Z 1 Z v 3 4 2 d cos α sin α Jt (s(v sin α) = dy J2t (2sy) 4v 0 −1 Z v Z v 3 1 2 dy y J2t (2sy) + 5 dy y 4 J2t (2sy), + 3 2v 0 4v 0 Z v Z v 1 dy y 2 J2s (2sy) = 1 − 2 dy J2s (2sy) 4s 0 0 v2 ′ v (2sv), + 2 J2s (2sv) − J2s 4s 2s

4.4 Gyrosynchrotron emission

Z

v 0

Z v 1 9 dy J2s (2sy) dy y 4 J2t (2sy) = 1 − 2 1− 2 4s 4s 0 9 v v2 ′ + 1 + 3v 2 − 2 J (2sv) − J (2sv) . 2s 4s 4s2 2s 2s

161

(4.4.13)

(6)

However, Is is required below only in the highly relativistic limit, and then the approximation Is(6) (v ′ ) ≈ cos2 α0 Is(3) (v ′ ) (4.4.14) follows from (4.4.12) by writing cos α = cos α0 + (cos α − cos α0 ), and noting that cos α − cos α0 is of order 1/γ, allowing one to replace the factor cos2 α inside the integral by a factor cos2 α0 to lowest order in the expansion in 1/γ. 4.4.4 Approximations to Bessel functions Three different approximations to the Bessel functions apply in the nonrelativistic (cyclotron), mildly relativistic (gyrosynchrotron) and highly relativistic (synchrotron) regimes (Trubnikov 1958). In the cyclotron regime, only the leading terms in the power series are retained, in the gyrosynchrotron regime the Carlini approximation is made, and in the synchrotron regime the Airyintegral approximation is made. The first of these approximations corresponds to (sv ′ )2 (sv ′ )2s ′ 1− + ··· , s = ω/Ω0 , (4.4.15) J2s (2sv ) = (2s)! 2s + 1 which is valid for sv ′ ≪ 1. The Carlini approximation corresponds to 1/2

J2s (2sx) =

ζ 4πs

Z=

ζ −1 ζ +1

Z 2s

1/2

(2 + 3v ′2 )ζ 3 1− + ··· , 8s

e1/ζ ,

ζ = (1 − v ′2 )−1/2 ,

s=

ωζ 2 , Ω0 γ

(4.4.16)

and is valid for s ≫ sc with s ≫ 1. The Airy integral approximation corresponds to J2s (2sx) =

(1 − v ′2 )1/2 √ K1/3 (R), π 3

R = 2s(1 − v ′2 )3/2 /3 = s/sc ,

sc = 3(1 − v ′2 )−3/2 /2,

(4.4.17)

and is valid for (1 − v ′ ) ≪ 1. The Carlini and Airy integral approximations to the five integrals that appear in (4.4.13) are given in Table 4.4.4, with J2s (2sv ′ ) = (ζ/4πs)1/2 Z 2s in the Carlini approximation and with sc = 3/2(1 − v ′2 )3/2 . The Carlini approximation overlaps the other two: for v ′2 ≪ 1 (4.4.16) reduces to (4.4.15)

162

4 Gyromagnetic processes Integral (1)

Is (v ′ ) (2)

Is (v ′ ) (3)

Is (v ′ )

Carlini J2s (2sv ′ ) 2s(1 − v ′2 )1/2 J2s (2sv ′ ) 2s2 (1 − v ′2 )

Airy Z integral ∞ 1 √ dt K1/3 (t) 2sπ 3 R Z ∞ (1 − v ′2 ) √ dt K5/3 (t) − K2/3 (R) 2sπ 3 R

(1 − v ′2 )1/2 J2s (2sv ′ ) (1 − v ′2 ) √ 2sv ′2 2sπ 3 J2s (2sv ′ ) sv ′ J2s (2sv ′ ) 2s2 v ′ (1 − v ′2 )1/2

Z

∞

R

dt K5/3 (t) + K2/3 (R)

(1 − v ′2 )1/2 (4) √ Is (v ′ ) K1/3 (R) sπ Z3 ∞ 1 (5) dt K1/3 (t) √ Is (v ′ ) 2s2 π 3 R Table 4.1. Carlini and Airy-integral approximations to the five integrals (4.4.13). In the Carlini approximation J2s (2sv ′ ) is approximated as in (4.4.16).

with (2s)! approximated by (4πs)1/2 (2s/e)2s using Stirling’s formula, and for 2s ≪ ζ, (4.4.16) overlaps with the limit R ≪ 1 of (4.4.17). Modifications to the Carlini approximation, to optimize the interpolation between the other two approximations, were developed by [2]. In deriving the gyrosynchrotron emissivity below the Wild-Hill approximation is used; the relevant expressions for the integrals are given in Table 4.4.5. 4.4.5 Gyrosynchrotron emissivity The emissivity in the gyrosynchrotron case is given by making the Carlini approximation or the Wild-Hill approximation to (4.4.7) with (4.4.8)–(4.4.9). A further approximation is to neglect the terms involving the operator dˆ (cf. Robinson and Melrose 1984). This approximation involves neglecting the derivative, φ′ (α), of the pitch angle distribution, although any anisotropy is still included through the function φ(α0 ) in (4.4.4). With this approximation (4.4.7) reduces to η¯(ω, θ, γ) =

Z 2s ζ 3/2 a3 sc 1/6 q 2 ω 2 γ sin2 α0 φ(α0 ) 1 + 3s 8π 2 ε0 Ω0 (|k|2 /ω 2 ) sin2 θ (4πs)1/2 2s −1/3 2 0.85sc 1 sin α0 T cos θ 1+ × L sin θ + 1 + T2 ζ2 s 2 2 2 4 2 2 sin α0 (ζ − sin α0 )T sin θ + , (4.4.18) + ζ 2ζ(s + sc )

with s = γω(1 − cos α0 cos θ)/Ω0 and sc = 3ζ 3 /2. The unmodified Carlini approximation to (4.4.18) corresponds to the limit sc /s → 0.

4.4 Gyrosynchrotron emission Integral (1)

Is (v ′ ) (2)

Is (v ′ ) (3)

Is (v ′ ) (4)

Is (v ′ ) (5)

Is (v ′ )

approximation a1 −1/2 1 Z 2s 3 + 2s (4πs)1/2 2s 2sc 2s 1 Z a2 −5/6 3 + 2s (4πs)1/2 4s2 2sc a3 1/6 Z 2s 3 1 + ′2 1/2 2sv 2sc 2s (4πs) Z 2s 3 a4 −1/6 1 + ′ 1/2 sv 2sc 2s (4πs) 3 1 Z 2s a5 −1/2 + 2s (4πs)1/2 2s2 v ′ 2sc

163

constant a1 = 1.4324 a2 = 2.2179 a3 = 13.5890 a4 = 0.5033 a5 = a1

Table 4.2. Wild-Hill approximations to the five integrals (4.4.13), with Z defined by (4.4.16). The Carlini approximation is given by setting ai , for all i to zero, and the Airy-integral approximation is given by setting sc → ∞, Z → 1, v ′ → 1.

4.4.6 Gyrosynchrotron emission: thermal For a thermal spectrum of mildly relativistic particles, (2.3.1) with ρ ≫ 1 implies N (γ) =

nργ(γ 2 − 1)1/2 −ργ e , K2 (ρ)

K2 (ρ) ≈

π 2ρ

1/2

e−ρ .

Inserting (4.4.19) into the expression for the volume emissivity, Z ∞ J(ω, θ) = dγ N (γ) η¯(ω, θ, γ),

(4.4.19)

(4.4.20)

1

with η¯(ω, θ, γ) given by (4.4.7). The integral over energy in (1.3.5) is carried out by the method of steepest descents, cf. (4.3.25)–(4.3.26). Specifically, the approximation used is Z

1

∞

dγ G(γ) exp[−F (γ)] =

2π F ′′ (γ0 )

1/2

G(γ0 ) exp[−F (γ0 )],

(4.4.21)

where G(γ) is slowly varying and F (γ) has a well defined minimum, being sharply peaked about some value γ = γ0 determined by F ′ (γ0 ) = 0, where a prime denotes differentiation. In the present case, one has 2 ζ −1 + , F (γ) = ρ(γ − 1) − s ln ζ +1 ζ 2 1/2 ω (1 − n2 v 2 cos2 θ) ωζ 2 , ζ= s= . (4.4.22) Ω0 γ ω 2 (1 − n2 v 2 )

164

4 Gyromagnetic processes

The effect of the medium is relatively unimportant in most cases of most practical interest, and then one may set n = 1. In this case, γ0 is the solution of ζ −1 κ(ζ 2 − 2 cos2 θ) 2 2κ F ′ (γ) = ρ 1 − ln + − = 0, ζ 2 − cos2 θ ζ +1 ζ ζ(ζ 2 − 1) κ=

ω sin2 θ , ρΩ0

ζ = (γ 2 sin2 θ + cos2 θ)1/2 ,

and one has F ′′ (γ0 ) =

4ρκ γ0 sin2 θ . (ζ02 − 1)2 ζ0

(4.4.23)

(4.4.24)

The solution of (4.4.23) for ζ0 simplifies in two limits: 1/3 2ω for κ ≫ 1, (1 + 9κ/2)−1/3 , (4.4.25) γ02 − 1 = ζ0 = (4κ/3) ρΩ0 1+κ for κ ≪ 1. where the final expression is an interpolation (Petrosian 1981). The resulting expression for the volume emissivity (4.4.20) for the thermal distribution (2.3.1) in the mildly relativistic case is 1/2 q 2 nρω(ζ02 − 1) sin6 α0 (γ02 − 1)γ05 J (ω, θ) = e−ρ(γ0 −1) hαβ , 4πsζ04 64π 3 ε0 cκ1/2 sin3 θ 2 −iη cos θ cos θ (ζ02 − sin2 α0 )ζ0 sin4 θ 11 22 12 , 1, (h , h , h ) = + , (4.4.26) ζ02 ζ0 2s sin4 α0 αβ

with s = ωζ02 /Ω0 γ0 . The result (4.4.26) simplifies in two important limits. First, for ζ0 ≫ 1, which corresponds to highly relativistic particles. One has ζ ≈ γ sin θ and κ ≫ 1. Then (4.4.26) reduces to q 2 nρ3/2 Ω0 ζ 3 32π 5/2 ε0 c sin4 θ 9 −1/3 3 cos2 θ −1/3 αβ ρ P 1/3 + × exp ρ − P − P h , sin θ 20 8 cos θ sin2 θ cos2 θ ζ 3 , + , 1, −iη (h11 , h22 , h12 ) = γ2 ζ2 2s ζ

J αβ (ω, θ) =

P =

9ω sin2 θ 9κ , = 2 2ρΩ0

s=

γω sin2 θ , Ω0

(4.4.27)

with ζ = ζ0 = γ0 sin θ given by (4.4.25). Second, in the limiting case ζ0 − 1 ≈ κ ≪ 1 one has s = ω/Ω0 , ζ = γ and exp[−F (γ0 )] = ( 12 eκ)ω/Ω0 , with e = exp(1). Then (4.4.26) simplifies to

4.4 Gyrosynchrotron emission

165

ω/Ω0 1/2 eω sin2 θ ω q 2 nω hαβ , 2ρΩ0 (2π)1/2 8π 2 ε0 c Ω0 sin4 θ 11 22 12 2 (h , h , h ) = cos θ + , 1, −iη cos θ . (4.4.28) ρ

J αβ (ω, θ) =

When the longitudinal component, L, of the polarization is important, one uses the transfer equation (??) with the volume emissivity, Jo,x (ω, θ) given by (4.4.28) with hαβ replaced by " # 2 To,x sin4 θ 1 2 ho,x → (To,x cos θ + Lo,x sin θ − η) + , (4.4.29) 2 1 + To,x ρ with η = −1 for electrons. For a thermal distribution, the absorption coefficient is directly proportional to the volume emissivity. This is given by (2.4.12) for a given mode, and the corresponding result (Kirchhoff’s law) is (in ordinary units) γ αβ (ω, θ) =

ρ(2πc)3 ω 2 n2 ∂(ωn)/∂ω

J αβ (ω, θ) , mc2

(4.4.30)

with ρ = mc2 /T and with n → 1 here. 4.4.7 Gyrosynchrotron emission: power-law A power-law energy spectrum in the mildly relativistic case is of the form  for γ − 1 ≤ γ1 , 0 −δ (4.4.31) N (γ) = (δ − 1)n γ − 1 for γ − 1 > γ1 ,  γ1 γ1 where γ1 and δ are constants. In discussing emission and absorption in this case, for simplicity the refractive index is set equal to unity and any pitchangle anisotropy is ignored. The volume emissivity and the absorption coefficient are then given by    αβ  Z ∞ 1 J (ω, θ)     = n(δ − 1)γ1δ−1 dγ(γ − 1)−δ  (2π)3 c (δ + 2)γ  hαβ . αβ 1+γ1 γ (ω, θ) ω2m γ 2 − 1 (4.4.32) The method of steepest descent is used to evaluate the γ-integral in an analogous fashion to the thermal case. It is convenient to introduce parameters a and b in writing 2 tζ 2 ζ−1 ln + , F (γ) = a ln(γ − 1) + b ln γ − ζ+1 ζ γ sin2 θ t=

ω sin2 θ , Ω0

ζ = γ 2 sin2 θ + cos2 θ

1/2

.

(4.4.33)

166

4 Gyromagnetic processes

One has a+b=

δ+1 δ+2

for J αβ , for γ αβ .

(4.4.34)

One finds F ′ (γ) =

b (ζ 2 − 2 cos2 θ) ζ −1 2 2t a + −t 2 . (4.4.35) ln + − 2 2 γ−1 γ ζ − cos θ ζ +1 ζ ζ(ζ − 1)

An approximate solution of F ′ (γ0 ) = 0 is (γ0 + 21 )2 − 1 =

4ω . 3Ω0 (a + b) sin θ

(4.4.36)

To a good approximation one has F ′′ (γ0 ) =

(a + b)2 3Ω0 sin θ . 2ω

(4.4.37)

The resulting expressions for the volume emissivity and the absorption coefficient for the power law energy spectrum (4.4.13) is, setting n → 1,  αβ  √ 1−δ J (ω, θ) γ0 + 1 3 Ω0 γ0 − 1 N (δ − 1)  = Z02s A 2 sin3 θ 8 ω γ0 γ1 αβ γ (ω, θ)   1   ×  (2πc)3 (δ + 2)γ1 γ0  hαβ , (4.4.38) ω 2 mc2

γ02 − 1

with

2 cos2 θ cos θ ωγ0 sin2 θ + , , 1, −iη , s= (h , h , h ) = 2 γ0 3(a + b + 2) γ0 Ω0 1/2 ζ0 − 1 ζ02 = γ02 sin2 θ + cos2 θ, (4.4.39) e1/ζ0 , Z0 = ζ0 + 1 11

22

12

and with a + b given by (4.4.34).

4.5 Synchrotron emission

167

4.5 Synchrotron emission Synchrotron emission is gyromagnetic emission by highly relativistic particles. In this section synchrotron emission and absorption are treated as the highly relativistic limit of the treatment of gyromagnetic emission and absorption in §4.4. The case of a power-law energy distribution of particles is of most interest in practice. The case of a relativistic thermal distribution is also discussed here: it is of formal interest because it can be treated using synchrotron theory and also using Trubnikov’s response tensor, allowing the two methods to be comparing in the synchrotron limit. 4.5.1 Strong and weak Faraday rotation In order to discuss synchrotron absorption, one needs to determine which of two different procedures to use to treat the transfer of polarization. The two procedures are: (i) separate into natural modes, with one intensity, one volume emissivity and one absorption coefficient for each mode, and (ii) assume two degenerate states of transverse polarization, with the intensity and polarization described by the Stokes parameters, and with the volume emissivity and the absorption coefficient written as polarization tensors. The latter procedure corresponds to the weak-anisotropy approximation, §??. In the present context, these two limits are sometimes referred to as strong and weak Faraday rotation, respectively. With the first procedure, let the modes be identified as one of the two magnetoioinic modes, labeled o and x. the transfer equations the two modes are given by dIo,x = Jo,x − µo,x Io,x , (4.5.1) ds where s denotes distance along the ray path, and µo,x = γo,x /vgo,x is the absorption per unit length. The other procedure is to use the transfer equation in terms of polarization tensors, which is valid only for transverse waves. The transfer equation in terms of the Stokes parameters is given by (??), which is generalized here to dSA /ds = JA + rAB SB − µAB SB ,

(4.5.2)

where A = (I, Q, U, V ) denotes the Stokes parameters, JA describes spontaneous emission and the other two terms are as in (??). The emission coefficients, JA , and absorption coefficients, µAB , are constructed by writing synchrotron emissivity and the absorption coefficient as polarization tensors, and translating into the notation used in (4.5.2). Provided that the longitudinal part is negligible, L → 0, (4.4.7) may be written as a polarization tenor, η¯ → η¯αβ , by writing H → H αβ , with the components of H αβ identified by writing (4.4.8) in the form

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4 Gyromagnetic processes

H=

1 T 2 H 11 + 2iT H 12 + H 22 . 1 + T2

(4.5.3)

The translation from the form involving polarization tensors to the form (4.5.2) involving the Stokes parameters follows by using the Pauli matrices, written down in § 2.5.9 of volume I as 1 0 1 0 µν µν σI = , σQ = , 0 1 0 1 0 −i 0 1 µν µν , (4.5.4) , σV = σU = i 0 1 0 where µ, ν running over the two transfer components, with X µν µν SA σA , SA = σA Iµν . I µν = 12

(4.5.5)

A

In the synchrotron case, the two transverse polarizations are chosen perpendicular and parallel to the projection of the magnetic field onto the transverse plane, and the the components JU and µU are zero. The interrelation between these two formalisms may be understood as follows. The intensities Io,x in (4.5.1) are specific linear combinations of I, Q, U, V , and there is a total of four linearly independent combinations of I, Q, U, V . The intensities Io,x may be written in terms of the moduli squared of the amplitudes for the two modes. The other linearly independent combinations of I, Q, U, V may then be chosen to be effectively the real and imaginary parts of the cross-correlation function between the amplitudes for the two modes. The transfer equation (4.5.2) retains information on these crosscorrelation functions whereas (4.5.1) does not. In order for (4.5.2) to reduce to (4.5.1), these cross-correlation functions must be negligible. This is the case if Faraday rotation is strong. Faraday rotation arises from the terms rAB in 2 2 (4.5.2), with (rQ + rU + rV2 )1/2 = ∆k, where ∆k is the difference in wavenumber between the two modes. The limit of strong Faraday corresponds to a large number of Faraday rotations, so that the cross-correlation function between the two modes averages to zero in any observation. One is to use (4.5.1) in the limit of strong Faraday rotation, and (4.5.2) must be used in any case where the cross-correlation is of interest. The cross-correlation function describes linear polarization in the simplest 2 2 case. This is for ω ≫ ωp when the magnetoionic waves have rV2 ≫ rQ + rU , except for a very small range of angles near θ = π/2, so that the natural modes are circularly polarized, and the cross-correlation between them corresponds to the combinations Q ± iU of linear polarizations. In the limit of strong Faraday rotation the linear polarization cannot be detected, for example, due to many rotations across the bandwidth of observation, or due to differential Faraday rotation from different points in the source. Then (4.5.2) reduces to (4.5.1) due to the cross-correlation being zero.

4.5 Synchrotron emission

169

F (R) 0.8

0.4

0 0

1

2

3

R

Fig. 4.3. The function F (R) defined by (4.5.8).

4.5.2 Synchrotron emissivity The emissivity in the synchrotron case is given by making the Airy-integral approximation to (4.4.7) with (4.4.8). In this limit one has s = γω sin2 θ/Ω0 and α0 = θ. The resulting expression for the average emissivity for synchrotron emission is √ 2 3q Ω0 ξ sin θφ(θ) αβ αβ F (ω, θ, γ), (4.5.6) η¯ (ω, θ, γ) = 64π 3 ε0 γ with Z ∞ F 11,22 (ω, θ, γ) = R dt K5/3 (t) ∓ R K2/3 (R), R

Z ∞ 2iη cot θ (2 + g(θ)) dt K1/3 (t) + 2R K1/3 (R) , F (ω, θ, γ) = − 3nξ R 12

R=

tan θφ′ (θ) , φ(θ)

ω 2s , = ωc 3ξ 3 sin3 θ

g(θ) =

ωc = 32 Ω0 ξ 2 sin θ,

ξ = (1 − n2 v 2 )−1/2 .

(4.5.7)

with F 21 = −F 12 , and with e1 = (cos θ, 0, − sin θ), e2 = (0, 1, 0). In this case the two terms involving dˆ in (4.4.8) are of order 1/γ smaller that the leading terms, and are neglected to lowest order in 1/γ. The off-diagonal terms are of order 1/γ compared with the diagonal terms, so that the term involving dˆ needs to be retained in F 12 . The s-derivative in dˆ follows from the explicit s-dependence and s∂R/∂s = R in the final entry in table 4.4.4. The functional dependence on the parameter R in the total emissivity (summed over the two states of polarization) is described by a function F (R), which for small and large arguments is given by Z ∞ k ⊥ 1 dt K5/3 (t). (4.5.8) F (R) = 2 [F (R) + F (R)] = R R

Limiting cases for F (R) are

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4 Gyromagnetic processes

 # 2/3 1/3 "  4π R R  1  + ··· R ≪ 1, 1 − 2 Γ (1/3)   √3Γ (1/3) 2 2 F (R) = 1/2   55 πR   e−R 1 + + ··· R ≫ 1.  2 72R (4.5.9) In between these limiting cases, there is a maximum at F (0.29) = 0.92. The function F (R) is plotted in figure 4.3. A simple analytic approximation to it is F (R) ≈ 1.8R0.3 e−R . 4.5.3 Synchrotron emission: power-law A power-law energy spectrum for highly relativistic particles particles is of the form Kε−a for ε1 ≤ ε ≤ ε2 , (4.5.10) N (ε) = 0 otherwise. The energy-integrals in the expression (2.5.1) with (4.5.6) for the synchrotron emission coefficient may be evaluated in closed form for ε1 = 0, ε2 = ∞ using the standard integral Z ∞ 1+µ−ν 1+µ+ν dx xµ Kν (ax) = 2µ−1 a−µ−1 Γ Γ , (4.5.11) 2 2 0 and identities satisfied by the Γ -function: Γ (1 + x) = xΓ (x),

Γ (1 − x)Γ (x) =

π . sin(πx)

(4.5.12)

One obtains J αβ (ω, θ) = A(θ) j αβ (a)

2ω 3Ω0 sin θ

−(a−1)/2

,

(4.5.13)

with

√ 2 3q Ω0 sin θφ(θ) , A(θ) = Km 64π 3 ε0 c 3a + 7 3a − 1 a + 5/3 11 2/3 (a−3)/2 2 Γ j (a), Γ , j 22 (a) = j 11 (a) = a+1 12 12 2/3 −1/2 (a−2)/2 2 a + 2 + g(θ) 2ω 12 j (a) = 2iη cot θ 3Ω0 sin θ 3a 3a + 8 3a + 4 ×Γ Γ , (4.5.14) 12 12 −a+1

with j 21 (a) = −j 12 (a). The fact that the frequency dependence of the 12term differs from the 11- and 22-terms by a factor ∼ (ω/Ω0 sin θ)−1/2 may be understood by noting that this factor results from the first order term in an expansion in 1/γ with ω ∼ Ω0 γ 2 sin θ.

4.5 Synchrotron emission

171

4.5.4 Synchrotron absorption The synchrotron absorption coefficient is given by (4.4.11) with v → 1 in the synchrotron limit. For an isotropic distribution one has Z (2π)4 cφ(θ) d ln[N (γ)/γ 2 ] αβ αβ , (4.5.15) γ (ω, θ) = − dγ η (ω, θ, γ) ω2m dγ with the synchrotron emissivity, η¯αβ (ω, θ, γ), given by (4.5.6). Thus, (6.3.1) becomes √ 2 Z (2π)4 c 3q Ω0 ξ sin θφ(θ) αβ d[N (γ)/γ 2 ] αβ 2 γ (ω, θ) = − 2 dγ γ F (ω, θ, γ) , ω m 64π 3 ε0 γ dγ (4.5.16) with F αβ (ω, θ, γ) given by (4.5.7). For an anisotropic distribution, the derivative with respect to α gives the derivative of the pitch angle distribution at α = θ, φ′ (θ) = dφ(θ)/dθ, and only the 12-term is nonzero up to first order in 1/γ. This additional contribution is √ Z Z 2iη(2π)4 cφ′ (θ) dγN (γ) 3q 2 Ω0 sin θφ(θ) ∞ 12 dt K1/3 (t), γα (ω, θ) = − 3ω 2 m γ 64π 3 ε0 c R (4.5.17) with R = ω/ωc, with ωc given by (4.5.7). The absorption coefficient may be evaluated explicitly for the power-law distribution (2.5.29), giving √ −a/2 2ω (2πc)3 K(mc2 )−a 3q 2 Ω0 sin θφ(θ) αβ αβ , γ (ω, θ) = jab (a) ω2 64π 3 ε0 c 3Ω0 sin θ (4.5.18) αβ αβ with jab (a) = (a + 2)j (a + 1) for the 11- and 22-components and with 12 jab (a) = (a + 3) a + 2 + g(θ) j 12 (a + 1)/ a + 3 + g(θ) . In the case of strong Faraday rotation, let the two modes be labeled ± with polarization vectors e± = (T± e1 + ie2 )/(T±2 + 1)1/2 with T+ T− = −1. The absorption coefficients, γ± or µ± = γ± /c, for the two modes are related to γ αβ by γ± (ω, θ) =

T±2 γ 11 (ω, θ) + 2iT±γ 12 (ω, θ) + γ 22 (ω, θ) . T±2 + 1

(4.5.19)

In the approximation in which the natural modes are the magnetoionic modes in the circularly polarized limit, one has T± = ±η cos θ/|η cos θ|, with the +mode being the o-mode for an electron gas, η = −1. 4.5.5 Razin suppression The presence of the plasma causes a suppression of synchrotron emission (Razin 1960). The Razin effect applies to any emission mechanism by highly

172

4 Gyromagnetic processes

relativistic particles. In the context of sion effect may be seen by noting that radiation falls off exponentially for R ωc = (3/2)Ω0 ξ 2 sin θ and with 2 2 −1/2

ξ = (1 − n v )

synchrotron radiation, the suppresthe (4.5.9) implies that synchrotron > ∼ 1 with, from (??), R = ω/ωc ,

≈γ

γ 2 ωp2 1+ ω2

!−1/2

.

(4.5.20)

The medium is unimportant for ω ≫ γωp , when one has ωc = (3/2)Ω0 γ 2 sin θ and R ∝ ω, implying that synchrotron emission falls off exponentially at high frequency. However, for ω < ∼ γωp one has ξ ≈ ω/ωp and R ∝ 1/ω, so that the emission also falls off exponentially at low frequency. The characteristic frequency below which this suppression effect occurs is determined by setting ω = γωp = ω = ωc , and is called the Razin-Tsytovich frequency, ωRT = 2ωp2 /3Ωe sin θ.

(4.5.21)

More generally, any form of emission by a highly relativistic particle in a plasma is suppressed at ω < ∼ γωp . The Razin effect also applies to emission by mildly relativistic particles. Consider gyromagnetic emission at the sth harmonic in a medium with refractive index n. The emission coefficient depends on n through a multiplicative factor of n2s−1 , which corresponds to 22s -multipole emission. For large s, and for n = (1 − ωp2 /ω 2 )1/2 with ω 2 ≫ ωp2 , one has 2

2

(1 − ωp2 /ω 2 )s−1/2 ≈ e−sωp /ω ,

(4.5.22)

which implies suppression for sωp2 /ω 2 > ∼ 1. With ω ≈ sΩe for cyclotron emis2 sion at the sth harmonic, this also implies suppression for ω < ∼ ωp /Ωe . 4.5.6 Possibility of maser synchrotron emission Synchrotron absorption cannot be negative under realistic conditions. The absorption coefficient may be written Z (2π)4 cφ(θ) d N (γ) αβ γ (ω, θ) = − dγ γ 2 η¯αβ (ω, θ, γ) ω2m dγ γ 2 Z (2π)4 cφ(θ) N (γ) d 2 αβ γ η¯ (ω, θ, γ) , (4.5.23) = dγ ω2m γ 2 dγ where the second form follows from the first by a partial integration. Absorption can be negative only if at least one of the two eigenvalues of γ αβ is negative. To lowest order in 1/γ the eigenvalues are 12 (γ 11 ± γ 22 ). When the Razin effect is ignored, ξ → γ, it is straightforward to carry out the derivative in the second form of (4.5.23), using (4.5.6) with (4.5.7), and one finds that the eigenvalues cannot be negative.

4.5 Synchrotron emission

173

This proof that synchrotron absorption cannot does not apply when the Razin effect is important, and it is then possible in principle for absorption 2 to be negative for ω < ∼ γωp . One also requires d[N (γ)/γ ]/dγ > 0. However, there is no known case where the Razin effect is important in the synchrotron limit, and no known case where the condition d[N (γ)/γ 2 ]/dγ > 0, and it is very implausible that both these necessary conditions could be satisfied simultaneously. 4.5.7 Synchrotron absorption: thermal Synchrotron absorption by a (highly relativistic) thermal distribution of particles may be treated both directly and using the antihermitian part of Trubnikov’s response tensor. The volume emissivity and the absorption coefficient for a thermal distribution are related by Kirchhoff’s law (2.5.20), and only one of them need be calculated. The absorption coefficient is Z (2π)3 nρ2 ∞ αβ γ (ω, θ) = 2 dγ γ(γ 2 − 12 )1/2 e−ργ η¯αβ (ω, θ, γ), (4.5.24) ω K2 (ρ) 1 where the thermal distribution (2.3.1) is assumed, and where the synchrotron emissivity, η¯αβ , is given by (4.5.7). The highly relativistic limit corresponds to ρ ≪ 1 and one may assume γ ≫ 1 in the integrand in (4.5.24) with the lower limit of the integration approximated by zero. For simplicity, only the two linearly polarized components for θ = π/2 are considered here, and the refractive index is assumed to be unity. On inserting the expression (??) for the emissivity, and noting R ∝ 1/γ 2 , one may partially integrate and introduce the variable x = 1/ρ2 γ 2 to write Z ∞ Z 2 −ργ dγγ e R dt K5/3 (t) ± K2/3 (R) R Z ∞ √ ′ (νx) , (4.5.25) = dx e−1/ x K5/3 (νx) ± K2/3 0

2

with ν = 2ωρ /3Ω0 . The resulting integral simplifies in two limits:  Γ (5/3)   for ν ≪ 1,   21/3 (νx)5/3 (1, 3) ′ K5/3 (νx) ± K2/3 (νx) = 1/2 −νx  2π e   (1, 3νx) for ν ≫ 1,  νx 3νx

(4.5.26)

where the recursion relation (A.1.14), (A.1.15) are used. The remaining integral is then performed using the method of steepest descents. The limit ν ≪ 1 in (4.5.26) corresponds to low frequencies, specifically to frequencies below the characteristic frequency ω ∼ Ω0 ρ2 of emission by a thermal particle with γ ∼ 1/ρ. In this limit (4.5.24) reduces to

174

4 Gyromagnetic processes

γ 11,22 (ω, π/2) =

3πωp2 −1/3 1, 3 , 4Ω0 4P 5

(4.5.27)

with P = 9ω/2ρΩ0 . In the high frequency limit ν ≫ 1, (4.5.27) is replaced by 1/3 3πωp2 ρ2 e−ρP 1/3 . γ 11,22 (ω, π/2) = √ 1, ρP 4 2Ω0 ρP 4/3

(4.5.28)

In (4.5.27) and (4.5.28) the factor K2 (ρ) in the normalization of the J¨ uttner distribution is approximated by 2/ρ2 , which applies for ρ ≪ 1. 4.5.8 Synchrotron absorption: Trubnikov’s form The foregoing results may be derived in a different manner by starting from the antihermitian part of Trubnikov’s response tensor, given by (??). Before applying this appraoch to the highly relativistic regime, it is instructive to apply it to the nonrelativistic and mildly relativistic cases. In this limit (??) gives √ 2 5/2 √ πωp ρ 11 22 γ (ω, θ) + γ (ω, θ) = eρ− r0 , ′′ 1/2 Ω0 (r0 r0 ) 2 ω sin θ(1 − cos Ω0 ξ0 ) − ρ2 cot2 θ, r0 = Ω0 r0′′ = −2

ω2 sin2 θ(1 − cos Ω0 ξ0 ). Ω02

(4.5.29)

For ω/ρΩ0 ≫ 1, r0′ = 0 has the approximate solution # 1/3 " 2/3 3 3 sin θ sin θ 1− Ω0 ξ0 = i + ··· , sin θ P P 20 sin2 θ

(4.5.30)

with P = 9ω/2ρΩ0. On substituting (4.5.30) into (4.5.29), and separating into the two modes for θ = π/2, one finds γ

11,22

(ω, π/2) =

√

πωp2 ρ1/2 3 −ρ e Ω0 2P

P 1/3 −1+9/20P 1/3

1 , 1 , (4.5.31) ρP 1/3

which reproduces the absorption coefficient implied by (??) with (??) and (??) for θ = π/2. The result (4.5.31) was derived by [3]. In treating the highly relativistic limit ρ ≪ 1, the asymptotic approximation is made to the Macdonald functions in (??): K2 r(ξ) ≈

2 1 − + ···, r2 (ξ) 2

K3 r(ξ) ≈

8 1 − + · · · . (4.5.32) r3 (ξ) r(ξ)

4.5 Synchrotron emission

175

The resulting approximation to the synchrotron absorption coefficient for a thermal distribution is Z ∞ ωp2 ρ2 1 1 ξ 4 ω 2 Ω02 11,22 γ (ω, π/2) = , + 6 dξ 4 , 4 K2 (ρ) −∞ r (ξ) r (ξ) r (ξ) 1 2 2 4 1/2 ω Ω0 ξ , (4.5.33) r(ξ) ≈ ρ2 − 2iρωξ − 12

with θ = π/2 assumed here. Trubnikov [3] evaluated the integrals in terms of the residue at the pole r(ξ) = 0, which for ρ2 ≪ 1 occurs at ξ1 = 2i(3ρ/Ω0 ω sin2 θ)1/3 . The specific integrals are ! 1/3 ! Z ∞ 1/r4 (ξ) (3π/2ρ4 Ω0 ) sin2 θ/4P 5 dξ = , (4.5.34) ξ 4 /r6 (ξ) −∞ (35 π/4ρ6 Ω05 )(23 sin4 θ P 11 )1/3

with P = (9ω/2ρΩ0 ) sin2 θ. Thus, (4.5.33) with (4.5.34) reproduces (4.5.27) for θ = π/2 and K2 (ρ) = 2/ρ2 .

176

4 Gyromagnetic processes

4.6 Thomson scattering in a magnetic field Inclusion of the magnetic field in the theory of scattering of waves by particles affects both the waves and the particles. The waves are the natural modes of the magnetized plasma, and waves in one mode may be scattered into waves in the same mode or in another mode. The spiraling motion of a scattering particle needs to be taken into account, and this introduces an additional length into the problem: the gyroradius of the scattering particle. Two limiting cases are for magnetized particles, when the gyroradius is smaller than other relevant lengths, and unmagnetized particles, when the gyroradius is smaller than other relevant lengths. Scattering by unmagnetized particles may be approximated by ignoring the spiraling motion and treating the unperturbed motion of the scattering particles as rectilinear. In cases where the dispersion of the plasma is weak and the electrons are strongly magnetized, there is a resonance in the scattering associated with the cyclotron frequency. 4.6.1 Scattering by a magnetized particle The theory of scattering of waves by unmagnetized particles, presented in §5.5 of Volume I. The generalization to include the effect of the magnetostatic field involves replacing the first order current for an unmagnetized particle by that for a magnetized particle. The resulting expression for the scattering probability is wMP (s, p, k, k ′ ) =

′

a ˜MP (s, k, k , p) = αµ

×G

q 4 RM (k)RP (k ′ ) |˜ aMP (s, k, k ′ , p)|2 ε20 m2 γ ωM (k)ωP (k ′ ) ×2πδ (kM u)k − (kP′ u)k − sΩ0 ,

e∗Mµ (k)eP ν (k ′ )

e

iηsψ

∞ X

eiηt(ψ+ψ

′

(4.6.1)

)

t=−∞

(s − t, kM , u)ταβ (kM u)k − (s − t)Ω0 G∗βν (t, kP′ , u) 2m µνρ ′ ′ ′ − Π (kM , kP , kM − kP )Uρ (s, kM − kP ) , q

(4.6.2)

with Gµν (s, k, u) given by (2.1.52), τ µν (ω) given by(2.1.15) and U µ (s, k, u) by (2.1.28) with (2.1.29. The term involving Π µνρ is due to nonlinear scattering. The resonance condition in (4.6.2) may be interpreted either from a purely classical viewpoint or from a semiclassical viewpoint. The classical interpretation is that the difference between the frequencies of the scattered and unscattered waves in the rest frame of the gyrocenter of the scattering particle is an integral multiple of the gyrofrequency. The semiclassical interpretation is in terms of conservation of energy and momentum, and follows from (4.1.10)– µ (4.1.12) by replacing the single wave quantum, kM , emitted by emission of ′ the beat disturbance kM − kP .

4.6 Thomson scattering in a magnetic field

177

As in the unmagnetized case, in a nonrelativistic plasma, nonlinear scattering is important for wavelengths of order the Debye length or longer. Thomson scattering and nonlinear scattering interfere for electrons, leading to a suppression of scattering by electrons at long wavelengths, when nonlinear scattering by ions dominates. The distinction between magnetized and unmagnetized particles is made to justify two opposite limiting approximations to the Bessel functions in U µ (s, k − k ′ ), given by (2.1.28) with (2.1.29). The particles are magnetized when the argument (k − k′ )⊥ R with R is the gyroradius of the particle. The power series expansion of the Bessel functions then converges rapidly and they may be approximated by the leading term in the expansion, and only the terms with s = 0, ±1 contribute significantly. In the opposite limit, (k − k′ )⊥ R ≫ 1, high harmonics of the Bessel functions dominate. This limit corresponds to large gyroradius of the scattering particle, and it is well approximated by assuming an infinite gyroradius, which corresponds to the unmagnetized case. A characteristic feature of scattering in a magnetic field is the appearance of resonances. The quantity ταβ (ω), with ω = (kM u)k − (s − t)Ω0 , is defined by (2.1.15); ταβ (ω) is singular at the cyclotron frequency, ω = Ω0 , of the scattering particle. This can lead to a greatly enhanced scattering cross section, referred to as resonant scattering, at the cyclotron frequency, provided that the wave dispersion is not strongly modified by the plasma. 4.6.2 Quasilinear equations for scattering Quasilinear equations for scattering by a magnetized particle may be derived using a semiclassical approach, in a way that is closely analogous to the unmagnetized case discussed in Volume 1. These equations include the kinetic equations for both the scattered and unscattered waves: Z 3 ′ ∞ Z X d4 p d k DNM (k) γwMP (s) NP (k ′ ) − NM (k) F (p) = 4 3 Dt (2π) (2π) s=−∞ (k − k ′ )uk ∂ ′ ′ α ∂ +NM (k)NP (k ) + (k − k )k α F (p) , (4.6.3) u⊥ ∂p⊥ ∂pk Z ∞ Z X d3 k DNP (k ′ ) d4 p γwMP (s) NP (k ′ ) − NM (k) F (p) =− 4 3 Dt (2π) (2π) s=−∞ (k − k ′ )u ∂ k ∂ ′ F (p) . (4.6.4) + (k − k ′ )α +NM (k)NP (k ) k u⊥ ∂p⊥ ∂pα k The kinetic equation for the particles due to wave-particle scattering is described by the quasilinear equation with the coefficients (4.1.21), (4.1.22) replaced by

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4 Gyromagnetic processes

A⊥ (p)

!

=−

XZ

d3 k γwMP (s) NM (k) − NP (k ′ ) 3 (2π)

sΩ/v⊥

!

, (k − k ′ )µk (4.6.5) for the terms that describe the effect of spontaneous emission alone, and     (sΩ/v⊥ )2 D⊥⊥ (p) Z 3   µ  X  sΩ d k ′   Dk⊥ (p)  = γw (s) N (k)N (k ) (k − k ′ )µk  MP M P ,    3 (2π) v⊥ s µν µ Dkk (p) (k − k ′ )k (k − k ′ )νk (4.6.6) for the diffusion coefficients, with wMP (s) = wMP (s, k, k ′ , p). Aµk (p)

s

4.6.3 Scattering cross section In describing Thomson scattering and Compton scattering, it is conventional and convenient to introduce the differential scattering cross section. Such a cross section is written down here for scattering in a magnetized plasma. However, the general form is cumbersome due to both the cumbersome description of the interaction itself, as described by the probability (4.6.1) with (4.6.2), and the cumbersome details of the wave energetics involved. To define a differential scattering cross section, one needs to determine the energy flux in the scattered radiation, per unit solid angle about the ray direction, in terms of the energy flux in the incident radiation per unit solid angle about its ray direction. The energy flux is along the group velocity, which is at an angle, θr , to the magnetic field that is different from the wave-normal angle θ. One has v gM (k) =

∂ωM (k) = (sin θr cos ψ, sin θr sin ψ, cos θr )vgM . ∂k

(4.6.7)

For simplicity in writing, the argument of the group speed, vgM , and a label on the ray angle, θr , denoting the mode, M , are omitted. For the mode P , the group speed is written v ′ gP , where the prime denoted that the argument depend on the primed variables. The cross section depends on the Jacobian ∂ cos θr /∂ cos θ of the transformation from the wave-normal to the ray angle. The differential scattering cross section is identified as X Z ∞ dω ω 2 n2 ∂(ωnM ) ∂ cos θ ∂ cos θ′ M ΣMP = wMP , (4.6.8) 3 ω′ v′ (2π) ∂ω ∂ cos θr ∂ cos θr′ 0 gP s where arguments are omitted for simplicity in writing. The integral over ω is performed over the δ-function in the probability (4.6.1). The cross section is too cumbersome to be useful in most cases where the plasma dispersion is important. A subtle point concerns the cancelation of two factors nM ∂(ωnM )/∂ω in the cross section, one from the group velocity, vgM , and one from the ratio of

4.6 Thomson scattering in a magnetic field

179

electric to total energy, RM in the probability wMP . (A similar cancelation occurs for n′P ∂(ω ′ n′P )/∂ω ′ .) At sufficiently high frequencies, nM ∂(ωnM )/∂ω approaches unity, vgM approaches the speed of light and RM approaches one half. In the opposite limit, near a resonance, nM ∂(ωnM )/∂ω becomes very large, and vgM , RM become small. The cancelation of these two factors in the cross section may be interpreted as the strength of the coupling becoming weak, due to small RM , being offset by the energy flux in the waves becoming small, due to small vgM , allowing a long time for the interaction. The cross section is rarely useful in such cases where the plasma dispersion has a large effect, and the cross section is used only sparingly in the following discussion. 4.6.4 Magnetized and unmagnetized particles The general form of the probability (4.6.1) with (4.6.2) is too cumbersome to be of practical used one needs to make various simplifying assumptions to reduce it to a directly useful form. One complicating feature is the sum over Bessel functions. This may be simplified in two limiting cases, referred to as magnetized and unmagnetized particles, respectively. For unmagnetized particles, the gyroradius of the scattering particle is effectively assumed infinite, in which case the motion of the particle approximated by constant rectilinear motion, and Thomson scattering in a magnetic field is replaced by its unmagnetized counterpart. Formally, this limit corresponds to large argument of the Bessel functions, (k − k′ )⊥ R ≫ 1, when the sum is dominated by high harmonics, of order this argument, s ∼ (k − k′ )⊥ R. For magnetized particles the small gyroradius approximation is made, (k− k′ )⊥ R ≪ 1, such that only the leading terms in the power series expansion of the Bessel functions need be retained. The first order current is approximated by (2.1.56). The corresponding approximation to the probability (4.6.1) is q4

RM (k)RP (k ′ ) |˜ aMP (k, k ′ , p)|2 2πδ (kM − kP′ )uk , ′ ωM (k)ωP (k ) (4.6.9) where the nonlinear scattering term is neglected, and with wMP (p, k, k ′ ) =

ε20 m2 γ

a ˜MP (k, k ′ , p) = e∗Mµ (k)eP ν (k ′ )Gαµ (kM , uk )ταβ (kM uk )Gβν (kP′ , uk ). (4.6.10) The right hand side of (4.6.10) may be rewritten using (2.1.57). Significant further simplification occurs only when further physical approximations are made. If the scattering particle is nonrelativistic, provided that the refractive index of neither wave is large, one has kM uk ≈ ωM (k), kP′ uk ≈ ωP (k ′ ), so that the δ-function in (4.6.9) implies that there is a negligible change in frequency. For nonrelativistic particles, the form (4.6.10) with (2.1.57) may be approximated by a ˜MP (k, k ′ , p) = e∗Mµ (k)eP ν (k ′ )τµν (kM uk ),

(4.6.11)

180

4 Gyromagnetic processes

with kM uk = kP′ uk . Further simplifications involve approximations to the wave properties. Specific cases discussed here are the scattering of magnetoionic waves by nonrelativistic electrons, and the scattering of the perpendicular and parallel modes of the birefringent vacuum in strongly magnetized very low density plasmas. 4.6.5 Scattering of magnetoionic waves High frequency waves in most magnetized plasmas are well approximated by treating the plasma as a cold electron gas, so that the waves are described by the magnetoinic theory §3.2.1. It is convenient to change the labeling of the modes M, P → σ, σ ′ . The wave properties in (4.3.1) include the polarization parameters Tσ , Lσ , and the refractive index nσ , and are given by (3.2.5)– (3.2.9). On inserting the wave properties (4.3.1) into the probability (4.6.9) with M, P → σ, σ ′ , to avoid loss of generality one needs to allow the two waves to be in different azimuthal planes, and this is achieved by assuming the wave normal direction to be κ = (sin θ cos ψ, sin θ sin ψ, cos θ), so that one has t = (cos θ cos ψ, cos θ sin ψ, − sin θ),

a = (− sin ψ, cos ψ, 0),

(4.6.12)

and similarly for κ′ , t′ , a′ in terms of θ′ , ψ ′ . The probability (4.6.9) depends on the azimuthal angle, ψ − ψ ′ between the scattered and unscattered wave, and assuming azimuthal symmetry in the particle and wave distributions, this dependence is of no interest. Using the wave properties (4.3.1), assuming the scattering particle to be at rest (p = 0) and averaging over azimuthal angles, the probability (4.6.9) reduces to [5] hwσσ′ (k, k′ )i =

q4 fσσ′ (ω, θ, θ′ ) 2πδ(ω − ω ′ ), 4ε20 m2

(4.6.13)

where the angular brackets denote the average over azimuthal angle. The dependence on the wave properties is included in −1 ∂ ∂ ′ 2 2 ′ ′ ′ fσσ (ω, θ, θ ) = (1 + Tσ ) nσ (ωnσ ) (1 + Tσ′ ) nσ (ωnσ ) ∂ω ∂ω 1+Y2 [(aσ aσ′ + 1)2 + (aσ + aσ′ )2 ] × 2(1 − Y 2 )2 2Y 2 ′ ′ ′ (aσ aσ + 1)(aσ + aσ ) + (bσ bσ ) , (4.6.14) + (1 − Y 2 )2 with Y = Ωe /ω and where the parameters describing the longitudinal and transverse components of the polarization appear in aσ = Lσ sin θ + Tσ cos θ, and similarly for the primed quantities.

bσ = Lσ cos θ − Tσ sin θ,

(4.6.15)

4.6 Thomson scattering in a magnetic field

181

4.6.6 Scattering of high-frequency waves At sufficiently high frequencies, the magnetoionic waves become nearly circularly polarized, except for a small range of angles about perpendicular propagation, with refractive indices close to unity. A generalization that is important in very low density plasmas with ωp ≪ Ωe , is when the frequency is much greater than the plasma frequency, ωp , but not necessarily small in comparison with the cyclotron frequency, Ωe . This limiting case is described by expanding in X ≪ 1 in the formulae in §3.2.1 that describe the magnetoionic theory. This gives 1

Tσ ≈ − 2

Y sin2 θ + σ( 14 Y 2 sin4 θ + cos2 θ)1/2 , cos θ

Lσ ≈ 0,

n2σ ≈ 1.

(4.6.16) Further simplification occurs for Y ≪ 1, when one has Tσ = −σ cos θ/| cos θ|, corresponding to circular polarization. In this limit, (4.6.14) simplifies to 1 [(1 + cos2 θ)(1 + cos2 θ′ ) + 2 sin2 θ sin2 θ′ + 4σσ ′ | cos θ cos θ′ |]. 8 (4.6.17) The sign σσ ′ is equal to +1 if the scattered wave is in the same mode as the unscattered wave, and equal to −1 is the mode changes. Scattering in which there is no change in mode is preferred. In the high-frequency, the scattering is equivalent to Thomson scattering in the absence of a magnetic field. This may be seen by considering isotropic, unpolarized initial radiation, and averaging over the angular distribution of the scattered radiation. The average over polarizations implies the term involving σσ ′ in (4.6.17) gives zero, and the averages over cos θ and cos θ′ imply hfσσ′ (ω, θ, θ′ )i = 4/3. The scattering cross section then reduces to the Thomson cross section. fσσ′ (ω, θ, θ′ ) ≈

4.6.7 Resonant Thomson scattering A specific case of interest in astrophysics is the scattering of waves with frequencies of order the cyclotron frequency by nonrelativistic electrons in a plasma with ωp ≪ Ωe . The factor (1 − Y 2 )2 in the denominator in (4.6.14) suggests that the scattering cross section diverges diverges ∝ 1/(1 − Y )2 ∝ 1/(ω − Ωe )2 for ω → Ωe . The enhanced scattering associated with this factor becoming very large is referred to as resonant scattering. Three aspects of resonant scattering are discussed here: first, the wave dispersion is determined by the cold plasma, then the effect of thermal motions is taken into account, and finally the extremely low density case is discussed, where the wave dispersion is determined by the birefringent vacuum. In the case where the wave dispersion is due to a cold plasma, as the resonance is approached, the axial ratios, Tσ , for the two modes approach

182

4 Gyromagnetic processes

T+ = −1/ cos θ, T− = cos θ, which corespond to the o and x modes, respectively. One has ao = 0, ax = cos2 θ. In this case, (4.6.14) implies foo′ = fox′ = fxo′ = 0, and fxx′ (ω, θ, θ′ ) ≈

(1 + cos2 θ cos2 θ′ )2 + (cos2 θ + cos2 θ′ )2 Ωe2 (4.6.18) 2 8(ω − Ωe ) (1 + cos2 θ)(1 + cos2 θ′ )

near the resonance at ω = Ωe . In this case, only the x mode is involved in resonant scattering. The enhancement associated with resonant scattering is limited by the dispersion itself: sufficiently near the resonance, the approximations (4.6.17) in which the refractive index is set to unity and the longitudinal part of the polarization is neglected is not justified. Near the resonance or has n2x ≈ 1 −

X , 1−Y

Lx ≈

X sin θ , 1−Y

(4.6.19)

and both diverge at the resonance. From (4.6.19) it follows that the neglect of the terms proportional to X is invalid for 1 − Y < ∼ X. This suggests that the maximum enhancement is by a factor ∼ 1/4(1 − Y )2 ∼ 1/4X 2 = (Ωe /ωp )4 /4. However, this estimate neglects the derivatives of the refractive index in (4.6.14) ∂ X , (4.6.20) nx (ωnx ) ≈ 1 + ∂ω 2(1 − Y )2

suggesting that the neglect of the terms involving X is valid only for 2(1 − Y 2 ) ≪ X, and that the enhancement is limited to a much smaller factor ∼ 1/4(1 − Y )2 ∼ 1/2X = (Ωe /ωp )2 /2. Note that the factor (4.6.20) does not appear in the cross section (??), due to the cancelation of two effects, and the implied large enhancement of the cross section for X < 1 − Y < (X/2)1/2 needs to be interpreted with care. Resonant scattering can give a large enhancement only if thermal effects are neglected. Thermal effects modify the dispersion near the cyclotron resonance, as discussed in §??, and this provides another limit on the enhancement associated with resonant scattering. To be consistent, when thermal effects are included in the wave dispersion, they also need to be included in the scattering itself. Specifically, one needs to average the probability for scattering over the thermal distribution of particles assumed to determine the wave dispersion. Averaging the enhancement factor for resonant scattering over a thermal distribution leads to an enhancement factor Z ∞ 2 2 dvz e−vz /2Ve Ωe2 ωe2 = − [1 − φ(ye )], (4.6.21) kz2 Ve2 (2π)1/1 Ve −∞ (ω − Ωe − kz vz )2 √ 2 Ry 2 with ye = (ω − Ωe )/ 2 |kz |Ve , and where φ(y) = 2ye−y 0 dt et is a form of the nonrelativistic plasma dispersion function. The maximum value of the integral (4.6.21) is for y of order unity. This implies that the maximum enhancement factor in resonant scattering is limited by thermal effects to of order c2 /Ve2 in ordinary units.

References

183

The other case of interest is where the density of the scattering electrons is so low that their contribution to the wave dispersion can be neglected in comparison with the contributions of the birefringent vacuum, as discussed in §??. The two modes, labeled ⊥, k, have T⊥ = 0, Tk = ∞. In this case, (4.6.14) gives       0 f⊥⊥′ 1 2  f⊥k′      0 cos2 θ = 1+Y  . +  (4.6.22)  fk⊥′  2(1 − Y 2 )2  cos2 θ′    0 2 2 ′ 2 2 ′ f k k′ cos θ cos θ sin θ sin θ

It follows that there is a resonance at the cyclotron frequency in all four scattering channels. Such resonant scattering is thought to play a role in pair production in pulsars [refs]. The scattering particles are highly relativistic and they scatter thermal photons from the neutron-star surface into high energy photons. In the rest frame of the scattering particle the initial and final frequencies satisfy ω ≈ ω ′ ≈ Ωe , and the boost in frequency occurs for the scattered photons in a forward cone on transforming to frame in which the scattering particle is highly relativistic.

References 1. 2. 3. 4. 5.

Schott (1912) Wild, Hill (1971) Trubnikov (1958) Sazonov (1969) Melrose, Sy (1972)

5 Magnetized Dirac electron

In this chapter Dirac’s equation is solved for an electron in the presence of a magnetostatic field. A complication is that Dirac’s equation, and its solutions, depend both on an arbitrary choice of gauge and on the choice of the spin operator. It is possible to separate the wave function into a gauge- and spin-dependent factor and a reduced wave function that satisfies a reduced form of Dirac’s equation. In generalizing QED to include the magnetic field exactly, a complication is that the conventional momentum representation for the Feynman amplitudes does not exist, because momentum perpendicular to the magnetic field is not conserved. However, the separation of the wave function allows an analogous separation of the electron propagator and the vertex functions, such that the gauge-dependent part can be treated separately; the gauge-independent part leads to a ‘reduced’ theory, involving the remaining dynamical variables, that is closely analogous to the momentum-space representation in the unmagnetized case. The gauge-dependent part (partially) describes the location of the center of gyration of the electron, and how it changes in a QED interaction, and such information is rarely of interest, and is simply ignored when using the reduced theory. Dirac’s equation in a magnetostatic field is written down in §5.1, and solutions are found for a convenient but implicit choice of spin operator. Eigenfunctions for well-defined spin operators are derived in §5.2. The electron propagator in a magnetostatic field is written down in §5.3, evaluated explicitly for the magnetized vacuum. In §5.4, the vertex function is written down and factorized into a gauge- and spin-dependent part; the gauge-independent part is evaluated explicitly for the different spin eigenfunctions. The reduced, gauge-independent formalism is developed in §5.5. Feynman rules for QPD processes in a magnetic field are summarized in §5.6.

186

5 Magnetized Dirac electron

5.1 Dirac wave functions in a magnetostatic field Explicit solutions of Dirac’s equation in the presence of a magnetostatic field, B, depend on the choice of gauge for the vector potential, A(x), for B, and on the choice of spin operator. However, the energy eigenvalues are independent of both choices. In this section solutions of Dirac’s equation are derived in the Landau gauge, and for an implicit choice of spin operator. 5.1.1 Review of Dirac’s equation for B = 0 The Dirac wave function, Ψ (x), has four complex components, which are written as a column matrix. Observable quantities are represented by operators which are 4 × 4 matrices. The Dirac matrices, γ µ are four such matrices that are assumed to transform as a 4-vector under a Lorentz transformation, and which satisfy γ µ γ ν + γ ν γ µ = 2g µν , (5.1.1) where it is implicit that the unit 4 × 4 matrix multiplies 2g µν on the right hand side. The covariant form of Dirac’s equation is (i/∂ − m) Ψ (x) = (ˆ/p − m) Ψ (x) = 0,

(5.1.2)

where the slash notation is defined by A / = γ µ Aµ ,

/∂ = γ µ ∂µ ,

(5.1.3)

for any 4-vector Aµ . The Dirac Hamiltonian is identified as ˆ =α·p ˆ + βm, H

α = γ 0 γ,

β = γ0,

(5.1.4)

ˆ = −i∂/∂x. The Dirac adjoint of the wave function is defined by with p Ψ (x) = Ψ † (x)γ 0 ,

(5.1.5)

and the adjoint of the Dirac equation in the form (5.1.2) becomes Ψ (x) (ˆ/p − m) = 0,

(5.1.6)

where the operators operate to the left. A specific choice for the Dirac matrices needs to be made for the purposes of detailed calculations, and here the standard representation is chosen. It corresponds to     1 0 0 0 0 0 0 1 0 1 0 0  0 0 1 0   γ0 =  γ1 =   0 0 −1 0  ,  0 −1 0 0  , 0 0 0 −1 −1 0 0 0     0 0 1 0 0 0 0 −i  0 0 0 −1   0 0 i 0   (5.1.7) γ3 =  γ2 =   −1 0 0 0  .  0 i 0 0, 0 1 0 0 −i 0 0 0

5.1 Dirac wave functions in a magnetostatic field

187

A convenient way of writing these and other 4× 4 matrices is in terms of block matrices. Let 0 and 1 be the null and unit 2 × 2 matrices. One writes σ 0 0 1 Σ= , ρx = , 0 σ 1 0 1 0 0 −i1 , (5.1.8) , ρz = ρy = 0 −1 i1 0 where the 2 × 2 matrices 0 1 σx = , 1 0

σy =

0 −i i 0

,

σz =

1 0 0 −1

,

(5.1.9)

are the usual Pauli matrices. In this representation one has γ µ = [ρz , iρy Σ],

α = ρx σ,

β = ρz .

(5.1.10)

5.1.2 Dirac’s equation in a magnetostatic field The minimal coupling procedure for including an electromagnetic field with 4-potential A(x) in Dirac’s equation is to replace pˆµ by pˆµ + eAˆµ (x), where −e is the charge on an electron. The Dirac Hamiltonian (A.3.30) becomes ˆ = α · [p + eA(x)] + βm − eφ(x). H

(5.1.11)

A variety of choices is possible for Aµ = [φ, A] for a uniform, magnetostatic field, B, and all involve a dependence on at least one component of the 4-vector x. The choices made in practice involve the Coulomb gauge (A0 (x) = 0), but this does not determine A uniquely. Let the magnetostatic field be along the z-axis. One choice of gauge is A = (0, Bx, 0),

(5.1.12)

which is called the Landau gauge. Other choices of gauge are related to (5.1.12) by adding the gradient of a scalar to A. One alternative choice of gauge is A = (−By, 0, 0),

(5.1.13)

and another is the cylindrical gauge A = 12 (−By, Bx, 0) = 12 B̟(− sin φ, cos φ, 0),

(5.1.14)

with ̟ = (x2 + y 2 )1/2 and x = ̟ cos φ, y = ̟ sin φ. The Landau gauge gauge is adopted for most purposes in the following discussion, with some results also written down for the cylindrical gauge. The introduction of a magnetostatic field leads to the Hamiltonian depending on a spatial coordinate, with the specific coordinate depending on

188

5 Magnetized Dirac electron

the choice of gauge. This leads to a conceptual complication: our description of the system depends in a nontrivial way on the choice of gauge. In the absence of the field, there are plane wave solutions that depend on the components of x in the form exp(−iP x), where the components of P µ are constants of the motion. In the presence of a magnetostatic field, the component of P µ conjugate to the component of x that appears in the Hamiltonian is not conserved. With the choice of the Landau gauge (5.1.12), one is free to seek solutions of the form exp(−iEt + iPy y + iPz z), where E, Py , Pz are constants of the motion, but the momentum, Px say, conjugate to x is not a constant of the motion. Alternatively, with the choice (5.1.14), one is free to seek solutions of the form exp(−iEt + iPz z + iPφ φ), where E, Pz are the momentum Pφ conjugate to φ are constants of the motion, but the momentum conjugate to the coordinate ̟ is not conserved. (It is possible to choose the temporal gauge, with A = −Bt dependent on time, and then Px , Py , Pz are constants of the motion, but the energy, P 0 = E, is not conserved.) From these remarks it is clear that the interpretation of the momentum components perpendicular to B requires care. For the Landau gauge, Py is interpreted as specifying the x-component of the center of gyration, and then the uncertainty principle implies that we have no information on the value of the conjugate momentum, Px . With the choice (5.1.14), it is the radial distance of the center of gyration from a particular field line that is specified, and we then have no information on the conjugate (radial) momentum. The detailed discussion below is for the Landau gauge (5.1.12), and some specific results for the cylindrical gauge (5.1.14) are noted. 5.1.3 Construction of the wave functions One is free to assume a wave function of the form Ψ (t, x) = f (x) e−iEt+iPy y+iPz z = f (x) e−iǫ(εt−py y−pz z) ,

(5.1.15)

where ǫ = ±1 is the sign of the energy, whose magnitude is ε. The function f (x) is a column matrix whose components are denoted by f1 . . . f4 . On inserting the trial solution (5.1.15) into Dirac’s equation in the form ∂ ˆ i − H Ψ (t, x) = 0, (5.1.16) ∂t ˆ given by (5.1.11), (5.1.12) in the Coulomb gauge, one requires with H  ˆ1   f (x)  O −ǫε + m 0 ǫpz 1 ˆ2   f2 (x)   0 −ǫε + m O −ǫp z    (5.1.17)   f3 (x)  = 0,  ǫpz ˆ1 O −ǫε − m 0 ˆ2 f4 (x) O −ǫpz 0 −ǫε − m ∂ ∂ ˆ ˆ O1 = −i + ǫpy + eBx , O2 = −i − ǫpy − eBx . (5.1.18) ∂x ∂x

5.1 Dirac wave functions in a magnetostatic field

189

It is convenient to write

so that (5.1.17) reduces to

ǫpy ξ = (eB)1/2 x + , eB

(5.1.19)

(−ǫε + m)f1 + ǫpz f3 − i(eB)1/2 (ξ + d/dξ)f4 = 0,

(−ǫε + m)f2 − ǫpz f4 + i(eB)1/2 (ξ − d/dξ)f3 = 0,

(−ǫε − m)f3 + ǫpz f1 − i(eB)1/2 (ξ + d/dξ)f2 = 0,

(−ǫε − m)f4 − ǫpz f2 + i(eB)1/2 (ξ − d/dξ)f1 = 0.

(5.1.20)

Operating on the first and third of these equations with (ξ − d/dξ) and on the second and fourth with (ξ + d/dξ), the four first order equations are replaced by two second order equations: 2 ε2 − m2 − p2z d 2 + − (ξ + 1) f1,3 = 0, dξ 2 eB 2 d ε2 − m2 − p2z 2 + − (ξ − 1) f2,4 = 0. (5.1.21) dξ 2 eB Equations (5.1.21) are of the same form as Schr¨odinger’s equation for a simple harmonic oscillator. The solutions are simple harmonic oscillator wave functions, which exist only for discrete energy eigenvalues (n + 12 )¯ hω for an oscillator with frequency ω. The differential equations (5.1.21) have normalizable solutions only if the constant n, defined by n=

ε2 − m2 − p2z , 2eB

(5.1.22)

has non-negative integral values. It is convenient to introduce another nonnegative integer, l, by writing 2n ∓ 1 = 2l + 1.

(5.1.23)

The interpretation of n and l is simplest in the nonrelativistic limit: n is the quantum number that determines the perpendicular energy of the particle, and it is composed of a gyrational motion, which is simple harmonic motion with energy (l + 12 )Ω, and a spin part, 12 sΩ, with s = ±1. In the relativistic case the corresponding contributions are to p2⊥ and they are (2l + 1)eB and seB, respectively. The normalized solutions are the harmonic oscillator wave functions 2 1 vn (ξ) = √ n 1/2 Hn (ξ) e−ξ /2 , ( π2 n!)

(5.1.24)

where Hn is a hermite polynomial. The differential operators in (5.1.20) become the raising and lowering operators that satisfy

190

5 Magnetized Dirac electron s = -1 4m

s = +1

n = 7 n = 6 n = 5

3m

n = 4 n = 3 n = 2

2m n = 1

m

n = 0

Fig. 5.1. The energy eigenvalues for B/Bc = 1, pz = 0 for n = 0, 1, . . . , 7. The uneven spacing between the levels is referred to as the anharmonicity. The levels for spin states s = +1, s = −1 are degenerate except for the ground state, n = 0, which has s = −1.

√ d ξ+ vn (ξ) = 2n vn−1 (ξ), dξ

p d ξ− vn (ξ) = 2(n + 1) vn+1 (ξ). dξ (5.1.25) A general solution of (5.1.20) may be written in the form   C1 vn−1 (ξ)  C2 vn (ξ)   (5.1.26) f (x) =   C3 vn−1 (ξ)  , C4 vn (ξ)

where C1 , . . . , C4 are constants. For convenience, so that (5.1.26) includes the ground state n = 0, it is assumed that v−1 (ξ) is identically zero. All the states except the ground state are doubly degenerate, as may be seen by writing (5.1.23) in the form n = l + 12 (1 + s), with s = ±1 as the spin eigenvalue. The ground state, n = 0, has l = 0, s = −1, and states with n > 1 are doubly degenerate with s = ±1, l = n − 21 (1 + s). The particle energy eigenvalues are ε = εn (pz ), with εn (pz ) = (m2 + p2z + 2neB)1/2 ,

n = l + 12 (1 + s).

(5.1.27)

With the sign ǫ included in P 0 = ǫε, the energy eigenvalues for positrons are the same as for electrons: ε = εn (pz ). Note that this fixes an ambiguity in the choice of sign of the spin of the positron relative to the electron: the ground state is l = 0, s = −1 for both an electron and a positron. For convenience in notation, the abbreviation εn (pz ) → εn is used when no confusion should result. The energy eigenvalues are illustrated in Fig. 5.1, showing the two branches with s = −1 and s = +1. The spacing between the energy eigenvalues decreases as n increases. This is a relativistic effect. In contrast, in the nonrelativistic case, the difference between neighboring eigenvalues is ¯hΩ independent of n, as for a simple harmonic oscillator. As a consequence the relativistic

5.1 Dirac wave functions in a magnetostatic field

191

dependence of the energy spacing on n is sometimes referred to as the anharmonicity. 5.1.4 Magnetic moment of the electron The energy eigenvalues (5.1.27) may be written εn = (m2 + p2z + p2n )1/2 , with pn = (2neB)1/2 , and the two contributions to n = l + 12 (1 + s) may be interpreted as an orbital part, described by l = 0, 1, 2, . . ., and a spin part, described by s = ±1. The orbital part describes the perpendicular motion, which is simple harmonic motion. The remaining part is interpreted as the magnetic energy, µ · B, due to a magnetic dipole µ in the magnetic field. Dirac’s theory thus predicts a magnetic moment (in SI units) ¯e h (5.1.28) 2m for the electron. In the nonrelativistic Pauli-Schr¨odinger theory the gyromagnetic ratio of the electron, that is the ratio of the magnetic moment to the spin, is undetermined. Dirac’s theory, in its simplest form, predicts the gyromagnetic ratio of the electron to have the value of 2, which is very close to the experimental value, and this prediction was one of the major successes of the theory. When radiative corrections to QED are included the small experimental difference from 2 is explained with high accuracy. µB =

5.1.5 Johnson-Lippmann wave functions It is possible to write down four independent solutions of Dirac’s equation without identifying the spin operator explicitly. The procedure used here follows Johnson and Lippmann [1]. One may construct two independent eigenstate for the doubly degenerate energy eigenvalues by choosing the first two columns of the matrix of coefficients in (5.5.11). This gives        ǫεn + m 0 C1   C2   1 − s  ǫεn + m   0    = cn  1 + s    (5.1.29)  C3   2  ǫpz  + 2  −ipn  , ipn −ǫpz C4 with the normalization coefficient identified as cn = 1/[2ǫεn (ǫεn + m)V ]1/2 . The four solutions, written Ψqǫ (t, x) with q denoting the quantum numbers pz , n, s collectively, are

e−iǫεt+iǫpy y+iǫpz z Ψqǫ (t, x) = [2ǫεn (ǫεn + m)V ]1/2      (ǫεn + m)vn−1 (ξ) 0  1 − s  (ǫεn + m)vn (ξ)  1 + s  0 +  . (5.1.30)   ×   2  ǫpz vn−1 (ξ) 2  −ipn vn−1 (ξ)  ipn vn (ξ) −ǫpz vn (ξ)

192

5 Magnetized Dirac electron

The solutions in the form (5.1.30) were written down by [1], and are referred to here as the Johnson-Lippmann wave functions. Although the four solutions do not correspond to any sensibly defined spin operator they may be interpreted loosely in terms of spin up and down (s = ±1) for a nonrelativistic electron. Recall that in the nonrelativistic Schr¨odinger-Pauli theory, the spin is independent of the other terms in the Hamiltonian. Spin-orbit coupling, which is an intrinsically relativistic effect, disappears in the nonrelativistic limit of Dirac’s theory so that all definitions of the spin operator become equivalent to the Schr¨odinger-Pauli theory. When one is interested in spin-dependent effects, the choice (5.1.30) is not appropriate, and a specific choice of spin operator needs to be made (§5.2). The choice (5.1.30) may be used when one is not interested in the spin and where either a sum or an average over the spin states is performed. 5.1.6 Orthogonality and completeness relations The orthogonality relation between the wave functions for different eigenstates is of the general form Z (5.1.31) d3 x [Ψq (t, x)]† Ψq′ (t, x) = δqq′ , where q and q ′ denote two sets of eigenvalues collectively. The completeness relation for the wave functions is X Ψq (t, x)[Ψq (t, x′ )]† = δ 3 (x − x′ ). (5.1.32) q

In the present case some of the quantum numbers are discrete (ǫ, l, s) and some are continuous (py , pz ). The sum and the Kronecker-δ are appropriate for discrete quantum numbers, and for continuous quantum numbers, these are replaced by integrals and Dirac δ-functions, respectively. To rewrite the sums as integrals, and Kronecker δs as Dirac δs, one needs to take the normalization conditions into account and ensure that the resulting integrals and Dirac δs are dimensionless. Consider the case where the particle is confined to a large but finite box, in which case all the quantum numbers are discrete. Let the sides of the box be of length Lx , Ly , Lz in the x, y, z directions, respectively. The eigenvalues py , pz are discrete, with values py = ny 2π/Ly , pz = nz 2π/Lz with ny , nz = 0, ±1, ±2, . . .. The sum over all states includes sums over ny , nz . To identify the corresponding integrals and Dirac δ-functions in the continuous limit, the box is allowed to extend to infinity. The basic identification is that the difference between ny , ny + 1 correponds to a difference δpy = 2π/Ly in py , and the difference between nz , nz + 1 correponds to a difference δpz = 2π/Lz in pz . The sum over states becomes

5.1 Dirac wave functions in a magnetostatic field

X

=

q

X

∞ X

ǫ,s=± n=0

Ly Lz

Z

dpy 2π

Z

dpz . 2π

193

(5.1.33)

The Kronecker-δ that expresses the orthogonality of the states becomes δqq′ = δss′ δnn′

2π 2π δ(py − p′y ) δ(pz − p′z ). Ly Lz

(5.1.34)

These results apply for the Landau gauge. There are analogousR results for other choices of gauge. In particular, for the cylindrical gauge, P Ly dpy /2π in (5.1.33) and (2π/Ly )δ(py − p′y ) in (5.1.34) are replaced by r and δrr′ , where r is the radial quantum number introduced in (5.2.26) below. 5.1.7 Normalization of the wave function The conventional normalization of the Dirac wave function in QED is to one charge in the volume V . The normalization is determined by integrating the probability density over V and setting the result equal to unity, as in (5.1.31) with q ′ = q. A normalization is implicit in (5.1.30), and evaluating the integral in (5.1.31) the factor on the right hand side of (5.1.31) is (1/eB)1/2 Ly Lz /V . There are three possible choices: accept that the normalization in (5.1.30) corresponds to 1/Lx(eB)1/2 charges in the volume V , adjust the normalization in (5.1.30) so that it corresponds to one charge in the volume V for arbitrary Lx , or choose Lx = (1/eB)1/2 . In the cylindrical gauge, the natural normalization of the wave functions corresponds to 1/AeB charges in the volume V = ALz with A = πR2 where R is the radius of the normalization cylinder, and three analogous choices are possible. The third of these choices is made here in both case. Hence, to preserve the normalization to one charge in the volume V , the following choices are made: Lx =

1 eB

1/2

in these two cases, respectively.

,

A = πR2 =

1 , eB

(5.1.35)

194

5 Magnetized Dirac electron

5.2 Specific spin eigenfunctions One specific solution of the Dirac wave function for an electron in a magnetostatic field is written down in (5.1.30), referred to as the Johnson-Lippmann wave functions. These wave functions do not correspond to any physically relevant spin operator. In this section solutions are written down for two specific spin operators: the helicity and the parallel (to B) component of the magnetic-moment operator. These solutions are derived in the Landau gauge (5.1.12); a solution is also written down for the choice of the cylindrical gauge (5.1.14). 5.2.1 Helicity eigenstates in a magnetic field ˆ , in the absence of a magnetic field is the timeThe helicity operator, σ · p component of a 4-vector. This operator commutes with the Hamiltonian and hence is a constant of the motion. As a consequence there are well-defined simultaneous eigenstates of both the Hamiltonian and the helicity operator. When a magnetostatic field is included, using the minimal coupling assumption, the helicity operator becomes ˆ = σ · (ˆ h p + eA).

(5.2.1)

The helicity is a constant of the motion in the presence of a magnetic field, provided that there is no electric field. Hence we can construct simultaneous eigenstates of this operator and of the Hamiltonian. Evaluating the helicity operator in the Landau gauge gives ǫpz Xˆ+  Xˆ− −ǫpz σ · [ˆ p + eA] =   0 0 0 0 

 0 0 0 0  , ˆ ǫpz X+  Xˆ− −ǫpz

√ Xˆ± = −i eB

∂ ± ξ . (5.2.2) ∂ξ

Let the eigenvalues of the helicity operator be ±h, with the magnitude, h, to be determined. The sign of the spin eigenvalue depends on the spin quantum number, σ = ±1. The sign σ = −1 is required for the ground state, and one finds this condition is satisfied only if the helicity eigenvalue is written as σǫP h with P = pz /|pz |. Hence, the eigenvalue equation is σ · [ˆ p + eA]Ψqǫ (t, x) = σǫP h Ψqǫ (t, x).

(5.2.3)

The only change from the Johnson-Lippmann wave functions is in the coefficients Ci in (5.1.26). For the helicity states, in place of (5.1.29), the coefficients Ci are determined by (5.2.3), which is regarded as an eigenvalue equation. The explicit form of this equation is

5.2 Specific spin eigenfunctions





195



C1 ǫpz − σǫP h −ipn 0 0   C2  ipn −ǫpz − σǫP h 0 0    = 0.   C3  0 0 ǫpz − σǫP h −ipn C4 0 0 ipn −ǫpz − σǫP h (5.2.4) The determinant of the matrix of coefficients in (5.2.4) is (h2 − p2n − p2z )2 . Setting this to zero, it follows that the eigenvalues of the helicity operator are doubly degenerate with eigenvalues of magnitude   

h = (p2n + p2z )1/2 = (ε2n − m2 )1/2 .

(5.2.5)

If the eigenvalues were not degenerate, the eigenfunctions could be constructed from the inverse of the square matrix in (5.2.4), but the degeneracy precludes this procedure because the inverse of the matrix of coefficients (5.2.4) is singular. Simultaneous eigenfunctions of the helicity and energy may be constructed by starting with an arbitrary linear combinations of the doubly degenerate Johnson-Lippmann wave functions (5.1.30). The ratio of the coefficients in the combination is found by applying the helicity operator and requiring that the eigenvalues be σP h. The solution is determined only to within an arbitrary phase factor for each eigenfunction. For any choice of spin operator, the ground state (n = 0) wave function has the same form as for the Johnson-Lippmann wave functions, and as already remarked, this criterion requires the presence of the sign ǫP in (5.2.3). The helicity eigenfunctions may be written in a variety of equivalent forms, by making particular choices of the overall and relative phases of the different eigenfunction, and by using the identity pn = [(h + σ|pz |)(h − σ|pz |)]1/2 .

(5.2.6)

Specific simultaneous eigenfunctions of the helicity operator (eigenvalues σǫP h) and of the Hamiltonian (eigenvalues ǫεn ) correspond to     [εn + ǫm]1/2 (h + σ|pz |)1/2 C1  C2  iσǫP [εn + ǫm]1/2 (h − σ|pz |)1/2  1   = ,  (5.2.7)  C3  [2h2εn V ]1/2  σP [εn − ǫm]1/2 (h + σ|pz |)1/2  1/2 1/2 C4 hel iǫ[εn − ǫm] (h − σ|pz |) with εn given by (5.1.27), with h given by (5.2.5) and with P = pz /|pz |. 5.2.2 Magnetic moment eigenstates The magnetic-moment operator in the absence of a magnetic field is discussed is ˆ = mσ − iγ × p ˆ, µ (5.2.8)

In the presence of a magnetic field the minimal coupling assumption implies that the magnetic-moment operator and its z-component in the Landau gauge are

196

5 Magnetized Dirac electron

ˆ = mσ − iγ × (ˆ µ p + eA),

m 0 0  0 −m −Xˆ− µ ˆz =   0 −Xˆ+ m Xˆ− 0 0 

 Xˆ+ 0  , 0  −m

(5.2.9)

respectively, with Xˆ± defined by (5.2.2). The simultaneous eigenvalues of the operator (5.2.9) and the Hamiltonian are found in the same way as for the helicity operator. Suppose that eigenvalues of µ ˆz are sλ, with λ yet to be determined. The operator Xˆ± has eigenvalues ∓ipn , and in place of (5.2.4), one finds    m − sλ 0 0 −ipn C1  0    −m − sλ −ip 0 n    C2  = 0. (5.2.10)  0   C3  ipn m − sλ 0 ipn 0 0 −m − sλ C4 The determinant of the matrix of coefficients give (λ2 − m2 − p2n )2 . Hence, there are degenerate eigenfunctions with eigenvalues sλ, s = ±1, with λ = ε0n , ε0n = (m2 + p2n )1/2 = (ε2n − p2z )1/2 .

(5.2.11)

The eigenfunctions are linear combinations of the Johnson-Lippmann wave functions. The ground state (n = 0, s = −1) must be the same (to within an arbitrary phase) for all choices of spin operator. One finds that simultaneous eigenfunctions of the Hamiltonian and magnetic moment operators correspond to     aǫs bs C1   C2  1   −is a−ǫs b−s  ,   =  C3  V 1/2  a−ǫs bs  is aǫs b−s C4 mm 0 1/2 1/2 εn + sm εn ± εn , b = . (5.2.12) a± = P± s 2ε0n 2ε0n where the subscript ‘mm’ denotes the magnetic moment eigenfunctions, and where the identities pz = 2εn aǫs a−ǫs , pn = 2ε0n bs b−s are used. The sign P± = 21 (1 + P ) ± 21 (1 − P ),

P = pz /|pz |,

(5.2.13)

is equal to unity for a+ and to P for a− . The overall phase of either eigenfunction is arbitrary, and so are the relative phase of the four eigenfunctions; these phases are chosen for convenience in writing down the form (5.2.12). An alternative form for the solutions (5.2.12), written down by Sokolov and Ternov [2], involves the sum and difference of (1 ± pz /εn )1/2 in place of the a± . The identities ε0n = (εn ± pz )1/2 (εn ∓ pz )1/2 ,

εn ± ε0n = 21 [(εn + pz )1/2 ± (εn − pz )1/2 ]2 , (5.2.14)

5.2 Specific spin eigenfunctions

197

relate the two notations. The choice of eigenfunctions made by Sokolov and Ternov [2] is     B1 (A1 + A2 ) C1  C2   1   −iB2 (A1 − A2 )  ,   = √ (5.2.15)   C3  B1 (A1 − A2 )  2 2V C4 mm iB2 (A1 + A2 ) 1/2 ǫpz , 1+ εn 1/2 sm B1 = 1 + 0 , εn

A1 =

1/2 ǫpz A2 = sǫ 1 − , εn 1/2 sm B2 = s 1 − 0 . εn

(5.2.16)

The relative phases of the four eigensolutions are again chosen for convenience in writing. The magnetic moment eigenfunctions and the Johnson-Lippmann wave functions are equivalent for nonrelativistic electrons. This may be seen by setting ǫ = 1 in (5.2.12), and making the nonrelativistic approximation in the form p2 p2 + p2n , ε0n = m + n . (5.2.17) εn = m + z 2m 2m When only first order terms in pz /m, pn /m are retained, the wave functions are equivalent. This justifies the use of the Johnson-Lippmann wave functions for nonrelativistic electrons. For positrons the Johnson-Lippmann wave functions do not correspond to any physically relevant spin eigenfunctions, and they should not be used even for nonrelativistic positrons. 5.2.3 Eigenstates in the cylindrical gauge Suppose that in place of the Landau gauge (5.1.12) one chooses the cylindrical gauge (5.1.14), viz. A = 12 (−By, Bx, 0) = 12 B̟(− sin φ, cos φ, 0),

̟ = (x2 + y 2 )1/2 , (5.2.18)

with x = ̟ cos φ, y = ̟ sin φ. In this case, z and φ are ignorable coordinates. In place of (5.1.15) an appropriate trial wave function is Ψ (t, x) = g(̟, φ) exp(−iǫεt + iǫpz z). In place of (5.1.17), Dirac’s equation gives ˆ1   g1 (̟, φ)  D −ǫε + m 0 ǫpz ˆ2    0 −ǫε + m D −ǫpz    g2 (̟, φ)  = 0,     ǫpz ˆ g3 (̟, φ)  D1 −ǫε − m 0 ˆ2 g4 (̟, φ) D −ǫpz 0 −ǫε − m 

(5.2.19)

198

5 Magnetized Dirac electron

∂ i ∂ − + 12 eB̟ , ∂̟ ̟ ∂φ ∂ i ∂ iφ 1 ˆ D2 = −ie + − eB̟ . ∂̟ ̟ ∂φ 2

ˆ 1 = −ie−iφ D

(5.2.20)

The dependence on φ is satisfied by the choice g1 (̟, φ) = g1 (̟) exp[i(a − 1)φ],

g2 (̟, φ) = g2 (̟) exp[iaφ],

g3 (̟, φ) = g3 (̟) exp[i(a − 1)φ],

g4 (̟, φ) = g4 (̟) exp[iaφ],

with a = 0, ±1, ±2, . . .. In place of (5.1.20) one finds d a + + 21 eB̟ g4 = 0, (−ǫε + m)g1 + ǫpz g3 − i d̟ ̟ d a−1 1 (−ǫε + m)g2 − ǫpz g4 − i − − 2 eB̟ g3 = 0, d̟ ̟ a d + + 21 eB̟ g2 = 0, (−ǫε − m)g3 + ǫpz g1 − i d̟ ̟ d a−1 1 (−ǫε − m)g4 − ǫpz g2 − i − − 2 eB̟ g1 = 0. d̟ ̟ In place of (5.1.21) one finds 2 1 d (a − 1)2 d 2 2 1 2 + + eB(2n − a) − 4 e B ̟ g1,3 = 0, − d̟2 ̟ d̟ ̟2 2 d 1 d a2 2 2 1 2 + + eB(2n − a + 1) − 4 e B ̟ g2,4 = 0. − d̟2 ̟ d̟ ̟2

(5.2.21)

(5.2.22)

(5.2.23)

The normalizable solutions of (5.2.23) are generalized Laguerre polynomials Lνn (x), specifically the functions 1

Jνn (x) = [n!/(n + ν)!]1/2 exp(− 21 x) x 2 ν Lνn (x).

(5.2.24)

In place of (5.1.26) one obtains the solutions   r C1 Jn−r−1 ( 12 eB̟2 ) ei(n−r−1)φ r  C2 Jn−r ( 21 eB̟2 ) ei(n−r)φ  , g(̟, φ) =  r  C3 Jn−r−1 ( 21 eB̟2 ) ei(n−r−1)φ  r C4 Jn−r ( 21 eB̟2 ) ei(n−r)φ

(5.2.25)

with the so-called radial quantum number identified as r = n − a.

(5.2.26)

The construction of specific spin eigenstates involves only the determination of the ratios of the Ci , and these are the same for all choices of gauge, including the Landau and cylindrical gauges. Hence, the values (5.1.29), (5.2.7) and (5.2.12) for the Ci , for the Johnson-Lippmann, helicity and magneticmomentum eigenfunctions, respectively, also apply for the choice of the cylindrical gauge.

5.3 Electron propagator in a magnetostatic field

199

5.3 Electron propagator in a magnetostatic field The electron propagator is derived in this section both for an arbitrary magnetized electron gas, and for the magnetized vacuum. The explicit form of the propagator depends on the choices of gauge and of spin operator. However, the gauge-dependent part can be combined into a single phase factor. Explicit evaluation is possible for the propagator in the magnetized vacuum, which can be written in several superficially different forms, one of which involves a single integral over elementary functions. 5.3.1 Statistically averaged electron propagator The statistically averaged electron propagator in coordinate space is Z X dE −iE(t−t′ ) ǫ ′ ǫ ′ ¯ Ψq (x)Ψ q (x ) e G(x, x ) = 2π ǫq 1 − nǫq nǫq × . (5.3.1) + E − ǫ(εq − i0) E − ǫ(εq + i0) In (5.3.1) the electrons (ǫ = 1) and positrons (ǫ = −1) have energy eigenvalues εq and occupation numbers nǫq , where q describes any appropriate set of quantum numbers for an electron in a magnetostatic field. On inserting the wave functions given in §5.2 for a particular choice of gauge and of spin operator into (5.3.1), one obtains an explicit form for the propagator. Although the resulting expression for the propagator is gauge dependent, the gaugedependent part may be separated out into a single multiplicative function, φ(x, x′ ). To see this, it is helpful to consider the explicit form in both the Landau and cylindrical gauges. In the Landau gauge, the sum over states includes an integral over py , which is evaluated using Z ′ 2 1 dpy vn (ξ)vn (ξ ′ ) eiǫpy (y−y ) = φ(x, x′ ) e−R /4 Ln (R2 /2), 1/2 (eB) R2 = eB[(x − x′ )2 + (y − y ′ )2 ],

(5.3.2)

where Ln is the Laguerre polynomial of order n. All the gauge dependence appears in the function (5.3.3) φ(x, x′ ) = exp −ieB 12 (x + x′ )(y − y ′ ) . The phase factor (5.3.3) applies in the Landau gauge Aµ = (0, 0, Bx, 0), and it may be written " # Z x′ ′ ′′ µ ′′ φ(x, x ) = exp −ie dxµ A (x ) , (5.3.4) x

200

5 Magnetized Dirac electron

where the integral is along the straight line between the two end points. (For example, write x′′µ = xµ + α(x′ − x)µ , xµ = (t, x, y, z), etc., and integrate over 0 ≤ α ≤ 1.) A generalization, introduced Schwinger [4], is ) ( Z x′ 1 µ ′′ ′ µν ′ ′′ 1 µν ′′ , (5.3.5) φ(x, x ) exp −ie dxµ A (x ) + 2 F xν − 2 iexµ xν F x

which applies along an arbitrary path between the end points. In the analogous derivation in the cylindrical gauge, (5.1.14), the sum is over the radial quantum number r, and is performed using 2 ∞ X ′ eB̟2 eB̟′2 R 1 2 r r Jn−r Jn−r ei(n−r)(φ−φ ) = φ(x, x′ )e− 4 R Ln , 2 2 2 r=0 (5.3.6) with ̟2 = x2 + y 2 , ̟′2 = x′2 + y ′2 . The gauge-dependence in (5.3.6) is written in terms of the factor φ(x, x′ ), again given by (5.3.4) but now for the cylindrical gauge. 5.3.2 Gauge-independent form for the propagator

The sum over states in the expression (5.3.1) for the propagator gives Z ′ ∞ X X dpz eiǫpz (z−z ) eB ǫ ′ ′ ǫ A(x) + m) φ(x, x ) Ψq (x)Ψ q (x ) = (i/∂ + e/ 2π 2π 2ǫεn q n=0 2 P± = 12 (1 ± Σz ), ×e−R /4 P+ Ln−1 (R2 /2) + P− Ln (R2 /2) , (5.3.7)

with Ln (R2 /2) assumed to be identically zero for n < 0. The differential operator (i/∂ + e/ A + m) operates on all quantities to its right when (5.3.7) is inserted into (5.3.1). This differential operator is gauge-dependent, and it may be commuted with the gauge-dependent factor, φ(x, x′ ), (i/∂ + e/ A(x) + m) φ(x, x′ ) = φ(x, x′ )(i/∂ + e/b(x − x′ ) + m), bµ (x) = (0, 12 B × x).

(5.3.8)

The remaining operator, (i/∂ + e/b(x) + m), which operates on all quantities to its right when (5.3.7) is inserted into (5.3.1), is independent of the choice of gauge. Hence, all the gauge dependence remains in the phase factor φ(x, x′ ). In general, the occupation numbers in (5.3.1) depend on n, pz and are different for electrons (ǫ = 1) and positrons (ǫ = −1), so that no further evaluation of (5.3.7) is possible in general. 5.3.3 G´ eh´ eniau form for the electron propagator In vacuo the occupation numbers in (5.3.1) are zero, and the sum over n and the integral over pz in (5.3.7) may be evaluated explicity. The first step is to perform the integral over E in (5.3.1), which gives a step function

5.3 Electron propagator in a magnetostatic field

Z

201

′

′ dE e−iE(t−t ) = −iǫ H ǫ(t − t′ ) e−iǫεn (t−t ) . 2π E − ǫ(εq − i0)

(5.3.9)

The multiplicative factor of ǫ cancels with a corresponding factor in (5.3.7). The integral over pz and the sum over n can now be performed explicitly. The integral over pz is rewritten using the identity Z ∞ Z t − t′ ρ2 λ (ε0n )2 dpz −iǫ[εn (t−t′ )−pz (z−z′ )] dλ , e = exp −iǫ + εn λ |t − t′ | 2 2λ 0 ρ2 = (t − t′ )2 − (z − z ′ )2 ,

(ε0n )2 = m2 + 2neB.

(5.3.10)

In view of the step function in (5.3.9), the integral is nonzero only for ǫ(t − t′ )/|t − t′ | = 1, and the exponent in (5.3.10) simplifies accordingly. The sum over n is performed using a generating function for the Laguerre polynomials, ∞ X

n=0

1

eiαn Ln (R2 /2) = i

2 1 1 e 2 iα eR /4 e− 2 i cot 2 α . 2 sin 12 α

(5.3.11)

with α = −eB/λ. The resulting expression for the propagator for the magnetized vacuum is Z ∞ dλ 1 − iΣz tan(eB/2λ) G(x, x′ ) = −φ(x, x′ )(i/∂ + e/b(x − x′ ) + m) 8π 2 (2λ/eB) tan(eB/2λ) 0 ieB[(x − x′ )2 + (y − y ′ )2 ] iλ im2 + [(z − z ′ )2 − (t − t′ )2 ] − × exp . 4 tan(eB/2λ) 2 2λ (5.3.12) Then (5.3.12) gives the G´eh´eniau form propagator [5, 6, 7, 4, 8] G(x, x′ ) = φ(x, x′ )∆(x − x′ ),

(5.3.13)

with the gauge-independent part, ∆(x), given by Z ∞ iλ(x2 )k eB −ieB(x2 )⊥ dλ im2 ∆(x) = − , B(λ, x) exp − − 16π 2 0 λ 4 tan(eB/2λ) 2 2λ (5.3.14) with (x2 )⊥ = −x2 − y 2 , (x2 )k = t2 − z 2 , and with B(λ, x) = [(γx)⊥ 12 eB cot(eB/2λ) + λ(γx)k + e/b(x)][cot(eB/2λ) − iΣz ]   (λt + m)C− 0 −λzC− −R− C+   0 (λt + m)C+ −R+ C− λzC+ , =   λzC− R− C+ (−λt + m)C− 0 R+ C− −λzC+ 0 (−λt + m)C+ R± = 21 eB(x + iy),

C± = cot(eB/2λ) ± i.

(5.3.15)

To ensure convergence of the integral, λ is to be interpreted as (1 + i0)λ.

202

5 Magnetized Dirac electron

5.3.4 Alternative derivation of the propagator The propagator in a magnetized vacuum can also be constructed by solving the inhomogeneous Dirac equation with the external electromagnetic field included, that is, by solving (i/∂ + e/ A(x) − m)G(x, x′ ) = δ 4 (x − x′ ).

(5.3.16)

It is convenient to introduce a new function S(x, x′ ) by writing G(x, x′ ) = (i/∂ + e/ A(x) + m)S(x, x′ ), such that (5.3.16) is replaced by µ D Dµ + m2 − eS µν Fµν (x) S(x, x′ ) = δ 4 (x − x′ ), Dµ = ∂ µ − ieAµ (x),

S µν Fµν = iα · E − σ · B.

(5.3.17)

(5.3.18)

The form (5.3.18) applies for an arbitrary static electromagnetic field. On specializing to a static magnetic field and choosing the Landau gauge (5.1.12), (5.3.18) reduces to " # 2 ∂2 ∂ ∂2 ∂2 2 − 2− 2− + ieBx + m + eσ · B S(x, x′ ) = δ 4 (x − x′ ). ∂t2 ∂x ∂z ∂y (5.3.19) A solution of (5.3.19) is found by firstly solving for the corresponding Klein-Gordon equation, which is similar to (5.3.19) but with the term −eσ· B omitted. Let the solution of the Klein-Gordon equation be G0 (x, x′ ). One has # " 2 ∂ ∂2 ∂2 ∂2 2 − 2− 2− + ieBx + m G0 (x, x′ ) = δ 4 (x − x′ ). (5.3.20) ∂t2 ∂x ∂z ∂y The solution, S(x, x′ ), of (5.3.19) follows from the solution of (5.3.20) for G0 (x, x′ ) by formally replacing m2 by m2 + eσ · B. The solution of the inhomogeneous equation (5.3.20) for G0 (x, x′ ) is constructed by first considering the solutions of the homogeneous equation. With a trial solution of the form f (x) exp[−i(Et − Py y − Pz z)], one finds that solutions exist only for E = ǫεn , εn = [m2 + p2z + (2n + 1)eB]1/2 for n ≥ 0, with Py = ǫpy , Pz = ǫpz and with the solution for a given n having f (x) = vn (ξ), with vn (ξ) the simple harmonic oscillator function (5.1.26), and with ξ given by (5.1.19). The identity ∞ X

n=0

allows one to write

vn (ξ)vn (ξ ′ ) = δ(ξ − ξ ′ )

(5.3.21)

5.3 Electron propagator in a magnetostatic field

δ 3 (x − x′ ) = (eB)1/2

∞ X

vn (ξ)vn (ξ ′ )

n=0

Z

203

dpy dpz iǫ[py (y−y′ )+pz (z−z′ )] e , (5.3.22) (2π)2

with ξ = (eB)1/2 (x − ǫpy /eB), ξ ′ = (eB)1/2 (x′ − ǫpy /eB). In order that the time dependence satisfy (5.3.20) one requires that G0 (x, x′ ) be continuous at t = t′ with a discontinuous first derivative, ∂G0 (x, x′ ) 3 ′ (5.3.23) ′ = δ (x − x ), ∂t t=t as required by the integral of (5.3.20) over t. On integrating (5.3.23) over time the choice of the sign of (t − t′ )/|t − t′ | determines whether the propagator is in its retarded ((t − t′ )/|t − t′ | > 0), advanced ((t − t′ )/|t − t′ | < 0) or Feynman (ǫ(t − t′ )/|t − t′ | > 0) forms. Here we require the Feynman form, which is ∞ X G0 (x, x ) = iǫH ǫ(t − t ) (eB)1/2 ′

′

n=0

Z

′

dpy dpz vn (ξ)vn (ξ ′ ) (2π)2 ǫεn ′

′

×e−iǫ[εn (t−t )−py (y−y )−pz (z−z )] .

(5.3.24)

The py -integral is the same as in (5.3.2), and the pz -integral is closely analogous to (5.3.10). The sum over n is performed using (5.3.11). The resulting expression is Z ∞ 2 eB e−im /2λ dλ G0 (x, x′ ) = − φ(x, x′ ) 2π 8π 2 (2λ/eB) sin(eB/2λ) 0 ′ 2 ′ 2 ieB[(x − x ) + (y − y ) ] iλ ′ 2 ′ 2 × exp + [(z − z ) − (t − t ) ] . 4 tan(eB/2λ) 2

(5.3.25)

The propagator has a physical intepretation as the propagator for a charged (q = −e) spinless particle that satsifies the Klein-Gordon equation. The electron propagator is obtained from (5.3.25) by noting that the replacement m2 → m2 + eΣ · B, with Σ · B = Σz B, converts the scalar function into a 4 × 4 matrix by introducing an additional factor e−ieΣz B/2λ = cos(eB/2λ) − iΣz sin(eB/2λ)

(5.3.26)

into the integrand. The additional factor (5.3.26) converts G0 (x, x′ ), as given by (5.3.25), into S(x, x′ ), and then the propagator (5.3.12) follows from (5.3.17) and (5.3.8). 5.3.5 Spin projection operators The Dirac matrices P± , introduced in (5.3.7), play the role of projection operators onto the eigenstates of Σz . With P± = 12 (1 ± Σz ) and Σz2 = 1, one has

204

5 Magnetized Dirac electron

(P± )2 = P± ,

P+ P− = 0,

They projection operators  1 0 0 0 0 0 P+ =  0 0 1 0 0 0

P+ + P− = 1,

P+ − P− = Σz .

(5.3.27)

have the standard matrix representations    0 0 0 0 0 0 1 0 0 0 ,  P− =  (5.3.28)  0 0 0 0. 0 0 0 0 0 1

These projection operators commute with the components Dirac matrices µ γkµ , but not with γ⊥ . Specifically, the relations γkµ Σz = Σz γkµ , imply

γkµ P± = P± γkµ ,

µ µ γ⊥ Σz = −Σz γ⊥ ,

(5.3.29)

µ µ γ⊥ P± = P∓ γ⊥ .

(5.3.30)

5.4 Vertex function in a magnetic field

205

5.4 Vertex function in a magnetic field In this section, the vertex function for an electron in a magnetic field is evaluated explicitly for the three choices of spin wave functions (JohnsonLippmann, helicity and magnetic moment) made in §5.2. 5.4.1 Definition of the vertex function In developing QED or QPD for an electron gas in a magnetostatic field it is convenient to use the vertex formalism. This formalism is based on the fact that a Dirac matrix, γ µ say, always appears in matrix multiplication along an electron line between a Dirac wave function and an adjoint Dirac wave function. Consider an electron line from x′ to x. Let the vertices correspond to γ µ at x′ and γ ν at x. With the propagator in the form (5.3.1), one may associate the adjoint wave function, Ψ ǫq (x′ ), with γ µ and the wave function Ψqǫ (x) with γ ν . (Note that the time-dependence of the wave function, e−iǫεq , is omitted in defining the vertex function.) The other electron line joining the vertex at x′ corresponds to either an initial electron, a final positron or to another propagator, and in all three cases there is another wave function, ′ Ψqǫ′ (x′ ) say, associated with it. In an analogous manner, the electron line joining the vertex at x corresponds to a final electron, an initial positron or to another propagator, and in all three cases there is another wave function, ′′ Ψ ǫq′′ (x) say, associated with it. It follows that all wave functions and adjoint wave functions are paired together with a γ-matrix. One then uses (5.4.1) with (??) to express such products in terms of a (coordinate-space) vertex ′ functions of the form Ψ ǫq′ (x)γ µ Ψqǫ (x). A momentum-space representation of the vertex function is introduced by the Fourier transforming: Z ′ ǫ′ ǫ µ (5.4.1) [γq′ q (k)] = d3 x e−ik·x Ψ qǫ′ (x)γ µ Ψqǫ (x), The evaluation of this function depends on the specific choice of gauge and of the spin operator. The vertex function factorizes into a gauge-dependent factor and a gaugeindependent part ′ ′ ′ [γqǫ′ ǫq (k)]µ = dǫq′ǫq (k) [Γqǫ′ qǫ (k)]µ , (5.4.2) ′

where the factor dqǫ ′ǫq (k) contains all the gauge-dependent factors. The gauge′ independent vertex function, [Γqǫ′ qǫ (k)]µ , remains dependent on the choice of ′ spin operator. The normalization of the factor dǫq′ǫq (k) is determined by requiring that it satisfy the identity X ′ ′′ ′′ ′ dǫq′ǫq′′ (k2 ) dǫq′′ǫq (k1 ), (5.4.3) dǫq′ǫq (k1 + k2 ) = q˜′′

where the sum over q˜′′ is over the gauge-dependent quantum number (p′′y or r′′ ) and p′′z .

206

5 Magnetized Dirac electron

5.4.2 Symmetry properties of the vertex function The reality condition for Fourier transforms implies that the vertex function satisfies the identity ′ ǫǫ′ µ [γqǫ′ ǫq (k)]∗µ = [γqq (5.4.4) ′ (−k)] . Both the gauge-dependent and the gauge-independent parts in (5.4.2) separately satisfy this property: ′

′

[dǫq′ǫq (k)]∗ = dǫǫ qq′ (−k),

′

′

ǫǫ µ [Γqǫ′ qǫ (k)]∗µ = [Γqq ′ (−k)] .

(5.4.5)

A further possible symmetry property follows by changing the signs ǫ, ǫ′ , but such symmetry depends on the choice of spin eigenfunctions. There is no such symmetry for the Johnson-Lippmann wave functions, cf. (??), and the symmetry properties can be seen by inspection of (??) for the helicity states, and by inspection of (5.4.18) for the magnetic-moment states. The vertex function also satisfies the relation ′

′

kµ [Γqǫ′ qǫ (k)]µ = (ω − ǫεq + ǫ′ ε′q′ )[Γqǫ′ qǫ (k)]0 .

(5.4.6)

The right hand side of (5.4.6) is zero only when the resonance condition, ω − ǫεq + ǫ′ ε′q′ = 0, is satisfied. 5.4.3 Landau and cylindrical gauges To illustrate the factorization (5.4.2), consider its explicit forms in the Landau gauge and in the cylindrical gauge. The wave functions in the Landau gauge are given by (5.1.30). Let the components of k be written k = (k⊥ cos ψ, k⊥ sin ψ, kz ) = |k|(sin θ cos ψ, sin θ sin ψ, cos θ).

(5.4.7)

The azimuthal angle ψ may be set to zero without loss of generality, provided that only one wave is involved. Then in (5.4.1) with (5.4.7), the integrals over y and z are trivial. The integral over x reduces to a standard integral [3] Z ∞ ′ ′ dx e−ikx x vn′ (ξ ′ )vn (ξ) = (eB)−1/2 eikx (ǫpy +ǫ py )/2eB −∞

′

2 ×{ieiψ }n−n Jnn′ −n (k⊥ /2eB),

(5.4.8)

with ky = ǫpy − ǫ′ p′y = kx tan ψ and where the function Jνn (x) is defined by (5.2.24). The functions Jνn (x) play an important role in the theory, and their properties are summarized in §A.1.6. The gauge-dependent factor for the Landau gauge is identified as ′ ′

′

dǫq′ǫq (k) =

eikx (ǫpy +ǫ py )/2eB 2πδ(ǫpy − ǫ′ p′y − ky ) 2πδ(ǫpz − ǫ′ p′z − kz ). (5.4.9) V (eB)1/2

5.4 Vertex function in a magnetic field

207

In the cylindrical gauge, the definition (5.4.1) of the vertex function is unchanged, but it is to be evaluated in cylindrical polar coordinates ̟, φ, z, rather than in cartesian coordinates. The normalization factor changes in accord with (5.1.35). On inserting the wave functions (5.2.19) with (5.2.25) into (5.4.1), the integral over z is trivial, and the remaining integrals are of the form Z 2π Z ∞ ′ ′ ′ r d̟̟e−i[k⊥ ̟ cos(ψ−φ)+(n −n−r +r)φ] Jnr′ −r′ ( 12 eB̟2 ) Jn−r dφ ( 12 eB̟2 ) 0

0

′ ′ 2π 2 2 = {−ie−iψ }r−r Jrr′ −r (k⊥ /2eB) {−ie−iψ }n −n Jnn′ −n (k⊥ /2eB). (5.4.10) eB

The gauge-dependent factor for the cylindrical gauge is identified as ′

′ dǫq′ǫq (k)

2 2π {−ie−iψ }r−r Jrr′ −r (k⊥ /2eB) = 2πδ(ǫpz − ǫ′ p′z − kz ). V eB

(5.4.11)

5.4.4 Gauge-dependent factor along an electron line The multiplicative property (5.4.3) implies that the gauge-dependence involves only the initial and final states. It is straightforward to show this for the explicit forms, (5.4.9) for the Landau gauge, and (5.4.11) for the cylindrical gauge. Consider two vertices along an electron line. The Dirac matrices are written according to matrix multiplication in the direction opposite to the arrow on the electron line. Suppose the initial, intermediate and final states are labeled with quantum numbers q, q ′′ , q ′ , respectively. The sum over the intermediate state gives X ′ ′′ X ′ ′′ ′′ ′ ′′ ′′ ′′ dǫq′ǫq′′ (k2 ) dǫq′′ǫq (k1 ) [Γqǫ′ qǫ′′ (k2 )]ν [Γqǫ′′ qǫ (k1 )]µ . [γqǫ′ ǫq′′ (k2 )]ν [γqǫ′′ ǫq (k1 )]µ = q′′

q′′

(5.4.12) The sum separates into into a gauge-dependent part and a gauge-independent part. For the Landau gauge, the gauge-dependent part reduces to Z Z X ′ ′′ dp′′y ′′ dp′′z ǫ′ ǫ′′ ǫ ǫ ǫ′′ ǫ dq′ q′′ (k2 ) dǫq′′ǫq (k1 ), (5.4.13) dq′ q′′ (k2 ) dq′′ q (k1 ) = Ly Lz 2π 2π ′′ q

where the sum over q ′′ reduces to the integrals over p′′y , p′′z . On inserting the explicit form (5.4.9), the integrals are trivial. The result may be written in the form X ′ ′′ ′′ ′ dǫq′ǫq′′ (k2 ) dqǫ ′′ǫq (k1 ) = ei(k1 ×k2 )z /2eB dǫq′ǫq (k1 + k2 ), (5.4.14) q′′

where Lx = 1/(eB)1/2 is used. The result (5.4.14) is independent of the choice of gauge.

208

5 Magnetized Dirac electron

The result (5.4.14) generalizes to an arbitrary number of vertices along an electron line. Specifically, for n vertices along a line, with 3-momentum ki emitted at the ith vertex, one has X ′ n−1 [γqǫ′ ǫqnn (kn )]νn [γqǫnnqǫn−1 (kn−1 )]νn−1 . . . [γqǫ22qǫ11 (k2 )]ν2 [γqǫ11qǫ (k1 )]ν1 q1 ,...,qn

P ′ ′ P i (ki ×kj )z /2eB X = dǫq′ǫq ( i ki ) e i<j [Γqǫ′ qǫnn (kn )]νn q1 ,...,qn

n−1 (kn−1 )]νn−1 . . . [Γqǫ22qǫ11 (k2 )]ν2 [Γqǫ11qǫ (k1 )]ν1 . ×[Γqǫnnqǫn−1

(5.4.15)

It follows that the choice of gauge affects the description only of the initial and final states. 5.4.5 Vertex function for arbitrary spin states ′

The form for the vertex function [Γqǫ′ qǫ (k)]µ , defined by (5.4.2) with (5.4.1), follows by assuming the wave function to be of the form (5.1.26). Specifically, Ψqǫ (x) is assumed to be the column matrix C1 vn−1 (ξ), C2 vn (ξ), C3 vn−1 (ξ), ′ C4 vn (ξ), and Ψ qǫ′ (x) is assumed to be the row matrix C1′∗ vn′ −1 (ξ), C2′∗ vn′ (ξ), ′∗ ′ −C3 vn −1 (ξ), −C4′∗ vn′ (ξ). The integrals follow from (5.4.8), which implies     Jnn′ −n vn′ (ξ ′ )vn (ξ) Z ∞  vn′ −1 (ξ ′ )vn−1 (ξ)    Jnn−1 ′ −n iψ n−n′    dx e−ikx x   vn′ −1 (ξ ′ )vn (ξ)  = D{ie }  ieiψ Jnn′ −n−1  , −∞ vn′ (ξ ′ )vn−1 (ξ) −ie−iψ Jnn−1 ′ −n+1 ′ ′

D = (eB)−1/2 eikx (ǫpy +ǫ py )/2eB ,

(5.4.16)

2 where the argument of the J-functions is x = k⊥ /2eB. The factor D is incorǫ′ ǫ porated into the gauge-dependent factor dq′ q (k) in (5.4.2). Explicit evaluation of the vertex function then gives h n−1 ′∗ ′∗ ′∗ ′∗ n Γ µ = c′∗ c ′ n n (C1 C1 + C3 C3 )Jn′ −n + (C2 C2 + C4 C4 )Jn′ −n ,

(C1′∗ C4 + C3′∗ C2 )ieiψ Jnn′ −n−1 + (C2′∗ C3 + C4′∗ C1 )(−ie−iψ )Jnn−1 ′ −n+1 ,

−i(C1′∗ C4 + C3′∗ C2 )ieiψ Jnn′ −n−1 + i(C2′∗ C3 + C4′∗ C1 )(−ie−iψ )Jnn−1 ′ −n+1 , i ′∗ ′∗ n (C1′∗ C3 + C3′∗ C1 )Jnn−1 ′ −n − (C2 C4 + C4 C2 )Jn′ −n , ′

c∗n′ = (−ie−iψ )n ,

cn = (ieiψ )n .

(5.4.17)

The general form (5.4.17) may be used to write down explicit forms for the vertex function for the three choices of spin eigenfunctions discussed in §5.2. Only the magnetic moment eigenfunctions are used explicitly here. The explicit form for the vertex function for the magnetic-moment eigenstates (5.2.12) is

5.4 Vertex function in a magnetic field

′

′

′

ǫ ǫ(+)

[Γqǫ′ qǫ (k)]µ = (ieiψ )n−n Aq′ q ǫ′ ǫ(−)

−Aq′ q ǫ′ ǫ(±)

Aq′ q

209

(0)

Jq′ q (k), ǫ′ ǫ(−)

(+)

Jq′ q (k), −iAq′ q

= a′ǫ′ s′ aǫs ± a′−ǫ′ s′ a−ǫs ,

′

′ (0) (−) Jq′ q (k), Bqǫ′ ǫq Jq′ q (k) ,

Bqǫ′ ǫq = a′ǫ′ s′ a−ǫs + a′−ǫ′ s′ aǫs ,

(0)

(±)

′ ′ n Jq′ q (k) = b′s′ bs Jnn−1 ′ −n (x) + s sb−s′ b−s Jn′ −n (x),

′ iψ n Jq′ q (k) = s′ b′−s′ bs e−iψ Jnn−1 Jn′ −n−1 (x), ′ −n+1 (x) ± sbs′ b−s e 0 1/2 1/2 εn ± ε0n εn + sm a± = P± , , bs = 2εn 2ε0n

(5.4.18)

with P± defined by (5.2.13). 5.4.6 Gauge-invariance condition The vertex function in the form (5.4.18) is used widely below, and to illustrate its properties it is instructive to derive the identity (5.4.6) explicitly, Specifically, the identity implies ′

′

ǫ′ ǫ(+)

kµ [Γqǫ′ qǫ (k)]µ = (ω − ǫεq + ǫ′ ε′q′ ) b′ (−ie−iψ )n ieiψ )n Aq′ q

(0)

Jq′ q (k). (5.4.19)

Noting the implicit relation kz = ǫpz − ǫ′ p′z , the identity (5.4.19) requires ′ ǫ′ ǫ(+) (0) (ǫεq − ǫ′ ε′q′ )Aq′ q − (ǫpz − ǫ′ p′z )Bqǫ′ ǫq Jq′ q (k) ǫ′ ǫ(−) (−) (+) . − k⊥ Aq′ q cos ψ Jq′ q (k) + i sin ψ Jq′ q (k) = 0.

(5.4.20)

The first term in (5.4.20) may be rewritten using a2ǫs + a2−ǫs = 1, pz = 2εn aǫs a−ǫs , and similarly for the primed variables. One finds the identity ǫ′ ǫ(+)

(ǫεq − ǫ′ ε′q′ )Aq′ q

′

ǫ′ ǫ(−)

− (ǫpz − ǫ′ p′z )Bqǫ′ ǫq = (sε0n − s′ ε0n′ )Aq′ q

.

(5.4.21)

The second term in (5.4.20) is rewritten using the properties of the Jfunctions. First note that the quantity inside the square brackets is indepen(+) dent of ψ, and equal to Jq′ q (k) with ψ = 0. Next, use the identities (A.1.30) and (A.1.31) to write n−1 n k⊥ Jnn−1 ′ −n+1 (x) = pn′ Jn′ −n (x) − pn Jn′ −n (x), n k⊥ Jnn′ −n−1 (x) = −pn Jnn−1 ′ −n (x) + pn′ Jn′ −n (x),

(5.4.22)

Using pn = 2ε0n bs b−s , and similarly for the primed variables, it is straightfor(0) ward to show that second term in (5.4.20) is proportional to Jq′ q (k), with the coefficient equal to minus the right hand side of (5.4.21). This completes the proof.

210 C 00 C 01 C 02

5 Magnetized Dirac electron (ǫ′ ε′n′ ǫεn + ǫ′ ǫ p′z pz + m2 ) Jnn−1 ′ −n

2

+ Jnn′ −n

2

n + 2pn′ pn Jnn−1 ′ −n Jn′ −n

iψ n−1 Jn′ −n+1 + Jnn′ −n e−iψ Jnn′ −n−1 −pn′ ǫεn Jnn−1 ′ −n e

n−1 −iψ n Jn′ −n−1 −ǫ′ ε′n′ pn Jnn′ −n eiψ Jnn−1 ′ −n+1 + Jn′ −n e

iψ n−1 Jn′ −n+1 − Jnn′ −n e−iψ Jnn′ −n−1 ipn′ ǫεn Jnn−1 ′ −n e

n−1 −iψ n Jn′ −n−1 +iǫ′ ε′n′ pn Jnn′ −n eiψ Jnn−1 ′ −n+1 − Jn′ −n e

2

2

C 03

(ǫ′ ε′n′ ǫpz + ǫεn ǫ′ p′z ) Jnn−1 ′ −n

C 11

(ǫ′ ε′n′ ǫεn − ǫ′ p′z ǫpz − m2 ) Jnn−1 ′ −n+1

2

+ Jnn′ −n−1

2

(ǫ′ ε′n′ ǫεn − ǫ′ p′z ǫpz − m2 ) Jnn−1 ′ −n+1

2

+ Jnn′ −n−1

2

+ Jnn′ −n

n +2pn′ pn cos(2ψ) Jnn−1 ′ −n+1 Jn′ −n−1

C 22

n −2pn′ pn cos(2ψ) Jnn−1 ′ −n+1 Jn′ −n−1

C 33 C 12

(ǫ′ ε′n′ ǫεn + ǫ′ p′z ǫpz − m2 ) Jnn−1 ′ −n

2

+ Jnn′ −n

−i(ǫ′ ε′n′ ǫεn − ǫ′ p′z ǫpz − m2 ) Jnn−1 ′ −n+1

n +2pn′ pn sin(2ψ) Jnn−1 ′ −n+1 Jn′ −n−1

C 13 C 23

2

2

n − 2pn′ pn Jnn−1 ′ −n Jn′ −n

− Jnn′ −n−1

−iψ n−1 Jn′ −n+1 + Jnn′ −n eiψ Jnn′ −n−1 −pn′ ǫpz Jnn−1 ′ −n e

2

n−1 iψ n Jn′ −n−1 −ǫ′ p′z pn Jnn′ −n e−iψ Jnn−1 ′ −n+1 + Jn′ −n e

−iψ n−1 Jn′ −n+1 − Jnn′ −n eiψ Jnn′ −n−1 −ipn′ ǫpz Jnn−1 ′ −n e

n−1 iψ n Jn′ −n−1 −iǫ′ p′z pn Jnn′ −n e−iψ Jnn−1 ′ −n+1 − Jn′ −n e

′

Table 5.1. The components of Cnǫ ′ǫn (ǫ′ p′k , ǫpk ) 2 lated; Jνn denotes Jνn (k⊥ /2eB).

µν

, denoted C µν = C ∗νµ , are tabu-

5.4.7 Sum over initial and final spin states In many applications one is not interested in the spin of the particles, and it is appropriate to average over the spin states. The transition rate for any specific process involving an electron involves the outer product of a vertex function and its complex conjugate, or a number of such outer products, each of which may be treated separately. It is useful to write µν X ′ Cn′ n (ǫ′ p′k , ǫpk ) µ ǫǫ′ ν X ǫ′ ǫ µ ǫ′ ǫ ∗ν Γq′ q (k) Γq′ q (k) , Γqǫ′ qǫ (k) Γqq = = ′ (−k) ′ ′ 2ǫ ǫεn′ εn ′ ′ s,s

s,s

(5.4.23) µ ′ ′ with p′µ k = [εn′ , 0, 0, pz ], pk = [εn , 0, 0, pz ]. Given a specific choice of spin operator, explicit evaluation of the sums in (5.4.23) is straightforward but tedious. The result is summarized in Table 5.1. A covariant form of the result in Table 5.1 is derived using the Ritus method in §5.5. It is convenient to write the result in terms of Pkµ =

5.4 Vertex function in a magnetic field

211

[ǫεn , 0, 0, ǫpk], Pk′µ = [ǫ′ ε′n′ , 0, 0, ǫ′ p′k ]. ǫ′ ǫ ′ µν Cn′ n (Pk , Pk ) = ′µ ν 2 n 2 Pk Pk + Pkµ Pk′ν − (P ′ P )k − m2 gkµν (Jnn−1 ′ −n ) + (Jn′ −n ) µν 2 n 2 (P ′ P )k − m2 (Jnn−1 −g⊥ ′ −n+1 ) + (Jn′ −n−1 ) 2 2 n +if µν (P ′ P )k − m2 (Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) µ ν µ ν n−1 n n +pn′ pn gkµν 2Jnn−1 ′ −n Jn′ −n + (e+ e+ + e− e− )Jn′ −n+1 Jn′ −n−1 n−1 n ν n ν −pn Pk′µ Jnn−1 ′ −n Jn′ −n−1 e+ + Jn′ −n Jn′ −n+1 e− µ µ n−1 n n −pn Pk′ν Jnn−1 ′ −n Jn′ −n−1 e− + Jn′ −n Jn′ −n+1 e+ n−1 ν n n ν −pn′ Pkµ Jnn−1 ′ −n Jn′ −n+1 e− + Jn′ −n Jn′ −n−1 e+ µ µ n−1 n n (5.4.24) −pn′ Pkν Jnn−1 ′ −n Jn′ −n+1 e+ + Jn′ −n Jn′ −n−1 e− ,

2 where, for simplicity in writing, the argument x = k⊥ /2eB of the Jµ functions is omitted in (5.4.24). The 4-vectors e± , whose components are eµ± = e∓iψ [0, 1, ±i, 0], may be written in the frame-independent form eµ± = µ µ µ µν µ (k⊥ ± ikG )/k⊥ , with k⊥ = g⊥ kν = k⊥ [0, cos ψ, sin ψ, 0], and kG = f µν kν = k⊥ [0, − sin ψ, cos ψ,′ 0]. νµ The tensor Cnǫ ′ǫn (Pk′ , Pk ) satisfies the identities

0ν µν ′ ′ = (ω − ǫεn + ǫ′ ε′n′ ) Cnǫ ′ǫn (Pk′ , Pk ) , kµ Cnǫ ′ǫn (Pk′ , Pk ) µν ′ µ0 ′ = (ω − ǫεn + ǫ′ ε′n′ ) Cnǫ ′ǫn (pz , k) . kν Cnǫ ′ǫn (Pk′ , Pk )

(5.4.25)

To establish the identities (5.4.25) are satisfied for the form (5.4.24) one needs to use the relations (5.4.22) and Pk′µ − Pkµ + kkµ = [ω − ǫεn + ǫ′ ε′n′ , 0].

212

5 Magnetized Dirac electron

5.5 Ritus method and the vertex formalism Three complicating factors in QED for a magnetized system, compared with an unmagnetized system, are 1) momentum is not conserved at the microscopic level, 2) the detailed results depend on the choice of the gauge used to describe the magnetic field, and 3) the detailed theory depends on the choice of spin operator. The fact that momentum is not conserved precludes a momentum-space representation, which is an essential ingredient in conventional QED for unmagnetized electrons. In most applications, the choice of gauge is of no physical relevance: specifically, the choice of gauge leads to a quantum number (py in the Landau gauge) that partly describes the location of the center of gyration of the particle, and in a homogeneous system this location is of no physical relevance. The choice of spin operator is important for some purposes, but in many applications the spin is of no interest and it is desirable to have a theory that applies directly to unpolarized particles, as is possible in the unmagnetized case. There are two related formalisms that allow these complications to be partially overcome, specifically allowing a gauge-independent, momentum-spacelike representation. These are the vertex formalism, based on the vertex function introduced in §5.4, and the Ritus method [9, 10]. The idea that underlies the Ritus method is to isolate the gauge dependence and the dependence of the theory on the choice of the spin operator from the other dependences. The remaining ‘reduced’ wave functions lead to a ‘reduced’ propagator whose Fourier transform does exist. The theory is then analogous in structure to the unmagnetized case. However, the calculation of traces in the Ritus formalism involves new features compared with the unmagnetized case. In contrast, the idea that underlies the vertex method is to associate all the spatial dependence with a vertex function, rather than the electron propagator, so that the spatial Fourier transform of the vertex function exists; the propagator is reduced to a function that depends on the time difference between two vertices and is independent of the positions of the two vertices. 5.5.1 Factorization of Dirac’s equation The Dirac wave function in a magnetic field may be factorized into a function that depends on the choice of gauge for the vector potential, A(x), and a part that is a gauge-independent Dirac spinor. The gauge-independent part may be further factorized into a spin-independent part, and a Dirac spinor that corresponds to the column matrix (C1 , C2 , C3 , C4 ) introduced in (5.1.26) and (5.1.29), and evaluated for specific spin operators in §5.2. This separation was first introduced by Ritus [9, 10] cf. also [11, 12]. The Dirac wave function is written as the product Ψqǫ (x) = e−iǫ(εn t−pz z) Vgǫ (x, n, pz )ϕǫs (n, pz ),

(5.5.1)

5.5 Ritus method and the vertex formalism

213

where Vgǫ (x, n, pz ) is a diagonal matrix, and where g denotes a gaugedependent quantum number. For an arbitrary choice of gauge, Dirac’s equation separates into two equations. One of these is the ‘reduced’ Dirac equation ǫ µ (5.5.2) [/ Pnǫ − m]ϕǫs (n, pz ) = 0, Pn = (ǫεn , 0, pn , ǫpz )

where ϕǫs (n, pz ) is a reduced wave function. The other equation determines the diagonal matrix Vgǫ (x, n, pz ) in (5.5.1). This equation depends explicitly on the choice of gauge for A. For an arbitrary gauge this equation is ∂ ∂ αx −i Vgǫ (x, n, pz ) = pn αy Vgǫ (x, n, pz ), + eAx + αy −i + eAy ∂x ∂y (5.5.3) with pn = (2neB)1/2 . The two equations (5.5.2) and (5.5.3) are discussed separately below, starting with the gauge-dependent part. 5.5.2 Gauge-dependent part of the wave function

After multiplying by αy , (5.5.3) becomes ∂ ∂ + eAx + −i + eAy Vgǫ (x, n, pz ) = pn Vgǫ (x, n, pz ). −iΣz −i ∂x ∂y (5.5.4) In the Landau gauge (5.1.12), the gauge-dependent quantum number corresponds to g → ǫpy . One seeks a solution of the form Vgǫ (x, n, pz ) ∝ eiǫpy y . Noting that the solution (5.1.26) may be written in the form      C1 vn−1 (ξ) 0 0 0 C1 vn−1 (ξ)     C2 vn (ξ)   0 v (ξ) 0 0 n   C2  , (5.5.5)   f (x) =   C3 vn−1 (ξ)  =  0 0 vn−1 (ξ) 0   C3  C4 0 0 0 vn (ξ) C4 vn (ξ)

one identifies the column matrix (C1 , C2 , C3 , C4 ) with ϕǫs (n, pz ) in (5.5.1). This leads to the identification   vn−1 (ξ) 0 0 0  0 vn (ξ) 0 0  . (5.5.6) Vgǫ (x, n, pz ) = eiǫpy y   0 0 vn−1 (ξ) 0  0 0 0 vn (ξ)

For some purposes it is more convenient to write (5.5.6) in the form P± = 12 (1 ± Σz ). Vgǫ (x, n, pz ) = eiǫpy y P+ vn−1 (ξ) + P− vn (ξ) ,

(5.5.7) The spin projection operators, P± , are introduced in (5.3.7), and their properties are summarized in (5.3.27), (5.3.28), (5.3.30). The result corresponding to (5.5.6) for the cylindrical gauge is

214

5 Magnetized Dirac electron

Vrǫ (x, n, pz ) = ei(n−r)φ+iǫpz z   r Jn−r−1 e−iφ/2 0 0 0 r   0 Jn−r eiφ/2 0 0 , × r −iφ/2   0 0 Jn−r−1 e 0 r iφ/2 0 0 0 Jn−r e

(5.5.8)

where the argument 12 eB̟2 of the J-functions is omitted for simplicity in writing. In this case the additional quantum number, g → r, is the radial quantum number (5.2.26), which is discrete. For an arbitrary choice of gauge, Vgǫ (x, n, pz ) satisfies orthogonality and completeness relations, Z 2π δ(p′z − pz ) δn′ n δg′ g , (5.5.9) d3 x Vgǫ†′ (x, n′ , p′z ) Vgǫ (x, n, pz ) = Lz X Z dpz Lz Vgǫ (x′ , n, pz ) Vgǫ† (x, n, pz ) = δ 3 (x′ − x). (5.5.10) 2π g,n

The sum over g becomes the integral over Ly dpy /2π for the Landau gauge and a sum over the radial quantum number for the cylindrical gauge. 5.5.3 Reduced form of Dirac’s equation

The reduced form (5.5.2) for Dirac’s equation follows from Dirac’s equation and the identity (5.5.4). The resulting equation for the coefficients C1 , . . . , C4 is    ǫεn − m 0 −ǫpz ipn C1   C2   0 ǫε − m −ip ǫp n n z    = 0.  (5.5.11)   C3   −ǫpz ipn ǫεn + m 0 C4 −ipn ǫpz 0 ǫεn + m

The square matrix in (5.5.11) may be written in the form ǫεn − ǫαz pz − αy pn − βm, and after multiplying by β = γ 0 , this may be written in the more concise form (/ Pnǫ −m)ϕǫs (n, pz ) = 0, reproducing (5.5.2) with the reduced wave function identified as   C1  C2   ϕǫs (n, pz ) =  (5.5.12)  C3  . C4 5.5.4 Reduced wave functions The specific form for the reduced wave function (5.5.12) depends on the choice of spin operator. For the Johnson-Lippmann wave functions it is given by (5.1.29), for the helicity eigenstates it is given by (5.2.7), and for the magneticmoment eigenstates it is given by (5.2.12). In particular, the eigenvalue equation for the reduced magnetic-moment wave function is

5.5 Ritus method and the vertex formalism

[mσ − iǫγ × Πǫn (pz )]z ϕǫs (n, pz ) = sλϕǫs (n, pz ), with, in the standard representation,   m 0 0 −ipn  0 −m −ipn 0  , [mσ − iǫγ × Πǫn (pz )]z =   0 ipn m 0  ipn 0 0 −m

215

(5.5.13)

Πǫn (pz ) = (0, ǫpn , pz ).

(5.5.14) Solving (5.5.13) with (5.5.14) reproduces (5.2.12). The normalization for the reduced wave functions is closely analogous to the normalization for (unmagnetized) free particles. Specifically, one has ′

′

mδ ǫǫ δss′ . ǫεn V (5.5.15) When summing over intermediate states, or summing or averaging over initial or final polarization states for an electron (or positron), the sum is over the outer product of a wave function and its Dirac conjugate. This sum gives ′

ǫ ϕǫ† s (n, pz ) ϕs′ (n, pz ) =

X

s=±

δ ǫǫ δss′ , V

′

ϕ¯ǫs (n, pz ) ϕǫs′ (n, pz ) =

ϕǫs (n, pz ) ϕ¯ǫs (n, pz ) =

P/nǫ + m , 2ǫεn V

(5.5.16)

µ with Pnǫ = (ǫεn , 0, pn , ǫpz ), and with, by analogous to (5.5.11),   ǫεn (pz ) + m 0 −ǫpz ipn   0 ǫεn (pz ) + m −ipn ǫpz . P/nǫ + m =    ǫpz −ipn −ǫεn (pz ) + m 0 ipn −ǫpz 0 −ǫεn (pz ) + m (5.5.17) In the magnetized case, the operator P/nǫ + m plays a role somewhat analogous to ǫ/p + m in the unmagnetized case. 5.5.5 Reduced propagator in the Ritus method Although the electron propagator has no momentum-space representation in a magnetic field, the Ritus method leads to a reduced propagator that plays a similar role to a momentum-space propagator. The absence of a formal momentum-space propagator is due to the coordinate-space form of the propagator depending separately on the space-time points, x, x′ , and the Fourier transform can be performed only for functions that depend on x − x′ , rather than on x, x′ separately. In a magnetized vacuum, the form (5.3.8) for the propagator shows that it may be separated into a gauge-dependent function, φ(x, x′ ), times a reduced propagator that depends only on the difference x−x′ , which can be Fourier transformed.

216

5 Magnetized Dirac electron

Using the Ritus method, a momentum-space representation for the statistically average electron propagator is possible provided that the electrons are unpolarized. On factorizing the wave functions in (5.3.1) using (5.5.1), one has ′

Ψqǫ (x)Ψ ,ǫq (x′ ) = eiǫpz z Vgǫ (x, n, pz )ϕǫs (n, pz )ϕ¯ǫs (n, pz )Vgǫ† (x′ , n, pz ) e−iǫpz z . (5.5.18) ′ In the Ritus method, the factors eiǫpz z Vgǫ (x, n, pz ) and Vgǫ† (x′ , n, pz ) e−iǫpz z are treated separately. The remaining factors, ϕǫs (n, pz )ϕ¯ǫs (n, pz ), are incorpo¯ q (E), which is defined by writing (5.3.1) rated into a reduced propagator, G with (5.5.18) in the form X 1 Z dE ′ ¯ x′ ) = e−iE(t−t ) G(x, V 2π ǫq ′

¯ q (E) Vgǫ† (x′ , n, pz ) e−iǫpz z . ×eiǫpz z Vgǫ (x, n, pz ) G

(5.5.19)

The reduced propagator, which may be interpreted as the propagator in the Ritus method, is identified as X nǫq 1 − nǫq ǫ ǫ ǫ ¯ . (5.5.20) + ϕs (n, pz )ϕ¯s (n, pz ) Gq (E) = E − ǫ(εq − i0) E − ǫ(εq + i0) s The factorization process is useful only if the reduced propagator is independent of the choice of spin operator, and this requires that the electrons be unpolarized. Then the spin appears only in ϕǫs (n, pz )ϕ¯ǫs (n, pz ), and the sum over s is performed using (5.5.15). The propagator in the Ritus formalism, ¯ ǫ (E) → G ¯ ǫ (E, pz ), becomes rewritten G q n ǫ P/n + m 1 − nǫn (ǫpz ) nǫn (ǫpz ) ǫ ¯ Gn (E, pz ) = , (5.5.21) + 2ǫεn E − ǫ(εn − i0) E − ǫ(εn + i0) µ ¯ ǫ (E, pz ), dewith Pnǫ = (ǫεn (pz ), 0, pn , ǫpz ). The reduced propagator, G n pends only on the quantum numbers ǫ, n, pz and the energy E in the internal particle line. 5.5.6 Propagator in the vertex formalism The vertex formalism provides a different way of achieving a momentumspace-like representation. The essential idea in the vertex formalism is that the wave functions are incorporated into vertex functions, such that all the dependence on x appears in the vertex function at x, and all the dependence on x′ appears in the vertex function at x′ . It is then straightforward to carry out Fourier transforms with respect to all space variables, as discussed in detail in §5.4. Energy is conserved at each vertex, and the Fourier transformed with respect to time at each of the vertices gives δ-functions, as in the unmagnetized case.

5.5 Ritus method and the vertex formalism

217

In contrast with the Ritus method, where the reduced wave functions are retained in the reduced electron propagator, in the vertex formalism the propagator is independent of the details of the states, and depends only on the (virtual) energy of the electron (or positron). Specifically, let Gqǫ (E) be the propagator in the vertex formalism in the absence of any statistical average, and let G¯qǫ (E) be the statistical average of this propagator. Starting from ¯ x′ ), given by (5.3.1), the wave functions Ψqǫ (x), Ψ ǫq (x′ ) the propagator G(x, are deleted (and transferred to the vertex functions) so that the propagator becomes Z X dE −iE(t−t′ ) ¯ǫ ¯ x′ ) = Ψqǫ (x)Ψ ǫq (x′ ) G(x, e Gq (E), (5.5.22) 2π ǫq In the absence of any statistical average one identifies Gqǫ (E) =

1 , E − ǫ(εq − i0)

(5.5.23)

where q denotes the quantum numbers for the electron in the intermediate state, with only n, pz relevant to the energy εq → εn (pz ). The statistical average over unpolarized particles, with q identified as n, pz , gives G¯nǫ (E, pz ) =

1 − nǫn (ǫpz ) nǫn (ǫpz ) + . E − ǫ(εn (pz ) − i0) E − ǫ(εn (pz ) + i0)

(5.5.24)

Comparison of (5.5.20), (5.5.21) with (5.5.23), (5.5.24) shows that the propagator in the vertex formalism corresponds to deleting the factor [/ Pnǫ + m]/2ǫεn associated with the sum over the spins of the outer product to the reduced wave functions in the reduced propagator. In the vertex formalism, the reduced wave functions are included in the vertex functions, and not in the propagators. 5.5.7 Vertex matrix in the Ritus method ′

The vertex function [γqǫ′ ǫq (k)]µ factorizes into gauge-dependent and a gauge′ independent parts, [Γqǫ′ qǫ (k)]µ , as shown explicitly in (5.4.2). In the Ritus method, a further factorization of the gauge-independent vertex function facilitates a momentum-space-like representation of the theory [9, 10, 11, 12]. This factorization allows one to separate the reduced wave functions from ′ [Γqǫ′ qǫ (k)]µ ; the reduced wave functions are included in the propagator, (5.5.20) or (5.5.21). The remaining quantity is a Dirac matrix, which is the vertex ma′ trix in the Ritus method. The vertex function [Γqǫ′ qǫ (k)]µ is the matrix element, over the reduced wave functions of this reduced vertex matrix in the Ritus method. The factorization of the wave function, in the form (5.5.1), allows one to make the factorization (5.4.1) of the vertex function explicit by writing

218

5 Magnetized Dirac electron ′

′

[Γqǫ′ qǫ (k)]µ = V ϕ¯ǫs′ (n′ , p′z ) Jnµ′ n (k⊥ ) ϕǫs (n, pz ),

(5.5.25)

which defines the vertex matrix Jnµ′ n (k⊥ ). For example, the form (5.4.17) for the vertex function for an arbitrary choice of spin operator may be written in the form (5.5.25) by identifying the reduced wave function as the column matrix (C1 , C2 , C3 , C4 ) and the adjoint wave function as (C1′∗ , C2′∗ , −C3′∗ , −C4′∗ ). In writing down an explicit form for Jnµ′ n (k⊥ ) it is useful to project onto µ µν the k and ⊥-subspaces, by writing γkµ = gkµν γν , γ⊥ = g⊥ γν , so that these µ µ 0 3 1 2 correspond to γk = (γ , 0, 0, γ ), γ⊥ = (0, γ , γ , 0). One has k

µ ⊥ Jnµ′ n (k⊥ ) = γkµ Jn′ n (k⊥ ) + γ⊥ Jn′ n (k⊥ )

(5.5.26)

k

where Jn′ n (k⊥ ), Jn⊥′ n (k⊥ ) are diagonal 4 × 4 matrices:  n−1  Jn′ −n 0 0 0  0 J n′ ′ 0 0  k n −n , Jn′ n (k⊥ ) = (−ie−iψ )n −n   0  0 Jnn−1 0 ′ −n 0 0 0 Jnn′ −n

(5.5.27)

′

Jn⊥′ n (k⊥ ) = (−ie−iψ )n −n   −ie−iψ Jnn−1 0 0 0 ′ −n+1   0 ieiψ Jnn′ −n−1 0 0  , (5.5.28) ×   0 0 −ie−iψ Jnn−1 0 ′ −n+1 0 0 0 ieiψ Jnn′ −n−1

2 where the arguments of the J-functions is k⊥ /2eB. k µ µ The matrices γk , Jn′ n (k⊥ ) commute, but the matrices γ⊥ , Jn⊥′ n (k⊥ ) do not commute. If one writes the matrix products in (5.5.26) in the opposite order, one needs to replace Jn⊥′ n (k⊥ ) by

′ J˜n⊥′ n (k⊥ ) = (−ie−iψ )n −n   iψ n ie Jn′ −n−1 0 0 0   0 0 0 −ie−iψ Jnn−1 ′ −n+1  . (5.5.29) × iψ n   0 0 ie Jn′ −n−1 0 0 0 0 −ie−iψ Jnn−1 ′ −n+1

The matrix (5.5.29) also appears in the symmetry relations k ∗ ⊥ ∗ k ⊥ ˜ Jnn′ (−k⊥ ) = Jn′ n (k⊥ ) , Jnn . ′ (−k ⊥ ) = Jn′ n (k ⊥ )

Using the relations (5.5.30) in (5.5.26), one finds ∗ ∗ µ ˜⊥ µ µ k Jn′ n (k⊥ ) . + γ⊥ Jnn ′ (−k⊥ ) = γ k Jn′ n (k ⊥ )

(5.5.30)

(5.5.31)

An alternative way of writing Jnµ′ n (k⊥ ) is in terms of the projection matrices P± introduced in (5.5.7). One has

5.5 Ritus method and the vertex formalism

219

′

n Jnµ′ n (k⊥ ) = (−ie−iψ )n −n γkµ Jnn−1 ′ −n (x) P+ + Jn′ −n (x) P− µ iψ n +γ⊥ − ie−iψ Jnn−1 Jn′ −n−1 (x) P− . (5.5.32) ′ −n+1 (x) P+ + ie

These matrices satisfy the identities P± γkµ = γkµ P± ,

µ µ P± γ⊥ = γ⊥ P∓ .

(5.5.33)

Using these relations, an alternative form of (5.5.32), with the matrix products written in the opposite order, is µ ′ n Jnµ′ n (k⊥ ) = (−ie−iψ )n −n Jnn−1 ′ −n (x) P+ + Jn′ −n (x) P− γk µ iψ n . (5.5.34) + − ie−iψ Jnn−1 Jn′ −n−1 (x) P+ γ⊥ ′ −n+1 (x) P− + ie

5.5.8 Calculation of traces using the Ritus method

An example where the Ritus method may be usedis in an alternative derivaµν ′ tion in the sum (5.4.23), leading to the quantity Cnǫ ′ǫn (pz , k) , defined by (5.4.23) as the sum over spins of the outer product of the vertex function with itself. The Ritus approach gives " ′ # ǫ′ ǫ µν P/nǫ′ + m ν P/nǫ + m µ Jnn′ (−k⊥ ) , (5.5.35) Cn′ n (pz , k) = Tr Jn′ n (k⊥ ) 2ǫ′ ε′n′ 2ǫεn ′ µ with Pnǫ′ = [ǫ′ ε′n′ , 0, pn′ , ǫ′ p′z ], ε′n′ = εn′ (p′z ), and similarly for the unprimed µ quantity Pnǫ . The evaluation of (5.5.35) is analogous to the corresponding unmagnetized case, where one needs to evaluate the trace of the product (ǫ/p + m)γ µ (ǫ′ / p′ + m)γ ν . However, new features appear associated with the projection operators, and in order to calculate (5.5.35), the general problem of evaluating traces that include P± needs to be addressed. In the absence of projection operators, the standard technique for calculating traces is based on the identities Tr γ µ γ ν = 4g µν , Tr γ µ γ ν γ ρ γ σ = 4 g µν g ρσ − g µρ g νσ + g µσ g ρν , (5.5.36) together with the trace of any product of an odd number of γ-matrices being zero. One way of deriving (5.5.36) involves choosing 16 independent matrices to span the Dirac spin space, for example 1, γ µ , [γ µ , γ ν ], γ 5 γ µ , γ 5 , and noting that only the unit matrix has a nonzero trace, equal to 4. If one includes a projection operator anywhere in the sequence of γ-matrices, this projects the unit matrix onto a 2-dimensional subspace, so that the trace of the unit matrix is reduced to 2. This reduces the value of the traces in (5.5.36) by one half. However, the projection introduces a second matrix with a nonzero trace: Σz has zero trace, but P± Σz = ±P± has trace ±2. With γ 1 γ 2 = −iΣz = −γ 2 γ 1 , this introduces and additional contribution whenever two of the indices in

220

5 Magnetized Dirac electron

(5.5.36) take on the values 1 and 2. It is useful to distinguish between k- and µ ⊥-components, achieved here by writing γkµ = [γ 0 , 0, 0, γ 3], γ⊥ = [0, γ 1 , γ 2 , 0]. Including a projection operator in the first of (5.5.36) gives i h µν µ ν ± if µν ], (5.5.37) Tr [γ⊥ γ⊥ P± ] = 2 g⊥ Tr γkµ γkν P± = 2gkµν ,

with f µν , defined by (1.1.13) in terms of the Maxwell tensor for the background magnetic field, having nonzero components f 12 = −f 21 = −1. An alternative expression is µ ν Tr [γ⊥ γ⊥ P± ] = −2eµ∓ eν± ,

eµ± = e∓iψ (0, 1, ±i, 0).

(5.5.38)

The second of (5.5.36) becomes Tr γ µ γ ν γ ρ γ σ P± = 2 g µν g ρσ − g µρ g νσ + g µσ g ρν ± 2i g µν f ρσ + f µν g ρσ (5.5.39) −g µρ f νσ − f µρ g νσ + g µσ f ρν + f µσ g ρν . The projection operator may be moved to any other location, using the commutation relations P± Mk = Mk P± , with Mk = γ 0 or γ 3 , or P± M⊥ = M⊥ P∓ , with M⊥ = γ 1 or γ 2 . µν ′ The evaluation of Cnǫ ′ǫn (pz , k) involves using the Ritus method reduces to evaluate the trace in (5.5.35). The result is written down in (5.4.24).

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221

5.6 Feynman rules for QPD in a magnetized plasma QED processes in the unmagnetized case are usually treated in momentum space, with 4-momentum conserved at each vertex in a Feynman diagram. The momentum-space representation cannot be used to treat QED processes in the presence of a magnetic field: 3-momentum perpendicular to B is not conserved in general. An alternative approach, referred to as the vertex formalism, does generalize in a straightforward way to the magnetized case. The development of the theory is requires a specific choice of gauge, with the Landau and cylindrical gauges being the conventional choices. The gaugedependent aspects of the theory concern only the centers of gyration and in a homogeneous system one usually either ignores such effects or averages over them. The resulting theory is gauge-independent, involving only reduced wave functions, gauge-independent vertex functions and reduced propagators. This gauge-independent formalism is summarized in terms of rules for QED processes in a uniformly magnetized plasma. 5.6.1 Transition rate The differential transition rate in the S-matrix formalism is given by |Sfi |2 Y Df , T →∞ T

wi→f = lim

(5.6.1)

f

where T is the normalization time. In the case of an unmagnetized system, 4-momentum is conserved, and it is conventional to write the scattering matrix as Sfi = δfi + i(2π)4 δ 4 (pf − pi ) Tfi . In a magnetic field, perpendicular 3-momentum is not conserved and this procedure cannot be used. Energy and parallel momentum are conserved in a magnetostatic field, and one may incorporate this into the theory by writing the S-matrix element in the form Sfi = δfi + i 2π δ(Ef − Ei ) Tfi , (5.6.2) where Ei , Ef are the energies of the initial and final states, respectively. On inserting (5.6.2) into (5.6.1), the square of δ(Ef −Ei ) gives T 2πδ(Ef −Ei ), and the factor of T cancels with that in the transition rate (5.6.1). The differential transition rate becomes Y Df . (5.6.3) wi→f = 2π δ(Ef − Ei ) |Tfi |2 f

The explicit form of the matrix element Tfi depends on the choice of gauge. ′ All the gauge-dependent details appear only is a single factor, dǫq′ǫq (K), associated with each electron line, between states q and q ′ with K the total momentum transfer along the line. One may discard the information on the locations of the gyrocenters, and replace the square of this factor in accord with (5.6.11).

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5.6.2 Average over the position of the gyrocenter Major simplification occurs if the system is homogeneous such that no physical process depends on the position of the gyrocenter of a particle. One can then average over the gyrocenter, removing all gauge-dependent factors. That the gauge-dependent quantum number py in the Landau gauge, and r in the cylindrical gauge, is interpreted as partially specifying the position of the center of gyration of the particle. These identifications are confirmed by evaluating the mean value of x in the Landau gauge and of ̟2 = x2 + y 2 in the cylindrical gauge. For the Landau gauge, on inserting the wave functions (5.1.30) into Z hxi = d3 x x [Ψqǫ (t, x)]† Ψqǫ (t, x), (5.6.4) the y- and z-integrals are trivial. The integral over x is rewritten as an integral over ξ, which reduces to Z ∞ ǫpy . (5.6.5) hxi = dξ x vn2 (ξ) = − eB −∞ Thus hxi is proportional to py , justifying the interpretation of py as specifying the x-coordinate of the center of gyration. Specifying py implies that one has no information about the y-position of the gyrocenter In the cylindrical gauge the mean value of the square of the radial distance (̟2 = x2 + y 2 ) is [2] 1 h̟2 i = (2n + 2r + 1). (5.6.6) eB In the non-quantum limit h̟2 i is equal to R2 + a2 , where R is the radius of gyration and a is the radial distance of the center of gyration from the z-axis. The radius of gyration in the non-quantum limit is R = (2n/eB)1/2 , and hence the radial distance of the center of gyration from the z-axis is related to (2r/eB)1/2 . This justifies the interpretation of r as specifying the radial position of the center of gyration. For most purposes the position of the center of gyration is irrelevant and one either ignores it or averages over it. In the Landau gauge this average involves the operation Z Lx /2 Z 2π dpy ˆav = 1 O , (5.6.7) dx → Lx −Lx /2 eBLx 2π where (5.6.5) is used. The coefficient on the right hand side of (5.6.7) reduces to 2π/(eB)1/2 with the normalization condition, Lx = 1/(eB)1/2 , chosen here. Similarly, in the cylindrical gauge the average involves the operation Z 2π Z ∞ 2π X ˆav = 1 dφ d̟̟ → , (5.6.8) O A 0 AeB r 0

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where the integrand is assumed to be independent of φ, and where A is the normalization area. As with (5.6.7), the coefficient on the right hand side of (5.6.8) reduces to 2π when the normalization conditions (5.1.35) is chosen. 5.6.3 Reduced density of final states A sum over final states appears in evaluating the transition rate for processes in QED or QPD, and in the presence of a magnetic field, this sum depends on the choice of gauge. In the absence of a magnetic field, the density of final states factor for an electron, denoted by a prime, includes a sum over the spin, s′ and an integral over V d3 p′ /(2π)3 , where p′ is the final 3-momentum. In the presence of a magnetic field, there are sums over s′ and the principal quantum number, n′ , an integral over Lz dp′z /(2π), and a sum or integral over the gaugedependent quantum number. For the Landau gauge there is an integral over Ly dp′y /(2π) and for the cylindrical gauge there is a sum over the final radial quantum number r′ . Usually one is not interested in the location of the center of gyration, and averages over it. Suppose that one uses a reduced formalism or that one has averaged over the position of the center of gyration, so that the part of the sum of final states that relates to the center of gyration is trivial. For the Landau gauge this corresponds to the integral over Ly dp′y /(2π). The integral over p′y may be rewritten as eB times the integral over the x′ -position of the center of gyration, and this integral gives the normalization length Lx . Hence, the integral over Ly dp′y /(2π) is replaced by eBLx Ly /2π. This argument is equivalent to ˆav → 1. For the averaging over the center of gyration using (5.6.7) with O cylindrical gauge, the gauge-dependent part of the sum over final states is the ˆav → 1 implies sum over r′ , and an analogous argument using (5.6.8) with O that this sum is replaced by eBA/2π in the density of final states factor. In both cases, the sum over the density of final states for an electron or positron corresponds to X eBV Z dp′ X z Df = . (5.6.9) 2π 2π ′ ′ ′ q

n ,s

The density of final states factor for a wave quantum is V d3 k′ /(2π)3 , unchanged by the inclusion of the magnetic field. 5.6.4 Neglect of gauge-dependent factors

Provided one is not interested specifically in the location of the gyrocenter, in treating QED processes in a magnetic field, one may simply ignore the gaugedependent factors. This leads to a reduced form for QED for magnetized electrons that is closely analogous to QED in the unmagnetized case. This may be seen by considering the various possible cases that arise. If there is only one electron line, it begins and ends in either the initial or final state, leaving three possibilities: it corresponds to an electron (or a positron) in

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the initial state and an electron (or a positron) in the final state, a pair in the initial state or a pair in the final state. It suffices to consider in detail only the case of an electron line from the initial state to the final state in establishing the reduced formalism. One can then argue that all other cases may be considered in an analogous manner, such that the gauge-dependent factors need never appear in the theory. ′ A gauge-dependent factor dǫq′ǫq (K) appears in Sfi associated with each electron line in the corresponding Feynman diagram, where q, q ′ correspond to the initial and final states of the electron, and where K is the net momentum transfer along the line. This factor appears squared in the transition rate ∝ |Sfi |2 . In the Landau gauge, the y-component (as well as the z-component) of 3-momentum is conserved at each vertex, and hence between the initial ′ and final states. In the factor dǫq′ǫq (K), this is expressed through the factor 2πδ(ǫpy − ǫ′ p′y − Ky ). After squaring, one has ′

|dǫq′ǫq (K)|2 =

Ly Lz 2πδ(ǫpy − ǫ′ p′y − Ky ) 2πδ(ǫpz − ǫ′ p′z − Kz ). V 2 eB

(5.6.10)

For an electron (rather than a positron) line, the unprimed state is the initial state and the primed state is the final state, and the density of final states factor is Ly Lz dp′y dp′z /(2π)2 . The p′y - and p′z -integral may be performed over the δ-functions in (5.6.10). The resulting factor is (Ly Lz )2 /V 2 eB = 1/L2x eB, which is equal to unity for the choice 1/Lx(eB)1/2 made here. It follows that ′ the factor |dǫq′ǫq (K)|2 may be replaced by unity, provided that it is implicit that the y- and z-components of momentum are conserved. Provided that on is not interested in the change in the position of the gyrocenter, one may simply ′ ignore the gauge-dependent factor by replacing |dǫq′ǫq (K)|2 by unity. However, this also discards information on the z-component of momentum, and it is usually appropriate to retain information on the z-component of momentum explicitly. This is achieved by replacing (5.6.10) and the density of final states factor for the line by Z dp′z 2π eBV ǫ′ ǫ 2 ′ ′ ¯ ¯ |dq′ q (K)| = 2πδ(ǫpz − ǫ pz − Kz ), Df = , (5.6.11) V eB 2π 2π ¯ f reproduces (5.6.9). respectively, where the expression for D The other possibilities for a single electron line may be treated in a similar manner. For a positron (ǫ = ǫ′ = −1) line, the foregoing argument applies, with primed and unprimed quantities interchanged. For a single electron line corresponding to a pair in the initial state, there is no density of states factor associated with particles in the final state. In this case, one averages over the position of the gyrocenter of either the initial electron or the initial positron. The factor (5.6.10) is replaced by unity, and both pz and p′z are assumed to be specified as part of the initial conditions, determining the momentum, Kz , in the final state. The first of the replacement s(5.6.11) also applies to this case, and the density of states factor is not relevant. For a single electron

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line corresponding to a pair in the final state, Kz is specified by the initial conditions, and there are integrals over both pz and p′z in the density of final states factor. One of these, say p′z , is performed over the δ-function in (5.6.11), and the other corresponds to the undetermined pz of the other particle in the final state. Hence, with appropriate re-interpretations, (5.6.11) applies to all cases for a single electron line. When there are two or more electron lines, the result (5.6.11) applies separately to each line. For example, consider Møller scattering. where there are two electrons in the initial state, and two electrons in the final state. In the absence of a magnetic field, the details of the scattering depend on the relative positions of the two particles; for example, the scattering angle increases as the distance of closest approach decreases. Nevertheless, in a momentum-space description, the relative position is not specified specifically: the momentum transfer is a free parameter, and it determines both the scattering angle and the distance of closest approach. Thus, the relative position of the scattering particles is not specified independently, but is determined by the theory itself. Similarly, although the relative position of the gyrocenters affects the scattering, in the momentum-space-like description, this relative position is determined by other parameters and does not need to be specified independently. The foregoing discussion of the treatment of the gauge-dependent factor that contains information on the gyrocenters applies separately to each electron line. 5.6.5 Rules for an unmagnetized system Before considering the rules for the magnetized case, it is appropriate to review the rules for the unmagnetized case, formulated in §7.1 of volume 1. The rules for Feynman diagrams are: (i) The initial state is to the right of the diagram and the final state is to the left. For a given process (specified initial and final states) all diagrams with the specified number and kind of particles and wave quanta in the initial and final states are to be drawn. (ii) An electron is represented by a solid line with an arrow pointing from right to left and a positron is represented by a solid line with an arrow pointing from left to right. The direction of the arrow along a solid line is continuous. (iii) A photon (any wave quantum) is represented by a dashed line. (iv) An electron and a photon line join at an electron-photon vertex, which has a 4-tensor index (µ, ν, . . .) and a space-time point associated with it. (v) The nth order nonlinear response of the medium is represented by an (n + 1)-photon vertex, which is a circle with n + 1 photon lines joining onto it. (vi) Any photon line begins or terminates at a vertex, either joining an electron-positron line at electron-photon vertex, or a m-photon vertex.

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(vii) An m-photon vertex represents a statistical average over an m-sided closed particle loop, and closed particle loops are omitted in diagrams in QPD so that their effect is not counted twice. (viii) The order of a diagram is equal to the number of its vertices in the absence of m-photon vertices. An m-photon vertex contributes m − 2 to the order. (ix) For diagrams in momentum space all lines are labeled with the 4momentum of the particles, rather than the vertices being labeled with the space-time points. 4-momentum is conserved at a vertex. (x) The integral d4 P/(2π)4 over any undetermined 4-momentum, P , in a closed loop or associated with an external field Aµ (P ) is to be performed. (xi) An interaction with an external field is described by a vertex with the photon line replaced by a squiggly line joined to an “x” that denotes the source of the external field. In the generalization to the magnetized case, the main change is the electron wave function, which correspond to free electrons in the unmagnetized plasma, are replaced by the wave functions for electrons in a magnetostatic field. For a free particle, the only relevant quantum numbers are the components of the 4-momentum P µ = [ǫε, ǫp], and for a magnetized electrons the set of quantum numbers includes n, pz , ǫ, s and a gauge-dependent quantum number that partially determines the center of gyration of the particle. The rules for for constructing Sfi in coordinate space are essentially unchanged from the unmagnetized case. Rule 1 The contributions from all diagrams with the specified number and kind of initial and final particles are to be added in determining Sfi . Rule 2 Each electron-photon vertex corresponds to a factor ie γµ , where µ is the 4tensor index associated with the vertex. The integrals are to be performed over the space-time coordinates associated with each vertex. Rule 3 An incoming electron line corresponds to Ψq+ (x) e−iεq t , and incoming positron −

line to Ψ q (x) e−iεq t , an outgoing electron line to Ψq+ (x) eiεq t , and an outgoing −

positron line to Ψ q (x) eiεq t , where q denotes the quantum numbers.

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227

Rule 4 An incoming photon line in the mode M , joining at a vertex labeled (x, µ), corresponds to a factor aM eµM e−ikM x , with aM = [µ0 RM /V ωM ]1/2 , and an ikM x outgoing photon line to aM e∗µ . In an interaction with an external field M e µ µ A (x), a factor A (x) is included in place of these photon factors. Rule 5 An internal electron-positron line pointing from x1 to x2 corresponds to the propagator iG(x2 , x1 ). An internal photon line between vertices (x1 , µ) and (x2 , ν) corresponds to the propagator −iDµν (x2 − x1 ). Rule 6 The Dirac spinors are written according to matrix multiplication along the direction opposite to the arrow along each solid line. An extra minus sign is to be included for each closed electron-positron loop. The overall phase of the amplitude is unimportant, but two diagrams that differ only by the interchange of two external electron-positron lines must have opposite signs. Rule 7 An m-photon vertex corresponds to a factor Z Z 4 d4 km−1 i(k0 x0 +···+km−1 xm−1 ) d k0 i · · · e − d4 x0 · · · d4 xm−1 m (2π)4 (2π)4 ×(2π)4 δ 4 (k0 + · · · + km−1 )Π (m−1)µ0 ...µm−1 (k0 , . . . , km−1 ).

(5.6.12)

5.6.6 Modified rules for magnetized systems The rules for construction of the probability per unit time, wi→f , in momentum space (for an unmagnetized system) cannot be applied to the magnetized case because momentum perpendicular to the magnetic field is not conserved on the microscopic level. Although there is no momentum space representation in the same sense as in an unmagnetized system, there are two methods that allow a momentum-space-like formalism to be developed: the vertex formalism and the Ritus method. A feature common to both approaches is that the gauge-dependent part of each Dirac wave function is factorized out, and treated separately from the remaining, or reduced wave function. In a Feynman diagram the resulting gauge-dependent factors along any electron line combine into a single factor that depends only on the initial and final states. Except in intrinsically inhomogeneous systems, the gauge-dependence appears only in a phase-like factor, in the sense that it does not appear in the transition

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5 Magnetized Dirac electron

rate for the process. In the formalism developed here, this phase-like factor is combined with a δ-function that expresses conservation of parallel momentum at each vertex, with the combined factor for a specific line expression conservation of parallel momentum along that line. In the vertex formalism, all the dependence on the space-components, x, of the wave functions at the vertex at a space-time point x are included in a vertex function, which is Fourier transformed in space. The vertex func′ tion, [Γqǫ′ qǫ (k)]µ , is a matrix element that depends on reduced wave functions associated with the incoming and outgoing electron lines at the vertex; the propagator, Gnǫ (E, pz ), is independent of these wave functions. In the Ritus method, the vertex matrix, γ µ say, is replaced by a Dirac matrix, denoted Jnµ′ n (k⊥ ) here, which depends explicitly on the principal quantum numbers, n, n′ , of the electron (or positron) in the outgoing and incoming electron lines, and on the perpendicular momentum k⊥ carried off in the line. (The parallel momentum in the photon line is included in the δ-function that expresses overall conservation of parallel momentum.) The reduced wave functions are included in the reduced electron propagator, here denoted Gǫn (E, pz ). The Ritus method is then analogous to the conventional momentum-space treatment of QED or QPD, with each vertex function by Jnµ′ n (k⊥ ) and each electron propagator by Gǫn (E, pz ). Rule 8a With Tfi defined by (5.6.2), the transition probability per unit time is given by (5.6.3), viz. Y Df . wi→f = 2π δ(Ef − Ei ) |Tfi |2 f

′ dǫq′ǫq (K),

There is a gauge-dependent factor, associated with each electron line in Tfi and the density of final states factor for an electron or positron also depends on the choice of gauge. When one is not interested in the locations ′ of the gyrocenters, these gauge-dependent factors are ignored: |dǫq′ǫq (K)|2 and the density of states factor are replaced according to (5.6.11), viz. Z ′ dp′z 2π ¯ f = eBV |d¯ǫq′ǫq (K)|2 = 2πδ(ǫpz − ǫ′ p′z − Kz ), D . V eB 2π 2π In the following, this replacement is assumed to be made and the overbar is omitted on Df . It is often convenient to assume that the integral over the δ-function is performed, so that ǫ′ p′z = ǫpz − Kz is implicit. Rule 8b ′

For each electron line, iTfi contains a product of Dirac matrices, with [Γqǫ′ qǫ (k)]µ associated with a vertex with 4-tensor index µ, between states ǫ, q and ǫ′ , q ′ ,

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229

and with k the 3-momentum emitted. The set q represents the quantum numbers n, s, pz , and the sum over these quantum numbers is performed for each internal electron line. Rule 8c For incoming photon line, iTfi contains a factor aM = [µ0 RM /V ωM ]1/2 and a polarization vector eµM , for a photon in the mode M at a vertex with 4-tensor index µ. For each outgoing photon line, Tfi contains a factor aM e∗µ M. Rule 9a In the vertex formalism the matrix element of the product of Dirac matrices, representing vertices and propagators, along an electron line is replaced by a ′ product of vertex functions, of the form ie[Γqǫ′ qǫ (k)]µ , and reduced propagators. The vertex functions are matrix elements that are independent of the Dirac algebra. An internal electron line is represented by a factor iGqǫ (E), wit the reduced propagator given by Gqǫ (E) =

1 , E − ǫ(εn − i0)

(5.6.13)

with εn = (m2 + p2z + 2neB)1/2 . After statistical averaging, and with a minor change in notation, the propagator (5.6.13) becomes 1 − nǫn (ǫpz ) nǫn (ǫpz ) + E − ǫ(εn − i0) E − ǫ(εn + i0) 1 − iǫ[1 − 2nǫn (ǫpz )]πδ(E − ǫεn ), =℘ E − ǫεn

G¯nǫ (E, pz ) =

(5.6.14)

where ℘ denoted the principal value. The reduced propagator is a scalar that is also independent of the Dirac algebra. In the vertex formalism, the order in which the the functions are written is unimportant, although for bookkeeping purposes it is usually convenient to follow the order in which the functions appear along the electron line. R One integrates, dE/2π, over any undetermined energy, E, in a closed loop. Rule 9b The Ritus method differs from the vertex formalism in that the reduced wave functions, ϕǫs (n, pz ), are associated with the propagators rather than the vertex functions. A vertex between unprimed and primed electron states is represented by a factor (which involves Dirac matrices) Jnµ′ n (k⊥ ), which is defined such that its matrix element with respect to the reduced wave functions is

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5 Magnetized Dirac electron

ǫ′ ǫ q′ q

[Γ (k)]µ , as written down in (5.5.25). An internal electron line is represented by a factor (also a Dirac matrix) iGnǫ (E, pz ), which involves the sum over spin states of the outer product of the reduced wave function and its Dirac conjugate. This sum is performed using (5.5.15), giving Gnǫ (E, pz ) =

1 P/nǫ + m , 2ǫεn E − ǫ(εn − i0)

(5.6.15)

µ with Pnǫ = (ǫεn , 0, pn , ǫpz ). When a statistical average is performed, (5.5.23) is replaced by P/nǫ + m 1 − nǫn (ǫpz ) nǫn (ǫpz ) ǫ ¯ Gn (E, pz ) = , (5.6.16) + 2ǫεn E − ǫ(εn − i0) E − ǫ(εn + i0) where the occupation number, nǫn (ǫpz ), is assumed independent of the choice of spin operator. Rule 10 An internal photon line corresponds to a factor −iDµν (k). Explicit forms for Dµν (k) are gauge-dependent. For a medium with linear response 4-tensor Π µν (k), the propagator in the G-gauge, with gauge condition Gµ Aµ (k) = 0, is Gα Gβ λµανβ (k) Dµν (k) = µ0 . (5.6.17) (Gk)2 λ(k) where λ(k)k µ k ν and λµνρσ (k) are the first and second order, respectively, matrices of cofactors of Λµν (k) = k 2 g µν − k µ k ν + µ0 Π µν (k). The temporal, Coulomb and Lorenz gauges correspond to Gµ = [1, 0], Gµ = [0, k] and Gµ = k µ , respectively. (c) An m-photon vertex corresponds to a factor −

i (m−1)µ0 µ1 ...µm−1 (k0 , k1 , . . . , km−1 ), Π m

where Π (m−1)µ0 µ1 ...µm−1 (k0 , k1 , . . . , km−1 ) is the (m − 1)th order nonlinear response 4-tensor. The convention is adopted that positive frequencies correspond waves in the initial state and negative frequencies to waves in the final state. 5.6.7 Probability of a process Any specific process may be described in terms of a probability of transition, which is defined such that crossing symmetries are built into it. For an emission process, the probability of transition is the probability of spontaneous emission, which is identical to the probability of stimulated emission and the probability of true absorption, so that detailed balance applies trivially. The

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231

probabilities for crossed processes to emission, including one-photon pair creation and decay, are obtained from the probability of emission by changing the sign of the appropriate 4-momenta, and it is convenient to include the signs explicitly in the probability. Two other features of the probability are that it is defined (a) such that it is independent of the normalization volume, V , and (b) such that conservation of 3-momentum is implicit, whereas conservation of energy is explicit in a δ-function. The probability of transition is identified from a QPD calculation by identifying the differential rate per unit time, wi→f , for the process with the same quantity expressed in terms of the probability of transition. ′ The probability of spontaneous emission simplest example. Let wqǫ′ ǫq (k), with ǫ′ = ǫ = 1, denote the probability of spontaneous emission of a photon in the mode M in the range d3 k/(2π)3 by an electron in an initial state q, with transition to a final state q ′ . The differential rate of transition is 3 3 wq++ which is identified with the transition rate, wi→f , for the ′ q (k) d k/(2π) emission process calculated using the QPD rules in order to identify wq++ ′ q (k). For emission by an electron, the density of final states factor is the product of V d3 k/(2π)3 for the photon, and (V eB/2π)(dp′z /2π) for the final electron. Conservation of (the z-component of ) momentum is explicit in the transition rate, wi→f , through a δ-function, specifically, (2π/V eB) 2πδ(p′z − pz + kz ) here. As already remarked, conservation of momentum is implicit in the probability of transition, and this is achieved by noting that this factor in the transition rate combines with the sum over the final states, through the factor (V eB/2π)(dp′z /2π) for the final electron, to give unity, specifically, Z dp′z 2πδ(p′z − pz + kz ) V eB = 1. 2π 2π V eB Both factors are omitted in the probability. The rules for the transition rate in QPD then imply the identity ′

ǫ ǫ 2 ′ wMqq ′ (k) = V |aM (k) Tfi | 2π δ(ǫ εq ′ − ǫεq + ωM ),

(5.6.18)

with ǫ′ = ǫ = 1. The probability is such that it does not depend on the normalization volume, V , due to the explicit power of V (from the density of final states factor V d3 k/(2π)3 ) canceling with a factor 1/V in |aM (k)|2 . This implies that, in ordinary units, the probability of emission has the dimensions ǫ′ ǫ 3 3 L3 T −1 , so that wMqq has the dimensions T −1 . ′ (k)d k/(2π) The transition rate calculated in QPD assumes a given initial electron, so that it corresponds to the rate of emission of photons by the electron. For a distribution of initial electrons, the relevant quantity of interest is the rate the distribution of electrons emits photons per unit volume. This rate per unit volume is found by multiplying the transition rate per electron by (eB/2π)(dpz /2π)nq , and integrating over pz . The result may be used to identify the probability averaged over the initial distribution of electrons; this ++ + averaged probability is wMqq ′ (k)nq . In the second quantization formalism

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5 Magnetized Dirac electron

(§8.2 of volume 1), the occupation number n+ q arises from a statistical average of the outer product of the creation and annihilation operators, a ˆ+† ˆ+ q a q , + +† and the statistical average of the operators in the opposite order, a ˆq a ˆq , gives 1 − n+ q for fermions. The latter is relevant to the effect of electrons in the final state, which suppress the emission due to the Pauli exclusion principle. Similarly, the presence of photons in the final state gives a statistical contribution 1 + NM (k), where the unit term is attributed to spontaneous emission and the other term is interpreted as stimulated emission. The total probability for emission, after statistically averaging over the electrons and photons, is ++ + + wMqq ′ (k)nq (1 − nq ′ )[1 + NM (k)].

Detailed balance requires that the probability (5.6.18) also apply to the absorption process, in the sense that the probability of true absorption is equal to the probability of spontaneous emission times the occupation number, NM (k), of the photons. In the absorption process, an electron in the initial state q ′ absorbs a photon with frequency ωM (k), with transition to the final state q. The matrix element Tfi for this process is the complex conjugate of that for the emission process, and hence the differential rate per unit time, wi→f , for the absorption process differs from that for the emission process only through the density of final states. For the absorption process there is only one density of final states factor, (V eB/2π)(dpz /2π) for the final electron. As for spontaneous emission, the integral over this factor combines with the δ-function that expresses momentum conservation, specifically (2π/V eB) 2πδ(p′z − pz + kz ), and neither appears explicitly in the probability. In the transition rate, wi→f , it is implicit that there is a single photon in the initial state. The absorption rate for a distribution of photons is found by multiplying by [V d3 k/(2π)3 ]NM (k) and integrating. To identify the probability of true absorption one equates this absorption rate to the probability time NM (k)d3 k/(2π)3 . The resulting expression for this probability of absorption is identical to the probability of spontaneous emission, (5.6.18) with ǫ′ = ǫ = 1, confirming that detailed balance is built into the theory. The statistical average over a distribution of electrons and photons gives the averaged probability for true absorption as ++ + + wMqq ′ (k)nq ′ (1 − nq )NM (k).

Kinetic equations for the photons and electrons can be written down in §6.1.7 using these statistically averaged probabilities. The probability of spontaneous emission by a positron is found from the expression (5.6.18) by setting ǫ′ = ǫ = −1. In this case the role of the initial and final states is interchanged, such that q ′ is interpreted as the initial state, and q is interpreted as the final state. The other processes described by the probability (5.6.18) have ǫ′ = −ǫ = ±1 and correspond to one photon pair creation and annihilation. For pair annihilation, the differential transition rate, wi→f , is for a given electron and a given positron in the initial state and in

5.6 Feynman rules for QPD in a magnetized plasma

233

order to identify the probability of pair annihilation one includes distributions of electrons and positrons. The differential transition rate per unit volume if found from the transition rate for a single initial pair by multiplying by a − ′ factor (eB/2π)(dpz /2π)n+ q for the electrons and a factor (eB/2π)(dpz /2π)nq′ 3 3 for the positrons, in addition to the density of final states factor V d k/(2π) , times 1 + NM (k) to take account of induced emission. The probability for pair annihilation so identified is given by (5.6.18) with ǫ = +1, ǫ′ = −1. The only other first order processes in QPD are photon splitting and the cross process of coalescence of two photons into one. The probability for these processes is the same as in the unmagnetized case, which is also the same as in a semiclassical treatment. The properties of the wave modes and the detailed expression for the quadratic and cubic response tensors are affected by the presence of the magnetic field, but otherwise the QPD theory of processes involving only photons is the same as the semiclassical theory for wave-wave interactions. 5.6.8 Probabilities for second-order processes Second order processes involve four external lines, and these are either two electron lines and two photon lines, four electron lines or four photon lines. Processes with two electron lines and two photon lines correspond to Compton scattering and related crossed processes, including double emission and two-photon pair creation and annihilation. Processes involving four electron lines, correspond to electron-electron scattering and related crossed processes, including electron positron scattering and the trident process (emission of a pair by and electron). Processes involving four photon lines correspond to photon-photon scattering and related crossed processes, and the QPD theory is equivalent to the semiclassical theory for such processes. The argument leading to the identification of the probability for Compton scattering is closely analogous to the argument leading to the expression (5.6.18) for the probability of spontaneous emission. Given the matrix element Tfi for Compton scattering, the probability is ′

′ ′ ǫ ǫ 2 ′ 2 wMM ′ qq ′ (k, k ) = V |aM (k) aM ′ (k ) Tfi | 2π δ(ǫ εq ′ − ǫεq + ωM − ωM ′ ), (5.6.19) with ǫ′ = ǫ = +1 and where the labels M ′ , M denote the modes of the unscattered and scattered photons, respectively. The factor V 2 cancels with factors 1/V in |aM (k)|2 and |aM ′ (k′ )|2 , such that the probability is independent of V . Conservation of (the z-component of) 3-momentum is implicit. In ordinary units, the probability of Compton scattering has the dimensions L6 T −1 . For electron-electron scattering, with the initial states labeled q1 , q2 and final states labeled q1′ , q2′ , the probability is given by

wq1 q2 q1′ q2′ = 2πδ(εq1 + εq2 − εq1′ − εq2′ ) |Tfi |2 ,

(5.6.20)

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5 Magnetized Dirac electron

where conservation of the z-component of momentum, p1z +p2z −p′1z −p′2z = 0, is implicit. In ordinary units, this scattering probability also has the dimensions L6 T −1 .

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Johnson and Lippmann (1949) Sokolov & Ternov (1968) Gradshteyn & Ryzhik (1965) Schwinger (1951) G´eh´eniau (1950) G´eh´eniau, Demeur & G´eh´eniau (1951) Katayama (1951) K¨ all´en (1958) Ritus (1970) Ritus (1972) Parle (1987) C. Graziani: Astrophys. J. 412, 351 (1958)

6 Quantum theory of gyromagnetic processes

First order processes are described by a Feynman diagram with a single electron-photon vertex. In the presence of a static magnetic field, such a diagram describes six allowed processes: emission and absorption of a photon by an electron or a positron, and one-photon pair creation or annihilation. These are referred to here as the first-order gyromagnetic processes. An important feature of the quantum theory of such gyromagnetic processes is that the electron (or positron) is in a discrete Landau levels both in the initial state and in the final state. In any emission, absorption or scattering process, the electron jumps from one discrete Landau level to another. General results for gyromagnetic processes are written down in §6.1, including the probability for gyromagnetic processes, and kinetic equations that describe the effects of gyromagnetic emission and absorption and pair creation. The theory is applied to the quantum theory of cyclotron emission in §6.2 and to the quantum theory of synchrotron emission in §6.3. One-photon pair creation and annihilation are discussed in §6.4.

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6 Quantum theory of gyromagnetic processes

6.1 Gyromagnetic emission and pair creation In this section the general probability for first-order gyromagnetic processes are identified, and kinetic equations are written down for each of the processes. Some general results relating to the kinematics of gyromagnetic processes are also derived, including explicit solutions of the resonance condition. 6.1.1 Probability for first-order processes The probability for gyromagnetic emission follows directly from (5.6.18) and the other rules. One finds ′

ǫǫ wMqq ′ (k) =

2 ′ e2 RM (k) ∗ eMµ (k)[Γqǫ′ qǫ (k)]µ 2πδ ǫεq − ǫ′ εq′ − ωM (k) , (6.1.1) ε0 |ωM (k)|

where ǫ, ǫ′ label the signs of the particle energies, and q, q ′ denote the other quantum numbers label of the particle states. The label M for the wave mode of the wave quantum is omitted below where no confusion should result. The sets q and q ′ are chosen here to be the parallel momenta, pz , the principal quantum number, n, and a spin quantum number, s. Conservation of parallel momentum is implicit in (6.1.2) and sometimes it is relevant to make this explicit by including a factor Z dp′z 2πδ(ǫpz − ǫ′ p′z − kz ) 2π in the probability. The gauge-dependent quantum numbers do not appear in ′ (6.1.2) in the vertex function, [Γqǫ′ qǫ (k)]µ , which is, however, dependent on the choice of the spin operator. For the choice of the magnetic-moment operator, the vertex function is given by (5.4.18). The spin-dependent part may be isolated by writing the vertex function in the form (5.5.25). The dimensions of the probability are L3 T −1 . To exhibit these dimensions explicitly, one includes relevant powers of c and h ¯ . Omitting all arguments k, and writing (6.1.2) in ordinary units gives ′

ǫǫ wMqq ′ = ′

2 ′ e2 c2 RM ∗ eMµ (k)[Γqǫ′ qǫ ]µ 2πδ ωM − (ǫεq − ǫ′ εq′ )/¯h , ε0 ¯ h|ωM |

(6.1.2)

where [Γqǫ′ qǫ ]µ is dimensionless, and with ǫ′ p′z = ǫpz − ¯hkz implicit. Gyromagnetic emission by an electron corresponds to ǫ = ǫ′ = +1 in (6.1.2). The unprimed state is the initial state and the primed state is the final state. Gyromagnetic emission by a positron corresponds to ǫ = ǫ′ = −1. The roles of the initial and final states are reversed, with the unprimed state being the final state and the primed state being the initial state. One may rewrite the probability for a positron in the same form as for an electron, so that the unprimed state is the initial state and the primed state is the final

6.1 Gyromagnetic emission and pair creation

237

state, by setting ǫ = ǫ′ = −1 in (6.1.2), interchanging primed and unprimed quantities, and changing the sign of k, using ωM (−k) = −ωM (k), RM (−k) = RM (k),

eMµ (−k) = e∗Mµ (k), ′

′

[Γqǫ′ qǫ (−k)]µ = [Γqǫ′ qǫ (k)]∗µ .

(6.1.3)

−− The only net change in (6.1.2) is the change Γq++ ′ q (k) for an electron to Γq ′ q (k) for a positron, as one expects. If the particles are unpolarized, the emission by a positron is the same as the emission from an electron provided one makes the replacement e∗Mµ (k) → eMµ (k) in (5.4.2), which corresponds to reversing the handedness of the waves. This is as one expects on the basis of a classical picture in which the positrons spiral about the magnetic field lines in the opposite sense to the electrons, so that the handedness of the radiation that a positron emits is opposite to the handedness of the emission from an electron.

6.1.2 Unpolarized particles When one is not interested in the spins of the particles, it is appropriate to average over initial spins and sum over final spins. Sums over spins are performed for the outer product of a vertex function and its complex conjugate in (5.4.23). For an arbitrary wave mode, this gives X e∗Mµ (k)[Γqǫ′′qǫ (k)]µ 2 = 2ε′n′ εn Cnǫ′′ǫn (pz , k) , M s,s′

ǫ′ ǫ ′ µν Cn′ n (pz , k) M = e∗Mµ (k)eMµ (k) Cnǫ ′ǫn (pz , k) ,

(6.1.4)

′ µν with Cnǫ ′ǫn (pz , k) given explicitly by (5.4.24). For unpolarized particles, (6.1.2) is replaced by

µ0 e2 RM (k) ǫ′ ǫ Cn′ n (pz , k) M 2ε′n′ εn 2πδ ǫεn − ǫ′ ε′n′ − ωM (k) , 2|ωM (k)| (6.1.5) with εn = εn (pz ) = (m2 + p2z + p2n )1/2 , p2n = 2neB, and with ε′n′ = εn′ (p′z ). A factor 2 in the denominator in (6.1.5) applies to emission or absorption, and arises from averaging over the initial spins and summing over the final spins of the particle. For one-photon pair production one sums over the spins of both particles, so that the factor 1/2 is omitted in (6.1.5), and for pair decay into one photon one averages over the spins of both particles, so that the factor 1/2 is replaced by 1/4 in (6.1.5). ′

ǫ ǫ w ¯Mqq ′ (k) =

6.1.3 Reduction to the nonquantum limit It is of interest to derive the nonquantum limit from the relativistic quantum case. Specifically, consider the reduction of the probability (6.1.5) to its

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6 Quantum theory of gyromagnetic processes

semiclassical counterpart (4.1.9). The semiclassical result for gyromagnetic emission for an electron at harmonic a may be written wM (a, k, p) =

2 µ0 e2 RM (k) ∗ eMµ (k)U µ (s, kM ) 2πδ (kM u)k − aΩe , (6.1.6) γωM (k)

and the objective is to derive (6.1.6) from (6.1.2) with ǫ = ǫ′ = 1. One step is the replacement of the resonance condition, expressed through the δ-function in (6.1.5), by its classical counterpart, expressed through the δ-function in (6.1.6). The resonance condition ω − ǫεn + ǫ′ ε′n′ = 0,

(6.1.7)

needs to be expanded in powers of h ¯ . Including h ¯ explicitly gives ω → ¯hω, εn = εn (pz ) = (m2 c4 + p2z c2 + 2neB¯hc2 )1/2 , ε′n′ = εn′ (p′z ), ǫ′ p′z = ǫpz − ¯hkz . One assumes ¯h → 0 and n.n′ → ∞ with n¯h → p2⊥ /2eB and n − n′ = a remaining finite. The expansion in powers of h ¯ involves the Taylor series ˆ + 1D ˆ 2 + · · ·)εn , ε′n′ = (1 − D 2

ˆ = a ∂ + kz ∂ , D ∂n ∂pz

(6.1.8)

with ∂/∂n → (eB/p⊥ )(∂/∂p⊥ ). For gyromagnetic emission by an electron, to lowest order in h ¯ this gives ¯hω − εn + ε′n′ → ¯h[(ku)k − aΩ0 µ with pk = [εn , 0, 0, pz ] = muµk , Ω0 = eB/m. The other step is to relate the 4-vector U µ (a, k) in (6.1.6) to the vertex function in (6.1.2). This involves the large-n approximation to the J-functions: writing the argument as 2 2 x=h ¯ k⊥ c /2eB = z 2 /4n, where z = k⊥ p⊥ /eB is the argument of the Bessel functions in U µ (a, k). Then the J-functions are expanded in Bessel functions using (A.1.47), which is an expansion in z/n. The expansion gives Jνn

z2 4n

=

(n + ν)! n!nν

1/2 X ∞ j=0

bj

z j Jν+j (z), 2n

(6.1.9)

with b0 = 1, b1 = − 12 (ν + 1), and where Stirling’s approximation to the factor (n + ν)!/n!nν gives unity for n → ∞. The function Jνn−1 differs from Jνn only by a quantum correction: 2 2 z z k⊥ ′ J (z). (6.1.10) Jνn−1 − Jνn =− 4n 4n p⊥ ν To lowest order only the terms in the vertex function that involve no spin flip contribute, and these terms are independent of the choice of spin operator. For the magnetic moment operator, the vertex function (5.4.18) for an electron, ǫ = ǫ′ = 1, and for no spin flip, s′ = s = ±1, gives, to lowest order in h ¯, CHECK −a

6.1 Gyromagnetic emission and pair creation

239

′

[Γqǫ′ qǫ ]µ = (ieiψ )−a [a2+ + a2− ] [b2+ + b2− ] Ja (z), −[a2+ − a2− ] [b+ b− ][e−iψ Ja+1 (z) + eiψ Ja−1 (z)],

−i[a2+ − a2− ] [b+ b− ][e−iψ Ja+1 (z) + eiψ Ja−1 (z)], [2a+ a− ] [b2+ + b2− ] Ja (z) ,

a2+ + a2− = 1,

a2+ − a2− = ε0n /εn ,

b2+ + b2− = 1

a+ a− =

pz , 2εn

b+ b− = p⊥ /ε0n .

(6.1.11)

Comparison of (9.1.8) with the expression (2.1.28) with (2.1.29) for U µ (a, k) leads to the identification ′

[Γqǫ′ qǫ ]µ =

(ieiψ )−a µ U (a, k), γ

(6.1.12)

in the large-n limit. Alternatively, the form in (6.1.4) in which the average over spins is performed may be evaluated by making the large-n approximation to (5.5.35) with (??); this gives ++ µν Cn′ n (pz , k) 1 (6.1.13) = 2 U µ (a, k)U ∗ν (a, k), 2ε′n′ εn γ which also follows from (5.4.23) with (6.1.12). CHECK AND COMPLETE 6.1.4 Kinematics of gyromagnetic processes The conditions under which gyromagnetic emission is possible in general are determined by the resonance condition. The resonance condition (6.1.7) determines the properties of the initial and final particle for given ω, kz and given n, n′ . To find the values of pz , εn or p′z , ε′n′ for which resonance at given ω, kz , n, n′ is possible, one needs first to eliminate the square roots in (6.1.7). One way of achieving this is to note that all resonance conditions are included in D(ω, εn , ε′n′ ) = 0, with D(ω, εn , ε′n′ ) = (ω − ǫεn + ǫ′ ε′n′ )(ω + ǫεn − ǫ′ ε′n′ )

×(ω − ǫεn − ǫ′ ε′n′ )(ω + ǫεn + ǫ′ ε′n′ )

2 ′2 2 = ω 4 − 2ω 2 (ε2n + ε′2 n′ ) + (εn − εn′ ) .

(6.1.14)

Using ǫ′ p′z = ǫpz − kz , D(ω, εn , ε′n′ ) = 0 becomes (−4(ω 2 − kz2 ) times) a quadratic equation for either pz or p′z . The quadratic equation that determines pz is 2 p2z − 2ǫkz pz fnn′ − (ω 2 − kz2 )fnn ′ +

ω2 (ε0 )2 = 0, ω 2 − kz2 n

(6.1.15)

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6 Quantum theory of gyromagnetic processes

and the quadratic equation that determines p′z is ′ 2 2 2 p′2 z + 2ǫ kz pz fn′ n − (ω − kz )fn′ n +

ω2 (ε0 ′ )2 = 0, ω 2 − kz2 n

(6.1.16)

fn′ n = 1 − fnn′ .

(6.1.17)

with (ε0n )2 = m2 + 2neB and fnn′ =

(ε0n )2 − (ε0n′ )2 + ω 2 − kz2 , 2(ω 2 − kz2 )

It is convenient to write 2 ω − kz2 − (ε0n − ε0n′ )2 ω 2 − kz2 − (ε0n + ε0n′ )2 2 2 . gnn′ = gn′ n = 4(ω 2 − kz2 )2

(6.1.18)

The solutions of (6.1.15) may be written pz = pz± , p′z = p′z± , and they must be chosen such that one has ǫ′ p′z± = ǫpz± − kz . The relevant solutions are ǫpz± = kz fnn′ ± ωgnn′ ,

ǫ′ p′z± = −kz fn′ n ± ωgn′ n .

(6.1.19)

The energies εn (pz± ), εn′ (p′z± ) are determined by [εn (pz± )]2 = (ωfnn′ ± kz gnn′ )2 ,

[εn′ (p′z± )]2 = (ωfn′ n ∓ kz gn′ n )2 . (6.1.20)

The signs of the energies must be chosen such that the resonance condition ω − ǫεn (pz± ) + ǫ′ εn′ (p′z± ) = 0 is satisfied. The identity fnn′ + fn′ n = 1 implies the solutions ǫεn (pz± ) = ωfnn′ ± kz gnn′ ,

ǫ′ εn′ (p′z± ) = −ωfn′ n ± kz gn′ n .

(6.1.21)

6.1.5 Allowed resonances The quadratic equations (6.1.15) and (6.1.16) may be written in the forms 2 (pz − ǫkz fnn′ )2 = (p′z + ǫ′ kz fn′ n )2 = ω 2 gnn ′.

(6.1.22)

2 A necessary condition for resonance to be possible is that gnn ′ be non-negative. From (6.1.18), this condition requires either

ω 2 − kz2 ≤ (ε0n − ε0n′ )2 ,

(6.1.23)

which is the condition for gyromagnetic emission to be allowed, or ω 2 − kz2 ≥ (ε0n + ε0n′ )2 .

(6.1.24)

which is the condition for one-photon pair creation to be allowed. Resonance is forbidden for (ε0n − ε0n′ )2 < ω 2 − kz2 < (ε0n + ε0n′ )2 . An important example is gyromagnetic emission by an electron, ǫ = ǫ′ = 1 for ω 2 − kz2 > 0; one necessarily has ω 2 − kz2 > 0 for emission in vacuo. For

6.1 Gyromagnetic emission and pair creation

241

2 1

-1

0

2

4

6

Fig. 6.1. Initial and final resonance ellipse in ε0 = (m2 + p2⊥ )1/2 –pz space. The horizontal lines represent the physically allowed values p2⊥ = 2neB with n = 0–4. (Forgotten the details)

ω 2 − kz2 > 0 one may make a Lorentz transformation to the frame in which kz is zero, and in this frame the condition that the energies given by (6.1.21) be non-negative requires fnn′ > 0, fn′ n < 0, and the identity fn′ n = 1−fnn′ then implies that gyromagnetic emission is possible only for 0 < fnn′ < 1, and this requires 2(n − n′ )eB > ω 2 − kz2 . Thus for ω 2 − kz2 > 0 gyromagnetic emission is possible only for n > n′ . Gyromagnetic emission is allowed for ω 2 − kz2 ≤ 0, requiring n = n′ for ω 2 − kz2 = 0 and n < n′ for ω 2 − kz2 < 0; the latter is referred to as the anomalous Doppler effect. For one-photon pair creation, ǫ = −ǫ′ = 1, (6.1.24) requires ω 2 − kz2 > 0 and the same argument leads to the requirements fnn′ > 0, fn′ n > 0 for the energies to be positive; the identity fn′ n = 1−fnn′ then requires 0 < fnn′ < 1. For pair creation, there is no restriction on the sign of n − n′ for pair creation, but the magnitude is restricted by 2|n − n′ |eB < ω 2 − kz2 . 6.1.6 Graphical solutions of resonance condition A graphical solution of the resonance condition can be useful and informative. Consider momentum space, specifically p⊥ –pz space. The physically allowed states correspond to the discrete values p2⊥ = p2n = 2neB, n = 0, 1, . . ., which correspond to horizontal lines in p⊥ –pz space. It is found more convenient to choose the vertical axis to be ε0 = (m2 + p2⊥ )1/2 , so that the horizontal lines start with ε0 = m for n = 0 in ε0 –pz space, as illustrated in Fig. 6.1. The resonance condition may be plotted on the same space in two different ways, one corresponding to the initial conditions, and the other to the final conditions. Suppose that the forms (6.1.15) and (6.1.16) for the resonance condition are expressed in terms of ε0 by writing ε2n (pz ) = (ε0 )2 + p2z and ε2n′ (p′z ) = (ε0 )2 + p2z − 2ǫ′ pz kz + kz2 − 2(n − n′ )eB. Then (6.1.15) becomes (ε0 )2 (pz − pc )2 + 2 = 1, 2 2 2 ω fnn′ (ω − kz2 )fnn ′ and (6.1.16) becomes

pc = ǫ′ kz fnn′ ,

(6.1.25)

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6 Quantum theory of gyromagnetic processes

(ε0 )2 (p′z − p′c )2 + 2 =1 2 2 ω fn′ n (ω − kz2 )fn2′ n

p′c = −ǫkz fn′ n .

(6.1.26)

For ω 2 − kz2 > 0, (6.1.25) and (6.1.26) are ellipses, called the initial and final ellipse here. For given ω, kz , n, n′ , allowed initial conditions correspond to points in ε0 –pz space where the initial ellipse intersects the line corresponding to ε0 = ε0n . Similarly, allowed final conditions correspond to points in ε0 –pz space where the final ellipse intersects the line corresponding to ε0 = ε0n′ . An example of such a graphical plot is shown in Fig. 6.1 for gyromagnetic emission by an electron, ǫ = ǫ′ = 1. For given n − n′ , the initial and final ellipses depend on ω and kz . The initial ellipse intersects the line ε0 = ε0n only for (ω 2 − kz2 )fn2′ n > (ε0n )2 , and two resonances are possible when this condition is satisfied. The intersections of the initial ellipse with the line ε0 = ε0n correspond to the solutions pz± , given by (6.1.19), and the intersections of the final ellipse with line ε0 = ε0n′ correspond to the solutions p′z± . These two solutions coincide at threshold, where the initial ellipse is tangent to the line ε0 = ε0n and the final ellipse is tangent to the line ε0 = ε0n′ . These threshold conditions correspond to gnn′ = 0, as implied by (6.1.22). 6.1.7 Kinetic equations for gyromagnetic emission For a distribution of electrons emitting and absorbing photons through gyromagnetic transitions, the evolution of the system of photons and electrons is described by a set of kinetic equations. The kinetic equations for gyromagnetic emission in the non-quantum limit are written down in §4.1, notably the quasilinear equation (4.1.20) for the particles. The fully quantum counterparts of these equation are derived as follows. Consider the effect of transitions q → q ′ and the inverse transition q ′ → q on the occupation number, NM (k), of the wave quanta emitted in the first ++ transition and absorbed in the latter transition. The probability wMqq ′ (k) is the same for both transitions. Let the occupation number for electrons in the state q be n+ q . The rate of increase of NM (k) is determined by the probability per unit time of emission, + ++ + wMqq ′ (k) 1 + NM (k) nq (1 − nq ′ ),

minus the probability per unit time of absorption

++ + + wMqq ′ (k)NM (k)nq ′ (1 − nq ).

If more than one transition with given q and different q ′ gives emission of the same wave quanta (same k and same wave mode M ), then one is to sum over all such states q ′ . The resulting kinetic equation for the waves is DNM (k) X ++ + + + wMqq′ (k) n+ = q (1 − nq′ ) + NM (k)[nq − nq′ ] , Dt ′ qq

(6.1.27)

6.1 Gyromagnetic emission and pair creation

243

which may be written in the form DNM (k) = βM (k) − γM (k)NM (k), XDt X ++ ++ + + + βM (k) = wMqq γM (k) = − wMqq ′ (k) nq , ′ (k) [nq − nq ′ ], (6.1.28) qq′

qq′

where βM (k) is an emission coefficient (the rate per unit time that the photon occupation number increases due to spontaneous emission), and γM (k) is the absorption coefficient. The sums over the states q, q ′ in (6.1.28) warrant comment. The sum over q is over the initial electrons, and this includes the integral over (eB/2π)dpz /2π, as well as the sums over n, s. In applications one is often interested in specific transitions, for example, the cyclotron transition n = 1, s = −1 to n′ = 0, s′ = −1, and then the sum over q reduces to the integral over (eB/2π)dpz /2π alone. The sum over q ′ is over all possible final states that are allowed. This sum is usually redundant because the final state is uniquely determined. In the example of cyclotron emission n = 1, s = −1 to n′ = 0, s′ = −1, the final state is the ground state, and p′z is implicitly determined by conservation of momentum, so that the sum over q ′ can be omitted. The rate of change of the occupation number, n+ q , of the electrons is given by the difference between the probabilities per unit time of transitions to the state q, due to emission, q ′′ → q, from higher energy states, and absorption, q → q ′ , from lower energy states, minus the probabilities per unit time of the inverse transitions. This gives Z dn+ d3 k X ++ q + + + wMqq′ (k) n+ =− q (1 − nq′ ) + NM (k)(nq − nq′ ) 3 dt (2π) q′ X + ++ + + + − wMq′′ q (k) nq′′ (1 − nq ) + NM (k)(nq′′ − nq ) , (6.1.29) q′′

where q ′ denotes the quantum numbers n′ , s′ , p′z = pz − kz and q ′′ denotes n′′ , s′′ , p′′z = pz + kz . The kinetic equations (6.1.28), (6.1.29) reproduce their unmagnetized counterparts, when the states are identified according to q → p, q ′ → p − k, q ′′ → p + k. The kinetic equations for positrons follow from those for electrons by making the replacements: ++ −− wMq ′ q (k) → wMqq ′ (k),

− n+ q → nq ′ ,

− n+ q ′ → nq .

6.1.8 Kinetic equations for one-photon pair creation The kinetic equations for one-photon pair creation and decay follow from arguments similar to those for gyromagnetic emission. On including the occupation +− + − numbers, the probability of pair-annihilation transitions is wMqq ′ (k) nq nq ′ [1+

244

6 Quantum theory of gyromagnetic processes

+− NM (k)], and the probability of pair-creation transitions is wMqq ′ (k) NM (k)(1− − n+ )(1 − n ). A pair-annihilation transition increases the occupation numq q′ + ber, NM (k), and decrease the occupation numbers of both electrons, nq , and positrons, n− q , and a pair-creation transition has the opposite effect. Adding up the rates of change, the resulting kinetic equation for the photons is

DNM (k) X +− − − + wMqq′ (k) n+ = q nq′ − NM (k)(1 − nq − nq′ ) . Dt ′

(6.1.30)

qq

− The term involving n+ q nq′ describes the effect of pair annihilation, and the − term involving 1 − n+ q − nq′ describes the effect of pair creation. The corresponding kinetic equations for the electrons and positrons are Z dn+ d3 k X +− q − + − wMqq′ (k) NM (k)(1 − n+ = q − nq ′ ) − nq nq ′ , 3 dt (2π) ′ q

dn− q′ dt

=

Z

d3 k X +− − + − wMqq′ (k) NM (k)(1 − n+ q − nq ′ ) − nq nq ′ . 3 (2π) q

(6.1.31)

− As in (6.1.30). the term involving n+ q nq′ describes the effect of pair annihila− tion, and the term involving 1 − n+ q − nq′ describes the effect of pair creation. − The Pauli exclusion principle requires n+ q , nq′ ≤ 1, but this does not forbid − 1−n+ q −nq′ < 0, and which conditions the effect of pair creation would change sign, implying an exponential growth of photons and an exponential decay of the electrons and positrons.

6.1.9 Anharmonicity A feature of the relativistic quantum theory of gyromagnetic emission and absorption, compared with the classical theory, is that each transition, n → n′ for emission by an electron, occurs at a different frequency. This effect results from the combination of relativistic and quantum effects. Consider the nonrelativistic quantum limit: the transitions between electron states with quantum numbers n and n′ involves emission (or absorption) of a wave quantum that satisfies the resonance condition ω − kz vz = (n − n′ )Ωe . For given kz , the resonant frequency then depends on only the difference n − n′ , and not on n, n′ separately. This is due to the energy of the electron being a sum of the kinetic energy, p2z /2m, associated with the motion along the field line, and the energy hΩe (in ordinary units) associated with the motion about the magnetic (n+ 12 )¯ field lines, which is simple harmonic motion. (There is also an energy ± 21 ¯hΩe associated with the spin.) The energy eigenvalues are harmonically related, such that the difference between neighboring levels has a fixed value, h ¯ Ωe , and emission occurs only at (Doppler shifted) harmonics of this difference. When relativistic quantum effects are taken into account, this degeneracy between

6.1 Gyromagnetic emission and pair creation

245

10 (a)

n’ - n = 2

8 n’ - n = 1

6

n’ - n = 3

4 2 0 4 n’ - n = 2

(b) n’ - n = 3

2 n’ - n = 1

1

-2 0

1

2

Fig. 6.2. The dependence of the logarithm of the emission coefficient on ω/Ωe is shown for B/Bc = 0.5 for the first three harmonics. The calculations were performed for θ = π/2 and radiation polarized (a) perpendicular to B,√and (b)√along B. √ The thresholds for the lowest transitions correspond to ω/m = 2 − 1, 3 − 1, 4 − 1 for n′ − n = 1, 2, 3, respectively, with Ωe /m = B/Bc .

levels is broken. Specifically, for given pz and kz , the frequency determined by the resonance condition, (6.1.7), gives a different value of ω for every different choice of n and n′ . The resonant frequency depends on n or n′ , as well as on n − n′ . This effect is sometimes called anharmonicity: there is no harmonic relation between the frequencies for given n − n′ and different n. 6.1.10 Quantum oscillations Another characteristic property of the quantum theory of gyromagnetic emission and absorption, and of one-photon pair creation and annihilation in a magnetic field, is that the transition rates have square-root singularities at the thresholds where each new value of n, n′ becomes allowed. These thresh2 olds are determined by gnn′ = 0, with gnn ′ given by (6.1.18). The singular behavior arises in a general class of integrals, which includes the emission and absorption coefficients. The gyromagnetic emission and absorption coefficients (6.1.28) involve sums over the initial and final states, which include integrals over pz and p′z . These integrals are performed over two δ-functions in the probability (6.1.2). Suppose the integral over p′z , is performed over δ(ǫ′ p′z − ǫpz + kz ), as is implicit in (6.1.2). Then integral over pz is of the generic form

246

6 Quantum theory of gyromagnetic processes ǫǫ′ Inn ′

=

Z

′

ǫǫ ′ ′ dpz Fnn ′ (pz ) δ[ǫεn (pz ) − ǫ εn′ (pz ) − ω],

(6.1.32)

′

ǫǫ where Fnn ′ (pz ) is arbitrary. Both the emission and absorption coefficients (6.1.28) are of the form (6.1.32), as is any other integral that involves a sum of the probability (6.1.2) over the initial and final states. Performing the integral over the δ-function in (6.1.32) gives ′

ǫǫ Inn ′ =

X ±

−1 ǫpz ǫ′ p′z ǫǫ′ Fnn − ′ (pz± ) εn (pz ) εn′ (p′ ) , z ±

(6.1.33)

where ǫpz − ǫ′ p′z − kz = 0 and the resonance condition expressed by the δfunction in (6.1.32) are used. The solutions (6.1.19) and (6.1.21) imply −1 ǫpz εn (pz )εn′ (p′z ) εn (pz± )εn′ (p′z± ) ǫ′ p′z = − εn (pz ) εn′ (p′ ) kz ǫεn (pz ) − ωpz = (ω 2 − k 2 ) gnn′ . (6.1.34) z ± z ±

′ 2 The function gnn ′ , given by (6.1.18) has zeros for each value of n , n, implying that (6.1.34) have a square-root singularity at each threshold. This singular behavior is sometimes called quantum oscillations in the case of gyromagnetic emission. An example is illustrated in Fig. 6.2 for B/Bc = 0.5. For the emission and absorption coefficients (6.1.28) evaluation using (6.1.34) gives " 2 # X µ0 e3 B R e∗ · Γ++ q′ q (k) βM (k) = εn± ε′n′ ± nn± , (6.1.35) 2 2) g ′ (2π) ω(ω − k nn z ± M

" 2 # X µ0 e3 B R e∗ · Γ++ q′ q (k) γM (k) = − 2 2 (2π) ω(ω − kz ) gnn′ ±

M

εn± ε′n′ ± (nn± − n′n′ ± ), (6.1.36)

respectively, with εn± = εn (pz± ), ε′n′ ± = εn′ (p′z± ), nn± = εn (pz± ), n′n′ ± = nn′ (p′z± ), where subscript M indicates that R, e and ω are to be evaluated for the mode M , and where only the contributions for a specific transition n, s → n′ , s′ is retained explicitly. On physical grounds one expects that the quantum oscillations, being an intrinsically quantum phenomenon, to become unobservable for sufficiently nonrelativistic particles. As the classical limit is approached the square-root singularities, that lead to the spiky structure in Fig. 6.2, become densely packed, with a square-root singularity at the threshold for each new n, n′ . The quantum oscillations become increasingly unobservable when the peaks are washed out by line-broadening effects. In a nonrelativistic thermal plasma the most important line-broadening effect is Doppler broadening, and the quantum oscillations become unobservable when the Doppler width exceeds the separation between neighboring peaks in the quantum oscillations. The classical Doppler broadening is zero for perpendicular propagation, and then

6.1 Gyromagnetic emission and pair creation

247

the transverse Doppler effect (due to the spread in Lorentz factors) needs to be taken into account. In the absence of Doppler broadening, there is a natural linewidth for emission by an individual particle: the finite half-life of the upper state introduces an uncertainty in the energy of the transition, and hence of its frequency. When this linewidth exceeds the separation between the peaks associated with the quantum oscillations, the quantum oscillations become intrinsically unobservable [1].

248

6 Quantum theory of gyromagnetic processes

6.2 Quantum theory of cyclotron emission Gyromagnetic emission from nonrelativistic electrons is called cyclotron emission. In a relativistic quantum treatment there are subtleties in what constitutes the ‘nonrelativistic’ approximation, and to avoid confusion, the term ‘cyclotron’ approximation is used here. In practice, the most important approximation is that the generalized Laguerre polynomials, introduced through the functions Jνn (x) defined by (5.2.24), are approximated by the leading term in their expansion in powers of x. In this section, after a general introduction to the quantum theory of cyclotron emission, radiative transitions between low Landau levels for an electron in vacuo are discussed in detail, and some expressions for cyclotron emission in nonrelativistic plasmas are written down. 6.2.1 Cyclotron approximation A gyromagnetic transition by an electron may be defined as a transition between an initial state with quantum numbers q = pz , n, s, and a final state with quantum numbers q ′ = p′z , n′ , s′ , with p′z = pz − kz . The parallel motion is unimportant in classifying gyromagnetic transitions: one may make a Lorentz transformation to the frame pz = 0, and provided one has |kz | ≪ m in this frame, the parallel recoil motion is nonrelativistic. The perpendicular momentum is quantized, p⊥ = p2n = (2neB)1/2 , and the nonrelativistic condition p⊥ ≪ m requires 2nB/Bc ≪ 1. In superstrong fields, B/Bc > ∼ 1, even the first Landau level, n = 1, has a relativistic energy, and the nonrelativistic approximation is not valid. For weaker fields the nonrelativistic approximation places a limit on n ≪ Bc /2B. The transitions n, s → n′ , s′ may be described in terms of the ladders in Fig. 5.1. Each step in a ladder represents a state n. The left hand ladder corresponds to s = −1, and includes the ground state n = 0, and the right ladder corresponds to s = +1 and starts from n = 1. The normal Doppler effect is defined such that emission leads to a transition in which n decreases to n′ < n, and the anomalous Doppler effect to a transition in which n increases to n′ > n. One can classify transitions as non-spin-flip for s′ = s, spin-flip for s = 1, s′ = −1, in which the jump is from the right ladder to the left ladder, and reverse spin-flip for s = −1, s′ = 1, in which the jump is from the left ladder to the right ladder. The value of j = n − n′ determines the number of steps down which the electron jumps in a transition. The normal Doppler effect corresponds to j > 0, and is the only case allowed for ω 2 − kz2 > 0. The anomalous Doppler effect, in which the particle moves from a lower to a higher Landau level (n′ > n), is possible for kz2 > ω 2 . The case j = 0, in which n = n′ does not change, is sometimes referred to as the Cerenkov effect, although this can cause confusion with the unmagnetized case. Emission at j ≤ 0 is allowed only for waves with refractive index greater than unity, and only emission in vacuo is discussed in detail in the following. However, the cyclotron approximation is

6.2 Quantum theory of cyclotron emission

249

relevant not only to cyclotron emission and absorption, but also to higher order processes for nonrelativistic particles; in particular, Compton scattering and double emission, cf. §??, involve transitions to and from virtual intermediate states, including transitions with j = 0. A major simplification in the cyclotron approximation follows from the assumption that the argument of the J-functions in the vertex function (5.4.18) 2 is small, x = k⊥ /2eB ≪ 1. One has 2 −1 2 2 k⊥ 1 k⊥ k 2 sin2 θ B B k⊥ x= = . (6.2.1) = = 2eB 2mΩe 2 m Bc Ωe 2Bc In the cyclotron approximation, k⊥ , kz , ω are assumed of order the cyclotron frequency, Ωe , and then (6.2.1) implies that x is of order B/Bc . The formal expansion parameter may be regarded as B/Bc . This approximation obviously breaks down in supercritical magnetic fields, B/Bc > ∼ 1. The power series expansions of the J-functions converge rapidly for x ≪ 1, and each function may be approximated by its leading term. For example, for x ≪ 1, (A.1.45) implies that Jan (x) with n′ = n − j is approximated by 1/2 n! (−)j n j n−j xj/2 . (6.2.2) J−j (x) = (−) Jj (x) ≈ j! (n − j)! The approximation (6.2.2) applies for any integer values of n, j subject to the restrictions n ≥ 0, j ≤ n, which follow from Jνn (x) being identically zero for n < 0 or ν < −n. 6.2.2 Cyclotron approximation to the vertex function The foregoing approximations are applied to the vertex function in the form (5.4.18). This involves assuming that pz /m, pn /m are small and expanding in powers of them. The momentum, kz , of the wave quantum is assumed to be of the same order as the momentum of the particle, so that k⊥ /m, kz /m, p′z /m are of the same order as pz /m. For the quantity pn /m = (2nB/Bc )1/2 to be small for modest values of n, one also needs to make the weak-field approximation B ≪ Bc . The expression (5.4.18) for the vertex function involve factors a′± , a± and ′ b± , b± , as well as J-functions. For an electron, ǫ = ǫ′ = 1, the as and bs are approximated by a′+ ≈ a+ ≈ 1,

a′− ≈

p′z , 2m

a− ≈

pz , 2m

b′+ ≈ b+ ≈ 1, b′− ≈

pn′ , 2m

b− ≈

pn , 2m

(6.2.3)

where only terms up to first order in small quantities are retained. These approximations allow one to order the coefficients of the J-functions in (5.4.18) in powers of the small quantities.

250

6 Quantum theory of gyromagnetic processes

For a transition that involves a jump of j rungs down the ladder in Fig. 5.1, one has n′ = n−j. The leading J-functions are those with lower index n′ −n+ µ 1 = −(j − 1), and these appear only in the µ = 1, 2 components of [Γq++ ′ q (k)] . ′ For s = s these are the only terms that need be retained. However, for s 6= s′ , the coefficients of these terms are one order higher in the small quantities, and the µ = 3 component is then of the same order as the µ = 1, 2 components. (The µ = 0 component is redundant when one chooses the temporal gauge for the polarization 4-vector of the waves.) The relevant approximation to the vertex function (5.4.18) is different for transitions involving no spin flip (s = s′ ), those involving a spin flip (s = 1, s′ = −1) and those involving a reverse spin flip (s = −1, s′ = 1). There are three factors in the vertex function(5.4.18), and each of these needs to be evaluated for the four different combinations of spins for specified j = n − n′ . The first is the phase factor (ieiψ )j . The second factor involves the combination of as, which reduce to, for ǫ = ǫ′ = 1, a′s′ as − a′−s′ a−s = s 12 (1 + s′ s) + 12 (1 − s′ s) (p′z − pz )/2m, , a′s′ a−s + a′−s′ as = 12 (1 + s′ s) (p′z + pz )/2m + 21 (1 − s′ s).

(6.2.4)

The other factor involves the J-functions. Retaining only the lowest order terms in x1/2 , (B/Bc )1/2 for each value of s, s′ , one finds b′s′ bs Jnn−1 ′ −n

+s

′

sb′−s′ b−s Jnn′ −n

(−)j = j!

l! (l − j)!

  1 × −j(B/Bc )1/2 /(l + 1 − j)1/2   O(B/Bc , x)

1/2

xj/2

for s = s′ , for s = −s′ = 1,

for s = −s′ = −1,

′ ′ iψ n b′−s′ bs e−iψ Jnn−1 Jn′ −n−1 ′ −n+1 ∓ ss bs′ b−s e 1/2 l! (−)j−1 x(j−1)/2 = (j − 1)! (l − j)!  1/2  for s = s′ ,  (B/Bc ) ×e−iψ 1/(l + 1 − j)1/2   O(B/Bc , x) for s = −s′ = −1.

(6.2.5)

In the entries for the reverse spin flip (s = −1, s′ = 1) in (6.2.5) the leading terms cancel, and the remaining terms are of higher order than those retained in the following discussion. 6.2.3 Multipole expansion of the vertex function Let Γ++ n′ ,s′ ;n,s (k) denote the space components of the vertex function. For nonspin-flip transitions the leading terms give

6.2 Quantum theory of cyclotron emission

1/2

251

1/2

x(j−1)/2 (1, i, 0), (j − 1)! (6.2.6) 2 with n = l + (s + 1)/2. With x = k⊥ /2eB, it follows that (6.2.6) corresponds j−1 to a current that is proportional to k⊥ . One expanding a current in powers of |k|, terms that depend on |k|j−1 correspond to either 2j -electric multipole or 2j−1 -magnetic multipole. The non-spin-flip term (6.2.6) is the 2j -electric multipole term, with j = n − n′ = l − l′ , so that j corresponds to the change in the orbital quantum number. Specifically, the transition with s′ = s and j = l − l′ = 1 corresponds to electric dipole radiation, the transition with s′ = s and j = l − l′ = 2 corresponds to electric quadrupole radiation, and so on. For the direct spin-flip transition, s = 1 → s′ = −1, (6.2.6) is replaced by 1/2 (j−1)/2 l! x (kz , ikz , −k⊥ ) j i(j−1)ψ Γ++ (k) = (−i) e , n−j,−1;n,1 (l − j + 1)! (j − 1)! 2m (6.2.7) which is the 2j−1 -magnetic multipole term. In particular, the transition s = 1, s′ = −1 for j = n − n′ = 1 involves no change in the orbital quantum number, l′ = l, and this corresponds to magnetic dipole radiation. The reverse spin-flip transition, s = −1 → s′ = 1, is of higher order and is effectively forbidden in the cyclotron approximation. There is a contribution to the vertex function for n′ = n, that is for j = 0, which may be approximated by j i(j−1)ψ Γ++ n−j,s;n,s (k) = (−i) e

B 2Bc

l! (l − j)!

′ ′ 1 Γ++ n,s;n,s (k) = 2 (1 + s s) [(pz + pz )/2m] (0, 0, 1),

(6.2.8)

which contributes only when there is no spin flip. As already remarked, although transitions with j = 0 correspond to Cerenkov-like emission and not to gyromagnetic emission, the term (6.2.8) is needed for the treatment of higher order gyromagnetic processes for nonrelativistic particles. 6.2.4 Probability for cyclotron emission The probability for gyromagnetic emission, (6.1.2), becomes the probability for cyclotron emission when the cyclotron approximation is made, in particular, when the approximation (6.2.6), (6.2.7) is made to the vertex function. The probability depends on the properties of the emitted waves, and it is convenient to assume a general form that applies when spatial dispersion is unimportant. For a wave in a mode M , this corresponds to k⊥ = nM ω sin θ, kz = nM ω cos θ and eM =

LM κ + TM t + ia , 2 1 + L2M + TM

RM =

2 1 + L2M + TM 2 )n ∂(ωn )/∂ω , 2(1 + TM M M

(6.2.9)

with κ = (sin θ cos φ, sin θ sin ψ, cos θ), t = (cos θ cos φ, cos θ sin φ, − sin θ), a = (− sin ψ, cos φ, 0) here.

252

6 Quantum theory of gyromagnetic processes

For an electron, the probability for a non-spin-flip transition, n → n′ = n − j, is µ0 e2 (1 + LM sin θ + TM cos θ)2 2(j−1) 2(j−1) sin θ nM 2 ) 2ωnM ∂(ωnM )/∂ω(1 + TM 2−j ω 2(j−1) B l! 2πδ(εn − ε′n−j − ω), (6.2.10) × 2 (l − j)! [(j − 1)!] 2Bc m

nsf (k) = wMn(n−j)

where the nonrelativistic approximation is not made to the δ-function. The probability for a transition n → n′ = n − j with a spin-flip, s = 1 → s′ = −1, is 2 µ0 e2 (cos2 θ + TM ) 2j 2(j−1) θ 2 ) nM sin 2ωnM ∂(ωnM )/∂ω(1 + TM 1−j ω 2j B l! 2πδ(εn − ε′n−j − ω). × j−1 2 (l − j + 1)! [(j − 1)!]2 Bc m (6.2.11)

sf (k) = wMn(n−j)

The probabilities corresponding to (6.2.10), (6.2.11) for a positron may be obtained by replacing the polarization vector by it complex conjugate. The dependence of the emissivity (6.2.10) on refractive index is ∝ n2j−1 N for non-spin-flip transitions and ∝ n2j N for spin-flip transitions. These are the characteristic dependences on refractive index for 22j -electric multipole emission, and 22j -magnetic multipole emission, respectively. 6.2.5 Spontaneous gyromagnetic emission in vacuo An electron in a magnetic field emits gyromagnetic radiation spontaneously. The rate of transitions due to gyromagnetic emission is given by summing the probability (6.2.1) over the density of states of the emitted photon. For transverse waves in vacuo the resulting rate, after summing over the two states of polarization, is given by Z i d3 k 1 h ++ 2 |Γq′ q (k)|2 − |κ · Γ++ 2π δ(εn − ε′n′ − ω), Rq′ q = 12 µ0 e2 q′ q (k)| 3 (2π) ω (6.2.12) with q and q ′ denoting pz , n, s and p′z , n′ , s′ , respectively, with εn , ε′n′ denoting εn (pz ), εn′ (p′z ), respectively, and with p′z = pz − kz . The power radiated, Pq′ q , is given by an analogous expression with the energy, ω, per photon included in the integrand: Z i d3 k h ++ 2 2π δ(εq − εq′ − ω). (6.2.13) |Γq′ q (k)|2 − |κ· Γ++ Pq′ q = 12 µ0 e2 q′ q (k)| 3 (2π) Conservation of energy and parallel momentum imply that the energy and parallel momentum of the radiating particle change according to

6.2 Quantum theory of cyclotron emission

d dt

εq pz

=−

X q′

2 1 2 µ0 e

Z

d3 k (2π)3

1 cos θ

253

h i ++ 2 2 × |Γ++ 2π δ(εq − εq′ − ω). q′ q (k)| − |κ · Γq′ q (k)|

(6.2.14)

P In particular one has dεq /dt = − q′ Pq′ q . In vacuo, the integral over d3 k in (6.2.12) or (6.2.13) may be rewritten as an integral over frequency and solid angle. To perform the ω-integral over the δ-function, it is convenient to transform to the frame pz = 0. In this frame the resonant frequency is ω = ωnn′ (θ), with o 1 n 2 [m + 2eBn]1/2 − [m2 + 2eB(n cos2 θ + n′ sin2 θ)]1/2 . 2 sin θ (6.2.15) The rate (6.2.12) becomes ωnn′ (θ) =

1

i h 0 ++ 2 2 ωnn′ (θ)[εn − ωnn′ (θ)] d cos θ |Γ++ , q′ q (k)| − |κ · Γq′ q (k)| ε0n − ωnn′ (θ) sin2 θ −1 (6.2.16) with ε0n = (m2 + 2neB)1/2 . The power (6.2.13) reduces to an analogous expression with an extra power of ωnn′ (θ) in the integrand. Although neither the rate nor the power can be evaluated exactly in the quantum case, they can be evaluated in the cyclotron and synchrotron limits. Rq′ q

µ0 e2 = 4π

Z

6.2.6 Lorentz transformation to the laboratory frame It is sometimes convenient to continue to use the frame pz = 0 to derive explicit expressions, and then to generalize these expression to an arbitrary frame, pz 6= 0, by making a Lorentz transformation. Let quantities in the frame in which the particle has no parallel momentum be denoted by tildes (˜ pz = 0), and let the Lorentz transformation to the laboratory frame, where the parallel momentum is pz , have a velocity V and a Lorentz factor Γ = (1 − V 2 )−1/2 . One has εn (pz ) = Γ ε0n ,

pz = Γ V ε0n .

(6.2.17)

In terms of the pitch angle, α, defined by pz = p cos α,

pn = p sin α,

p = (p2z + p2n )1/2 ,

(6.2.18)

one has

1 , V = cos α. sin α The frequency and angle of emission transform according to Γ =

ω ˜=ω

1 − cos α cos θ , sin α

cos θ˜ =

cos θ − cos α . 1 − cos α cos θ

(6.2.19)

(6.2.20)

254

6 Quantum theory of gyromagnetic processes

Applying the Lorentz transformation to the expression (6.2.16) for Rq′ q and to the analogous expression for Pq′ q involves making the appropriate replacements on the right hand side, and noting that the rate, Rq′ q , transforms as a frequency and that the power, Pq′ q , is an invariant. Specifically, one has ˜ q′ q with R ˜ q′ q identified with the expression (6.2.16). Rq′ q = Γ R 6.2.7 Spontaneous cyclotron emission in vacuo Consider spontaneous emission by an electron in vacuo, as described in the general case by (6.2.3)–(6.2.5). In the cyclotron limit, the transition rate (6.2.3) with (6.2.6), (6.2.7) may be evaluated explicitly for transitions n → n′ = n − j both without and with a spin flip. The rates for emission of photons polarized perpendicular and parallel to the projection of B onto the transverse plane may also be evaluated explicitly. Let Rn→n−j denote the rate summed over the states of polarization, and let rn→n−j be the difference in the rates for perpendicular and parallel polarization, divided by the sum of these rates. One may interpret rn→n−j as giving a measure of the degree of polarization, with positive values (rn→n−j > 0) favoring emission of perpendicularly polarized photons. The actual degree of polarization depends on the angle of emission. For non-spin-flip transitions (s′ = s), one finds nsf Rn→n−j

2j+1 (j + 1)! ω 2j−1 l! = αc m (l − j)! (j − 1)! (2j + 1)! m nsf rn→n−j =

B Bc

2−j

j , j+1

, (6.2.21)

with n = l + 21 (1 + s) and where ω = εn − εn−j is the frequency of the emitted photons. The emission frequency at the jth harmonic is approximated by ω = jΩe , due to only the lowest order terms in B/Bc being retained in (6.2.21), with (εn − εn−j )/m = j(B/Bc ) to this order. For spin-flip transitions, s = 1, s′ = −1, (6.2.21) is replaced by sf Rn→n−j =

nsf ω 2 B −1 2Rn→n−j , n−j m Bc

sf rn→n−j =−

j , j+1

(6.2.22)

with l = n − 1 in this case. The rate (6.2.22) for the spin-flip transition is smaller than the rates (6.2.21) for the non-spin-flip transition by B/Bc ≪ 1. The rate for the reverse spin-flip transition, s = −1, s′ = 1, is smaller than the rate for the direct spin-flip transition by a further factor of order (B/Bc )2 , and is neglected here. The results (6.2.21), (6.2.22) imply that for nonrelativistic electrons initially in a high Landau level in a field with B/Bc ≪ 1, the branching ratio for jumps involving a jth-harmonic transition (j ≥ 2) with no spin flip depends on the (j +1)th power of B/Bc , and a transition with spin flip (s = 1, s′ = −1)

6.2 Quantum theory of cyclotron emission

255

is higher than that without spin flip by an extra factor of order B/Bc . Hence, the most probable sequence of transitions for B/Bc ≪ 1 is that the electron jumps stepwise down the ladder to l = 0. That is, the most probable sequence of transitions is a sequence with j = 1 for each transition and with no spin flip. For an electron initially with s = −1, the level l = 0 is the ground state, n = 0. For an electron with s = 1, the likelihood of a spin-flip transition is low until it reaches l = 0 (n = 1 in this case), where it remains until it jumps to n = 0 by a spin-flip transition. The probability of a reverse spin flip (s = −1, s′ = 1) is negligible in the cyclotron approximation. In the cyclotron approximation, retaining only the electric-dipole transition, the rate of change in the energy and parallel momentum, cf. (6.2.5), reduce to 3 dpz 4αc m2 B dεn l, =− = 0, (6.2.23) dt 3 Bc dt with n = l + 12 (1 + s). It follows from (6.2.23) with εn = m + (p2n + p2z )/2m that in the nonrelativistic limit the energy radiated comes from a decrease in pn at fixed pz . For j = 1 and with the approximation ω = Ωe , (6.2.21), (6.2.22) gives (in ordinary units) nsf R1→0

4αc mc2 = 3¯ h

B Bc

2

,

sf nsf R1→0 = R1→0

B . Bc

(6.2.24)

For typical strong fields available in the laboratory, say several tesla, one has nsf B/Bc ∼ 10−9 , and the lifetime (1/R1→0 ) of an electron in the first excited state is tens of minutes. For typical pulsar fields, B/Bc ∼ 0.1, the lifetime is extremely short, and all the electrons are expected to be in their ground states. It should be emphasized that this discussion of the spin-flip and non-spinflip transitions applies only when the spin operator is the magnetic-moment operator. If some other spin operator is chosen, then this discussion does not apply. For example, consider the helicity eigenfunctions (5.2.6). For a nonrelativistic electron, the helicity eigenfunction σ = 1 is a mixture of the s = 1, s = −1 states in the ratio cos αn /2 : sin αn /2, where αn is the counterpart of the classical pitch angle: pn /(p2n +p2z )1/2 = sin αn , pz /(p2n +p2z )1/2 = cos αn . As an electron radiates, n decreases, and hence pn decreases, whereas pz remains constant in the nonrelativistic approximation. Hence αn decreases as a result of the gyromagnetic emission. Suppose that there is no spin-flip in s, and that in a sequence of electric-dipole transitions pn decreases at fixed s. The linear relation between the σ- and s-states is a function of pn /pz and hence as pn changes, the mixture of σ states changes. However, there is no spin flip, in the sense that, by hypothesis, all transitions are electric dipole. ‘Spin flip’ should be used only to mean a change in the eigenvalue, s, of the magnetic-moment operator.

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6 Quantum theory of gyromagnetic processes

6.2.8 Cyclotron resonance condition The resonance condition in the probability (6.2.10) for cyclotron emission is in its exact form, εn − ε′n′ − ω = 0, and it is approximated by ω = jΩe , j = n − n′ , in say (6.2.23). Some care is required in identifying the cyclotron approximation to the resonance condition. Suppose one makes the nonrelativistic approximation, εn ≈ m + p2z /2m + nΩe , so that the resonance condition becomes ω − (n − n′ )Ωe − kz pz /m − kz2 /2m = 0, where the final term is due to the quantum recoil. However, the quantum recoil term is not given correctly by this strictly nonrelativistic derivation, and −kz2 /2m is replaced by +(ω 2 − kz2 )/2m in the relativistic approximation to a relativistically correct treatment of the recoil. Independent of the form of the spin flip, the resonance condition has only a single solution for pz , and there is no anharmonicity. However, there are two solutions, pz = pz± , to the exact resonance condition (6.2.10), and the anharmonicity leads to the quantum oscillations in the emission and absorption coefficients. In taking the cyclotron limit, one needs to identify how the two solutions for pz are replaced by one, and how the quantum fluctuations are smoothed out. The nonrelativistic resonance pz = m[ω − (n− n′ )Ωe )]/kz in the absence of the recoil follows from the exact solutions pz± as follows. First, one neglects quantum recoil terms (by restoring h ¯ and expanding in powers of it): the leading terms in (6.1.17) and (6.1.18) give fnn′ ≈ m

jΩe , − kz2

ω2

gnn′ ≈ m

(j 2 Ωe2 − ω 2 + kz2 )1/2 , ω 2 − kz2

(6.2.25)

respectively, with j = n − n′ . The two solutions (6.1.19) then give pz± ≈ m

kz jΩe ± ω(j 2 Ωe2 − ω 2 + kz2 )1/2 . ω 2 − kz2

(6.2.26)

On expanding in powers of ω − jΩe , the solution with kz ± |kz | = 0 reproduces the nonrelativistic result pz = m(ω − jΩe )/kz . The other solution is not in the nonrelativistic range, and is ignored. The expansion in ω − jΩe requires |ω − jΩe | ≪ kz2 /2ω, and hence requires |pz |/m ≪ |kz |/2ω. The neglect of the second solution is clearly not valid near threshold, where the two solutions coincide, with pz+ = pz− = mkz /(ω 2 − kz2 )1/2 , at (ω 2 − kz2 )1/2 = jΩe in the approximation (6.2.26). The second solution is nonrelativistic for sufficiently small kz ; its interpretation is straightforward in terms of the semirelativistic approximation to the resonance condition, as illustrated in Fig. 4.1. To include the quantum oscillations in the nonrelativistic limit, one retains the next order terms in the expansion of gnn′ in powers of B/Bc . This corresponds to replacing gnn′ in (6.2.25) by gnn′

1/2 m ′ B 2 2 2 2 ≈ 2 j Ωe 1 − (n + n ) − ω + kz . ω − kz2 Bc

(6.2.27)

6.2 Quantum theory of cyclotron emission

257

The threshold condition becomes (ω 2 − kz2 )1/2 = jΩe [1 − (n + n′ )B/2Bc ], which depends on both j = n − n′ and n + n′ so that it has a unique value for each transition n → n′ . 6.2.9 Cyclotron emission and absorption coefficients The gyromagnetic emission and absorption coefficients, (6.1.35) and (6.1.36) respectively, become cyclotron emission and absorption coefficients when the cyclotron approximation is made to them. Three separate approximations are involved. The approximation to the vertex function is straightforward, using (6.2.6), (6.2.7). The approximation made to the factor (ω 2 − kz2 )gnn′ in the denominator depends on the specific effects that one wishes to include. One uses (6.2.27) if the anharmonicity and the associated quantum oscillations are of interest, and (6.2.25) if they are ignored. The square-root singularity is either case. The simplest approximation to pz± is when one solution corresponds to pz = m(ω − jΩe )/kz and the other to a value where the occupation number in (6.1.35) and (6.1.36) are negligible. With these approximations, (6.1.35) and (6.1.36) are replaced by ∗ nn (pz ) eBµ0 e2 m ++ µ 2 R (k) e (k)[Γ (k)] , ′ M Mµ q q (2π)2 (j 2 Ωe2 − ω 2 + kz2 )1/2 pz =pzj ∗ [nn (pz ) − nn (p′z )] eBµ0 e2 m µ 2 eMµ (k)[Γ ++ γM (k) = − R (k) (k)] , ′ M q q (2π)2 (j 2 Ωe2 − ω 2 + kz2 )1/2 pz =pzj (6.2.28) βM (k) =

respectively, with pzj = m(ω − jΩe )/kz , p′zj = pzj − kz . The quantum recoil is included simply by replacing pzj by pzj = (m/kz )[ω − jΩe + (ω 2 − kz2 )/2m]. The foregoing procedure needs to be modified near threshold, at pz+ = pz− = mkz /ω, where both solutions, pz± , need to be included. This is discussed below for a particular case of a thermal-like distribution. 6.2.10 Bi-Maxwellian distribution By way of illustration, consider the emission and absorption coefficients (6.2.28) for a bi-Maxwellian distribution, which is a nondegenerate, thermallike distribution with different temperatures, Tk and T⊥ , parallel and perpendicular to the magnetic field, respectively. The occupation number for such a distribution is 2

nn (pz ) = A e−na−pz /2mTk ,

A=

(2π)3/2 ne Bc tanh(a/2), Bm3 (Tk /m)1/2

(6.2.29)

with a = (m/T⊥ )(B/Bc ). The emission and absorption coefficients follow from (6.2.28) simply by interpreting nn (pzj ), nn (pzj ) − nn (p′zj ), in terms of the distribution function (6.2.29). The emission coefficient then contains a

258

6 Quantum theory of gyromagnetic processes

factor exp[−m(ω − jΩe )2 /kz2 Tk ] that determines the line profile, at the jth harmonic. The Doppler width is of order |kz |(Tk /m)1/2 . In the denominator, one may make the approximation (j 2 Ωe2 − ω 2 + kz2 )1/2 ≈ |kz |. Near threshold, both solutions pz± /m ≈ jΩe kz /(ω 2 − kz2 ) ± ωgnn′ , need to be included. Consider the expression (6.2.25) for gnn′ . The foregoing treatment 1/2 breaks down for kz2 /2ω < ∼ |ω − jΩe | ∼ |kz |(Tk /m) , which corresponds to a small range of angles about perpendicular propagation; emission for kz → 0 is possible only at ω < jΩe . In this range the two solutions may be approximated by pz± /m ≈ kz /ω±[2(jΩe −ω)/ω]1/2 . and, in the denominator, one may make the approximation (j 2 Ωe2 − ω 2 + kz2 )1/2 ≈ [2ω(jΩe − ω)]1/2 . The case where two nonrelativistic solutions need to be included is the counterpart of the case where the non-quantum resonance condition corresponds to a resonance ellipse that is approximately circular near the origin, as illustrated in Fig. 4.1 6.2.11 Cyclotron absorption in a completely degenerate plasma Cyclotron emission is not possible for a completely degenerate electron gas because all possible final states are filled. Cyclotron absorption is possible. Suppose that the states are filled up to the level n = nc , |pz | = pF . Then the absorption n → n + j is possible for all n + j > nc because all such states are empty. The absorption coefficient in this case follows from (6.2.28) with nn (pz ) = 0, nn′ (p′z ) = 1. One has, for any particular transition, n′ , s′ → n, s, RM (k) ∗ eBµ0 e2 ωm2 ++ µ 2 e (k)[Γ (k)] , q′ q (2π)2 (ω 2 − kz2 ) gnn′ ωM (k) Mµ pz =pz± ± (6.2.30) where H± = H(nc − n′ )H(n − nc )H(pF − |p′z± |)H(|pz± | − pF ) denotes the combination of step functions that express the requirement that the initial state be occupied and the final state be empty before the transition. γM (k) =

X

H±

6.3 Quantum theory of synchrotron emission

259

6.3 Quantum theory of synchrotron emission In the quantum generalization of the theory of synchrotron radiation (§4.5), the central assumptions are that the resonance condition is approximated by assuming that the particles are highly relativistic and that the transition involves very high harmonics such that the harmonic number, n−n′ , is treated as a continuous variable. In the non-quantum theory the assumptions justify an Airy-integral approximation to the Bessel functions that appear in the theory, and in the quantum case they justify an Airy-integral approximation to the J-functions. The intrinsically quantum modifications to synchrotron emission discussed here are the quantum recoil, which is important when an emitted photon carries away a large fraction of the initial energy of the electron, quantum fluctuations in the orbit, the effect of synchrotron emission on the spin of the electrons, and effects associated with superstrong fields. 6.3.1 Semi-quantitative discussion of quantum effects To distinguish intrinsically quantum effects, it is helpful to consider a semiclassical treatment of synchrotron emission, in which only the language is quantum mechanical. (SI unit with h ¯ and c included explicitly are used for this semi-quantitative discussion.) The (classical) power radiated in synchrotron emission in the frame pz = 0 follows by making the ultrarelativistic approx2 imation to (4.2.11), giving P = e2 Ωe2 γ⊥ /6πε0 c. The electron loses energy systematically, at the rate ε˙ = −P . The energy of a typical photon is h ¯ hωi 2 . The with, from Fig. 4.3, hωi ≈ 0.29ωc and with, from (4.5.7), ωc = 23 Ωe γ⊥ typical rate photons are emitted is N˙ ph P/¯hhωi. Interestingly, in the classical interpretation of synchrotron emission in terms of a pulse of radiation seen by the observer each gyration, cf. Fig. 2.2, the number of photons per pulse is N˙ ph /Ωe , which is less than unity, ∼ 4παc ∼ 0.1. Although this implies that such pulses are unobservable (as pulses), nevertheless, the classical description remains valid. A basic feature of the quantum theory of gyromagnetic emission is the quantization of the perpendicular motion of the electron. In the frame with 2 pz = 0, the initial energy of the electrons is ε0n = mc2 γ⊥ , with γ⊥ = 1+ 2 1/2 (pn /mc) , pn = (2neB¯ h) . The synchrotron approximation requires the perpendicular motion to be ultrarelativistic, pn ≫ mc, which corresponds to n ≫ Bc /2B.

(6.3.1)

The synchrotron approximation also requires that the electron remain highly relativistic after emission of the photon, n′ ≫ 1, Bc /2B, and that the harmonic number be large, n − n′ ≫ 1, Bc /2B. In supercritical fields, B > ∼ Bc , the synchrotron requirement, n ≫ 1, is stronger than ultrarelativistic requirement, pn ≫ mc, and n − n′ ≫ 1 is not necessarily satisfied. The quantum recoil affects synchrotron emission substantially when the energy of the emitted photon carries away most of the energy of the initial

260

6 Quantum theory of gyromagnetic processes

electron [2, 3]. Consider the classical synchrotron spectrum, Fig. 4.3: there is a high-frequency tail that includes photons of sufficiently high energy that this condition is satisfied. In a quantum treatment, the classical spectrum is truncated above the frequency corresponding to h ¯ ω = mc2 γ⊥ . This effect becomes important when the energy of a photon near the peak of the spectrum becomes of order mc2 γ⊥ ; the synchrotron spectrum is strongly suppressed 2 when the photon energy at the peak, ∼ ¯hΩe γ⊥ , exceeds mc2 γ⊥ . Equating these two energies implies that this quantum effect is important for 3 n> ∼ (Bc /B) ,

1/2 γ⊥ > ∼ (Bc /B) .

(6.3.2)

ε ≈ mc2 (Bc /B)5/2 .

(6.3.3)

The final energy of the electron, ε′ = ε − ¯hω, satisfies ε′ ≪ ε, implying n′ ≪ n for n ≫ (Bc /B)3 . A more subtle classical effect is related to spatial diffusion across the field lines. Such diffusion occurs in the non-quantum limit, but its description of this effect is gauge dependent. Diffusion along the x-axis may be described using the Landau gauge (5.1.13) and radial diffusion away from a central field line may be described using in the cylindrical gauge (5.1.14). In the Landau gauge, emission of a photon with given ky implies a jump in the center of gyration of ∆x = h ¯ ky /eB. The mean square displacement of the center of gyration increases at a rate dh(∆x)2 i/dt ∼ N˙ ph (¯ hky /eB)2 . In the cylindrical gauge, the radial position is described by a discrete radial quantum number, defined by (5.2.26), and an analogous dispersion occurs in this parameter. The semi-classical description is based on the quasilinear approximation, in which the motion is assumed to be a classical spiraling along a field line. A quantum effect associated with this effect is referred to as a quantum ‘broadening’ of the classical orbit by this diffusion [4]. The condition for this to occur in the Landau gauge is (∆x)2 ∼ ¯h/eB with ky ∼ hωi/c corresponding to emission near the peak of the classical spectrum. In the cylindrical gauge, the condition is that the emission of such a photon causes the radial quantum number to change by unity [4]. This occurs for 5 n> ∼ (Bc /B) ,

This broadening effect, which has been confirmed experimentally [6, 7], is not discussed further here. Effects associated with the spin of the electron are obviously intrinsically quantum mechanical. Unlike the cyclotron case, where transitions involving a spin flip are suppressed relative to those that do not involve a spin flip, in the quantum theory of synchrotron emission transitions with and without spin flip are of the same order for a highly relativistic electron, but they do not occur at exactly the same rate. As a result of such transitions the electrons tend to become polarized, in the spin state that has the lowest transition rate for emission [5, 6]. The spin also has a subtle effect on the form of the synchrotron spectrum, and can be seen by comparing synchrotron emission by a spinless particle with that by an electron.

6.3 Quantum theory of synchrotron emission

261

6.3.2 Quantum synchrotron parameter In the classical theory of synchrotron emission, the frequency dependence is described by a parameter R = ωωc , with the characteristic frequency of synchrotron emission given by (4.5.7). It is convenient to introduce a parameter ξ by writing this definition in the form ωc =

3Ωe γ 2 sin α = ξ(mc2 /¯h)γ⊥ , 2m

ξ=

3Bγ⊥ . 2Bc

(6.3.4)

It is convenient to develop the quantum theory of synchrotron emission in the frame in which the initial electron has no motion along the field lines, pz = 0. The initial energy of the electron is then εn = ε0n = mc2 γ⊥ ; n is assumed large and continuous, and is re-expressed in terms of γ⊥ . The suppression due to quantum effects is significant when the the final energy of the electron, ′ , is much smaller than the initial energy, due to the energy of εn′ ≈ mc2 γ⊥ the photon being comparable with this initial energy, h ¯ ω ∼ mc2 γ⊥ . This is described by the parameter y, defined by ξy =

¯hω ¯hω = ′ , mc2 γ⊥ − ¯hω mc2 γ⊥

(6.3.5)

such that the allowed range of frequencies corresponds to 0 < y < ∞. The recoil implies that, in the frame pz = 0, the final momentum is p′z = −¯hkz = −(¯ hω/c) cos θ, which needs to be included in the detailed theory but which is unimportant in the definition (6.3.5). The suppression of synchrotron emission at high frequencies due to quantum effect is characterized by the product of parameters, ξy. To lowest order in the expansion in small parameters, ξy is related to the initial and final quantum number by √ n √ = 1 + ξy, (6.3.6) n′ such that one has √ √ n − n′ hω ¯ √ = = ξy, ′ mc2 γ⊥ n′

¯ω h = mc2 γ⊥

√

√ ξy n − n′ √ = . 1 + ξy n

(6.3.7)

The classical limit applies for ξy ≪ 1, and the extreme quantum corresponds to ξy ≫ 1. 6.3.3 Sum over final states Reverting to natural units, the probability for synchrotron emission follows from the probability (6.2.1) for gyromagnetic emission (with ǫ = ǫ′ = 1) when one assumes highly relativistic electrons and makes relevant to the vertex function. For emission in vacuo one has RM (k) → 1/2, ωM (k) → ω, and there are

262

6 Quantum theory of gyromagnetic processes

two transverse states of polarization for the photons. Complete information on the polarization is retained by writing the probability as a (second-rank) polarization tensor which spans the two transverse components. In describing the emitting electron, the quantum numbers of the initial state are n, pz , s. The parameter n is written in terms of the Lorentz factor, γ⊥ = ε0n /m, of the perpendicular motion, with pz = 0 by choice of frame. The rate at which transitions occur, which is the rate at which photons are emitted, is found by summing over the density of final states. For the electron this involves integral over p′z , which is performed implicitly in using the probability (6.2.1), and sums over n′ , s′ . The sum over n′ is replaced by an integral which is performed over the δ-function in (6.2.1). One has Z εn′ εn − ω mγ⊥ 1 dn′ δ(εn − εn′ − ω) = = = . (6.3.8) eB eB eB 1 + ξy There are four possible transitions for the spin states, s = ±1, s′ = ±1, and the transition rates for all four are different. The integral over the density of final states for the photon is written Z 1 Z mγ⊥ Z dω ω 3 d3 k → 2π d cos θ , 3 (2π) (2π)3 −1 0 and the integral over ω may be rewritten using (6.3.5): Z ∞ Z mγ⊥ dy ξy dω = ξmγ⊥ , ω = mγ⊥ . 2 (1 + ξy) 1 + ξy 0 0

(6.3.9)

6.3.4 Synchrotron approximation to the resonance condition As in the non-quantum case, important properties of synchrotron emission may be inferred from the resonance condition by appealing to known relativistic effects, simplified by choosing the frame pz = 0. Rather than use the quantum gyroresonance condition in the standard form (6.1.7), it is more convenient to use the form pz = pz± , given by (6.1.19), so that the resonance condition becomes p2z± = 0 in the frame pz = 0. Assuming emission in vacuo, the resonance condition in this form implies 2 2 2 gnn ′ = fnn′ cos θ,

(6.3.10)

2 with fnn′ , gnn ′ given by (6.1.17), (6.1.18), respectively. The assumption of 2 emission in vacuo implies ω 2 −kz2 = k⊥ , so that one may write ω 2 −kz2 = 2eBx. Then (6.1.17), (6.1.18) give

fnn′ =

(x+ x− )1/2 + x , 2x

2 gnn ′ ≈

(x − x− )(x − x+ ) , 4x2

√ √ x± = ( n ± n′ )2 .

(6.3.11)

6.3 Quantum theory of synchrotron emission

263

2 ′2 where (ε0n ± ε0n′ )2 in (6.1.18) are expanded in m2 /p2n = 1/γ⊥ , m2 /p2n′ = 1/γ⊥ , and only the leading term retained. The frame pz = 0 corresponds to pitch angle α = π/2, and the property that the emission is confined to a small 2 range of angles about θ = α implies that cos2 θ ≈ (θ − π/2)2 is of order 1/γ⊥ . 2 This implies that the right hand side of (6.3.10) is of order 1/γ⊥ , so that the left hand side must be of the same order. To zeroth order this requires x = x− . In the synchrotron approximation one sets x = x− except where the difference x − x− appears explicitly. The zeroth order resonance condition for synchrotron emission becomesx = x− or ω 2 = (pn − pn′ )2 . The difference x − x− is negative, and it is more convenient to express the difference in terms of the positive quantity 1 − x/x− . There are two contributions from the resonance condition in the form (6.3.10). One is from the exact 2 2 2 0 0 2 form for gnn is approximated ′ , given by (6.1.18), where ω − kz − (εn − εn′ ) by 2eB(x − x− ) to zeroth order. Retaining the next order terms gives m2 2 2 0 0 2 . (6.3.12) ω − kz − (εn − εn′ ) ≈ 2eB x − x− 1 − pn pn′

The other contribution comes from the term proportional to cos2 θ in (6.3.10). The resulting expansion gives r x n 1 1 1 1− = , = 2 + (θ − α)2 . (6.3.13) ′ 2 2 x− n ζ ζ γ⊥ The correction (6.3.13 ) appears explicitly in the Airy-integral approximation to the function Jνn (x). In an arbitrary frame, with pz 6= 0, the parameter cos2 θ in (6.3.13) may be reinterpreted as (θ − α)2 in the frame in which the pitch angle is α. 6.3.5 Airy-integral approximation to the functions Jνn (x) The synchrotron approximation to the functions Jνn (x) is based on an Airyintegral approximation [8, 5, 9, 6]. Some details of a derivation of this approxn′ imation are summarized in Appendix J. In brief, the function u = Jn−n ′ (x) satisfies the differential equation d2 (x1/2 u) − φ(x) (x1/2 u) = 0, dx2 q q (x − x− )(x − x+ ) 1 ± n + n′ + φ(x) = , x = ± 2 4x2

(6.3.14) 1 2

2

.

(6.3.15)

In the synchrotron limit one has n, n′ ≫ 1, so that the terms 12 are negligible, giving φ(x) → (x − x− )(x − x+ )/4x2 , which implies φ(x) → gnn′ in view of (6.1.21). Airy’s differential equation and its relevant solution are

264

6 Quantum theory of gyromagnetic processes

d2 w(z) − zw(z) = 0, dz 2

1 z 1/2 K1/3 (ρ), π 3

2 3/2 z . 3 (6.3.16) The differential equation (6.3.14) is transformed into the differential equation (6.3.16) by writing Z x− 2 ρ = z 3/2 = dx′ [φ(x′ )]1/2 , ρ′ = z 1/2 z ′ = −[φ(x)]1/2 , (6.3.17) 3 x w(z) = Ai (z) =

ρ=

where the prime denotes a derivative with respect to x, and with the second derivative, z ′′ , neglected. In (6.3.17) one makes the approximation φ(x) ≈ (x − x− )(x− − x+ )/4x2− , and the integral is then elementary. One finds 1/2 x . 1 − 1/2 x− x− (6.3.18) A normalization constant is determined by considering the limit x → 0 [6]. The resulting Airy-integral approximation is 1/2 1 x n′ √ Jn−n K1/3 (ρ), (6.3.19) (x) = 1 − ′ x− π 3 √ √ which applies for x < x− = ( n − n′ )2 . The corresponding result for n′ dJn−n ′ (x)/dx is also needed below. This follows from (6.3.19) using the identity d 1/3 [ρ K1/3 (ρ)] = −ρ1/3 K2/3 (ρ), (6.3.20) dρ which implies 1/2 d n′ x 1 1− K2/3 (ρ), (6.3.21) Jn−n′ (x) = −ρ′ √ dx x− π 3 ρ=

3/2 x 2 1/2 , (nn′ )1/4 x− 1− 3 x−

ρ′ = −

(nn′ )1/4

with ρ′ given by (6.3.18). Four specific J-functions are required to treat gyromagnetic emission: n−1 n−1 n n n′ Jn−n ′ (x), Jn−n′ (x), Jn−n′ +1 (x), Jn−n′ −1 (x). The others are related to Jn−n′ (x) using the properties of the J-functions given in Appendix J, specifically, ′ x d n n +n−x n n−1 Jn′ −n (x) = Jn′ −n (x) − J ′ (x) , 2x dx n −n (nn′ )1/2 x 1/2 n′ − n + x d n n n Jn′ −n (x) + J ′ (x) , Jn′ −n−1 (x) = n′ 2x dx n −n x 1/2 n′ − n − x d n n Jnn−1 (x) = J (x) − J (x) . (6.3.22) ′ ′ −n+1 n′ −n n 2x dx n −n In the Airy-integral approximation one sets x = x− except where 1 − x/x− appears explicitly. Hence, the coefficients in (6.3.22) are approximated by

6.3 Quantum theory of synchrotron emission ′

n + n − x− = 1, 2(n′ n)1/2

′

n − n + x− = −1, 2(n′ x− )1/2

265

′

n − n − x− = −1. (6.3.23) 2(nx− )1/2

n−1 n−1 n n Thus the functions Jn−n ′ (x), Jn−n′ (x), Jn−n′ +1 (x), Jn−n′ −1 (x) are all equal in magnitude to a first approximation in the extreme relativistic limit, and only the differences resulting from the terms involving the derivatives in (6.3.22) need be retained in distinguishing between them. One finds  n       Jn′ −n 1 0 ′ n −n 1/2 n−1  J ′  (1 + ξy)  1   −ξy  K2/3   nn −n  = (−) √ ,   K1/3 +    Jn′ −n−1  −1 1 + ξy ζ π 3ζ −1 −1 Jnn−1 ′ −n+1 (6.3.24) where the arguments x of the J-functions and ρ = (y/2)(γ⊥ /ξ)2 of the Macdonald functions are omitted, with y defined by (6.3.4) and (6.3.6).

6.3.6 Approximation to the vertex function The synchrotron approximation to the vertex function (5.4.18) involves the Airy-integral approximations to the J-functions and appropriate approximations to the parameters a± , a′± , b±s , b′±s′ . For pz = 0, implying p′z = −kz , the parameters a± , a′± that appear in the vertex function (5.4.18) reduce to a+ ≈ 1,

a′+ ≈ 1,

a− ≈ 0,

a′− ≈ − 12 ξy cos θ.

(6.3.25)

The leading approximations to the other parameters that appear in the vertex function (5.4.18) are b′s′ bs ≈ 1 +

1 [s + s′ (1 + ξy)]. 2γ⊥

(6.3.26)

Combining these various approximations, the synchrotron approximation to the space components of the vertex function (5.4.18) for pz = 0 becomes 1/2 1 + s′ s n−n′ (1 + ξy) √ Γ++ (k) = (−i) 2K1/3 (z), ′ qq 2 2π 3 ζ ′ K2/3 (z) s ξy i − , −2ξy cos θ K1/3 (z) K1/3 (z) + (2 + ξy) γ⊥ ζ ′ ′ K2/3 (z) s 1−ss K1/3 (z) − + 0, −isξy cos θ K1/3 (z), +ξy , (6.3.27) 2 γ⊥ ζ where only the lowest order terms are retained in an expansion in 1/γ⊥ , 1/ζ, cos θ. Denoting the space components of (6.3.27), by superscripts x, y, z, k ++ y the two transverse components correspond to Γs⊥′ s = [Γqq ′ ] (k) and Γs′ s = ++ x ++ z cos θ[Γqq ′ ] (k) − sin θ[Γqq ′ ] (k). Specifically, the transverse components are

266

6 Quantum theory of gyromagnetic processes ⊥,k

′

Γs′ s = (−i)n−n

(1 + ξy)1/2 (1 + t2 ) ⊥,k √ 2 γs′ s , 2π 3 γ⊥

(6.3.28)

with t = γ⊥ cos θ, γ⊥ /ζ = 1 + t2 , and with γs⊥′ s = i 21 (1 + s′ s) − s′ ξy K1/3 (z) + (2 + ξy)(1 + t2 )K2/3 (z) − 21 (1 − s′ s) ξy t K1/3 (z) , k

γs′ s = 21 (1 + s′ s) (2 + ξy) t K1/3 (z)

+ 21 (1 − s′ s) ξy s′ K1/3 (z) − (1 + t2 )K2/3 (z) . (6.3.29)

6.3.7 Transition rate for synchrotron emission ⊥,k

Consider the rate of transitions, Rs,s′ , between an initial state with quantum numbers n, s, pz = 0, and final states n′ , s′ , p′z = −kz due to synchrotron emission of photons with ⊥ and k polarizations. The initial state is described by the perpendicular Lorentz factor, γ⊥ , and the sum over n′ is replaced by an integral. The rate follows by integrating the probability (6.1.2) over the density of final states for the photon. The integral over n′ is performed over the δ-function, as in (6.3.8), and the integral over ω is replaced by an integral over y using (6.3.9). Z ∞ Z ∞ 3µ0 e2 m B dy y ⊥,k ⊥,k 2 2 Rs′ s = (6.3.30) |γ ′ |2 dt (1 + t ) 3 32π Bc 0 (1 + ξy)3 s s 0 ⊥,k

with γs′ s (k) given by (6.3.29). The integrand in (6.3.30) is an even function of cos θ, and is dominated by the region cos θ < ∼ 1/γ⊥ , allowing the integral over cos θ to be rewritten as 2 times the integral over t = γ⊥ cos θ between 0 and ∞. The corresponding power radiated is found by including an extra 2 factor of ω = mγ⊥ ξy/(1 + ξy) in the integrand in (6.3.30): Z ∞ Z ∞ dy y 2 9µ0 e2 m B ⊥,k ⊥,k 2 2 2 Ps′ s = mγ⊥ dt (1 + t ) |γs′ s |2 . (6.3.31) 3 4 64π Bc (1 + ξy) 0 0 The following integrals are required in (6.3.30) or (6.3.31) [5]: Z ∞ Z ∞ π dx K5/3 (x) + K2/3 (y) , dt(1 + t2 )2 [K2/3 (ρ)]2 = √ 2 3y y 0 Z ∞ Z ∞ π 2 2 2 dx K5/3 (x) − K2/3 (y) , dt t (1 + t )[K1/3 (ρ)] = √ 2 3y y 0 Z ∞ Z ∞ π 2 2 dt(1 + t )[K1/3 (ρ)] = √ dx K1/3 (x), 3y y 0 Z ∞ π (6.3.32) dt(1 + t2 )3/2 K1/3 (ρ)K2/3 (ρ) = √ K1/3 (y), 3y 0

6.3 Quantum theory of synchrotron emission

with ρ = 12 y(1 + t2 )3/2 , and where the identity Z ∞ Z ∞ dx K1/3 (x) = − dx K5/3 (x) + 2K2/3 (y), y

267

(6.3.33)

y

is used. The transition rate (6.3.30) becomes √ Z ∞ dy 3 µ0 e2 m B ⊥,k ⊥,k Rs′ s = G′ , 32π 2 Bc 0 (1 + ξy)3 s s and the power radiated (6.3.31) becomes √ Z ∞ 2 dy y B 3 3 µ0 e2 m2 γ⊥ ⊥,k ⊥,k G′ , Ps′ s = 2 64π Bc 0 (1 + ξy)4 s s

(6.3.34)

(6.3.35)

where the integral (6.3.32) imply Z ∞ ξ2y2 ⊥ ′ 2 1 1 Gs′ s = 2 (1 + s s) (1 + 2 ξy) dx K5/3 (x) − K2/3 (y) + 12 (1 − s′ s) 4 y Z ∞ Z ∞ × dx K5/3 (x) + K2/3 (y) + 2 dx K1/3 (x) + 2sK1/3 (y) , y

k Gs′ s

y

Z ′ 2 1 1 = 2 (1 + s s) (1 + 2 ξy)

∞

y

ξ2y2 + 2

Z

y

∞

dx K5/3 (x) + K2/3 (y)

dx K1/3 (x) − sξy(1 + 12 ξy)K1/3 (y)

+ 12 (1 − s′ s)

ξ2 y2 4

Z

∞

y

dx K5/3 (x) − K2/3 (y) .

(6.3.36)

The rate of transitions per unit frequency and the power per unit frequency follow from (6.3.34) and (6.3.35), respectively, by omitting the integral and using (6.3.9) to write dy = dω (1 + ξy)/mγ⊥ ξy. 6.3.8 Power emitted for unpolarized electrons When the polarization of the electrons is of no interest, one averages over the initial polarization, s, and sums over the final polarization, s′ . Denoting ⊥,k ¯ ⊥,k , P ⊥,k ¯ ⊥,k , the resulting quantities by an overbar, with Rs′ s → R s′ s → P ⊥,k ⊥,k ¯ in (6.3.30), (6.3.31), (6.3.36) gives Gs′ s → G Z ∞ ξ2 y2 ¯ ⊥,k = (1 + ξy) G dx K5/3 (x) + K2/3 (y) ∓ K2/3 (y). (6.3.37) 1 + ξy y On summing over the two states of polarization of the photon, (6.3.30) and (6.3.31) give the total transition rate and power radiated, respectively:

268

6 Quantum theory of gyromagnetic processes

Fig. 6.3. TBA

√ Z ∞ Z ∞ 3 µ0 e2 m B ξ2y2 dy R= K2/3 (y) , dx K5/3 (x) + 32π 2 Bc 0 (1 + ξy)2 1 + ξy y (6.3.38) √ Z ∞ Z ∞ 2 2 2 2 2 ξ y dy y 3 B 3 µ e m γ 0 ⊥,k ⊥ dx K (x) + Ps′ s = K (y) . 5/3 2/3 64π 2 Bc 0 (1 + ξy)3 y 1 + ξy (6.3.39) The final term in (6.3.39) is associated with spin-flip transitions. This term is absent when the theory is developed for spinless particles [6]. Its effect is illustrated in Fig. 6.3, where three cases are compared [10]: the curve labeled 2 corresponds to the quantum theory of synchrotron emission by an unpolarized electron, the curve labeled 3 corresponds to the non-quantum theory, and the curve labeled 1 corresponds to a spinless particle. The dashed curve is calculated using a model based on the Klein-Nishina cross-section for scattering virtual photons [10]. Comparing the quantum (ξy > ∼ 1 and non-quantum ξy → 0 limit, there are three notable differences. The first is an overall suppression for the quantum case compared with the non-quantum limit, showing that the quantum mechanical suppression is large only when the particle loses a large fraction of its energy in the emission of a single photon. The second is in the final term inside the square brackets in (6.3.39), which is absent ′ for ξy → 0. The coefficient of this term is ξ 2 y 2 /(1 + ξy) = ω 2 /m2 γ⊥ γ⊥ , implying that the synchrotron spectrum differs substantially from its non′ quantum form only for ω 2 ∼ m2 γ⊥ γ⊥ . A third difference is in the parameter ′ y = 2mω/eBγ⊥γ⊥ , which is larger in the quantum case compared with the classical case. These differences are substantial only as the quantum mechan′ ical cutoff ω → mγ⊥ , γ⊥ → 0 is approached. The power radiated (6.3.39), may be compared with the non-quantum limit, ξy ≪ 1, by expanding the integrand in powers of ξy and integrating over y using the integral

6.3 Quantum theory of synchrotron emission

Z

0

∞

dy y q−1 Kp (y) = 2q−1 Γ

q−p 2

Γ

q+p 2

.

269

(6.3.40)

The expansion to order (ξy)2 was evaluated by Sokolov and Ternov [6], whos also evaluated the power in the extreme quantum limit, ξy ≫ 1, finding √  55 3 56 8   ξ + ξ 2 + ξ 2 for ξ ≪ 1, 1 − 24 3 3 (6.3.41) P = Pcl  28/3 Γ (2/3)   for ξ ≫ 1, 9ξ 4/3

respectively, where Pcl is the power in the non-quantum limit. The final term, 8ξ 2 /3, arises from the final term in (6.3.39), and is the contribution from spin-flip transitions. 6.3.9 Change in the spin during synchrotron emission

The rate of transitions is different for the different spin states. Adding the contributions to the two polarizations, and expanding the integrand to second order in ξy, evaluation of the resulting integrals using (6.3.40) gives [6] √ 5µ0 e2 m B 1 + s′ s 16 3 25 √ Rs′ s = 1− ξ + ξ2 2 45 18 8π 3 Bc √ ! 2 √ ! ξ ξ 1 − s′ s 20 3 ′8 3 ′ ξ + 1−s . (6.3.42) −s 1 − 9 5 2 15 6 The final term in (6.3.42) describes the spin flips, and implies that the ratio of the rate of direct spin flips,√s = 1 → s′ √ = −1, to the rate of reverse spin flips, s = −1 → s′ = 1, is (15 + 8 3)/(15 − 8 3). It follows that after an arbitrarily long time, the occupation numbers for electrons in the states s = ±1 approach the ratio √ 15 + 8 3 n− √ . → (6.3.43) n+ 15 − 8 3 This corresponds to 96% of the electrons in the state s = −1. Sokolov & Ternov [5] showed that this limit is approached exponentially, ∝ exp(−t/τ ), on a timescale (in SI units) √ 3 3 5 3 e2 mc 2 B B −1 27 −1 2 τspin = γ = 10 . (6.3.44) s γ ⊥ ⊥ 8 4πǫ0 ¯ Bc Bc h2 It is relevant to compare this time with the classical synchrotron half-lifetime, τε = mc2 γ⊥ /Pcl in ordinary units. One has √ √ 5 3 15 3 B τε = = γ⊥ ξ, (6.3.45) τspin 16 Bc 8

with ξ given by (6.3.4). It follows that for ξ ≫ 1 the alignment of the spins in the state s = −1 occurs before the electron radiates away a signficant fraction of it energy. Alignment of the spins does not have time to occur for ξ ≪ 1.

270

6 Quantum theory of gyromagnetic processes

6.3.10 Gyromagnetic emission in supercritical fields The synchrotron approximation requires not only that the particles be highly relativistic but also that the quantum numbers n, n′ be large enough to be treated as continuous variables. The latter condition requires n, n′ ≈ 2 ′2 γ⊥ Bc /2B, γ⊥ Bc /2B ≫ 1. In a supercritical field, B ≫ Bc , there is a range of γ⊥ where the particles are highly relativistic but do not satisfy this second requirement. In this case, gyromagnetic emission must be treated in terms of transitions, n → n′ , between discrete levels. For small n, n′ this case is somewhat analogous to cyclotron emission in that it involves discrete lines. However, the main simplifying approximation in the cyclotron limit results from the argument, x, of the J-functions being small does not apply in this case. Explicit expressions for Jνn (x) for n = 0, 1, 2, 3 are written down in (A.1.41)–(A.1.44) in Appendix A.1.6. These functions are of the form xν/2 e−x/2 times a polynomial in x of order n, with x of order unity for small n, n′ . Unlike the cyclotron case, where the transition rates for transitions n − n′ > 1 and those with a spin flip occur are smaller that the transition rate n → n′ = n − 1 without a spin flip by a power of B/Bc for > 1. Unlike B/Bc ≪ 1, all these transition rates become comparable for B/Bc ∼ the synchrotron case, where emission by highly relativistic particles is strongly beamed, around sin θ = 1 in the frame pz = 0, there is no strong beaming for small n, n′ , and a synchrotron-like approximation cannot be developed on the basis of the angular distribution. The exact value of the frequency of emission in vacuo for pz = 0 in an arbitrary transition n → n′ is ωnn′ (θ), given by (6.2.15). In the cyclotron limit one expands the square roots in (6.2.15) assuming 2eB ≪ m2 , giving ωnn′ (θ) ≈ (n − n′ )Ωe , Ωe = eB/m, In the relativistic case, neglecting the terms m2 , (6.2.15) gives ωnn′ (θ) =

i (2eB)1/2 h 1/2 2 2 ′ 1/2 . n − (n cos θ + n sin θ) sin2 θ

(6.3.46)

2 In the synchrotron limit, relativistic √beaming implies cos θ ≪ 1 and (6.3.46) √ 1/2 ′ gives ωnn′ (θ) ≈ (2eB) ( n − n ), corresponding to the synchrotron approximation x ≈ x− . The synchrotron approximation is not valid for n′ ≪ n, specifically for n′ sin2 θ ≪ n cos2 θ, when (6.3.46) gives ωnn′ (θ) ≈ (2neB)1/2 /(1 + | cos θ|). A specific case that has been treated in detail is for transitions from large n to n′ ≪ n [11, 12, 7]. In this case, the emission is concentrated √ around θ = π/2, and the frequency (6.2.15) becomes ωn0 (π/2) ≈ (2eB)1/2 n, corresponding to a photon energy approximately equal to the initial electron energy. Such transitions have been treated using both the exact wave functions [11], and using a parabolic cylinder approximation to the J-functions [12]. The rate of transitions for n′ = 0 is, in ordinary units,

R0n =

αc m2 c4 B −Bc /B e , 2¯ hε Bc

(6.3.47)

6.3 Quantum theory of synchrotron emission

271

with the initial energy given by ε ≈ (2neB¯hc2 )1/2 . The result (6.3.47) and its generalization to nonzero n′ ≪ n was derived in Ref. [12], who also noted that the inverse process in which a photon is absorbed by an electron at rest with excitation to n ≫ 1 has a cross-section 1/2 2π 3/2 αc ¯h Bc σn0 = e−Bc /B . (6.3.48) mω B The cross section, which has a maximum at B = 2Bc , can be substantially larger than the Thomson cross section. Some other aspects of gyromagnetic emission in supercritical fields were reviewed in [7], who cited relevant earlier literature.

272

6 Quantum theory of gyromagnetic processes

6.4 One-photon pair creation One-photon pair creation and annihilation are allowed in a magnetized vacuum, as first discussed in the early 1950s [13, 8]. Pair creation and annihilation are related to gyromagnetic emission by crossing symmetries, and are included in the probability (6.2.1) for ǫ = −1, ǫ′ = 1. The analogy between one-photon pair creation and gyromagnetic emission is relatively close. For a nonrelativistic pair the analogy is with cyclotron emission (§??), and for an extremely relativistic pair the analogy is with synchrotron emission (§??). The photon involved is necessarily of high energy, ω > 2m, and the effect of an ambient plasma on the dispersion of the photons is usually negligible. Birefringence due to the magnetized vacuum can play a significant role, due to perpendicular and parallel polarized photons annihilating into pairs with different quantum numbers. One-photon pair creation is of particular interest in connection with pulsars. It is thought that a pulsar magnetosphere is populated by secondary pairs. The sequence of processes envisaged is that primary particles, which are accelerated to extremely high energies by an electric field parallel to the magnetic field, emit photons with ω ≫ m at very small angles to the magnetic field, ω sin θ ≪ 2m. The threshold for pair creation is reached as θ increases, due to k and B deviating increasingly from each other as a result of the curvature of the magnetic field lines. Once the threshold is exceeded the photon can decay into a pair. Just above threshold the pair is nonrelativistic, and the cyclotron-like approximation to pair creation. Otherwise, the pairs are highly relativistic, and the synchrotron-like approximation is appropriate. 6.4.1 Probability for pair creation and decay The probability for pair creation or decay follows from (6.2.1) by setting ǫ = 1, ǫ′ = −1: −+ wMqq ′ (k) =

2 µ0 e2 RM (k) ∗ eMµ (k)[Γq+′ q (k)]µ 2πδ εq + εq′ − ωM (k) . (6.4.1) |ωM (k)|

As already remarked, in most cases of interest the wave dispersion can be neglected (RM (k) → 21 , ωM (k) → ω = |k|) and the weak anisotropy limit applies with the polarization restricted to the transverse plane. The absorption coefficient follows from the probability by summing over the final states. The sum over final states, q, q ′ . includes sums over the discrete quantum numbers, n, s, n′ , s′ , and integrals over pz , p′z . As noted in connection with the definition (5.6.18) of the probability, the integral over (V eB/2π)(dp′z /2π) is performed over a δ-function, that is omitted from the probability, specifically over (2π/V eB) 2πδ(p′z − pz + kz ) here. The factor V in the integral over (V eB/2π)(dpz /2π) is incorporated in the definition of the probability, and is omitted in the sum over final state q.

6.4 One-photon pair creation

273

80

n 60

40

20

0 0

20

40

60

n’

80

Fig. 6.4. The threshold condition for pair creation is plotted for (ω 2 − kz2 )1/2 = 10 m, B = 0.5 Bc . The allowed integral values of n, n′ are below the curve, and the threshold for√a given n, n′ corresponds to the point lying on the curve, which is √ x2+ = ( n + n′ )2 = 81 in this case.

In treating pair creation it is often convenient to transform to the frame kz = 0, which plays a role analogous to the frame with pz = 0 in the case of gyromagnetic emission. In the frame kz = 0, the parallel momenta of the electron and positron are equal and opposite. The resonance condition in the form (6.1.19) implies that the parallel momenta are given by |pz± | = ωgnn′ . The threshold condition corresponds to gnn′ = 0, and pair creation is possible only above threshold, ω ≥ ε0n + ε0n′ . For given ω, B/Bc , this condition restricts the values of n, n′ . For example, for n′ = 0 the restriction implies n ≤ (ω/m−1)2 Bc /2B. The restriction is illustrated in Fig. 6.4, in an arbitrary frame (ω → (ω 2 − kz2 )1/2 ), for values such that for n′ = 0 the maximum is n = 81. A detailed investigation [9] leads to the following semi-quantitative conclusions for large n + n′ : the most probable transitions are for n, n′ near the boundary of the allowed region, the number of states n, n′ within the boundary is of order Nstates (ω, B) ≈ 8ω(ω + 4m)(ω − 2m)2 Bc2 /3m4 B 2 , and the photon energy tends to be shared roughly equally by the electron and positron (n ∼ n′ ) for B ≪ Bc , but unequally (n ≫ n′ or n′ ≫ n) for B > ∼ Bc . 6.4.2 Rate of pair creation The rate of pair creation corresponds to the absorption coefficient (per unit time) for the photons. It may be obtained directly by summing the probability (6.4.1), or from the kinetic equation (6.1.30). In the latter case, provided that the number of pairs is well below the degeneracy limit, the occupations numbers of the pairs may be neglected in (6.1.30), which then reproduces the result obtained directly from (6.4.1). The resulting expression for the rate of production of pairs with quantum numbers q = n, s, q ′ = n′ , s′ through decay

274

6 Quantum theory of gyromagnetic processes

of a photon in mode M is RqM′ q

eB = 2π

Z

dpz +− w ′ (k). 2π Mqq

(6.4.2)

The integral over pz is carried out over the δ-function, and (6.4.2) gives RqM′ q

∗ −+ µ 2 µ0 e3 B RM (k) eMµ (k)[Γq′ q (k)] εn′ εn , = 2 (k) − k 2 π ωM (k) ωM gnn′ z

(6.4.3)

where pz , p′z , εn , εn′ are to be evaluated at the ±- solutions (6.1.19)), (6.1.21)), and with gnn′ given by (6.1.18). An extra factor of 2 is included in (6.4.3) to take account of equal contributions from the two solutions, pz = pz± , of the resonance condition. One is usually justified in assuming that the wave dispersion is that of the vacuum (ωM (k) → ω = |k|, RM (k) → 1/2), and that the two modes (M →⊥, k) of the birefringent vacuum have polarization vectors e⊥ = a, ek = t, The projections onto the polarization vectors is particularly simple in the frame the in which the photon is propagating along the x-axis (kz = 0, ψ = 0), when e⊥ , ek are along the y- and z-axes, respectively. In this frame the only relevant components of the vertex function are k

z Γq′ q = [Γq−+ ′ q (k)] .

y Γq⊥′ q = [Γq−+ ′ q (k)] ,

(6.4.4)

The rate(6.4.3) becomes ⊥,k

Rq′ q =

µ0 e2 m2 B εn′ εn ⊥,k 2 |Γq′ q | , 2πω 3 Bc gnn′

(6.4.5)

⊥,k

with p′z = −pz for kz = 0, and with Γq′ q given by (6.4.4). It is usually convenient to calculate the transition rate in the frame kz = 0 and to make a Lorentz transformation to an arbitrary frame with kz 6= 0. The rate transforms like a frequency, as is apparent from the quantities on the right hand side of (6.4.3) which are invariants except for ε′n′ , εn , gn′ n which all transform as frequencies. Hence, denoting a quantity in the frame kz = 0 by a tilde, with ˜ ⊥,k R q′ q (ω) given by (6.4.5), with the frequency dependence made explicit, the ⊥,k ˜ ⊥,k decay rate in an arbitrary frame is R ′ (ω) = sin θ R ′ (ω sin θ). q q

q q

6.4.3 Rate of pair production near threshold The rate at which photons decay into pairs has singularities at the threshold, ω = ε0n + ε0n′ for creation of a pair with quantum numbers n, n′ and pz = −p′z = 0 (in the frame kz = 0). This is illustrated in Fig. 6.4. The (square-root) singularities correspond to the zeros of gnn′ in (6.4.5), and may be regarded the counterparts for pair creation of the quantum oscillations in gyromagnetic

6.4 One-photon pair creation

275

Fig. 6.5. Plots.

emission, illustrated in Fig. 6.2. Another notable feature in Fig. 6.4 is that the lowest threshold for the parallel and perpendicular polarizations are different. These features may be treated by approximating the rate by its form near each threshold, with the total absorption coefficient found by summing over all values of n, n′ and s, s′ . The relevant components of the vertex function are ′

′ ′ n Γq⊥′ q = −is′ (i)n−n (a′s′ a−s + a′−s′ as ) (b′−s′ bs Jnn−1 ′ −n+1 − s sbs′ b−s Jn′ −n−1 ), ′

k

′ ′ n Γq′ q = (i)n−n (a′s′ as + a′−s′ a−s ) (b′s′ bs Jnn−1 ′ −n + s sb−s′ b−s Jn′ −n ), 1/2 1/2 0 εn ± ε0n ) εn ± m . a± = , b± = 2εn 2ε0n

(6.4.6)

At threshold one has pz = −p′z = 0, and near threshold the small contributions from pz = −p′z 6= 0 are neglected. In this approximation, the factors involving the αs in (6.4.6) give a′−s′ as − a′s′ a−s ≈ s 12 (1 − s′ s),

a′−s′ a−s + a′s′ as ≈ 21 (1 + s′ s).

(6.4.7)

k

It follows that only Γq⊥′ q contributes for s = −s′ and only Γq′ q contributes for s = s′ . The resulting approximations to the components (6.4.6) of the vertex function near threshold give ′ n 2 |Γq⊥′ q |2 = 12 (1 − s′ s) |b′s bs Jnn−1 ′ −n+1 + b−s b−s Jn′ −n−1 | , k

′ n 2 |Γq′ q |2 = 12 (1 + s′ s) |b′s bs Jnn−1 ′ −n + b−s b−s Jn′ −n | ,

(6.4.8)

276

6 Quantum theory of gyromagnetic processes

Near the threshold ω = ε0n + ε0n′ , the expression (6.1.18) for gnn′ in the denominator in (6.4.5) may be approximated by gnn′ =

1/2 1 0 0 2 ε ε ′ ω − (ε0n + ε0n′ )2 . ω2 n n

(6.4.9)

An approximation to the rate (6.4.5) is found by setting ω equal to its threshold value, ε0n + ε0n′ , except in the expression (6.4.9) for gnn′ . The argument of the J-functions is approximated by x = (ε0n + ε0n′ )2 /2eB. Most interest is in the lowest thresholds, corresponding to n = 0 and n′ = 0, 1. (There is symmetry in the interchange of n, n′ .) In both cases, the only J-function required is J00 (x) = e−x/2 . The rate (6.4.5) gives, for n = n′ = 0 in ordinary units, mc2 e−2Bc /B αc mc2 B , ¯h Bc [(¯ hω)2 − (2mc2 )2 ]1/2

k

R0′ 0 =

(6.4.10)

and for n = 0, n′ = 1 in ordinary units, R1⊥′ 0 =

αc mc2 B h ¯ Bc

mc2 ε01

1/2

0

2

mc2 e−(Bc /B)(1+ε1 /mc ) , [(¯ hω)2 − (mc2 + ε01 )2 ]1/2

(6.4.11)

with ε01 = mc2 (1 + 2B/Bc )1/2 . Due to the exponential factors in (6.4.10), (6.4.11) and the formulae near other thresholds, one-photon pair creation is a strong effect only for B/Bc ∼ > 1. For B/Bc ∼ > 1 the thresholds at ω = ε0n + ε0n′ for different n, n′ are well separated, by an energy of order the rest energy times B/Bc . For B/Bc ≪ 1 the thresholds become densely packed, and the sums over n, n′ are replaced by integrals, leading to a synchrotron-like description of pair creation. 6.4.4 Relativistic pairs When a photon with ω sin θ ≫ 2m decays, the resulting pair is highly relativistic. As for synchrotron emission, in the extreme relativistic limit one treats n, n′ as continuous variables and makes an Airy integral approximation to the J-functions. The detailed analysis, in the frame kz = 0, is similar to that for synchrotron emission, with the important change being that in the expression 2 2 (6.1.19) for √ gnn ′ , one has x ≈ x+ , rather than x ≈ x− , with x = ω /2eB √ 2 ′ and x± = ( n ± n ) . The relativistic approximation for pair creation involves setting x = x+ , except where the difference appears explicitly, when it is written in terms of (x/x+ − 1), replacing the sums over n, n′ by integrals, and making the Airy-integral approximation to the J-functions. It is convenient to change the variables of integration from n, n′ to gnn′ , fnn′ and thence to hyperbolic functions of variables u, w. The hyperbolic functions are introduced by writing pz = m sinh u and the difference over the sum of the perpendicular momenta as tanh w, specifically,

6.4 One-photon pair creation

gnn′ =

m sinh u, ω

√ √ n − n′ √ = tanh w. √ n + n′

277

(6.4.12)

2 In the explicit expression (6.1.17) for gnn ′ with kz = 0, one approximates 0 1/2 0 ′ εn ≈ pn = ω(n/x) , εn′ ≈ pn′ = ω(n /x)1/2 , with x ≈ x+ , except in x 4m2 2 2 0 0 2 ω − (εn + εn′ ) ≈ 2eBx+ − 1 − 2 cosh w , (6.4.13) x+ ω

which follows by expanding in m2 /p2n , m2 /p2n′ . The resulting approximation 2 to gnn ′ is √ 4m2 x nn′ 2 2 √ − 1 − cosh w . (6.4.14) gnn ′ ≈ √ ω2 ( n + n′ )2 x+ The approximate form (6.4.14) implies 4m2 x − 1 ≈ 2 cosh2 u cosh2 w. x+ ω

(6.4.15)

In the relativistic approximation to fnn′ = (n − n′ + x)/2x it suffices to set x = x+ , ignoring the corrections of order (m/ω)2 : √ √ n n′ 1 √ = 2 (1 + tanh w), √ = 21 (1 − tanh w). fn′ n ≈ √ fnn′ ≈ √ n + n′ n + n′ (6.4.16) A fraction fnn′ of the photon energy goes into the electron, and the remaining fraction fn′ n = 1 − fnn′ goes into the positron. The integrals over n, n′ may be rewritten as integrals over u, w. The Jacobian of the transformation implies dn dn′ == gnn′

1 2

ω 3 B 2 c

m

B

du dw

cosh u , cosh2 w

(6.4.17)

with the range of integration is restricted to u, w > 0. The upper limits of integration, for n, n′ or u, w, are finite and large, and a standard approximation is to extend them to infinity. Thus, after integrating over n, n′ , the rate (6.4.5) becomes Z µ0 e2 m cosh u ⊥,k ⊥,k (6.4.18) du dw Rs′ s = εn′ εn |Γq′ q |2 , πω 2 cosh2 w where the result remains dependent on the spins of the electron and positron, and the total rate is found by summing over all four possible combinations s′ , s = ±1. The restriction to u > 0 corresponds to pz > 0, and the restriction to w > 0 corresponds to n > n′ . There is a symmetry in the interchange of the electron and positron, such that the interpretation of the unprimed and primed

278

6 Quantum theory of gyromagnetic processes

quantum numbers corresponding to the electron or positron is arbitrary. The right hand side of (6.4.18) is an even function of u, and the contribution from negative u is taken into account by multiplying including an extra factor of two in (6.4.18). The assumption w > 0 for the unprimed particle implies that the particle with spin s has pn > pn′ . Thus, in formulae that depend on the sign of s sinh w, s sinh w > 0 is interpreted as the particle with the larger perpendicular momentum having spin up and s sinh w < 0 is interpreted as the particle with the larger perpendicular momentum having spin down. 6.4.5 Airy-integral approximation The derivation of the Airy-integral approximation relevant to one-photon pair creation is analogous to the derivation of the Airy-integral (6.3.19) approximation for synchrotron emission. The main change is that the relevant zero of 2 φ(x) = (x − x− )(x − x+ √)/4x √changes from x = x− to x = x+ . The approximation for x ≈ x+ = ( n + n′ )2 , with x − x+ small and positive, leads to a Macdonald function K1/3 (ρ) with argument 2 ρ = [nn′ x2+ ]1/4 3

x −1 x+

3/2

,

√ ′ 1/2 nn x ρ = . −1 x+ x+ ′

(6.4.19)

which is the counterpart for pair creation of the result (6.3.18) for synchrotron emission. The Airy-integral approximation to the J-functions for pair creation is ′

Jnn′ −n (x) = An′ n K1/3 (ρ),

An′ n =

(−)n −n 2m √ cosh u cosh w. (6.4.20) ω π 3

where the argument, ρ, of the Macdonald functions is given by (6.4.19). It is convenient to introduce the parameter χ such that (6.4.19) becomes ρ=

2 cosh3 u cosh2 w, 3χ

χ=

ω B . 2m Bc

(6.4.21)

The corresponding approximation to the derivative of the J-functions is derived using (6.3.20) and ρ′ = (m/ω) cosh u: m d n Jn′ −n (x) = −An′ n cosh u K2/3 (ρ). dx ω

(6.4.22)

Approximations to the other J-functions that appear follow from (6.4.22) by using the relations (6.3.22). In place of (6.3.23), for x → x+ one finds n′ + n − x+ = −1, 2(n′ n)1/2

n′ − n + x+ = 1, 2(n′ x+ )1/2

In place of (6.3.24), for pair creation one finds

n′ − n − x+ = −1. (6.4.23) 2(nx+ )1/2

6.4 One-photon pair creation

 Jnn′ −n  J n−1   nn′ −n   Jn′ −n−1  Jnn−1 ′ −n+1 

with An′ n

279

0 1 2m a+ + a−  −1      cosh u  = An′ n   K2/3 (ρ) ,  K1/3 (ρ) + −a+ 1 ω a− −1 (6.4.24) given by (6.4.22), and with 

ζ± =







1 = cosh w(cosh w ± sinh w). 1 ∓ tanh w



(6.4.25)

6.4.6 Relativistic approximation to the vertex function The ‘relativistic’ approximation, which is actually the large-(n, n′) approximation, involves not only the Airy integral approximation to the J-functions, but also approximations to a± , a′± and b′± , b± in the components (6.4.6) of the vertex function. Only the leading terms, of zeroth and first order in m/ω, need be retained; these are a+ ≈ 1,

m m a− ≈ ζ− sinh u, a′− ≈ −ζ+ sinh u, ω ω h i m ω 1 + (sζ− + s′ ζ+ ) , (6.4.26) bs bs′ ≈ 2 cosh w ω a′+ ≈ 1,

with ζ± given by (6.4.25). The leading terms in the expansion of the components (6.4.6) of the vertex function follow from (6.4.24) and (6.4.26). They are of the form ′

⊥,k

(εn ε′n′ )1/2 Γq′ q ≈

(−i)n−n 2m2 ⊥,k √ cosh u cosh w γs′ s , ω π 3

and explicit evaluation gives γs⊥′ s = i 12 (1 + s′ s) sinh u cosh w K1/3 + 21 (1 − s′ s) s cosh w K1/3 + cosh u sinh w K2/3 , k γs′ s = 21 (1 + s′ s) − s cosh w K1/3 + cosh u cosh w K2/3 + 21 (1 − s′ s) − sinh u sinh w K1/3 ,

(6.4.27)

(6.4.28)

(6.4.29)

where the argument of the functions K1/3 (ρ), K2/3 (ρ) is given by (6.4.21), specifically by ρ = (4mBc /3ωB) cosh3 u cosh2 w. 6.4.7 Spin- and polarization-dependent decay rates The rate of photon decay for perpendicular- and parallel-polarized photons propagating across the magnetic field, reduces to Z ∞ Z µ0 e2 m3 Bc ∞ ⊥,k ⊥,k 3 dw |γs′ s |2 . (6.4.30) du cosh u Rs′ s = 3π 3 ω 2 B 0 0

280

6 Quantum theory of gyromagnetic processes

The spin-dependent factors arise from the squares of the vertex components (6.4.27)–(6.4.29), respectively: 2 2 + 12 (1 − s′ s) cosh2 w K1/3 |γs⊥′ s |2 = 12 (1 + s′ s) sinh2 u cosh2 w K1/3 2 + 2s cosh u cosh w sinh w K1/3 K2/3 , + cosh2 u sinh2 w K2/3 k 2 2 |γs′ s |2 = 21 (1 + s′ s) cosh2 w K1/3 + cosh2 u cosh2 w K2/3 2 . (6.4.31) −2s cosh u cosh2 w K1/3 K2/3 + 21 (1 − s′ s) sinh2 u sinh2 w K1/3

The argument of the Macdonald functions is given by (6.4.21), specifically, ρ = 12 y cosh3 u, y = (8m/3ω)(Bc /B) cosh2 w. The integral over u in (6.4.30) with (6.4.31) may be performed using (6.3.32) with t = sinh u and y = (8m/3ω)(Bc /B) cosh2 w. This gives Z ∞ µ0 e2 m2 dw ⊥,k ⊥,k √ (6.4.32) Ss′ s , Rs′ s = 2 8π 3 ω 0 cosh2 w with the spin- and polarization-dependence included in Z ∞ 2 2 ′ ⊥ 1 dx K5/3 (x) − cosh w K2/3 (y) Ss′ s = 2 (1 + s s) cosh w y Z ∞ dx K5/3 (x) + (5 cosh2 w − 1)K2/3 (y) + 12 (1 − s′ s) − (cosh2 w + 1) y +4s cosh w sinh w K1/3 (y) , k Ss′ s

Z 2 + s s) − cosh w

∞

dx K5/3 (x) + 5 cosh2 w K2/3 (y) −4s cosh2 w K1/3 (y) Z ∞ + 12 (1 − s′ s) sinh2 w dx K5/3 (x) − sinh2 wK2/3 (y) .

=

1 2 (1

′

y

(6.4.33)

y

6.4.8 Spin-summed decay rates The total rate of decay is found by summing over the spins, s′ , s = ±1, of the electrons an positron. For unpolarized photons one averages over the two polarizations. The resulting rates are denoted X ⊥,k ¯ = 1 (R⊥ + Rk ). R Rs′ s , (6.4.34) R⊥,k = 2 s′ ,s

The sums over s′ , s are evaluated using (6.4.31):

6.4 One-photon pair creation

281

Fig. 6.6. DH83 Fig. 5

X s′ ,s

X s′ ,s

2 2 , + cosh2 u sinh2 w K2/3 γs⊥′ s = 2 cosh2 u cosh2 w K1/3

k 2 2 . + cosh2 u cosh2 w K2/3 γs′ s = 2 (cosh2 w + sinh2 u sinh2 w)K1/3

(6.4.35)

Alternatively, the decay rates summed over the spin states may also be derived more directly using (6.4.5). The perpendicular- and parallel-polarized rate in (6.4.5) arise from the 22- and 33-components of (??). To lowest order in an expansion in m/ω the coefficients εn εn′ + m2 ± p2z and pn pn′ are equal, and n−1 2 the resulting combinations of J-functions, (Jnn′ −n−1 + Jnn−1 ′ −n+1 ) and (Jn′ −n + n 2 2 Jn′ −n−1 ) respectively, are of second order in m/ω, and proportional to K2/3 . n−1 n 2 There are contributions of the same order from (Jn′ −n−1 − Jn′ −n+1 ) and n 2 2 (Jnn−1 ′ −n − Jn′ −n−1 ) , which are proportional to K1/3 , with coefficients εn εn′ + m2 ± p2z − pn pn′ = m2 [1 ± sinh2 u + cosh2 u (2 cosh2 w − 1)], (6.4.36) respectively. The combinations (6.4.35) are reproduced. After integration over u, the spin-summed decay rate (6.4.34) becomes Z ∞ µ0 e2 m2 dw √ R⊥,k = 4π 2 3 ω 0 cosh2 w Z ∞ dx K5/3 (x) + (4 cosh2 w ± 1)K2/3 (y) , − y

y=

8m Bc 3ω B

cosh2 w

3. The decay rates (6.4.37) are illustrated in Fig. 6.6.

(6.4.37)

282

6 Quantum theory of gyromagnetic processes

An alternative derivation by [28] using Schwinger’s proper time technique gives Z 1 9 − v2 µ0 e2 m2 ⊥,k √ ± 1 K2/3 (y), (6.4.38) R = dv 3(1 − v 2 ) 4π 2 3 ω 0

with y = (8m/3ω)(Bc /B)(1 − v 2 )−1 . The equivalence of (6.4.37) and (6.4.38) may be shown by writing v = tanh w and using the identity Z ∞ Z ∞ Z ∞ dw dw 1 + 43 sinh2 w K2/3 (y), dx K5/3 (x) = 2 2 cosh w y cosh w 0 0 (6.4.39) with y = (8m/3ω)(Bc/B) cosh2 w. 6.4.9 Asymptotic approximation The result (??) simplifies when the Macdonald functions may be approximated by their asymptotic limit Kν (y) ∼ (π/2y)1/2 exp(−y). This corresponds to the limit in which the energies of the electron and positron are close to ω/2, which corresponds to the integrals being dominated by the region where tanh w is small, that is, to w ≪ 1. In this limit one may make the approximation sinh w ≈ w and cosh w ≈ 1, except in the exponent y ≈ (4/3χ)(1 + w2 ). In this limit, (6.4.37) gives a well-known result (ordinary units) √ 8mc2 Bc mc2 B 3 3 ⊥ k 1 ¯ √ exp − , (6.4.40) R = 2 (R + R ) = αc ¯h Bc 16 2 3¯ hωB which applies for h ¯ ω/2mc2(B/Bc ) ≪ 1. An accurate approximation that includes (6.4.40) and applies more generally is [15] √ mc2 B 3 3 2χ 2 4mc2 Bc ¯ √ K1/3 (χ), χ= , (6.4.41) R = αc h Bc 16 2 π ¯ 3¯ hωB The polarization- and spin-dependence is included in (6.4.32) with (6.4.33) and the leading terms that correspond to the approximation (6.4.40) give Rs⊥′ s = 12 (1 − s′ s)

¯ R [1 + shwi], 6

k

Rs′ s = 12 (1 + s′ s)(1 − s)

¯ R , 12

(6.4.42)

with hwi = (3¯ hωB/8πmBc )1/2 , and where terms hw2 i and higher are neglected. Thus, in this limit, perpendicular-polarized photons decay into pairs with opposite spins, with a small preference for the higher energy particle having spin up, and parallel-polarized photons decay into pairs with the same spin with a strong preference for spin down. These spin preferences are consistent with the preferences found near the thresholds n, n′ = 0, 1: the lowest threshold for parallel-polarized photons is n = n′ = 0 corresponds to both particles having spin down (s = s′ = −1), and the lowest threshold for perpendicularpolarized photons is n = 0, n′ = 1 or n = 1, n′ = 0, with s′ = −s the higher-energy particle having spin up (s′ = 1 or s = 1, respectively).

6.4 One-photon pair creation

283

6.4.10 Energy distribution of the pairs The decay rates (6.4.37) contain information on the energy distribution between the electron and positron. Writing these energies as εn → ε, ε′n′ → ω−ε, one has dw ω2 2dε , . (6.4.43) 4 cosh2 w = = ε(ω − ε) ω cosh2 w

Let dR⊥,k /d(ε) be the rate of decay with one of the particles in the energy range dε. Then (6.4.37) implies Z ∞ dR⊥,k ω2 µ0 e2 m2 √ = ± 1 K2/3 (y) , − dx K5/3 (x) + dε ε(ω − ε) 2π 2 3 ω 2 y y=

2mBc ω2 . 3ωB ε(ω − ε)

(6.4.44)

The distribution (6.4.44) averaged over the polarization of the photons has a functional form ∝ yK2/3 (y) that is well described by y 1/3 e−y ; which increases ∝ y 1/3 for y ≪ 1, has a maximum at y = 1/3 and decreases exponentially for y ≫ 1. Regarding y as a function of ε/ω for given 2mBc /3ωB, it follows from (6.4.43) that the minimum value of y is for ε = 12 ω (w = 0), when the energies of the electron and positron are equal. For 2mBc /3ωB ≫ 1, one has y ≫ 1 for all w ≥ 0, and the exponential decrease of the Macdonald functions with y implies that the distribution is sharply peaked around the minimum of y. Hence, for 2mBc /3ωB ≫ 1 decay in which the energies of the electron and positron are nearly equal is strongly favored. For 2mBc /3ωB ≪ 1 the minimum value of y is small. The maximum of the Macdonald functions corresponds to to 4 cosh2 w = ωB/2mBc. Hence, for 2mBc /3ωB ≪ 1, one of the two particles gains nearly all the photon energy, with the other having an energy ε ≈ 2mBc /B ≪ ω. 6.4.11 Lorentz transformation to an arbitrary frame The foregoing results for pair creation in the ultrarelativistic limit are derived in the frame in which the photon is propagating across the magnetic field lines (kz = 0, sin θ = 1). Results for an arbitrary direction of propagation of the photon are obtained by making a Lorentz transformation to a frame with kz 6= 0, sin θ 6= 1. The Lorentz transformation is closely analogous to that introduced above for synchrotron radiation, cf. (6.2.18). Specifically, the Lorentz factor for the transformation is 1/ sin θ, and the only changes to the foregoing formulas involve inclusion of factors of sin θ. The frequency, ω, and the Lorentz factor, γ, in the frame kz = 0 are replaced by ω sin θ and γ sin θ, respectively, in an arbitrary frame. Suppose one denotes quantities in the ˜ ⊥,k (˜ frame kz = 0 by tildes. For example, R⊥,k (ω) in (??) is replaced by R ω ). The transformation gives

284

6 Quantum theory of gyromagnetic processes

˜ ⊥,k (ω sin θ). R⊥,k (ω) = sin θ R

(6.4.45)

In the analogous relation for dR⊥,k /dε, multiplicative factors of sin θ cancel, and one simply replaces ω and ε, by ω sin θ and ε sin θ, respectively, on the right hand side of (6.4.44). The relativistic approximation, ω ≫ 2m in the frame kz = 0, is replaced by ω sin θ ≫ 2m in an arbitrary frame. 6.4.12 One-photon pair annihilation The inverse process of one-photon pair decay is annihilation of a pair into one photon. The kinetic equation (6.1.30) includes both pair creation and decay, and it implies that the rate of increase of the photon occupation number due to pair annihilation is related in a simple way to the absorption coefficient for pair annihilation. Specifically, to obtain DN (k)/Dt for pair annihilation, one repeats the foregoing calculation for the decay rate for pair creation after first − ′ ′ multiplying by the product of the occupation numbers, n+ s (n, pz )ns′ (n , pz ), of the electrons and positrons. A general expression for the rate of production of (linearly polarized) photons is " # − 2 + ′ X µ0 e3 B εn′ εn |Γq⊥,k ′ q | ns (n, pz )ns′ (n , kz − pz ) DN ⊥,k (k) = , Dt 4π ω 3 sin2 θ gnn′ n,n′ ,± ±

(6.4.46) ⊥,k where Gammaq′ q is defined by (6.4.4), and where ± indicates evaluation at the two solutions (??). The volume emissivity, J(ω, θ), is related to DN ((k)/Dt by ω 3 DN ⊥,k (k) , (6.4.47) J ⊥,k (ω, θ) = (2π)3 Dt with k⊥ = ω sin θ, kz = ω cos θ. For nonrelativistic pairs, annihilation for each set of quantum numbers n, n′ may be treated independently of other set of these quantum numbers. There is a threshold condition, gnn′ = 0, with gnn′ given by (6.4.9) in this case, and the rate (6.4.46) of anniliation of a pair for given n, n′ is singular at gnn′ = 0. As for creation of a pair, the annihilation of an electron and a positron with the same spin produces a k-polarized photon, and annihilation of an electron and a positron with the oppositee spins produces a ⊥-polarized photon. For the two lowest thresholds, n = 0 = n′ and n + n′ = 1, the rates of annihilation follows from (6.4.46) and the result (??) for creation of a pair: CHECK FACTORS OF 2 + µ0 e3 B sin θ e−x DN k (k) X n− (0, pz )n− = − (0, kz − pz ) ± , 2 2 2 1/2 Dt 2π(ω sin θ − 4m ) ± DN ⊥ (k) X µ0 e3 B sin θ e−x = Dt π[ω 2 sin2 θ − 4m2 (1 + B/Bc )]1/2 ±

6.4 One-photon pair creation

− + − × n+ − (0, pz )n+ (1, kz − pz ) + n+ (1, pz )n− (0, kz − pz ) ± ,

285

(6.4.48)

where in the latter case the particle in the first excited state must have spin up, s = 1, n = 1, s′ = −1, n′ = 0 or s = −1, n = 0, s′ = 1, n′ = 1. For given n, n′ , ω, θ, the values pz± are determined by (6.1.19), and the rate of annihilation is proportional to the occupation numbers of the electrons and positrons at pz = pz± , p′z = kz − pz± = pz∓ . In the case n = n′ = 0 the solutions (6.1.19) simplify to ω kz ± 2 2

ω 2 − kz2 − 4m2 ω 2 − kz2

1/2

,

ω kz ε0 (pz± ) = ± 2 2

ω 2 − kz2 − 4m2 ω 2 − kz2

1/2

.

pz± =

(6.4.49)

In (6.4.48) vacuum dispersion is assumed in writing ω 2 − kz2 = ω 2 sin2 θ. When treating pair annihilation it is appropriate to regard the energies, ε, ε′ , and momenta, pz , p′z , as given so that ω = ε + ε′ and kz = pz + p′z are determined by energy and momentum conservation. This interpretation is implicit in (6.4.48). In particular, for pairs in the ground state the first of (6.4.48) may be rewritten CHECK FACTORS OF 2 DN k (k) µ0 e3 B sin θ e−x ′ = n+ (0, pz )n− − (0, pz ) Dt π[ω 2 − kz2 − 4m2 ]1/2 −

1/2 ω = (m2 + p2z )1/2 + (m2 + p′2 , z )

kz = pz + p′z ,

(6.4.50)

with θ determined by writing sin2 θ = (ω 2 − kz2 )/ω 2 . The foregoing discussion applies to nonrelativistic pairs, and in the opposite limit of ultrarelativistic pairs quite different approximations are made. First, the sum over n, n′ and the factor 1/gnn′ are rewritten in terms of integrals over u, w, as in (6.4.17). The next step is to re-express the right hand side of (6.4.46) in a form analogous to the right hand side of (??), multiplied by the occupation numbers for the electrons and positrons. Next, the dependence of these occupation numbers on n, n′ , pz is re-expressed in terms of u, w. Formally, in the frame sin θ = 1, n, n′ are determined implicitly in terms of u, w by (??), (6.4.12), (6.4.15). To lowest order in the expansion in m/ω, n, n′ are related to w by n=

ω 2 sin2 θ (1 ± tanh w)2 , 8eB

n′ =

ω 2 sin2 θ (1 ± tanh w)2 . 8eB

(6.4.51)

Evaluating the ±-solutions (??) in the ultrarelativistic limit gives pz± = 12 (1 + tanh w)ω cos θ ± m sinh u.

(6.4.52)

286

6 Quantum theory of gyromagnetic processes

The ultrarelativistic approximation involves expanding in m/ω sin θ, and to lowest order one neglects the final term in (6.4.48) that contains the dependence on u. In this approximation, (6.4.51), (6.4.48) imply that the pitch angles of the particles, α = arctan[pz± /pn ], pn = (2neB)1/2 , satisfy α = θ to lowest order in m/ω sin θ. Hence, if one writes the occupation numbers in terms of polar coordinates, p, α, in momentum space, n± (n, pz ) → n± (p, α), then the occupation numbers are to be evaluated at p = pn , p′ = pn′ , with n, n′ given by (6.4.51), and at α = θ. The final step is to integrate over u. Because the occupation numbers are independent of u to lowest order, the integral over u is identical to that performed in the derivation of (6.4.37). Hence, using (6.4.37), the emission of photons due to annihilation of ultrarelativistic pairs reduces to CHECK NUMERICAL COEFFICIENT Z ∞ Z ∞ µ0 e2 m2 dw DN ⊥,k (k) √ − dtK5/3 (t) = Dt 2π 2 3 ω 0 cosh2 w y +(4 cosh2 w ± 1)K2/3 (y) [n+ (p+ , θ)n− (p− , θ) + n+ (p− , θ)n− (p+ , θ)],

(6.4.53)

with p± = 12 (1 ± tanh w) ω sin θ. The spin dependence, which is neglected in deriving (6.4.53); the dependence on the spins of the electron and positron may be inferred from the discussion of the spin-dependence of pair creation. There is a preference for given initial spins to lead to either perpendicular- or parallel-polarized photons, but there is no selection rule that forbids any of the possibilities. [16] corrected by [17]

References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13.

Schwinger A.A Sokolov, I.M. Ternov Synchrotron radiation, Pergamon Press, Oxford A.A Sokolov, I.M. Ternov Radiation from relativistic electrons, AIP, New York O.F. Dorofeyev, A.V. Borisov, V.Ch. Zhukovsky, Chapter 7 in V.A. Bordovitsyn (ed.) Synchrotron Radiation Theory and Its Development, World Scientific, Singapore, p. 347 (1999) N.P. Klepikov ZETP 26, 19 (1954) T.K. Daugherty, A.K. Harding Astrophys. J. 273, 781 (1983) R. Lieu, W.I. Axford Astrophys. J. 416, 700 (1993) D. White Phys. Rev. D, 9, 868 (1974) A.A. Sokolov, V. Ch. Zhukovskii, N.S. Nikitina, Phys. Lett, 43a, 85 (1973) J.S. Toll, PhD thesis, Princeton University (1952)

References 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

W. Tsai, T. Erber, Phys. Rev. D 10, 92 (1974) T. Erber, Rev. Mod. Phys., 38, 626 (1966) A.K. Harding, Astrophys. J. 3000, 167 (1986) M.G. Baring MNRAS 235, 51 (1988)

Q. Luo (1996) R.J. Stoenham, Optica Acta, 27, 537 (1980) R.J. Stoenham, Optica Acta, 27, 545 (1980)

287

7 Second order gyromagnetic processes

Second order processes are described by Feynman diagrams with two electronphoton vertices. Such processes include Compton scattering, two-photon emission, two-photon pair creation and annihilation, and electron-electron scattering. General properties of Compton scattering and the related processes of twophoton emission and absorption are summarized in §7.1. Specific results for Compton scattering are discussed for n, n′ = 0 in §7.2. Two-photon emission and two-photon pair annihilation and creation are discussed in §7.4. Electronion and electron-electron scattering are discussed in §7.5

290

7 Second order gyromagnetic processes

7.1 General properties of Compton scattering A general theory for Compton scattering by an electron in a strong magnetic field is developed in this section. The probability for Compton scattering is written down in a form that includes all processes related to Compton scattering by crossing symmetries: Compton scattering by a positron, two-photon pair creation and annihilation, and double emission and absorption. The resonance condition for Compton scattering is used to derive results for the kinematics of these processes. Kinetic equations that describe the effects of Compton scattering on the distributions of photons and electrons are written down. Some analytic results relating to the evaluation of the probability are derived by partially performing the sum over an intermediate state, q ′′ , specifically by performing the sums over s′′ , ǫ′′ explicitly. 7.1.1 Probability for Compton scattering The Feynman diagrams for Compton scattering in a magnetic field are the same as those for Compton scattering in the absence of a magnetic field, except for the labeling of the states. In Fig. 7.1, the relevant Feynman diagrams are drawn for Compton scattering by an electron in an initial state q = n, pz , s and a final state q ′ = n′ , p′z , s′ . Let the initial and final photons be in the modes M ′ and M , and have 4-momenta k ′ and k, respectively. Conservation of the parallel component of 4-momentum implies ǫpz − ǫ′ p′z + kz′ − kz = 0.

(7.1.1)

The condition (7.1.1) is implicit in the general expression for the probability for Compton scattering and related processes. The probability is written down in (5.6.19) in a generic form involving the transition matrix Tfi . On evaluating Tfi using the vertex formalism, the probability becomes ′

′ µν 2 e4 RM (k)RM ′ (k′ ) ∗ eMµ (k)eM ′ ν (k′ ) Mqǫ′ qǫ (k, k′ ) ′ 2 ′ |ε0 ωM (k)ωM (k )| ×2πδ[ǫεq − ǫ′ εq′ − ωM (k) + ωM ′ (k′ )], ′ µν ′ µν ′ µν = Mqǫ′ qǫ (k, k′ ) 1 + Mqǫ′ qǫ (k, k′ ) 2 + Mqǫ′ qǫ (k, k′ ) nl , ′ ′′ ′′ X [Γqǫ′ qǫ′′ (k)]µ [Γqǫ′′ qǫ (−k′ )]ν (k × k′ )z = , exp −i 2eB ǫεn − ǫ′′ ε′′q′′ + ωM ′ (k′ ) ′′ ′′

wqǫ′ǫqMM ′ (k, k′ ) =

ǫ′ ǫ µν Mq′ q (k, k′ )

ǫ′ ǫ µν Mq′ q (k, k′ ) 1

ǫ ,q

′ ′′

′′

X [Γqǫ′ qǫ′′ (−k′ )]ν [Γqǫ′′ qǫ (k)]µ ǫ′ ǫ (k × k′ )z ′ µν , Mq′ q (k, k ) 2 = exp i ǫεn − ǫ′′ ε′′q′′ − ωM (k) 2eB ′′ ′′ ǫ ,q

ǫ′ ǫ µν 2 ′ ′ Mq′ q (k, k′ ) nl = [Γqǫ′ qǫ (k − k′ )]θ Dθη (kM − kM ′) e ′ ′ ×Π ηµν (kM − kM ′ , −kM , kM ′ ).

(7.1.2)

7.1 General properties of Compton scattering (a)

k q'

q''

ν

(b)

k q

q'

ν

291

q

q''

μ

μ

k'

k'

k

k'

q'

q

(c)

Fig. 7.1. The three diagrams that contributions Compton scattering in a magnetic field.

The case ǫ = ǫ′ = 1 corresponds to Compton scattering by an electron. The various crossed processes correspond to changing the signs ǫ or ǫ′ or changing the signs of k or k′ . There are three contributions in (7.1.2), labeled 1, 2 and nl, and these correspond to the three diagrams in Fig. 7.1. The term labeled nl describes nonlinear scattering, which is associated with the quadratic response tensor, Π ηµν . There are contributions from the medium and from the magnetized vacuum to Π ηµν , with the former typically the more important [24]. Nonlinear scattering in a thermal plasma is important for relatively long wavelengths, of order the Debye length, whereas quantum effects tend to be significant only for very much shorter wavelengths. As a consequence, when nonlinear scattering is important, the non-quantum theory suffices, and it is not considered in the following discussion of the quantum theory of scattering. 7.1.2 Compton cross-section It is conventional to describe Compton scattering in terms of the scattering cross section. The relation between the cross section and the probability is discussed in §4.6.1, and given by (4.6.8), which translates into, in ordinary units, Z ∞ dω ω 2 n2M ∂(ωnM ) ∂ cos θ ∂ cos θ′ ++ ΣMM ′ = w′ (7.1.3) ′, ′ (2π)3 ω ′ vgM ∂ω ∂ cos θr ∂ cos θr′ q qMM ′ 0 where the probability (7.1.2) needs to be written in ordinary units. Omitting all arguments k, k′ , as in (7.1.3), (7.1.2) gives, in ordinary units ′

wqǫ′ ǫqMM ′ =

′ µν 2 e4 c4 RM RM ′ ∗ e eM ′ ν Mqǫ′ qǫ 2πδ[ωM − ωM ′ − (ǫεq − ǫ′ εq′ )/¯h], ε20 |ωM ωM ′ | Mµ (7.1.4)

292

7 Second order gyromagnetic processes

where the vertex functions are dimensionless in ′ ′′

µν ′ Mqǫ′ qǫ (k, k′ ) 1

′′

X [Γqǫ′ qǫ′′ (k)]µ [Γqǫ′′ qǫ (−k′ )]ν (k × k′ )z , exp −i¯ h = ǫεn − ǫ′′ ε′′q′′ + h 2eB ¯ ωM ′ ′′ ′′ ǫ ,q

(7.1.5) ′ µν with h ¯ included similarly in the analogous expression for Mqǫ′ qǫ 2 . In the approximation in which the refractive indices are approximated by unity, and the ray and wave normal angles are not distinguished, the labels M, M ′ in (7.1.3) may be interpreted as describing (transverse) polarization states, and (7.1.3) reduces to Z 1 ∞ dω ω 2 ++ w′ (7.1.6) ΣMM ′ = 4 ′. c 0 (2π)3 ω ′ q qMM On inserting the probability (7.1.4), the differential cross section (7.1.6) becomes ∗ e4 ω 2 e eM ′ ν Mqǫ′′qǫ µν 2 ΣMM ′ = (7.1.7) (4πε0 )2 ω ′2 Mµ

h is implicit. The total cross section is found by where ω − ω ′ = (εq − εq′ )/¯ integrating (7.1.7) over the solid angle of the scattered photon. 7.1.3 Kinematics of Compton scattering

The resonance condition in the probability (7.1.2) with (7.1.1) can be reduced to the same form as the gyroresonance condition in (6.1.2). This is achieved by writing Ω = ω − ω ′ , with ω = ωM (k), ω ′ = ωM ′ (k′ ), and K = k − k′ . The kinematics for gyromagnetic emission may then be used to describe the kinematics of Comtpon scattering. Specifically, the solutions of the gyroresonance condition found in §6.1.4 translate trivially into corresponding solutions for Compton scattering and crossed processes. In place of (6.1.17)–(6.1.19), (6.1.21), one has (ε0n )2 − (ε0n′ )2 + Ω 2 − Kz2 , Fn′ n = 1 − Fnn′ , 2(Ω 2 − Kz2 ) 2 Ω − Kz2 − (ε0n − ε0n′ )2 Ω 2 − Kz2 − (ε0n + ε0n′ )2 2 , = Gn′ n = 4(Ω 2 − Kz2 )2

Fnn′ = G2nn′

pz± = ǫ(Kz Fnn′ ± ΩGnn′ ),

p′z± = −ǫ′ (Kz Fn′ n ∓ ΩGn′ n ),

(7.1.8)

(7.1.9) (7.1.10)

εn′ (p′z± ) = −ǫ′ (ΩFn′ n ∓ Kz Gn′ n ), (7.1.11) respectively. The physically allowed regions correspond to G2nn′ ≥ 0, and are εn (pz± ) = ǫ(ΩFnn′ ± Kz Gnn′ ),

Ω 2 − Kz2 ≤ (ε0n − ε0n′ )2 ,

Ω 2 − Kz2 ≥ (ε0n + ε0n′ )2 .

(7.1.12)

7.1 General properties of Compton scattering

293

The inequality Ω 2 − Kz2 ≤ (ε0n − ε0n′ )2 applies to Compton scattering, with the equality being the threshold condition for Compton scattering at given n, n′ . The region Ω 2 − Kz2 ≥ (ε0n + ε0n′ )2 corresponds to an allowed process that is rarely discussed: decay of a photon into another photon and a pair. For the crossed processes corresponding to ω ′ , kz′ → −ω ′ , −kz′ , such that one has Ω = ω + ω ′ , Kz = kz + kz′ , the first of the regions (7.1.12) is where double emission is allowed, and the second region is where two-photon pair creation and decay are allowed. 7.1.4 Kinetic equations for Compton scattering The kinetic equations for Compton scattering, two-photon emission and absorption and two-photon pair creation and decay are derived in the same way as the kinetic equations for the single-photon processes, cf. §6.1.7. For Compton scattering of photons in the mode M ′ into photons in the mode M by an electron, the kinetic equations for the photons are Z Z 3 ′ dpz d k DNM (k) X eB ′ wq++ = ′ qMM ′ (k, k ) 3 Dt 2π 2π (2π) q′ q × nq [1 + NM (k)]NM ′ (k ′ ) − nq′ NM (k)[1 + NM ′ (k ′ )] , X eB Z dpz Z d3 k DNM ′ (k ′ ) w++ (k, k′ ) =− 3 q′ qMM ′ Dt 2π 2π (2π) q′ q × nq [1 + NM (k)]NM ′ (k ′ ) − nq′ NM (k)[1 + NM ′ (k ′ )] . (7.1.13)

The kinetic equation for the particles is Z Z 3 ′ X d3 k d k dnq ′ ++ wqq = ′′ MM ′ (k, k ) dt (2π)3 (2π)3 ′′ q × nq′′ [1 + NM (k)]NM ′ (k ′ ) − nq NM (k)[1 + NM ′ (k ′ )] X ′ ′ − wq++ ′ qMM ′ (k, k ) ) nq [1 + NM (k)]NM ′ (k ) q′

−nq′ NM (k)[1 + NM ′ (k ′ )] ,

(7.1.14)

where q ′ denotes the quantum numbers n′ , s′ , p′z = pz + kz′ − kz and q ′′ denotes n′′ , s′′ , p′′z = pz − kz′ + kz . 7.1.5 Sum over intermediate states: vertex formalism Explicit evaluation of the probability (7.1.2) for Compton scattering and related crossed processes is tedious. It can be helpful to start by carrying out the sum over intermediate states. The sum is over states labeled q ′′ = ǫ′′ , s′′ , n′′ ,

294

7 Second order gyromagnetic processes

′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ )

cǫq′ǫq (k, k ′ )

cǫq′ǫq (k, k ′ )

′

cǫq′ǫq (k, k ′ ) ′

′

cǫq′ǫq (k, k ′ ) ′

ǫ′ ǫ q′ q

c

(k, k ′ )

′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ )

cǫq′ǫq (k, k ′ )

ǫ′ ǫ q′ q

c

(k, k ′ )

′

′

cǫq′ǫq (k, k ′ ) ′

cǫq′ǫq (k, k ′ )

00

n′′

′+ [(ǫεq + ω ′ )A+ + (ǫpz + kz′ )Bq+′ q ]kq′+′ q + A− (mkq′− ′ q + pn′′ lq ′ q ) q′ q q′ q

01

) [(ǫεq + ω ′ )A− − (ǫpz + kz′ )Bq−′ q ]u′+ + A+ (mu′− + pn′′ t′+ q′ q q′ q q′ q q′ q q′ q

02

)} −i{[(ǫεq + ω ′ )A− − (ǫpz + kz′ )Bq−′ q ]u− + A+ (mu′+ − pn′′ t′+ q′ q q′ q q′ q q′ q q′ q

03

′+ ′− − ′+ ′′ [(ǫεq + ω ′ )Bq+′ q + (ǫpz + kz′ )A+ q ′ q ]kq ′ q + Bq ′ q (mkq ′ q + pn lq ′ q )

n′′ 10

n′′

n′′ 20

′+ ′− + ′+ − ′ ′′ [(ǫεq + ω ′ )A− q ′ q + (ǫpz + kz )Bq ′ q ]vq ′ q − Aq ′ q (mvq ′ q − pn rq ′ q )

′− + ′+ −i{[(ǫεq + ω ′ )A− + (ǫpz + kz′ )Bq−′ q ]vq′− ′ q − Aq ′ q (mvq ′ q − pn′′ tq ′ q )} q′ q

′+ [(ǫεq + ω ′ )Bq+′ q + (ǫpz + kz′ )A+ ]k′+ − Bq−′ q (mkq′− ′ q + pn′′ lq ′ q ) q′ q q′ q

n′′ n′′ 30 n′′

11

n′′

[(ǫεq + ω ′ )A+ − (ǫpz + kz′ )Bq+′ q ]p′+ − A− (mp′− − pn′′ qq′+ ′q) q′ q q′ q q′ q q′ q

22

[(ǫεq + ω ′ )A+ − (ǫpz + kz′ )Bq+′ q ]p′+ − A− (mp′− + pn′′ qq′+ ′q) q′ q q′ q q′ q q′ q

12

i{[(ǫεq + ω ′ )A+ − (ǫpz + kz′ )Bq+′ q ]p′− − A− (mp′+ − pn′′ qq′− ′q ) q′ q q′ q q′ q q′ q

n′′ 33

n′′

n′′ 21

′+ [(ǫεq + ω ′ )A+ + (ǫpz + kz′ )Bq+′ q ]kq′+′ q − A− (mkq′− ′ q + pn′′ lq ′ q ) q′ q q′ q

−i{[(ǫεq + ω ′ )A+ − (ǫpz + kz′ )Bq+′ q ]p′− − A− (mp′+ + pn′′ qq′− ′ q )} q′ q q′ q q′ q q′ q

) [−(ǫεq + ω ′ )Bq−′ q + (ǫpz + kz′ )A− ]u′+ + Bq+′ q (mu′− + pn′′ t′− q′ q q′ q q′ q q′ q

n′′

13

n′′ 31 n′′ 23

n′′

32

n′′

′+ [(ǫεq + ω ′ )Bq−′ q + (ǫpz + kz′ )A− ]v ′+ − Bq+′ q (mvq′− ′ q − pn′′ rq ′ q )} q′ q q′ q

′− ]v ′− − Bq+′ q (mvq′+ −i{[(ǫεq + ω ′ )Bq−′ q + (ǫpz + kz′ )A− ′ q − pn′′ rq ′ q )} q′ q q′ q

)} ]u′− + Bq+′ q (mu′+ − pn′′ t′− −i{[−(ǫεq + ω ′ )Bq−′ q + (ǫpz + kz′ )A− q′ q q′ q q′ q q′ q

′

µν

Table 7.1. Table of tensor components of cǫq′ǫq (k, k ′ ) n′′ , defined by (7.1.18), with A± = a′ǫ′ s′ aǫs ±a′−ǫ′ s′ a−ǫs , Bq±′ q = a′ǫ′ s′ a−ǫs ±a′−ǫ′ s′ aǫs , a′± = [(ε′n′ ±ε0n′ )/2εn′ )]1/2 , q′ q a± = [(εn ± ε0n )/2εn )]1/2 , and with the remaining quantities defined by Table 7.3.

and the sums over ǫ′′ , s′′ can be performed explicitly. The resulting cumbersome expressions are useful as the basis for various physical and mathematical approximations. The sums need to be performed for each of the two contributions labeled 1 and 2 to the matrix element in (7.1.2). The term labeled 1 corresponds to the photon k′ being absorbed at the first vertex, labeled ν, and the photon k being emitted at the second vertex, labeled µ, so that the intermediate state has ǫ′′ p′′z = ǫpz + kz′ . The term labeled 2 corresponds to the sequence of absorption and emission being interchanged, and then the intermediate state has ǫ′′ p′′z = ǫpz − kz . The sums over ǫ′′ , s′′ can be performed explicitly by first rationalizing the denominators is these two terms, so that ǫ′′ appears only in the numerator. Omitting the labels denoting the modes, the rationalization gives

7.1 General properties of Compton scattering

′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) d

ǫ′ ǫ q′ q

(k, k ′ )

′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) d

ǫ′ ǫ q′ q

(k, k ′ )

′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ ) ′

dqǫ ′ǫq (k, k ′ )

00

[(ǫεq − ω)A+ + (ǫpz − kz )Bq+′ q ]kq+′ q + A− (mkq−′ q + pn′′ lq+′ q ) q′ q q′ q

n′′

01

n′′ 10 n′′

02

n′′ 20

n′′ 03

−[(ǫεq − ω)A− + (ǫpz − kz )Bq−′ q ]vq+′ q + A+ (mvq−′ q − pn′′ rq+′ q ) q′ q q′ q ) −[(ǫεq − ω)A− − (ǫpz − kz )Bq−′ q ]u+ − A+ (mu− + pn′′ t+ q′ q q′ q q′ q q′ q q′ q + + + − − ′′ −i{−[(ǫεq − ω)A− q ′ q + (ǫpz − kz )Bq ′ q ]vq ′ q + Aq ′ q (mvq ′ q − pn rq ′ q )}

)} i{[(ǫεq − ω)A− − (ǫpz − kz )Bq−′ q ]u− + A+ (mu+ − pn′′ t− q′ q q′ q q′ q q′ q q′ q [(ǫεq − ω)Bq+′ q + (ǫpz − kz )A+ ]k+ − Bq−′ q (mkq−′ q + pn′′ lq+′ q ) q′ q q′ q

n′′

30

+ − − + ′′ [(ǫεq − ω)Bq+′ q + (ǫpz − kz )A+ q ′ q ]kq ′ q + Bq ′ q (mkq ′ q + pn lq ′ q )

n′′ 11

[(ǫεq − ω)A+ − (ǫpz − kz )Bq+′ q ]p+ − A− (mp− − pn′′ qq+′ q ) q′ q q′ q q′ q q′ q

n′′

22

[(ǫεq − ω)A+ − (ǫpz − kz )Bq+′ q ]p+ − A− (mp− + pn′′ qq+′ q ) q′ q q′ q q′ q q′ q

12

−i{[(ǫεq − ω)A+ − (ǫpz − kz )Bq+′ q ]p− − A− (mp+ − pn′′ qq−′ q ) q′ q q′ q q′ q q′ q

n′′ 33

n′′

21

n′′ 13 n′′

31

[(ǫεq − ω)A+ + (ǫpz − kz )Bq+′ q ]kq+′ q − A− (mkq−′ q + pn′′ lq+′ q ) q′ q q′ q

n′′

n′′ 23 n′′

32

n′′

i{[(ǫεq − ω)A+ − (ǫpz − kz )Bq+′ q ]p− − A− (mp+ + pn′′ qq−′ q )} q′ q q′ q q′ q q′ q )} [(ǫεq − ω)Bq−′ q − (ǫpz − kz )A− ]u+ − Bq+′ q (mu− + pn′′ t+ q′ q q′ q q′ q q′ q [−(ǫεq − ω)Bq−′ q − (ǫpz − kz )A− ]v + + Bq+′ q (mvq−′ q − pn′′ rq−′ q ) q′ q q′ q )} ]u− + Bq+′ q (mu+ − pn′′ t− i{[−(ǫεq − ω)Bq−′ q + (ǫpz − kz )A− q′ q q′ q q′ q q′ q −i{[−(ǫεq − ω)Bq−′ q − (ǫpz − kz )A− ]v − + Bq+′ q (mvq+′ q − pn′′ rq−′ q )} q′ q q′ q ′

Table 7.2. As for Table 7.1 but for dǫq′ǫq (k, k′ )

µν ′ Mqǫ′ qǫ (k, k′ ) 1

=

X n′′

µν

n′′

µν ′ ′ 1/2 ′ aǫq′ǫq (k, k′ ) 1,n′′ ei(xx ) sin(ψ−ψ )

2ǫ(εq ω ′ − pz kz′ ) + ω ′2 − kz′2 − 2n′′ eB

.

,

X µν ′ ′ ′′ ′ ∗ν ǫǫ′′ aǫq′ǫq (k, k′ ) 1,n′′ = [ǫεq + ǫ′′ ε′′q′′ + ω ′ ] [Γqǫ′ qǫ′′ (k)]µ [Γqq . ′′ (k )]

295

(7.1.15)

s′′ ,ǫ′′

µν ′ Mqǫ′ qǫ (k, k′ ) 2

=

X n′′

ǫ′ ǫ µν ′ 1/2 ′ aq′ q (k, k′ ) 2,n′′ e−i(xx ) sin(ψ−ψ )

−2ǫ(εq ω − pz kz ) + ω 2 − kz2 − 2n′′ eB

,

X ǫ′ ǫ µν ′′ ′ ′′ aq′ q (k, k′ ) 2,n′′ = [ǫεq + ǫ′′ ε′′q′′ − ω] [Γqǫ′′ qǫ′ (k′ )]∗ν [Γqǫ′′ qǫ (k)]µ .

(7.1.16)

s′′ ,ǫ′′

The two tensors (7.1.15), (7.1.16) are related by ǫ′ ǫ µν ′ νµ Mq′ q (k, k′ ) 2 = Mqǫ′ qǫ (−k′ , −k) 1 ,

ǫ′ ǫ µν ′ νµ aq′ q (k, k′ ) 2 = aǫq′ǫq (−k′ , −k) 1 . (7.1.17)

296

7 Second order gyromagnetic processes ′

′

kq±′ q

′ ′ n ′ n b′s′ bs Jnn′′−1 (x′ )Jnn−1 ′′ −n (x) ± s sb−s′ b−s Jn′′ −n′ (x )Jn′′ −n (x) −n′

lq±′ q

sb′s′ b−s Jnn′′−1 (x′ )Jnn′′ −n (x) ± s′ b′−s′ bs Jnn′′ −n′ (x′ )Jnn−1 ′′ −n (x) −n′

p± q′ q

b′s′ bs ei(ψ

qq±′ q

′

′

′

−ψ)

sb′s′ b−s ei(ψ

′

′

Jnn′′−1 (x′ )Jnn−1 ′′ −n+1 (x) −n′ +1 ±s′ sb′−s′ b−s e−i(ψ

+ψ)

−ψ)

′

Jnn′′ −n′ −1 (x′ )Jnn′′ −n−1 (x)

′

Jnn′′−1 (x′ )Jnn′′ −n−1 (x) −n′ +1 ±s′ b′−s′ bs e−i(ψ

′

′

′

′ +ψ)

′

Jnn′′ −n′ −1 (x′ )Jnn−1 ′′ −n+1 (x) ′

′

rq±′ q

′ ′ −iψ b′s′ bs eiψ Jnn′′−1 (x′ )Jnn−1 Jnn′′ −n′ −1 (x′ )Jnn′′ −n (x) ′′ −n (x) ± s sb−s′ b−s e −n′ +1

t± q′ q

′ ′ iψ n b′s′ bs e−iψ Jnn′′−1 (x′ )Jnn−1 Jn′′ −n′ (x′ )Jnn′′ −n−1 (x) ′′ −n+1 (x) ± s sb−s′ b−s e −n′

u± q′ q

sb′s′ b−s eiψ Jnn′′−1 (x′ )Jnn′′ −n−1 (x) ± s′ b′−s′ bs e−iψ Jnn′′ −n′ (x′ )Jnn−1 ′′ −n+1 (x) −n′

vq±′ q

sb′s′ b−s eiψ Jnn′′−1 (x′ )Jnn′′ −n (x) ± s′ b′−s′ bs e−iψ Jnn′′ −n′ −1 (x′ )Jnn−1 ′′ −n (x) −n′ +1

′

′

′

′

′

′

′

′

Table 7.3. Table of the combinations of J-functions that appear in Table 7.2, with b′s′ = (ε0n′ +s′ m)1/2 , bs = (ε0n +sm)1/2 . The combinations of J-functions in Table 7.1 ′ are modified, denoted by the prime, by (a) multiplication by (−)n −n , and (b) the ′ ′ interchanges x ↔ x , ψ ↔ ψ .

On inserting the explicit form (5.4.18) for the vertex function, it is straightforward but tedious to perform the sums in (7.1.15), (7.1.16) directly. It is convenient to write the result in the form ′ µν ǫ′ ǫ µν ′ ′ ′′ ′ aq′ q (k, k′ ) 1,n′′ = (ie−iψ )n (−ieiψ )n e−in (ψ −ψ) cǫq′ǫq (k, k′ ) n′′ , µν ′ ǫ′ ǫ µν ′ ′ ′′ ′ (7.1.18) aq′ q (k, k′ ) 2,n′′ = (−ie−iψ )n (ieiψ )n ein (ψ −ψ) dǫq′ǫq (k, k′ ) n′′ .

′ µν ′ µν The components of cǫq′ǫq (k, k′ ) n′′ and dǫq′ǫq (k, k′ ) n′′ are written down in Tables 7.1 and 7.2, respectively. The components are related by the symmetry property (7.1.17). 7.1.6 Sum over intermediate states: Ritus method

The Ritus (§5.5) provides an alternative way of evaluating the quan ′ methodµν tities Mqǫ′ qǫ (k, k′ ) 1.2 in the probability (7.1.2). To illustrate the use of this ′ µν method, consider the evaluation of Mqǫ′ qǫ (k, k′ ) 1 , which involves the product ′ ′′ ′′ [Γqǫ′ qǫ′′ (k)]µ [Γqǫ′′ qǫ (−k′ )]ν , and in the Ritus method this product becomes ′ V 2 ϕ¯ǫs′ (n′ , p′z ) Jnµ′ n′′ (k⊥ ) φǫs′′ (n′′ , p′′z ) φ¯ǫs′′ (n′′ , p′′z ) Jnν′′ n (−k′⊥ ) ϕǫs (n, pz ).

The sums are over s′′ , ǫ′′ , n′′ . The sum over s′′ is performed using (5.5.16), which gives

7.1 General properties of Compton scattering

297

′′

P/ ′′ǫ′′ + m) φǫs′′ (n′′ , p′′z ) φ¯ǫs′′ (n′′ , p′′z ) = n ′′ , ǫ εq′′ V s′′ ′ µν ′′ µ with Pn′′ǫ′′ = (ǫ′′ ε′′n′′ , 0, pn′′ , ǫ′′ p′′z ). In evaluating aǫq′ǫq (k, k′ ) 1,n′′ , as given by (7.1.15), the sum over ǫ′′ is performed using X

′′

X ǫ′′

[ǫεq + ǫ′′ ε′′q′′ + ω ′ ]

P/n′′ǫ′′ (p′′z ) + m 2ǫ′′ εq′′

= (ǫεq + ω ′ )γ 0 − pn′′ γ 2 − (ǫpz + kz′ )γ 3 + m,

(7.1.19)

with p′′z = ǫ′′ (ǫpz + kz′ ). The final step is to evaluate the matrix elements of the resulting product of Dirac matrices for the projected states. By way of illustration, consider the evaluation for µ = ν = 1. Evaluating the product γ 1 times (7.1.19) times γ 1 gives (ǫεq + ω ′ )γ 0 − (ǫpz + kz′ )γ 3 − m

and

− pn′′ γ 2 ,

for the k- and ⊥-parts, respectively. For an arbitrary choice of spin opera′ tor the reduced wave functions have the form (5.5.12), so that [ϕ¯ǫs′ (n′ , p′z )]± become the row matrices ϕ¯′+ = (C1′∗ , 0, −C3′∗ , 0), ϕ¯′− = (0, C2′∗ , 0, −C4′∗ ) and [ϕǫs (n, pz )]± become the column matrices ϕ+ = (C1 , 0, C3 , 0), ϕ− = (0, C2 , 0, C4 ). The matrix elements for the k- and ⊥-parts are nonzero only between ϕ¯′± , ϕ± and for ϕ¯′∓ , ϕ± , respectively. For example, the coefficient ′ ′ n−1 ′ ′ ′ of ei(ψ−ψ ) Jnn′′−1 −n′ −1 (x) Jn′′ −n+1 (x ) is the sum of (ǫεq + ω ), −(ǫpz + kz ), m times   +   ′∗  0 Aq′ q C1 C1 + C3′∗ C3 γ ′ ′  3 ′∗ ′∗    ϕ+ = C1 C3 + C3 C1 = bs′ bs Bq+′ q  , ϕ¯+ γ C1′∗ C1 − C3′∗ C3 1 A− q′ q ′

′

n ′ respectively, and the coefficient of ei(ψ+ψ ) Jnn′′−1 −n′ −1 (x) Jn′′ −n+1 (x ) is −pn′′ times −ϕ¯′+ γ 2 ϕ− = i(C1′∗ C4 + C3′∗ C2 ) = −sb′s′ b−s A− q′ q .

The explicit forms for the combinations of C ′∗ s and Cs are for the magnetic± moment eigenfunctions (5.2.12), with the A± q′ q , Bq′ q defined in Table 7.1. These ′ 11 results reproduce half the terms in the entry for cǫq′ǫq (k, k′ ) n′′ in Table 7.1, ′± ± ± ′ ′ with p′± q′ q , qq′ q given by ψ, x ↔ ψ , x in the entries pq′ q , qq′ q in Table 7.3. ǫ′ ǫ 11 The remaining terms in cq′ q (k, k′ ) n′′ come from the other nonzero matrix elements, ϕ¯′+ (γ 0 , γ 3 , 1)ϕ+ , ϕ¯′− γ 2 ϕ+ . Similar calculations reproduce the other µ, ν component in Tables 7.1 and 7.2. In comparing the Ritus method and the vertex formalism, neither has an obvious major advantage over the other in detailed calculations. The vertex formalism involves more direct calculations, but requires a specific choice of spin operator to obtain a specific form for the vertex function. In the vertex formalism, one necessarily calculates transition probabilities between specific

298

7 Second order gyromagnetic processes

spin states. One classifies the transitions as either non spin-flip, for s′ s = 1, or spin-flip, for s′ s = −1, and treats these separately. The Ritus method is closer to the conventional method used in the unmagnetized case to evaluate Feynman amplitudes and transition rates. In the unmagnetized case one is rarely interested in any spin-dependence, and one averages over the initial spins and sums over the final spins. This allows rules for evaluating transition rates to be formulated without requiring a specific choice of spin operator. This is possible in the magnetized case using the Ritus method, but the evaluation of the traces is much more cumbersome than in the unmagnetized case.

7.2 Compton scattering by an electron with n = 0

299

7.2 Compton scattering by an electron with n = 0 The general form of the probability for Compton scattering in a magnetic field is too cumbersome to be of direct use in specific applications, and it needs to be replaced by simpler approximate forms under appropriate conditions. In many cases one can justify using a simpler theory. On the one hand, if quantum effects are not important, one may reduce the probability (7.1.2) to its non-quantum counterpart (4.6.2), and use the classical theory for Thomson scattering in a magnetic field. On the other hand, if quantum effects are important, and the frequency is so high that magnetic effects are small, one may use the relativistic quantum theory for Compton scattering in the absence of a magnetic field. However, there are situations where both quantum and magnetic effects are important, and for these one needs to make other approximations to (7.1.2). An important special case where the magnetized quantum theory is essential is for scattering by an electron in its ground (Landau) state. In a superstrong magnetic field, gryomagnetic losses occur at a very high rate: according to (6.2.21) this rate is of order (in ordinary units) αc (mc2 /¯h)(B/Bc )2 ≈ 1019 (B/Bc )2 s−1 . As a consequence, one expects most electrons to be in their ground state, n = 0. The case of Compton scattering by an electron in the state n = 0 is discussed in this section. 7.2.1 Scattering probability for n = 0 The probability (7.1.2) for Compton scattering simplifies considerably for an electron with n = 0, when one has ǫ = ǫ′ = 1 and s = −1. One is free to choose the frame pz = 0, implying p′z = kz′ −kz , εn = m. The J-functions in Table 7.3 with upper index n − 1 are identically zero for n = 0, reducing the number of combinations that appear by one half. After omitting the labels M, M ′ referring to the modes, and assuming RM , RM ′ can both be approximated by 1/2, and neglecting nonlinear scattering, the probability reduces to ′ wq++ ′ 0 (k, k ) =

e4 4ε20 |ωω ′ |

∗ ′ ++ ei ej M ′ (k, k′ ) ij 2 2πδ(εq′ − m − ω ′ + ω), (7.2.1) q 0

′ ij = with εq′ = [m2 + 2n′′ eB + (kz′ − kz )2 ]1/2 . The components of Mq++ ′ 0 (k, k ) ++ ++ ′ ij ′ ij Mq′ 0 (k, k ) 1 + Mq′ 0 (k, k ) 2 follow from (7.1.15), (7.1.16), (7.1.18), which give, for n = 0, pz = 0, ++ ij X Mq′ 0 (k, k′ ) 1 = n′′

′ ij iF1 c++ q′ 0 (k, k ) n′′ e

2mω ′ + ω ′2 − kz′2 − 2n′′ eB ′

′′

with the phase factor eiF1 = (ie−iψ )n e−in

,

(ψ ′ −ψ) i(xx′ )1/2 sin(ψ−ψ ′ )

e

(7.2.2) and

300

7 Second order gyromagnetic processes ′

kq±′ 0

∓s′ b′−s′ Jnn′′ −n′ (x′ )Jn0′′ (x)

lq±′ 0

−b′s′ Jnn′′−1 (x′ )Jn0′′ (x) −n′

′

p± q′ 0

∓s′ b′−s′ e−i(ψ

qq±′ 0

−b′s′ ei(ψ

rq±′ 0

′

+ψ)

′

−ψ)

′

Jnn′′ −n′ −1 (x′ )Jn0′′ −1 (x)

′

Jnn′′−1 (x′ )Jn0′′ −1 (x) −n′ +1

′

′

∓s′ b′−s′ e−iψ Jnn′′ −n′ −1 (x′ )Jn0′′ (x) ′

t± q′ 0

∓s′ b′−s′ eiψ Jnn′′ −n′ (x′ )Jn0′′ −1 (x) ′

u± q′ 0

−b′s′ eiψ Jnn′′−1 (x′ )Jn0′′ −1 (x) −n′

vq±′ 0

−b′s′ eiψ Jnn′′−1 (x′ )Jn0′′ (x) −n′ +1

′

′

Table 7.4. The entries in Table 7.3 reduce to the entries shown for n = 0, with b′± = [(ε0n′ ± m)/2ε0n′ ]1/2 .

′ ij Mq++ ′ 0 (k, k ) 2 ′

′

=

X n′′

′′

++ ij dq′ 0 (k, k′ ) n′′ eiF2

−2mω + ω 2 − kz2 − 2n′′ eB

′

′ 1/2

,

(7.2.3)

′

with eiF2 = (−ie−iψ )n ein (ψ −ψ) e−i(xx ) sin(ψ−ψ ) . The components of ++ ij ′ ij cq′ 0 (k, k′ ) n′′ and d++ q′ 0 (k, k ) n′′ follow from Tables 7.1 and 7.2, respectively, with the combinations of J-functions in Table 7.3 simplifying to the forms given in Table 7.4. The combinations of J-functions in Table 7.4 have relatively simple forms for small values of n′ , specifically Jν0 (x)–Jν3 (x) are given by (A.1.41)–(A.1.44). For example, the exact form for Jν0 (x) is Jν0 (x) =

e−x/2 xν/2 , (ν!)1/2

(7.2.4)

Inserting (7.2.4) and the other relations into the entries in Table 7.4 leads to explicit forms for small values of n′ = 0. 7.2.2 Allowed transitions n = 0 → n′ ≥ 0 The possible final states, n′ , of the electron with n = 0 depend on the frequency, ω ′ , of the unscattered photon: the electron can be excited to the state n′ only if the photon energy exceeds that for gyromagnetic absorption 0 → n′ . This energy is equal to that for the inverse transition of gyromagnetic emission n′ → n = 0, with the proviso that the electron have pz = 0 in the state n = 0. The relevant frequency is determined by the zero of the denominator in (7.2.2), with n′′ replaced by n′ . This gives # " 1/2 m B ′ 2 ′ ′ −1 . (7.2.5) 2n sin θ ωn′ (θ ) = 1+ Bc sin2 θ′

7.2 Compton scattering by an electron with n = 0

301

For B/Bc ≪ 1, (7.2.5) reduces to ωn′ (θ′ ) ≈ n′ Ωe , reproducing the familiar frequency of n′ th harmonic gyromagnetic emission for B/Bc ≪ 1. In a general frame, in which the electron with n = 0 has arbitrary pz , (7.2.5) is replaced by ωn′ (θ′ ) =

[(ε0 − pz cos θ′ )2 + 2n′ eB sin2 θ′ ]1/2 − (ε0 − pz cos θ′ ) , sin2 θ′

(7.2.6)

with ε0 = (m2 + p2z )1/2 . The frequency (7.2.6) reduces to (7.2.5) for pz = 0. 7.2.3 Scattering for n, n′ = 0 For photons with ω ′ < ω1 (θ′ ), given by (7.2.5) with n′ = 1, scattering by an electron with pz = 0, n = 0, the final electron is in the ground Landau state, n′ = 0. This leads to a further simplification of the entries in Table 7.4, ++ ′ ij ± with s′ = −1 and lq±′ q = qq±′ q = u± q′ q = vq′ q = 0. As a result, c0′ 0 (k, k ) n′′ ′ ij and d++ 0′ 0 (k, k ) n′′ simplify such that each involves only four independent 3-tensors. A convenient notation for expressing this is in term of the 3-vectors e± = (1, ±i, 0),

b = (0, 0, 1).

(7.2.7)

Using (7.2.4) and Tables 7.1, 7.2 and 7.4, one finds, for n′′ = 0, ′

e−(x +x)/2 ′ ′ ω (ε0 + m) + kz′ (kz′ − kz ) bi bj , ′ ′ 1/2 [ε0 (ε0 + m)] ′ e−(x +x)/2 = ′ ′ − ω(ε′0 + m) + kz (kz′ − kz ) bi bj , [ε0 (ε0 + m)]1/2

′ ij c++ 0′ 0 (k, k ) 0 =

′ ij d++ 0′ 0 (k, k ) n′′

(7.2.8)

with ε′0 = [m2 + (kz′ − kz )2 ]1/2 and, for n′′ ≥ 1, ′

′ ij c++ 0′ 0 (k, k ) n′′

′

′′

e−(x +x)/2 (x′ x)n −1 1 = ′ ′ (n′′ − 1)! [ε0 (ε0 + m)]1/2 ′ ′ ′ ω (ε0 + m) + kz′ (kz′ − kz ) e−i(ψ−ψ ) ei+ ej− + ω ′ (ε′0 + m) + kz′ (kz′ − kz ) (x′ x/n′′ )1/2 bi bj ′ ′ −iψ i j +(kz′ − kz ) k⊥ eiψ ei+ bj + k⊥ e b e− ,

′ ij d++ 0′ 0 (k, k ) n′′ =

′′

(7.2.9)

e−(x +x)/2 (x′ x)n −1 1 (n′′ − 1)! [ε′0 (ε′0 + m)]1/2 ′ − ω(ε′0 + m) − kz (kz′ − kz ) ei(ψ−ψ ) ei− ej+ − ω(ε′0 + m) + kz (kz′ − kz ) (x′ x/n′′ )1/2 bi bj ′ iψ i j ′ −(kz′ − kz ) k⊥ e e− b + k⊥ e−iψ bi ej+ . (7.2.10)

302

7 Second order gyromagnetic processes

The probability (7.2.1) for scattering n = 0 to n′ = 0 in the frame pz = 0 reduces to ∞ ++ −i[n′′ (ψ ′ −ψ)+(xx′ )1/2 sin(ψ−ψ ′ ) ∗ ′ X [c0′ 0 (k, k′ )]ij e4 ′ n′′ e w0′ 0 (k, k ) = 2 e e i j 4ε0 |ωω ′ | −2mω + ω 2 − kz2 − 2n′′ eB n′′ =0 ij i[n′′ (ψ ′ −ψ)+(xx′ )1/2 sin(ψ−ψ ′ ) 2 ′ [d++ ′ (k , k)]n′′ e 2πδ(ε′0′ − m − ω ′ + ω), + 00 2mω ′ + ω ′2 − kz′2 − 2n′′ eB (7.2.11)

with ε′0′ = [m2 + (kz′ − kz )2 ]1/2 . Further simplification of the probability (7.2.11) occurs for x, x′ ≪ 1, which corresponds to the cyclotron-like limit discussed in §??. The resonance condition restricts the possible final frequencies, ω, for given ω ′ . For scattering by an electron with pz = 0, the recoil implies p′z 6= 0, and hence the energy of the final electron is greater than that of the initial electron. Energy conservation then requires ω < ω ′ . Using either the resonance condition in the δ-function in (7.2.11), or setting pz± = 0 and n = n′ = 0 in (7.1.10), one finds that for (kz − kz′ )2 ≪ m2 , the final frequency may be approximated by, in ordinary units, ¯hω ′ ′ ′ 2 ω =ω 1− (cos θ − cos θ ) , (7.2.12) mc2 where vacuum dispersion is assumed. 7.2.4 Scattering n = 0 → n′ ≥ 1

When the frequency of the initial photon satisfies ω ′ > ω1 (θ′ ), with ωn′ (θ′ ) given by (7.2.5), scattering can cause an electron in the ground level, n = 0, to jump to the first Landau level, n′ = 1. This results in a scattered photon with frequency ω ≈ ω ′ − ω1 (θ′ ). An independent subsequent process is possible: the electron in the first Landau level can radiate a cyclotron photon, jumping back to the ground state. This combination of scattering and emission may be regarded as a resonant form of photon splitting, in which the initial photon effectively splits into the final photon plus the emitted cyclotron photon. Similarly, for ωn0 +1 (θ′ ) > ω ′ > ωn0 (θ′ ), the final electron can be in any of the states n′ = 0, 1, . . . , n0 . The electron can then jump back to the ground state through gyromagnetic emission involving any sequence of allowed transitions. The probability for the scattering process is given by setting n′ = 1 in (7.2.1), which does not simplify in any obvious way. A simplification that occurs for n′ = 0 is that half the entries in Table 7.4 are identically zero, ± ′ specifically lq±′ q = qq±′ q = u± q′ q = vq′ q = 0, but for n ≥ 1 all eight entries in Table 7.4 are nonzero. As a consequence, the simplified forms (7.2.8)–(7.2.10) that involve only four 3-tensors formed from the vectors e± , b do not apply for n′ > 0. Moreover, the two possible spin states, s′ = ±1, requires that one

7.2 Compton scattering by an electron with n = 0

303

distinguish between transitions without spin flip, s′ = −1, and transitions with spin flip, s′ = 1. Significant simplification occurs if one has x, x′ ≪ 1, when a cyclotron-like approximation applies, as discussed in §??. 7.2.5 Resonant Compton scattering There are resonances in the probability (7.2.11) at the frequencies ωn′′ (θ′ ), where the denominator in the final term inside the modulus has zeros. The presence of a resonance implies enhanced scattering for ω ′ ≈ ωn′′ (θ′ ). Near a resonance one may approximate the probability (7.2.11) by retaining only the term that diverges at the resonance. Such resonances in Compton scattering are associated with virtual transitions. A transition n → n′ involves an intermediate state n′′ , such that the net transition may be regarded as a sum (over n′′ ) of virtual transitions n → n′′ → n′ . A resonance occurs when the transition n → n′′ is allowed, so that the net transition, n → n′ , becomes two sequential allowed transitions, gyromagnetic absorption n → n′′ and gyromagnetic emission n′′ → n′ . Consider the simplest case of resonant scattering: n = 0, n′ = 0, n′′ = 1. Formally, the probability (7.2.11) implies that the cross section becomes infinite at the resonance, but in practice the probability remains finite. In particular, the enhancement factor is limited by the natural line width for gyromagnetic emission from the state n′′ = 1 to n′ = 0. In the immediate vicinity of the resonance at n′′ = 1 one needs to take account of the virtual intermediate state becoming a real intermediate state. The conditions under which one should regard resonant scattering as an enhanced form of scattering and when one should regard it as a sequence of two independent processes, gyromagnetic absorption n = 0 → n′′ = 1 and gyromagnetic emission n′′ = 1 → n′ = 0, depends on the relative magnitude of two frequencies. One of these frequencies is the mismatch, ∆ω ′ = ω ′ − ωn′ ′′ , between the frequency of the intial photon and the resonant frequency, ωn′ ′′ (θ′ ), determined by (7.2.5). The other is the natural line width of the emitted radiation, which is determined by the inverse of the lifetime of the excited state. The process should be regarded as scattering if the frequency mismatch exceeds the line width and as absorption and re-emission if the line width exceeds the frequency mismatch. In the present case, the natural line width of the n′′ th Landau level is determined by the gyromagnetic decay rate, Γq′′ , for an electron in the state q ′′ (with q ′′ including n′′ ). The limitation on the enhancement due to the n′′ th resonance may be determined by making the replacement εq′′ → εq′′ + iΓq′′ in (7.1.15), implying that the minimum value of the resonant denominator is ≈ Γq′′ . For frequencies within Γq′′ of the resonance, the intermediate state is real, and the “resonant scattering” must be treated as a sequence of absorptions and re-emissions. The occupation number of electrons in the excited states is then nonzero, and is determined by the ratio of the probability per unit time of an absorption event to that state divided by the decay rate for that state.

304

7 Second order gyromagnetic processes

7.2.6 Resonant scattering versus absorption plus emission When the condition ω ′ = ωn′′ (θ′ ) is satisfied, the electron can absorb the photon, and jump from the state n = 0 to n′′ . Subsequent emission of a photon with the electron jumping either back to n′ = 0 or to some intermediate state 0 < n′ < n′′ is equivalent to resonant scattering. It is useful to define an effective scattering probability that includes both limits. The effective scattering probability is constructed by combining the probabilities per unit time for the two transitions, and dividing by the rate of decay of the intermediate state. This compound probability is X (7.2.13) wq′′ qM ′ (k′ ) wq′ q′′ M (k)/Γq′′ , w ¯q′ qMM ′ (k, k′ ) = q′′

where the sum is over all possible resonances, and where Γq′′ is the decay rate of the intermediate state. In the immediate vicinity of the resonance, specifically within a frequency less than Γq′′ of the resonant frequency, the effective scattering probability is given by (7.2.13). An interpolation between (7.2.13) in the core of the line, and (7.1.2) in the wings of the line is obtained by replacing the resonant factor by a Lorentzian line profile with the width determined by the lifetime of the intermediate state. Including the various crossed processes, an interpolation formula is [26] ′ Res wq++ ′ qMM ′ (k, k ) =

X e4 RM (k)RM ′ (k′ ) µ 2 ′ e∗Mµ (k) Γ ++ ′ q′ q′′ (k) 2 ε0 |ωM (k)ωM ′ (k )| q′′

′ ∗ν 2 × eM ′ ν (k′ ) Γq++ ′′ q (k )

πΓq′′ /2 [εq − εq′′ − ωM (k)]2 + (Γq′′ /2)2

×2πδ[εq − εq′ − ωM (k) + ωM ′ (k′ )];

(7.2.14)

which reduces to (7.2.13) in the core of the line, |εq − εq′′ − ωM (k)| < ∼ Γq′′ /2, and it provides an approximation to resonant scattering in the wings of the line, |εq − εq′′ − ωM (k)| > ∼ Γq′′ /2. In the far wings of the line, where resonant scattering is no longer dominant, the exact probability (7.1.2) needs to be used. An implication of (7.2.14) is that the maximum enhancement in resonant scattering is limited by the natural line width of the excited state to gryomagnetic emission. This quantum limitation tends to dominate over the non-quantum limitations, due to cold plasma and thermal effects §4.6.7, at low densities in very strong magnetic fields.

7.3 Scattering in the cyclotron and synchrotron limits

305

7.3 Scattering in the cyclotron and synchrotron limits Compton scattering by a magnetized electron simplifies in the cyclotron-like approximation, defined as the limit in which the power series of the J-functions converges rapidly. The cyclotron-like limit applies when the scattering electron is nonrelativistic. In the opposite limit when the scattering electron is highly relativistic, a synchrotron-like or inverse-Compton-like approximation is appropriate. 7.3.1 Cyclotron-like approximation Further simplification of the probability (7.2.11) occurs for x′ , x ≪ 1, when only J00 (x) ≈ 1 need be retained in (7.2.9), (7.2.10). An associated approximation is that the kinetic energy of the final electron is negligible, ε′0 → m, except where the difference ε′0 − m appears explicitly. In this approximation only n′′ = 1 contributes to the term along ei+ ej− , only n′′ = 0 contributes to the term along bi bj , and the remaining two terms are of higher order and are neglected. In this case (7.2.11) reduces to w0′ 0 (k, k′ ) =

e4 4ε20 m2 |ωω ′ |

∗ ′ ij 2 ei ej g 2πδ(ε′0 − m + ω ′ − ω),

(7.3.1)

with only n′′ = 0, 1 contributing in the sum over intermediate states, and with ′ ′ ′ ω (ε0 + m) + kz′ (kz′ − kz ) ei(ψ −ψ) ei− ej+ ij g = 2m(ω + Ωe ) − ω 2 + kz2 ′ ω(ε′0 + m) − kz (kz′ − kz ) e−i(ψ −ψ) ei+ ej− + 2m(ω ′ − Ωe ) + ω ′2 − kz′2 ω(ε′0 + m) + kz (kz′ − kz ) ω ′ (ε′0 + m) − kz′ (kz′ − kz ) i j + bb . (7.3.2) + 2mω ′ + ω ′2 − kz′2 2mω − ω 2 + kz2 The argument of the δ-function In (7.3.1) may be rationalized by writing (ω ′ − ω)2 − (kz′ − kz )2 ε′0 + m + ω ′ − ω ′ ′ ′ δ ω −ω− δ(ε0 − m + ω − ω) = , 2m 2m (7.3.3) which includes the quantum recoil explicitly. When the quantum recoil is neglected, the right hand side of (7.3.3) reduces to δ(ω ′ − ω), and g ij reduces to g ij =

′ ′ ω ω e−i(ψ −ψ) ei− ej+ + ei(ψ −ψ) ei+ ej− + 2bi bj . ω − Ωe ω + Ωe

(7.3.4)

The probability (7.3.1) then reduces to its nonquantum counterpart (4.6.13).

306

7 Second order gyromagnetic processes

7.3.2 Scattering in the birefringent vacuum In strong fields, where the electrons are predominantly in the state n = 0, the relevant wave modes are usually those of the birefringent vacuum. The scattering probabilities for the two modes of the birefringent vacuum are found by inserting the polarization vectors for these modes into (7.3.1). These are e⊥ = (− sin ψ, cos ψ, 0),

ek = (cos θ cos ψ, cos θ sin ψ, − sin θ),

(7.3.5)

with analogous expressions for the primed vectors. The projection of g ij onto the specific polarizations (7.3.5) gives      1 g⊥⊥ 0  cos θ′ cos θ   gk k   2 sin θ′ sin θ  X ω   . = + ±i cos θ g⊥k 0 ω ± Ωe ± ∓i cos θ′ gk⊥ 0 

(7.3.6)

On inserting (7.3.6) into (7.3.1) one may compare the rates of scattering for the different modes, ⊥→⊥, k, k→⊥, k. It is convenient to compare the cross sections for the scattering. The cross section is formally identified by (4.6.8). Here it suffices to note that for for Ωe , (7.3.6) with (7.3.1) reduces to the probability for Thomson scattering when the quantum recoil term is neglected in the resonance condition, implying ω = ω ′ . One may write the polarization-dependent and angle-dependent cross sections in terms of the Thomson cross section, σT = (8π/3)(µ0 e2 /4πm)2 :      1 σ⊥⊥ (θ′ , θ) 0     2 2 2 ′ 2 ′ ′ 1  cos θ cos θ   σkk (θ , θ)  9σT  sin θ sin θ  = +     ,   2 2 ′ 2 2 Y cos θ σ⊥k (θ , θ) 0  7  (1 − Y )   Y 2 cos θ′ σk⊥ (θ′ , θ) 0 (7.3.7) with Y = ω/Ωe . The total cross section implied by (7.3.7) reduces to the Thomson cross section after averaging over the initial and summing over the final states polarization, averaging over cos θ′ and cos θ, and setting Y → 0. A notable feature of (7.3.7) is the singularity that occurs in each cross section for Y = 1, referred to as resonant Compton scattering. Resonant scattering is of specific interest in connection with secondary pair production in the polar cap region of pulsars [18, 19, 22, 23]. An initial step in secondary pair production is the acceleration of primary particles to very high energies by an electric field parallel to the magnetic field. These primary particles radiate high energy photons, which produce the secondary pairs by one-photon decay. The suggested role of resonant scattering is in the production of the highenergy photons: the enhancement due to resonant scattering is invoked in the scattering by the primary particles of thermal photons from the surface of the neutron star. 

7.3 Scattering in the cyclotron and synchrotron limits

307

Fig. 7.2. TBA

Fig. 7.3. TBA

7.3.3 Inverse Compton emission The simplifying assumption made in §?? involves the initial electron, which is assumed to be in the ground state n = 0. Another way in which the general probability (7.1.2) simplifies is when the initial photon is propagating along the magnetic field. The argument, x′ , of the J-functions is then zero, leading to significant simplification in that the only functions that appear, Jνn (x), are of the same as the functions that appear in gyromagnetic emission. The special case of parallel propagation is relevant to Compton scattering by a highly relativistic electron, γ ≫ 1, that is, to inverse Compton scattering. If one considers the scattering in the frame in which the electron is at rest, pz = 0, the Lorentz transformation to this frame transforms the angle, θ′ , of propagation of nearly all photons into a small cone of angles, θ′ < ∼ 1/1/γ. This

308

7 Second order gyromagnetic processes

Fig. 7.4. TBA

allows one to treat inverse Compton scattering by considered the scattering of a parallel-propagating photon, θ′ = 0, by an electron with pz = 0 [21]. The approximation that all photons are propagating parallel to the magnetic field in the rest frame of a highly relativistic electron corresponds to setting sin θ′ = 0, and hence x′ = 0, so that one has Jνn (x′ ) = 1 for ν = 0 and Jνn (x′ ) = 0 for ν 6= 0. TBC

7.4 Two-photon processes

309

7.4 Two-photon processes Processes in which there are two photons in the initial or final state and no photons in the other state include double emission and absorption, and twophoton pair creation and annihilation. Two-photon gyromagnetic emission is allowed under the same conditions as one-photon gyromagnetic emission, and because it is a higher order process it occurs at a rate of order αc slower than the one-photon process. Nevertheless, two-photon can play a significant role in providing a source of additional photons in a plasma where resonant scattering is the dominant process. Similarly, two-photon pair creation or annihilation compete with one-photon pair creation or annihilation, and is important only when some special condition applies to favor it. 7.4.1 Kinetic equations for double gyromagnetic emission Consider the effects of two-photon emission between states q ′ , q involving photons in the modes M, M ′ . The kinetic equations for two-photon emission and absorption are Z Z DNM ′ (k ′ ) X eB dpz d3 k ++ wq′ qMM ′ (k, −k′ ) = 3 Dt 2π 2π (2π) q′ q nq [1 + NM (k)][1 + NM ′ (k ′ )] − nq′ NM (k)NM ′ (k ′ ) , Z Z 3 ′ dpz d k DNM (k) X eB ′ wq++ = ′ qMM ′ (k, −k ) 3 Dt 2π 2π (2π) q′ q nq [1 + NM (k)][1 + NM ′ (k ′ )] − nq′ NM (k)NM ′ (k ′ ) , Z Z 3 ′ hX d3 k d k dnq ′ ++ = wqq ′′ MM ′ (k, −k ) dt (2π)3 (2π)3 q′′ × nq′′ [1 + NM (k)][1 + NM ′ (k ′ )] − nq NM (k)NM ′ (k ′ ) X ′ ′ − wq++ ′ qMM ′ (k, −k ) ) nq [1 + NM (k)][1 + NM ′ (k )] q′

−nq′ NM (k)NM ′ (k ′ ) ,

(7.4.1)

where q ′ denotes the quantum numbers n′ , s′ , p′z = pz − kz′ − kz and q ′′ denotes n′′ , s′′ , p′′z = pz + kz′ + kz . The term on the right hand side of each of (7.4.1) that is independent of NM (k), NM ′ (k ′ ) describes the effect of spontaneous double emission. This term is intrinsically quantum mechanical. 7.4.2 Kinetic equations for two-photon pair creation The kinetic equations for the photons due to two-photon pair creation and annihilation are

310

7 Second order gyromagnetic processes

DNM (k) X = Dt ′ qq

Z

d3 k′ +− ′ − wq′ qMM ′ (k) n+ q nq′ 1 + NM (k) + NM ′ (k ) 3 (2π) − −NM (k)NM ′ (k′ )(1 − n+ (7.4.2) q − nq ′ ) ,

plus an analogous equation with primed and unprimed quantities interchanged. The corresponding kinetic equations for the electrons and positrons are Z 3 ′ X Z dn+ d k d3 k q ′ − + = wq+− ′ qMM ′ (k) NM (k)NM ′ (k )(1 − nq − nq ′ ) 3 3 dt (2π) (2π) q′ ′ − −n+ q nq′ 1 + NM (k) + NM ′ (k ) , Z 3 ′ X Z dn− d k d3 k q′ ′ − + wq+− = ′ qMM ′ (k) NM (k)NM ′ (k )(1 − nq − nq ′ ) 3 3 dt (2π) (2π) q ′ − −n+ (7.4.3) q nq′ 1 + NM (k) + NM ′ (k ) .

In practice one usually considers pair annihilation in the absence of photons (NM (k = 0, NM ′ (k′ ) = 0) and pair creation in the absence of other particles − (n+ q = nq′ = 0). 7.4.3 Double cyclotron emission

Two-photon emission is related to Compton scattering by making the crossing transformation k′ → −k′ , with ω ′ → −ω ′ , e′ → e′∗ , in the probability (7.1.2). This converts the initial photon into a final photon, so that there are two final photons and no initial photon. Consider an electron in an initial state q corresponding to a Landau level n. The electron can jump to lower Landau levels, n′ < n, through gyromagnetic emission of a single photon. However, it can also make the jump emitting two photons simultaneously, with the sum of the frequencies of the two photons being equal to that of the single photon in one-photon emission. These two processes compete, and the ratio of the rates of transition determines the relative probability of the transition occurring due to the two processes. This ratio is referred to as a branching ratio. The transition of most interest is that between the first excited state and the ground state. More generally, the probability for two-photon transition between any excited state, n′ , s′ , and the ground state, n = 0, is obtained from (??) with (??), for s′ = −1, or with (??) for s′ = 1, by making the replacements k → −k, ω → −ω, e∗ → e. The resulting probability corresponds to two-photon absorption by an electron initially at rest (pz = 0). The probability of two-photon emission between the same two states is given by the same expression. However, this corresponds to an inconvenient reference frame, specifically that in which the electron is at rest after emitting the two photons, and one needs to make a Lorentz transformation to derive the probability for two-photon emission in an arbitrary frame. In the nonrelativistic

7.4 Two-photon processes

311

case, which corresponds to |kz′ + kz | ≪ m, the differences between these two probabilities may be neglected. Only this simpler case is discussed in detail here. The resonance that applies when ω ′ satisfies (??) for Compton scattering continues to apply for double absorption. In addition, the crossing of the unprimed photon to the initial state for double absorption implies an analogous resonance for this photon. There are resonances for both ω, ω ′ for double absorption and for double emission. For n′′ = 1 the resonances correspond to either ω ′ or ω nearly satisfying the condition for single-photon gyromagnetic emission or absorption. The resonance then leads to a divergence in the probability, which corresponds to an “infra-red catastrophe” in the sense that it occurs when either ω/ω ′ or ω ′ /ω becomes arbitrarily small. The divergence does not occur in practice for several reasons, including the natural broadening of the transition associated with the lifetime of the excited state due to one-photon gyromagnetic emission. This makes double emission ill-defined when the frequency of the softer photon is less than the natural line width for one-photon gyromagnetic emission. The double transition of most interest is that from the first excited state, n′ = 1, s′ = −1, to the ground state, n = 0, in the nonrelativistic approximation, kz , k ′ z ≪ m, x, x′ ≪ 1. [26] hw0′ 1 (k, −k′ )i =

3 (2π)3 α2c c6 B h(y, −y ′ ) 4 45Ωe Bc y 2 y ′2 ¯h((kz + kz′ ))2 ×δ ω + ω ′ − Ωe − (kz + kz′ )vz + 2m

(7.4.4)

Two-photon emission is an intrinsically quantum effect, in the sense that the branching ratio for double- to single-photon emission is proportional to Planck’s constant. This dependence may be written in terms of the ratio B/Bc (= h ¯ eB/m2 c2 in SI units), so that one expects the branching ratio to be significant only for magnetic fields close to the critical field, B ∼ Bc . A detailed calculation of the branching ratio gives [26] TBA

(7.4.5)

7.4.4 Two-photon pair creation The probability of two-photon pair creation or annihilation is given by (7.1.2) with ε′ = −ε. The kinetic equations for these processes are written down in (7.4.2), (7.4.3). The discussion here of these processes follows the analogous discussion on the corresponding one-photon processes. The absorption coefficient for the photons involved in two-photon pair creation may be defined in an analogous way to that for one-photon pair

312

7 Second order gyromagnetic processes

creation, cf. (6.4.2). After integrating over the distribution of photons in the mode M ′ , the absorption coefficient for photons in the mode M becomes X eB Z dpz Z d3 k′ ′ ′ γM (k) = w+− (7.4.6) ′ ′ (k, k ) NM ′ (k ). 2π 2π (2π)3 q qMM ′ ′ n,n ,s,s

It is helpful to distinguish two cases. One case corresponds to a distribution of hard photons, with ω > ∼ 2m, producing pairs as they propagate through a distribution of soft photon. In this case, one may regard the soft photons are having a given distribution, and consider the absorption coefficient for the hard photons. In (7.4.6), the hard photons are identified with the label M and the soft photons with the label M ′ . The other case consists of a single distribution of hard photons, with ω > ∼ m, that annihilate with each other. In this case the two distributions are the same, M = M ′ , and an additional factor of 1/2 needs to be included, e.g., in the definition of the probability, to avoid double counting. Qualitatively, most of the properties of two-photon pair creation can be understood in terms of the foregoing discussions of the properties of one-photon pair creation and of Compton scattering. For nonrelativistic pairs, there is a sequence of thresholds corresponding to different values of n, n′ . As for onephoton pair creation, the integral over pz in (7.4.6) may be evaluated using the δ-function, to give an expression with a factor Gnn′ in the denominator, cf. (6.4.3). A notable difference between two-photon and one-photon expressions is associated with the fact that Gnn′ depends on the properties of the 2 photons through the invariant Ω 2 − Kz2 , which is ω 2 − kz2 ≈ k⊥ ≈ ω 2 sin2 θ in ′ 2 ′ 2 the one-photon case, and is (ω + ω ) − (kz + kz ) in the two-photon case. In the one-photon case, the absorption coefficient (6.4.3) is singular at the values of ω 2 sin2 θ that correspond to the zeros of gnn′ . As for Compton scattering, the integral over d3 k′ in (7.4.6) includes an integral over dkz′ , and the singular function Gnn′ is integrable. Thus, the singularities are smoothed out in a way that depends on the details of the kz′ -dependence of NM ′ (k′ ) in (7.4.6).

7.5 Electron-ion and electron-electron scattering

313

7.5 Electron-ion and electron-electron scattering Particle-particle scattering is a second order process in QED, In the absence of a magnetic field, Mott scattering provides a simple model for electronion scattering in which the ion is replaced by a fixed charge at the origin, which reduces the problem to a first order process in QED, but there is no such simplification for electron-electron scattering, called Møller scattering in QED. The generalization of these two processes to include a magnetized plasma are discussed in this section. 7.5.1 Collisional excitation by a classical ion In the discussion below emphasis is placed on collisionless processes. The general theory may also be used to treat the effects of collisions between particles. The simplest example is a collision between an electron and an ion, with the ion treated as an unmagnetized classical particle with charge Q and mass M . Two processes are of interest: non-radiative transitions and bremsstrahlung. Collisional excitation or de-excitation of an electron or positron through interaction with an external field is described by the same Feynman diagram for Mott scattering, cf. figure ??. The probability per unit time of an electron initially in the state q being in the state q ′ after interacting with the ion is Z 2 ′ e2 d4 k (Q)∗ wqQ′ q = lim Aµ (k)[Γqǫ′ qǫ (k)]µ 2πδ(ǫεq − ǫ′ εq′ − ω), (7.5.1) 4 T →∞ T (2π)

where T is a normalization time. The 4-potential A(Q) (k) of the classical ion is given by (??), viz. A(Q)µ (k) = eikx0 QDµν (k)uν 2πδ(kU ),

(7.5.2)

where U is the 4-velocity of the ion whose orbit is assumed to be of the from x = x0 + U τ , with x0 , U constant and with τ the proper time. The δ-function in the classical form (7.5.2) appears squared in the probability (7.5.1), where one writes [2πδ(kU )]2 = (T /γ)2πδ(kU ), where γ is the Lorentz factor of the particle. The normalization time, T , cancels with the factor 1/T in (7.5.1). The purely classical result can be generalized to a semi-classical form that takes the recoil of the ion into account. This generalization involves the replacement δ(kU ) →

M δ(E ′ − E − ω), E

(7.5.3)

in (7.5.1), with E = (2 +M P 2 )1/2 , P = Γ M V , Γ = 1/(1 − V 2 )1/2 , E ′ = (M 2 + P ′2 )1/2 , P ′ = P − k. One is free to choose the frame in which the ion is at rest, and then in the Coulomb gauge the 4-potential has only a time-component. This is

314

7 Second order gyromagnetic processes

A(Q)0 (k) = eikx0

Q2πδ(ω) , 4πε0 |k|2 K L (ω, k)

(7.5.4)

which is the electrostatic potential of the ion, with K L (ω, k) the longitudinal part of the dielectric tensor. In this case the (7.5.1) gives Z 2 2 e2 Q2 d3 k ǫ′ ǫ Q 0 |k|2 K L (0, k) 2πδ(ǫεq − ǫ′ εq′ ). wq′ q = [Γ ′ q (k)] q (4πε0 )2 (2π)3 (7.5.5) The δ-function in (7.5.5) can be satisfied only for ǫ′ = ǫ, and then (7.5.5) describes collisional excitation or de-excitation of an electron (ǫ = 1) or a positron (ǫ = −1) as a result of the interaction with the ion. 7.5.2 Probability for electron-ion bremsstrahlung The electron (or positron) may emit a photon as a result of the interaction with the ion. The Feynman diagrams for this form of bremsstrahlung are the same as those for bremsstrahlung in the absence of a magnetic field, cf. figure 5.10. The corresponding probability is Z 2 d4 k ′ ∗ µ0 e4 RM (k) ′ µν ∗ ′ ǫ′ ǫ (k)A (k )[M wq′ qM (k) = lim e ′ q (k, −k )] M ν q T →∞ T |ωM (k)| (2π)4 (7.5.6) ×2πδ(ǫεq − ǫ′ εq′ − ωM k − ω ′ ), ′

with [Mqǫ′ qǫ (k, k′ )]µν given by (??). For sccattering by an ion at rest, the probability (7.5.6) simplifies in an analogous way to the derivation of (7.5.5) from (7.5.1). The resulting probability is 2 Z 3 ′ e∗ (k)[M ǫ′′ ǫ (k, −k′ )]µ0 2 2 M q q d k e Q RM (k) wq′ qM (k) = 2 2 (4πε0 )2 |ωM (k)| (2π)3 |k| K L (0, k) ×2πδ(ǫεq − ǫ′ εq′ − ωM k).

(7.5.7)

For ǫ′ = ǫ (7.5.7) describes bremsstrahlung associated with collisional excitation or de-excitation of an electron (ǫ = 1) or a positron (ǫ = −1) by the ion. For ǫ′ = −ǫ = −1 (7.5.7) describes pair creation by a photon as a result of its passage past the ion. 7.5.3 Electron-electron scattering The foregoing discussion of scattering of an electron by an ion involves a major simplifying assumption: the ion is treated as a classical particle, so that only its Coulomb field need be considered. Scattering of an electron by a real particle requires that the particle be treated quantum mechanically. The processes of interest are electron-electron scattering, electron-positron scattering, and electron-ion scattering.

7.5 Electron-ion and electron-electron scattering

315

The Feynman diagrams for electron-electron scattering are the same as in the absence of a magnetic field, cf. figure 5.8. It is convenient to denote the initial states of the electrons by 1, 2 and the two final states by 3, 4. In full, 1 denotes a set of quantum numbers q1 = (p1z , p1y , n1 , s1 , and similarly for 2, 3, 4. It is then convenient to describe the scattering as 1, 2 → 3, 4. The two different diagrams, for given initial states 1, 2, are related by the final states of the two electrons being interchanged, 3 ↔ 4. The scattering amplitude for this process involves four vertex functions: for one diagram the vertex (−k)]ν appear, where k is the 4-momentum trans(k)]µ , [Γq++ functions [Γq++ 2 q4 1 q3 fer. It is convenient to simplify the notation by writing these as [Γ13 (k)]µ , [Γ24 (−k)]ν , respectively. For the other diagram the vertex functions that appear are [Γ14 (k′ )]µ , [Γ23 (−k′ )]ν , where k ′ is the 4-momentum transferred. This simplified notation is also applied to the energies, εq1 (p1z ) → ε1 , and so on. One is usually not interested in the location of the centers of gyration of the electrons, and the average over the initial positions and the sum over the final positions are performed. The average transition rate then becomes dp3z dp4z 2πδ(p1z + p2z − p3z − p4z ) 2πδ(ε1 + ε2 − ε3 − ε4 ) w ¯i→f = e4 2π Z 2π d4 k × 2πδ(p1z − p3z + kz ) 2πδ(ε1 − ε3 + ω) (2π)4 2 µ ν × Γ31 (−k) Γ42 (k) Dµν (k) Z d4 k 2πδ(p2z − p3z + kz ) 2πδ(ε2 − ε3 + ω) + (2π)4 2 µ ν × Γ32 (−k) Γ41 (k) Dµν (k) Z Z 2π d4 k d4 k ′ − 2πδ(p2z − p3z + kz ) 2πδ(ε2 − ε3 + ω) 4 eB (2π) (2π)4 ′ ×2πδ(p1z − p3z + kz′ ) 2πδ(ε1 − ε3 + ω ′ ) ei(k×k )z /eB µ ν ∗ α β × [Γ32 (−k) Γ41 (k) Dµν (k) [Γ31 (−k′ ) Γ42 (k′ ) Dαβ (k ′ ) + c.c. ,

(7.5.8)

where “c.c.” denotes complex conjugate. A case of particular interest is scattering of an electron initially in its ground state into an excited final state. This is relevant for cyclotron emission from the thermal plasmas in the polar cap of X-ray pulsars. A problem in understanding such cyclotron emission is the origin of the cyclotron photons. In such a strong magnetic field, the electrons all relax rapidly to their ground state, and the only possible source of cyclotron photons is some non-radiation process that excites the electron to a higher Landau level, when the resulting gyromagnetic emission produces cyclotron photons. This case was considered by Langer (1983) for the Johnson-Lippmann wavefunctions and by Storey

316

7 Second order gyromagnetic processes

& Melrose (1987) for the magnetic moment eigenfunctions. The transition probability in the latter case is α2c 2π dp3z dp4z 2πδ(p1z + p2z − p3z − p4z ) 2πδ(ε1 + ε2 − ε3 − ε4 ) eB 2π 2π ε1 + m ε2 + m (ε3 + ε03 )(ε03 + m ε4 + m 1 × 2ε1 2ε2 4ε3 ε03 2ε4 n! Z ∞ Z ∞ Z ∞ n −2x x e (xy)n/2 e−(x+y) 2 2 (R + Rb ) − 2 Ra Rb , dx 2 dx dy 2 (x + Q2 )2 a (x + Q2 )(y 2 + Q2 ) 0 0 0 (7.5.9) w ¯i→f =

where the electrons labeled 1,2,4 are in their ground state and that labele 3 is in its nth Landau level with ε03 = (m2 + 2neB)1/3 . In deriving (7.5.9), the photon propagator is assumed to have its vacuum form, Dµν (k) ∝ 1/k 2 . The functions Ra , Rb depend on the spin of the electron, with Ra A31 A42 − B31 B42 , = Rb nsf A32 A41 − B32 B41 (2neB)1/2 B31 A42 − A31 B42 Ra = , Rb sf B32 A41 − A32 B41 ε0n + m Aab = 1 +

pza pzb , (εa + ε0a )(εb + ε0b )

Bab =

pzb pza + , εa + ε0a εb + ε0b

(7.5.10)

with ε0a = m for a = 1, 2, 4, and where non-spin-flip (nsf) transition corresponds to the 3-state having s = −1 and the spin-flip (sf) transition corresponds to the 3-state having s = 1. The probability for scattering of an electron by an ion in the general case where the ion is treated quantum mechanically is closely analogous to the probability (7.5.8) for electron-electron scattering. One notable change is that there is only one Feynman diagram: the two diagrams for electron-electron scattering ensure that interchanging the two final electrons does not change the probability, whereas this is obviously not the case for an electron and an ion. Starting from (7.5.8) with only the first term inside the square brackets retained, one simply replaces the vertex function for say electrons 2 and 4 by vertex functions for the ions. If the ion is a fermion, its vertex function has the same form as for an electron, with the obvious changes in the mass and charge. If the ion is a boson one uses the appropriate vertex function derived in chapter 9. 7.5.4 Trident process A crossed process related to electron-electron scattering is the trident process, in which there is an electron in the initial state, and two electrons and a positron in the final state. This corresponds to gyromagnetic emission, with the emission being that of a pair rather than a photon.

References

317

References 1. 2. 3. 4. 5. 6. 7.

Schwinger A.A Sokolov, I.M. Ternov Synchrotron radiation, Pergamon Press, Oxford A.A Sokolov, I.M. Ternov Radiation from relativistic electrons, AIP, New York O.F. Dorofeyev, A.V. Borisov, V.Ch. Zhukovsky, Chapter 7 in V.A. Bordovitsyn (ed.) Synchrotron Radiation Theory and Its Development, World Scientific, Singapore, p. 347 (1999) N.P. Klepikov ZETP 26, 19 (1954) T.K. Daugherty, A.K. Harding Astrophys. J. 273, 781 (1983) R. Lieu, W.I. Axford Astrophys. J. 416, 700 (1993) D. White Phys. Rev. D, 9, 868 (1974) A.A. Sokolov, V. Ch. Zhukovskii, N.S. Nikitina, Phys. Lett, 43a, 85 (1973) J.S. Toll, PhD thesis, Princeton University (1952) W. Tsai, T. Erber, Phys. Rev. D 10, 92 (1974) T. Erber, Rev. Mod. Phys., 38, 626 (1966) A.K. Harding, Astrophys. J. 300, 167 (1986) M.G. Baring, MNRAS 235, 51 (1988)

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. J.K Daugherty, A.K. Harding, Astrophys. J. 309, 362 (1986) 20. A.K. Harding, J.K Daugherty, Astrophys. J. 374, 687 (1991) 21. R.L.Gonthier, A.K. Harding, M.G. Baring, R.C. Costello, C.L. Mercer, Astrophys. J. 540, 907 (2000) 22. C.D. Dermer, Astrophys. J. 360, 197 (1990) 23. Q. Luo, Astrophys. J. 468, 338 (1996) 24. R.J. Stoneham, Optica Acta, 27, 537 (1980) 25. R.J. Stoneham, Optica Acta, 27, 545 (1980) 26. 27. H. Herold, Phys. Rev. D 19, 2868 (1979) 28. K.A. Milton, W. Tsai, L.L DeRead, Jr, N.D. Hari Dass, Phys. Rev. D 10, 1299 (1974)

8 Magnetized vacuum

The magnetized vacuum has dispersive properties similar to a material medium. TBC

320

8 Magnetized vacuum

8.1 Linear response of the magnetized vacuum The linear response for a magnetized vacuum is referred to as the vacuum polarization tensor. TBA: historical review [1], As in the absence of a magnetic field, the expression for the vacuum polarization tensor is derived from the amplitude for the bubble diagram. This amplitude diverges and must be regularized to derive a physically meaningful expression for the response tensor. Two different forms of the vacuum polarization tensor are derived in this section. One derivation involves using the G´eh´eniau form (5.3.13) of the propagator for a magnetized electron. The other derivation is based on either the vertex formalism or the Ritus method, and involves a sum over particles states. A generalization of this method allows one to include the response of an electron gas, as described in detail in §9.1. 8.1.1 Unregularized tensor: G´ eh´ eniau form The vacuum response tensor in coordinate space follows from the amplitude for the bubble diagram and has the general form Π µν (x − x′ ) = −ie2 Tr [γ µ G(x, x′ )γ ν G(x′ , x)].

(8.1.1)

The response tensor in momentum space is obtained by Fourier transforming (8.1.1). The propagators may be written in the G´eh´eniau form (5.3.13), that is, G(x, x′ ) = −φ(x, x′ )∆(x − x′ ), where φ(x, x′ ) is a phase factor depends separately on x, x′ . In (8.1.1) the phase factors cancel. In the resulting expression, Π µν (x − x′ ) = −ie2 Tr [γ µ ∆(x − x′ )γ ν ∆(x′ − x)],

(8.1.2)

with ∆(x) given by (5.3.14). the trace over the Dirac matrices is dµν (λ, λ′ , x) = Tr [γ µ B(λ, x)γ ν B(λ′ , −x)],

(8.1.3)

with B(λ, x) given by (5.3.15) Z Z m2 1 1 e4 B 2 ∞ dλ ∞ dλ′ µν ′ d (λ, λ , x) exp −i + Π µν (x) = −i (4π)4 0 λ 0 λ′ 2 λ λ′ i ieB 2 eB eB × exp (x + y 2 ) cot + cot ′ + (λ + λ′ )(z 2 − t2 ) . (8.1.4) 4 2λ 2λ 2 It is convenient to change the variables of integration in (8.1.4) to 1 1 eB eB 1 eB eB 1 + ′ , β= − ′ , , λ′ = , λ= α= 2 λ λ 2 λ λ α+β α−β (8.1.5)

8.1 Linear response of the magnetized vacuum

321

and to Fourier transform before evaluating the trace in (8.1.3). The Fourier transform of (8.1.4) involves integrals that can be reduced to the form Z ∞ π 1/2 ib2 ±iaη 2 +ibη , (8.1.6) dη e = (1 ± i) exp ∓ 2a a −∞

and derivatives of such integrals. Specifically, writing Z 2 2 µ µ [J(k), J⊥ (k), Jk (k)] = d4 x[1, xµ⊥ , xµk ]ei[kx−a⊥ (x )⊥ −ak (x )k ] ,

(8.1.7)

and so on, using (8.1.6) one finds 2 i(k 2 )k i(k )⊥ iπ 2 + exp J(k) = , a⊥ ak 4a⊥ 4ak

(8.1.8)

µ µ with J⊥ (k) = −i∂J(k)/∂k⊥µ = (k⊥ /2a⊥ )J(k), Jkµ (k) = −i∂J(k)/∂kkµ = µ (kk /2ak )J(k). Integrals with higher powers of x⊥ , xk in the integrand are evaluated in the same manner, corresponding to the replacements

xµ⊥ →

µ k⊥ , 2a⊥

xµk →

kkµ

2ak

,

a⊥ = −

eB sin α , 2 cos α − cos β

ak =

eBα . α2 − β 2 (8.1.9)

The Fourier transform of (8.1.4) gives Z Z e2 m2 ∞ dα α α2 − β 2 k 2 α µν µν Π (k) = + i dβ d (k, α, β) exp − i (2π)2 0 α 0 L 4αL m2 2 2 2 α −β cos α − cos β k⊥ i + + , (8.1.10) 2L 2α sin α m2 with L = B/Bc , Bc = m2 /e. The tensor dµν (k, α, β) is defined by the replacements (8.1.5), (8.1.9 in (8.1.3). Specifically, in (8.1.3) one makes the replacement (α + β)(γk)k B(λ, x) k⊥ α+β α+β → − γ 1 cot + γ2 − iΣz , cot eB 4ak 4a⊥ 2 2 (8.1.11) with (γk)k = γ 0 ω − γ 3 kz , and with γ 1 , γ 2 the components of γ along eµ1 = [0, k⊥ /k⊥ ], eµ2 = [0, b × κ], respectively. The replacement for B(λ′ , −x) is given by (8.1.11) with k → −k, β → −β. Symmetry properties for the tensor dµν (k, α, β) can be inferred by inspection of (8.1.3), (8.1.4). The only part of the tensor that contributes to the integral (8.1.4) is symmetric and is an even function of k and β: dµν (k, α, β) = dνµ (k, α, β) = dµν (−k, α, β) = dµν (k, α, −β).

(8.1.12)

The tensor is also nongyrotropic: this implies that in a frame with B along the 3-axis and k in the 1-3 plane, the components d02 , d12 , d32 and d20 , d21 , d23

322

8 Magnetized vacuum

are all zero, where the arguments k, α, β are omitted. The response tensor must satisfy the charge-continuity and gauge-invariance conditions, and only the part of dµν that do not satisfy these conditions is to be retained. All these conditions are satisfied by retaining only the three invariant components for an appropriately chosen set of basis vectors. 8.1.2 Invariant components of the tensor A convenient choice of basis 4-vectors is the set introduced in (1.1.19): bµ1 = f µα kα = (0, ky , −kx , 0), bµ3 =

bµ2 = φµα kα = (kz , 0, 0, ω),

µα 2 2 2 (ωk⊥ , kx (k 2 )k , ky (k 2 )k , kz k⊥ ) k 2 g⊥ kα + k µ k⊥ = , 2 2 k k

(8.1.13)

2 with bµ4 = k µ = (ω, kx , ky , kz ), with k⊥ = kx2 + ky2 , (k 2 )k = ω 2 − kz2 . It is convenient to write

fiµν =

bµi bνi , b2i

f0µν =

3 X bµ bν i=1

i i b2i

= g µν −

kµ kν . k2

(8.1.14)

The physically acceptable components of the vacuum polarization tensor can be written in terms of three invariants, Πi (k), either with i = 1, 2, 3 or with i = 0, 1, 2. The latter is more convenient because the invariant Π0 (k), after regularization, must reduce to the unmagnetized vacuum polarization tensor in the limit B → 0. Thus the physically relevant part of the tensor is written in the form Π µν (k) =

2 X i=0

Πi (k)fiµν ,

f0µν =

3 X bµ bν i=1

i i b2i

= g µν −

kµ kν . k2

(8.1.15)

The unregularized form for Πi (k) follows from (8.1.10) and (8.1.15): Z Z e2 m2 ∞ dα α α α2 − β 2 k 2 Πi (k) = dβ d (k, α, β) exp −i +i i 2 (2π) 0 α 0 L 4αL m2 2 2 i α − β2 cos α − cos β k⊥ , (8.1.16) + + 2L 2α sin α m2 P2 with di defined by writing dµν = i=0 di fiµν . The calculation, which involves taking the trace in (8.1.3) with (8.1.11), gives cos β β tan β k 2 , d0 (k, α, β) = 1− 2 sin α α tan α m2 2 cos α − cos β β tan β cos β k⊥ d1 (k, α, β) = 1− , + 3 2 2 sin α α tan α m sin α 2 β tan β cos β ω 2 − kz2 α − β2 1 − d2 (k, α, β) = . (8.1.17) − 2α2 tan α 2 sin α α tan α m2

8.1 Linear response of the magnetized vacuum

323

The integral diverges in the limit α → 0 for d0 , but not for d1 , d2 . A physical result is obtained only after regularizing Π0 (k) to remove this divergence. 8.1.3 Regularization of the response tensor Regularization of the polarization tensor for the magnetized vacuum is based on the fact that the introduction of an external field into QED does not introduce any additional divergences. An unregularized expression for the tensor may be regularized by subtracting its zero-field limit from it, and adding the known regularized, zero-field vacuum polarization tensor to it [2, 7, 3, 1, 8] The regularization procedure involves subtracting the limit B → 0 from the di and adding the known zero-B expression. In taking the limit B → 0, it is helpful to incorporate L = B/Bc into new variables of integration α/L, β/L, before taking the limit L → 0. One finds that Π1 (k), Π2 (k) are non-divergent and do not require regularization. Regularization of Π0 (k) gives Z ∞ Z e2 k 2 α dα α α2 − β 2 k 2 reg Π0 (k) = dβ exp −i + i (2π)2 0 α 0 L 4αL m2 2 2 β tan β cos β i α − β2 cos α − cos β k⊥ × 1− exp + 2 sin α α tan α 2L 2α sin α m2 1 β2 − , (8.1.18) 1− 2 2α α The regularized, unmagnetized, vacuum polarization tensor needs to be added to (8.1.18) to obtain the full expression for reg Π0 (k). 8.1.4 Weak-field and strong-field limits In the weak field approximation, the power series expansions of the functions di in (8.1.17) converge rapidly. On retaining only the leading terms, the integrals to be evaluated are elementary, and one finds Π0 (k) e2 B 2 = − , k2 90π 2 Bc2

Π1 (k) 4e2 B 2 = , 2 k⊥ 45π 2 Bc2

Π2 (k) 7e2 B 2 = . (8.1.19) ω 2 − kz2 45π 2 Bc2

In the strong field limit B ≫ Bc , one has Π0 (k), Π1 (k) ∼ O exp(−B/Bc ), and Π2 (k) may be approximated by ( 2 e3 B −k⊥ /2eB −1 e Π2 (k) = 2π 2 s ) ω 2 − kz2 4m2 arctan +p . (8.1.20) 4m2 − (ω 2 − kz2 ) [4m2 − (ω 2 − kz2 )]1/2 (ω 2 − kz2 ) The result (8.1.20) simplifies to Π2 (k) ≈ (e2 /12π 2 )(B/Bc )(ω 2 −kz2 ) for 4m2 ≪ 2 ω 2 − kz2 , k⊥ ≪ 2eB.

324

8 Magnetized vacuum

8.1.5 Unregularized tensor: vertex formalism An alternative procedure for calculating the vacuum polarization tensor is based on the amplitude of the bubble diagram derived using the vertex formalism. This procedure has a straightforward generalization to include the response of an electron gas, as discussed in §9.1. The vacuum polarization tensor follows from the more general form (9.1.6) for the response tensor by setting the occupation numbers to zero. The resulting expression for unregularized vacuum polarization tensor is ′ µ ′ ∗ν Z 1 dpz 2 (ǫ′ − ǫ) Γqǫ′ qǫ (k) Γqǫ′ qǫ (k) e3 B X µν Π (k) = − , (8.1.21) 2π 2π ω − ǫεq + ǫ′ εq′ + i0 ′ ′ ǫ,q,ǫ ,q

Here, the sets of quantum numbers q ′ , q denote the spins, s′ , s, and the Landau ′ levels, n′ , n. sum over The µν s, s in (8.1.21) is of the form (5.4.23), which defines ǫ′ ǫ the tensor πn′ n (pz , k) , given explicitly by (5.4.24). Conservation of parallel momentum requires ǫ′ p′z = ǫpz − k, which is implicit in (8.1.21). Consider the hermitian part of (8.1.21). (The antihermitian part is discussed separately below.) The tensor (8.1.21) is unchanged by the relabeling ǫ′ ↔ −ǫ, n′ ↔ n, p′z ↔ pz . Using these symmetry properties, (8.1.21) gives ǫ µν Z ¯n′ n (pz , k) e3 B X dpz ǫ π µν Π (k) = , (8.1.22) 2π 2π ω − ǫ(εn + ε′n′ ) ′ ǫ,n,n

µν ǫ with p′z = kz − pz . The components of πn−ǫ that do not cancel ′ n (pz , k) follow from (5.4.24): µν ǫ 1 µν n−1 g (Jn′ −n+1 )2 + (Jnn′ −n−1 )2 π ¯n′ n (pz , k) =− ′ 2εn′ εn ⊥ µ ′ν n−1 2 n 2 ν + ε′n′ εn − p′z pz + m2 gkµν − [p′µ k pk + pk pk ] (Jn′ −n ) + (Jn′ −n ) µ ν µ ν n−1 n n , (8.1.23) +pn′ pn gkµν 2Jnn−1 ′ −n Jn′ −n + (e+ e+ + e− e− )Jn′ −n+1 Jn′ −n−1

µ µ ′ ′ ′ with p′µ k = [εn′ , 0, 0, pz , pk = [εn , 0, 0, pz , pz = −pz + ǫkz , and where e± is 2 defined by (5.5.38). The argument x = k⊥ /2eB of the J-functions is omitted in (8.1.23). On writing the tensor in the integrand as aµν +ǫbµν , the sum over ǫ reduces to X (ǫω + εn + ε′n′ )(aµν + ǫbµν ) = 2(εn + ε′n′ ) aµν + 2ω bµν . ǫ

8.1.6 Unregularized invariant components: vertex formalism TBA projecting onto the fiµν , giving

8.1 Linear response of the magnetized vacuum

Z e3 B X dpz ǫω + εn + ε′n′ −ǫ ǫ Πi (k) = π ′ (k) i , 2π 2π ω 2 − (εn + ε′n′ )2 n n ′

325

(8.1.24)

ǫ,n,n

′

′

with i = 0, 1, 2, and with the [πnǫ ′ǫn (k)]i constructed by projecting [πnǫ ′ǫn (k)]µν onto the basis vectors by analogy with (8.1.15). Using the shorthand notation ǫ πi = [πn−ǫ ′ n (k)]i , explicit evaluation gives 2 2 1 k2 1 2 + Jnn′ −n−1 [pn′ + p2n − (ω 2 − kz2 )] Jnn−1 π0 = ′ ′ −n+1 2 2 2 2εn′ εn ω − kz n , +2pn′ pn Jnn−1 ′ −n+1 Jn′ −n−1 π1 =

2 k⊥ 2pn′ pn n−1 π0 − ′ J ′ J n′ , 2 k εn′ εn n −n+1 n −n−1

π2 = −

2 ω 2 − kz2 ε′n′ pz + εn p′z n−1 2 ω 2 − kz2 Jn′ −n + Jnn′ −n . π − 0 k2 ωkz 2ε′n′ εn

(8.1.25)

The resulting expressions for the real parts of the Πi (k) are Z gi e3 B X dpz 1 , Re Πi (k) = − ′ 2 2π 2π εn′ εn ω − (εn + ε′n′ )2 ′ n,n

2

k n−1 n n (ε′ ′ + εn )pn (Jnn−1 ′ −n+1 Jn′ −n + Jn′ −n Jn′ −n−1 ), k⊥ n 2 g0 n + 4(ε′n′ + εn )pn pn′ Jnn−1 g1 = k⊥ ′ −n+1 Jn′ −n−1 , k2 g0 = −

g2 = −(ω 2 − kz2 )

ω 2 − kz2 g0 2 n 2 − (εn p′z − ε′n′ pz )[(Jnn−1 ′ −n ) + (Jn′ −n ) ], k2 kz (8.1.26)

2 where the argument k⊥ /2eB of the J-functions is omitted. The expression (8.1.26) for Π0 (k) diverges, and it needs to be regularized to obtain a physically meaningful result. The regularization procedure used above involves subtracting the B → 0 limit of g0 from g0 , and adding the unmagnetized form for Π0 (k) to the resulting expression. With the form (8.1.26), the sums over n, n′ complicate taking the limit B → 0. An alternative regularization procedure involves evaluating the antihermitian part, and using the KramersKronig relation to construct hermitian part from the antihermitian part.

8.1.7 Antihermitian part of the response tensor The vacuum response tensor is symmetric, and hence its antihermitian part is its imaginary part. The imaginary part of the form (8.1.10) is obtained by replacing the exponent function according to eiy → i sin y. The interpretation

326

8 Magnetized vacuum

of the result is not immediately obvious. The antihermitian part of the form (8.1.21), derived using the vertex formalism, of the response tensor has a straightforward interpretation in terms of one-photon pair creation. The antihermitian part, of (8.1.21) is found by retaining only the imaginary part of the resonant denominator. After summing over s′ , s and using the Plemelj formula (2.1.17), (8.1.21) gives Z e3 B X dpz −+ µν Π Aµν (k) = −iπ π ′ (k) δ(ω − εn + ε′n′ ). (8.1.27) 2π 2π n n ′ n,n

where only ǫ = 1, ǫ′ = −1 contributes, with p′z = −pz + kz , contributes for ω > 0. The antihermitian part is an odd function of ω, and the form for ω < 0 is given by minus the form (8.1.27). Carrying out the integral over the δ-function, for ω > 0 one finds µν e3 B X πn−+ ′ n (k) Aµν , (8.1.28) Π (k) = −i 4π (ω 2 − kz2 )gnn′ ′ n,n

which follows from (8.1.28) with the resonance condition implying the solutions (6.1.19) with (6.1.17), (6.1.18). The interpretation of (8.1.28) in terms of one-photon pair creation is confirmed by showing that it reproduces results derived in §6.4 by starting from the probability (6.4.1) for one-photon pair creation. For an arbitrary wave mode, M , the antihermitian part of the response tensor implies damping, with an absorption coefficient γM (k) = 2i

RM (k) Π A (kM ), ε0 ωM (k) M

(8.1.29)

A with ΠM (kM ) found by projecting Π Aµν (k) onto the polarization vector for µ the mode M , with kM = [ωM (k), k]. The absorption coefficient may be interpreted as the probability per unit time for a photon to be absorbed, and for pair creation, this is the probability per unit time that a pair is created. The latter probability is equivalent to the rate of pair production, written down in (6.4.2). Comparison of (8.1.29) and (8.1.28) with (6.4.2) and (6.4.1)shows that the two results are equivalent, and confirms the interpretation of (8.1.28).

8.1.8 Regularization: vertex formalism TBA The hermitian part of the response tensor is given in terms of the antihermitian part by the Kramers-Kronig relation Z ∞ dω ′ i Hµν Π (ω, k) = − ℘ Π Aµν (ω ′ , k). (8.1.30) ′ π ω − ω −∞

8.1 Linear response of the magnetized vacuum

327

8.1.9 Long-wavelength limit In the long-wavelength limit, k⊥ → 0, kz → 0, the general result (8.1.26) simplifies due to (??) and n−1 n Jν+1 (x) Jνn (x) Jνn−1 (x) Jν−1 (x) = − lim = n1/2 (δν,0 − δν,−1 ), 1/2 1/2 x→0 x→0 x x

lim

εn p′z + ε′n pz (ε0n )2 − p2z = , kz →0 kz2 εn

(8.1.31)

lim

with p′z = kz − pz , (ε0n )2 = ε2n − p2z . In this limit one finds Re Π1 (k) = 0 and e3 Bω 2 Re Π0 (k) = − π

Z

∞ dpz X 1 εn εn+1 + m2 + p2z , 2π n=0 εn εn+1 (εn + εn+1 ) ω 2 − (εn + εn+1 )2

e3 Bω 2 Re Π2 (k) = −Re Π0 (k) − 2π

Z

∞ ε2 − p2z dpz X gn 3 n2 . 2π n=0 εn (ω − 4ε2n )

(8.1.32)

For ω 2 ≪ 4m2 and B ≪ Bc , after an expansion in eB/ε2n , the sum over n may be performed using the Euler-Maclaurin summation formula. Regularization of the resulting expression gives [2] 2 4 e2 ω 2 e2 ω 2 B B e2 ω 4 + + ···, − 2 2 2 2 60π m 90π Bc 105π Bc 2 4 7e2 ω 2 B 13e2 ω 2 B Re Π2 (k) = − + ···. (8.1.33) 180π 2 Bc 630π 2 Bc Re Π0 (k) = −

8.1.10 Strong-B limit In the strong field limit, (8.1.26) is dominated by finds Z Re Π2 (k) e3 B =− ω 2 − kz2 2πkz

2 B ≫ Bc , for k⊥ ≪ m2 , ω 2 − kz2 ≪ m2 , the sum in ′ the term n = n = 0. Retaining only this term, one

dpz 1 ε0 (pz )p′z + ε0 (p′z )pz , 2π ε0 (pz )ε0 (p′z ) ω 2 − [ε0 (pz ) + ε0 (p′z )]2

(8.1.34)

with p′z = kz − pz . Expanding in kz and retaining only the lowest order term, (8.1.34) reduces to e2 B Re Π2 (k) = , (8.1.35) 2 2 ω − kz 12π 2 Bc which reproduces the ω 2 − kz2 ≪ 4m2 limit of (8.1.20).

328

8 Magnetized vacuum

8.2 Schwinger’s proper-time technique Schwinger [4] developed a powerful method, known as the proper-time technique, for treating QED processes in a static electromagnetic field. This theory not only provides an alternative method for treating QED processes in the magnetized vacuum, but also includes the effect of a static electric field. In particular, the proper-time technique leads to a generalization of the propagator G´eh´eniau form (5.3.13) for the electron propagator, to an arbitrary static field F µν , and this allows one to generalize the vacuum polarization tensor to include a static electric field. Schwinger also used the proper-time technique to derive a general of the Lagrangian for the electromagnetic field, which includes nonlinear corrections resulting from the interaction of the electromagnetic and Dirac fields. The generalized form, referred to as Heisenberg-Euler Lagrangian, was first derived in the 1930s [?, ?]; the original derivation by Weisskopf [?] is outlined at the end of this section. 8.2.1 Proper-time method In the proper-time technique the propagator, G(x, x′ ), is regarded as the maˆ |xi, of an operator, G, ˆ between space-time states, |xi, and trix element, hx′ | G ′ ˆ |x i that are eigenfunctions of the space-time operator, xˆ. The operator G satisfies (/ Π − m)G = 1, Π = p + eA, (8.2.1) where the hats on operators are omitted for simplicity. Equation (8.2.1) is chosen such that its matrix element, between |xi and |x′ i, gives the correct differential equation for the propagator: (i/∂ − e/ A(x) + m)G(x, x′ ) = δ(x − x′ ).

(8.2.2)

The solution of (8.2.1) is written G=

Π / +m = i(/ Π + m) Π / 2 − m2

Z

0

∞

2 ds exp i(/ Π − m2 )s].

(8.2.3)

The parameter s is (8.2.3) is intepreted as a proper time. The essence of the proper-time technique is to identify an equivalent dynamical system that determines how the operators x and Π evolve as a function of s. This equivalent dynamical system is identified by writing say G(x′ , x′′ ) as the matrix elements between eigenstates |x(0)′′ i and hx(s)′ | of the operator x(s), which is assumed to evolve with the parameter s. Let the Hamiltonian of this equivalent dynamical system be H, which is a function of the coordinate operator, x, and the momentum operator, Π, as well as of the parameter s. Assuming that s plays the role of a time, the evolution of the states in the Schr¨odinger picture leads to the requirement hx′ | exp[−iHs] |x′′ i = hx(s)′ |x(0)′′ i.

(8.2.4)

8.2 Schwinger’s proper-time technique

329

In effect, the operator exp[−iHs] corresponds to the S-matrix for the system, with t → s, t0 → 0, HI → H. The notation in (8.2.4) reflects the fact that x(s) is to be identified as x′′ for s = 0 and as x′ for arbitrary s. By inspection of (8.2.3), a choice of the Hamiltonian is H =Π / 2 = Π 2 + eS µν Fµν ,

Π µ = i[∂ µ − ieAµ (x)].

(8.2.5)

For a uniform static electromagnetic field, F µν , Hamilton’s equations for the operators x and Π give dΠ µ dxµ = −i[xµ , H] = 2Π µ , = −i[Π µ , H] = 2eF µν Πν , (8.2.6) ds ds respectively. Schwinger wrote these equations in the symbolic form dx/ds = 2Π, dΠ/ds = 2eF Π, and wrote down the solutions e2eF s − 1 Π(0). (8.2.7) eF The commutation relations [xµ (0), Π ν (0)] = −ig µν are written symbolically as [x(0), Π(0)] = −i. The commutation relation of the second of (8.2.7) with x(0) implies e2eF s − 1 . (8.2.8) [x(s), x(0)] = i eF The essential step in linking the proper time formalism to the propagator is to note that the factor involving Π / 2 → H in the integrand in (8.2.3) may be written as a matrix element of the position operator at different proper times: Z ∞ 2 G(x′ , x′′ ) = i(/ Π + m) ds e−im s hx(s)′ |x(0)′′ i 0 Z ∞ 2 =i ds e−im s [hx(s)′ |/ Π(s)|x(0)′′ i + mhx(s)′ |x(0)′′ i]. (8.2.9) Π(s) = e2eF s Π(0),

x(s) − x(0) =

0

The evaluation of the propagator is thereby reduced to the evaluation of the matrix element in the integrand of (8.2.9). This is determined by dhx(s)′ |x(0)′′ i = hx(s)′ |H|x(0)′′ i, ds hx(s)′ |Πµ (s)|x(0)′′ i = [i∂µ′′ − eAµ (x′ )]hx(s)′ |x(0)′′ i.

(8.2.10)

The relevant solutions are φ(x′ , x′′ ) −L(s) 1 i(x′ −x′′ )eF [coth(eF s)](x′ −x′′ ) −ieSF hx(s)′ |x(0)′′ i = −i e e4 e , 16π 2 s2 hx(s)′ |Π(s)|x(0)′′ i = 21 [eF coth(eF s) − eF ] (x′ − x′′ ) hx(s)′ |x(0)′′ i, sinh eF s , (8.2.11) L(s) = 12 tr ln eF s

with φ(x′ , x′′ ) given by (5.3.4), SF = S µν Fµν , and where tr denotes the trace over the 4-tensor indices, with tr F = 0 due to F µν being anitsymmetric.

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8 Magnetized vacuum

8.2.2 Propagator in a static magnetic field The form (8.2.9) with (8.2.11) for the propagator depends on the electromagnetic field, F , in a manner that can be made explicit by expressing the result in terms of the invariants S, P , defined by (1.1.30).To illustrate the steps involved in the calculation, it is convenient to start with a magnetostatic field, that is, for E = 0, S = −B 2 /2, P = 0. To show that Schwinger’s method reproduces the result (5.3.12), it is conveninet to start from the first form of (8.2.9). In a magnetostatic field the Maxwell tensor is of the form F µν = Bf µν ,

µν g⊥ = −f µ α f αν ,

µν gkµν = g µν − g⊥ .

(8.2.12)

The quantity L(s) in (8.2.11) is determined by the eigenvalues of F µν : two of the four eigenvalues are zero and the other two are ±iB. After summing over the contributions from each of these eigenvalues, one has e−L(s) = eBs/ sin(eBs). The next factor in (8.2.11) involves a 4-tensor eF s coth(eF s), which is defined formally in terms of the expansion in a power series in the 4-tensor eF s. In the case of a magnetostatic field, the Maxwell tensor and its powers are confined to the two-dimensional x-y space, whose metric tensor is µν g⊥ . The invariant xeF s [coth(eF s)]x separates into a trivial part, x2k , and a part (xeF s [coth(eF s)]x)⊥ that is evaluated by summing over the eigenvalues F → ±iB. Thus one finds xeF s [coth(eF s)]x = x2k + x2⊥ eBs cot(eBs).

(8.2.13)

One uses (8.2.13) in (8.2.11) with x → x′ − x′′ and (x′ − x′′ )2k = (t′ − t′′ )2 − (z ′ − z ′′ )2 , (x′ − x′′ )2⊥ = −(x′ − x′′ )2 − (y ′ − y ′′ )2 . The final factor in (8.2.11) is exp(−ieSF ), with S µν Fµν = −σ · B, that is S µν Fµν = −σz B for B along the z axis. Hence one has exp[−ieSF s] = exp[ieσz Bs] = cos(eBs) + iσz sin(eBs).

(8.2.14)

Combining these factors, (8.2.11) reproduces the propagator (5.3.12) with 1/2λ interpreted as Schwinger’s proper time, s. 8.2.3 Propagator for an electromagnetic wrench The generalization to an arbitrary static electromagnetic field, that is, to an electromagnetic wrench, involves assuming the invariant P 6= 0. In general the eignevalues, F ′ , of the electromagnetic field, F µ ν , satisfy det [F µ ν − F ′ δνµ ] = F ′4 − 2SF ′2 − P 2 = 0.

(8.2.15)

2 The four eigenvalues are F ′ = ±f± , with f± = S ± (S 2 + P 2 )1/2 and hence

8.2 Schwinger’s proper-time technique

f± =

1

21/2

(S + iP )1/2 ± (S − iP )1/2 .

331

(8.2.16)

In the general case of an electromagnetic wrench, one has e−L(s) =

(es)2 P , Im cosh(esX)

xeF [coth(eF s)]x = x2k eE coth(eEs) + x2⊥ eB cot(eBs), tr exp[−ieSF s] = 4Re cosh(esX),

(8.2.17)

where tr denotes the trace over the 4-tensor indices, and where one has X 2 = (B + iE)2 = −2S + 2iP,

(8.2.18)

with S, P defined by (1.1.30). The final result for the propagator is G(x′ − x′′ ) = φ(x′ , x′′ ) ∆(x′ − x′′ ), Z 2 −e2 EB ∞ e−im s ∆(x) = ds 16π 2 sinh(eEs) sin(2Bs) 0 # " −iσeBs eE(γx)k e eB(γx)⊥ e−αeEs + + m e−αeEs e−iσeBs × 2 sinh(eEs) 2 sin(eBs) " # ieEx2k ieBx2⊥ × exp − . (8.2.19) − 4 tanh(eEs) 4 tan(eBs) The gauge dependent factor, φ(x′ , x′′ ), is defined by (5.3.4), and for the choice Aµ (x) = (0, 0, Bx, Et),

(8.2.20)

it has the explicit form ′

′′

φ(x′ , x′′ ) = eie[E(t +t

)(z ′ −z ′′ )−B(x′ +x′′ )(y ′ −y ′′ )]/2

.

(8.2.21)

8.2.4 Nonlinear terms in the Lagrangian density Schwinger [4] used the proper time technique to rederive and generalize the Heisenberg-Euler Lagrangian. This is achieved by considering the energy assocaited with a variation, δA, in the electromagnetic field. This energy is Z Z δW (1) = d4 x δAµ (x) hJ µ (x)i = δ d4 x LI (x), (8.2.22) where LI (x) is the nonlinear correction to the Lagrangian and hJ µ (x)i is the vacuum expectation value of the current density. The calculation of hJ µ (x)i using the proper time technique involves first writing it in the form

332

8 Magnetized vacuum

hJ µ (x)i = ie tr γ µ hx|G|xi,

(8.2.23)

where G is the operator introduced in (8.2.1). An operator δA is also introduced such that its diagonal matrix elements give the variation in the electromagnetic field: hx|δAµ |x′ i = δ 4 (x − x′ ) δAµ (x).

(8.2.24)

Then using (8.2.3) and the identity −eδA = δ(/ Π + m), the variation (8.2.22) is written in the form Z ∞ 2 i ds −im2 s iΠ e e / s , δW (1) = ieTr γ δA G = δ 2 0 s

(8.2.25)

(8.2.26)

where Tr denotes summation over the diagonal terms with respect to both the Dirac spinor indices and the space-time coordinates. To within an additive constant, the Lagrangian density is identified as Z i ∞ ds −im2 s (1) / 2s L (x) = e hx|eiΠ |xi. (8.2.27) 2 0 s The proper time technique gives hx| exp(i/ Π 2 s)|xi = hx(s)|x(0)i, and an ex(1) plicit expression for L (x) follows from (8.2.11): Z ∞ ds −im2 s −L(s) 1 (1) e e tr eieSF s . (8.2.28) L (x) = 2 32π 0 s3 The total Lagrangian is obtained by subtracting the divergent term for F = 0, and also the terms of first order in F 2 , and adding the Lagrangian, S/µ0 , for the Maxwell field. The final result for the Lagrangian is S L(x) = µ0 Z ∞ ds −im2 s 1 Re cosh(esX) 2(es)2 S 2 e − 2 (es) P − 1 + , (8.2.29) 8π 0 s3 Im cosh(esX) 3 Z ∞ S 1 Re cosh(esX) 2(es)2 S ds −im2 s 2 L(x) = , − 2 e (es) P −1+ µ0 8π 0 s3 Im cosh(esX) 3 (8.2.30) with X 2 = (B + iE)2 , as in (8.2.17). The variable of integration interpreted as the proper time, s. The result (8.2.30) is Schwinger’s generalization of the Heisenberg-Euler expression. The lowest order terms in the expansion of (8.2.30) in powers of the fields are

8.2 Schwinger’s proper-time technique

L=

S e4 e6 + (4S 2 + 7P 2 ) + (8S 3 + 13SP 2 ) 2 4 µ0 360π m 630π 2 m8 e8 + (48S 4 + 88S 2 P 2 + 19P 4 ) + · · · . 945π 2 m12

333

(8.2.31)

The nonlinear terms may be rewritten in terms of the fine structure constant, αc = e2 /4πε0 m = µ0 e2 /4πm, and the critical magnetic field, Bc = m2 /e. For a magnetostatic field, the nonlinear terms are proportional to αc /µ0 times (B/Bc )4 , (B/Bc )6 , (B/Bc )8 , and so on. 8.2.5 Weisskopf ’s Lagrangian The earliest derivation of the nonlinear Lagrangian for the vacuum, by Weisskopf [?], was based on Dirac’s hole theory. The presence of a static electromagnetic field, F0µν , modifies the energy of the virtual particles, compared with the vacuum in the absence of the static field. It follows that the vacuum energy density for F0µν 6= 0 is different from that for F0µν = 0. Suppose that the additional vacuum energy is UV (F0 ), with UV (0) = 0 for F0 = 0. The change in the Lagrangian density is numerically equal to minus this energy density. The nonlinear corrections to the Lagrangian are included in LI = −UV (F0 ),

(8.2.32)

and these nonlinear terms give the Heisenberg-Euler Lagrangian. In Dirac’s theory the vacuum consists of empty positive-energy states and filled negative-energy states, with a positron identified as a hole in the negative-energy sea. The vacuum energy consists of the contributions from all the filled negative energy states. Inclusion of a static magnetic field changes the energy eigenstates and hence changes the vacuum energy. The energy eigenvalues are ±(m2 + p2z + 2neB)1/2 , with the state n = 0 being nondegenerate and the states n ≥ 1 being doubly degenerate. The vacuum energy for the filled negative energy states is identified as UV = −

eB 2π

Z

∞ dpz X gn (m2 + p2z + 2neB)1/2 2π n=0

(8.2.33)

where the degeneracy factor is g0 = 1, gn = 2 for n ≥ 1. The sum over n in (8.2.33) is performed using the Euler-Maclaurin summation formula Z ∞ X 1 ∞ 1 F (nb) = 2 [F (0) + F (∞)] + dx F (x) b 0 n=0 +

∞ i X B2k b2k−1 h (2k−1) F (∞) − F (2k−1) (0) , (2k)! k=1

(8.2.34)

334

8 Magnetized vacuum

where the Bn are the Bernouilli numbers, with B2 = 1/6, B4 = −1/30, and with F (n) (x) denoting the nth derivative of F (x). In this way one finds Z ∞ Z 1 dpz UV = − dx (m2 + p2z + x)1/2 2π 2π 0 ∞ X B2k (2eB)2k d(2k−1) (m2 + p2z + x)1/2 , (8.2.35) − (2k)! dx2k−1 x→0

k=1

The vacuum energy is divergent and needs to be regularized. Regularization of (8.2.35) requires that the divergent term independent of B be omitted, and that the term proportional to B 2 be replaced by the known correct value for the Lagrangian of the Maxwell field. The nonlinear terms that remain are Z ∞ 1 dpz X B2k b2k−1 d(2k−1) (m2 + p2z + x)1/2 reg UV = . (8.2.36) 2π 2π (2k)! dx2k−1 x→0 k=2

The integral over pz gives Z

dpz d2k−1 (m2 + p2z + x)1/2 Γ (2k − 2) = . 4(k−1) 2π dx2k−1 4πm x→0

The integral representation of the Γ -function, Z ∞ Γ (n) = dη η n−1 e−η ,

(8.2.37)

(8.2.38)

0

allows one to rewrite (8.2.36) with (8.2.37) in the form reg UV =

m4 8π 2

Z

dη

2k ∞ e−η X 22k B2k ηeB . η3 (2k)! m2

(8.2.39)

k=2

It is convenient to replace the variable of integration according to η → m2 η. After evaluating the sum in (8.2.39) explicitly, one obtains Z 1 dη −m2 η LI = − 2 e ηeB coth(ηeB) − 1 − 13 η 2 e2 B 2 . (8.2.40) 3 8π η

The result (8.2.40) is for a magnetostatic field. Weisskopf [?] evaluated the Lagrangian including an electrostatic field by assuming the electric field to be spatially periodic and allowing the wavelength to go to infinity. In the limit where the spatial period of the field is arbitrarily long, the result corresponds to (8.2.40) with B → iE.

8.3 Vacuum with a homogeneous electrostatic field

335

8.3 Vacuum with a homogeneous electrostatic field The theory of the QED in the vacuum with a homogeneous magnetostatic field may be extended to include the effects of a homogeneous electrostatic field, E 0 . In the presence of a homogeneous electrostatic field, the theory becomes intrinsically time dependent. Free electrons are systematically accelerated by the electric field, and the distribution function for such electrons is necessarily time dependent; this precludes generalizing the description of the response of an electron gas to E 0 6= 0. The response of the vacuum is attributed to virtual electrons, and this problem does not arise directly. However, it does arise indirectly due to their being a nonzero probability that a virtual pair can become a real pair by tunneling through the distance required to gain the rest energy from the potential electric field. As a consequence, a homogeneous electrostatic field is intrinsically unstable to decay into pairs. In this section, the electron propagation derived using proper-time technique is used to generalize the derivation of the polarization tensor in §9.5 to include an electrostatic field. The linear and nonlinear response tensors for ω ≪ m are also derived using the Heisenberg-Euler Lagrangian. Spontaneous pair creation E 0 6= 0 is treated both for B 0 = 0 and B 0 6= 0. 8.3.1 Response tensor for an electromagnetic wrench A derivation of the vacuum response tensor for an electromagnetic wrench is analogous to the derivation in §9.5 for a magnetostatic field with the propagator (8.2.19) with S 6= 0, P 6= 0 in place of the G´eh´eniau form (5.3.13), which corresponds to S < 0, P = 0. It is possible to choose a set of basis 4-vectors that generalize the set (8.1.13). The Maxwell 4-tensor and its dual for an electromagnetic wrench may be written   0 0 0 −E  0 0 −B 0   F µν = Bf µν + Eφµν =   0 B 0 0 , E 0 0 0   0 0 0 −B 0 0 E 0  F µν = −Ef µν + Bφµν =  (8.3.1)  0 −E 0 0  , B 0 0 0 respectively, where f µν , φµν are as defined for E = 0 by (1.1.4), (1.1.5) respectively. The resulting (unnormalized) basis 4-vectors, ˜bµi say, are ˜bµ = kα F αµ = Bbµ + Ebµ , 1 1 2

˜bµ = kα F αµ = −Ebµ + Bbµ , 2 1 2

(8.3.2)

However, unlike the magnetostatic case, an electromagnetic wrench, the response tensor is not diagonal with any choice of (real) basis 4-vectors, and the

336

8 Magnetized vacuum

form (??) needs to be generalized to include the off-diagonal terms. The only nonzero off-diagonal terms are along f3µν , with ˜bµ˜bν f˜2µν = 2 2 2 , ˜b 2

˜bµ˜bν f˜1µν = 1 2 1 , ˜b 1

˜bµ˜bν + ˜bµ˜bν 2 1 . f˜3µν = 1 2 2˜b1˜b2

(8.3.3)

The response tensor is of the form, cf. (8.1.15), Π µν (k) =

3 X

α ˜ i (k)f˜iµν ,

(8.3.4)

i=0

with f˜0µν = f0µν defined by (8.1.15). One finds α ˜ 0 (k) = Φ0 k 2 , α ˜ 1 (k) =

2 B 2 k⊥ + E 2 (ω 2 − kz2 ) 2 (B Φ1 + E 2 Φ2 + 2EBΦ3 ), (B 2 + E 2 )2

α ˜ 2 (k) =

2 B 2 (ω 2 − kz2 ) + E 2 k⊥ (E 2 Φ1 + B 2 Φ2 − 2EBΦ3 ), 2 2 2 (B + E )

α ˜ 3 (k) =

2EB[(B 2 − E 2 )Φ3 − EB(Φ1 − Φ2 )] 2 k , (B 2 + E 2 )2

(8.3.5)

with the quantities Φi defined in terms of an integral over functions φi : Z Z α 2 e4 EB ∞ φi e−im α dα Φi = dβ exp (2π)2 0 sin(eBα) sinh(eEα) 0 cosh(eEα) − cosh(eEβ) 2 cos(eBα) − cos(eBβ) 2 2 k⊥ + (ω − kz ) , i 2eB sin(eBα) 2eE sinh(eEα)

φ0 =

(8.3.6)

cos(eBβ) cosh(eEβ) tan(eBβ) tanh(eEβ) 1− , 2 tan(eBα) tanh(eEα)

φ1 = cosh(eEα) φ2 = cos(eBα)

cos(eBα) − cos(eBβ) + φ0 , sin2 (eBα)

cosh(eEα) − cosh(eEβ) − φ0 , sinh2 (eEα)

1 [1 − cos(eBα) cos(eBβ)][1 − cosh(eEα) cosh(eEβ)] φ3 = − 2 sin(eBα) sinh(eEα) + sin(eBβ) sinh(eEβ) .

(8.3.7)

The results (8.3.1)–(8.3.7) also applies to an arbitrary electromagnetic wrench, and includes the special cases of a magnetostaic field (B 6= 0, E = 0) and an electrostatic field (B = 0, E 6= 0).

8.3 Vacuum with a homogeneous electrostatic field

337

8.3.2 Linear and nonlinear response tensors for ω ≪ m The Heisenberg-Euler Lagrangain implies that the vacuum in the presence of a static electromagnetic field has a nonzero linear response, and a hierarchy of nonlinear responses. The Heisenberg-Euler Lagrangian applies only to static field. However, the electromagnetic field in low-energy photons, with ω ≪ m, are effectively static on the characteristic time scale of QED, which the light crossing time over a Compton wavelength. In the presence of low-frequency photons, one may separate the Maxwell tensor, F µν , into the homogeneous field, F0µν , and a radiation field, F1µν . By treating the radiation field as a perturbation in the Heisenberg-Euler Lagrangian, and expansion in F1µν leads to expressions for the linear and nonlinear response tensors for the vacuum for ω ≪ m. The existence of linear and nonlinear responses implies a nonzero interacR tion energy d4 x J(x)δA(x), where J(x) is the 4-current induced by a perturbation, δA(x), in the electromagnetic field. The electromagnetic field is separated into the static background field, and this perturbation, which is expressed in terms of its Fourier transform. The current is then identified as the sum of the linear and nonlinear responses to this perturbation: J µ (k) = Π µν (k)Aν (k) ∞ Z X + dλ(n) Π (n)µν1 ...νn (−k, k1 , . . . , kn )Aν1 (k1 ) . . . Aνn (kn ), n=2

(8.3.8)

where the convolution differential dλ(n) is defined by (1.2.6). The interaction energy is identified as Z Z d4 k µ J (k) δAµ (−k), (8.3.9) d4 x J(x)δA(x) = (2π)4 with J µ (k) identified as the response (8.3.8). One then has Z Z 1 d4 k µ 4 d x J(x)δA(x) = δ J (k) Aµ (−k)Aν (k) 2 (2π)4 Z ∞ X 1 dλ(n) Π (n)ν0 ν1 ...νn (k0 , k1 , . . . , kn ) + n + 1 n=2 (8.3.10) ×Aν0 (k0 )Aν1 (k1 ) . . . Aνn (kn ) , where the factor 1/(n + 1) arises from the dependence of the nth term on the right hand side on the (n + 1)th power of the wave amplitude, and due to the symmetry of the response tensors under arbitrary permutations of the suffices 0, . . . , n.

338

8 Magnetized vacuum

The interaction energy is also related to the Lagrangian of the field, Z Z d4 x J µ (x)δAµ (x) = δ d4 x L(x), (8.3.11) with L(x) → LI identified as the Heisenberg-Euler Lagrangian (8.2.30). The comparison involves separating F µν into a static part, and a fluctuating part determined by δA, expressing the latter in terms of its Fourier transform, and expanding in powers of the fluctuating field. This leads to the identifications µ0 Π µν (k) = kα kβ

∂ 2 LI , ∂Fαµ ∂Fβν

(8.3.12)

for the linear response tensor, and (−i)n+1 α0 α1 ∂ n+1 LI k0 k1 . . . knαn . n! ∂F α0 ν0 F α1 ν1 . . . F αn νn (8.3.13) for the nonlinear response tensors. The linear and nonlinear response tensors derived in this way apply only at low frequencies and long wavelengths, which corresponds to |ωi | ≪ m, |ki | ≪ m for all i = 0, . . . n. Π ν0 ν1 ...νn (k0 , k1 , . . . , kn ) =

8.3.3 Derivation from the Heisenbeg-Euler Lagrangian The low-frequency (ω ≪ m) limit of the vacuum polarization tensor may be derived using the Heisenbeg-Euler Lagrangian The procedure involves the following steps. First, the total field, including the induced field, is identified as ∂L F µν + M µν = , (8.3.14) Gµν = µ0 ∂Fµν with L identified as the Heisenberg-Euler Lagrangain The invariants involved are S = − 14 F µν Fµν = 12 (E 2 − B 2 ),

P = − 14 F µν Fµν = E · B.

(8.3.15)

and their derivatives are ∂S = −F µν , ∂Fµν

∂P = F µν , ∂Fµν

∂F αβ = g µα g νβ − g µβ g να , ∂Fµν

∂F αβ = ǫµναβ , ∂Fµν

(8.3.16)

where the definition F µν = 12 ǫµναβ Fαβ , of the dual is used, with the permutation symbol ǫµναβ completely antisymmetric with ǫ0123 = 1. Next, the Maxwell tensor, F µν , is assumed to consist of two parts: F µν = F0µν + δF µν ,

(8.3.17)

8.3 Vacuum with a homogeneous electrostatic field

where F0µν is the static field and Z d4 k ikx µ ν δF µν (x) = −i e k A (k) − k ν Aµ (k) , 4 (2π)

339

(8.3.18)

describes the wave field. One then expands in powers of δF µν (x). The term linear in δF µν (x) allows one to identify the fourth-rank susceptibility tensor, from which the require linear response tensor follows using (1.5.13): M µν = χµν ρσ δF ρσ ,

χµνρσ =

1 ∂ 2 LI , 2 ∂Fµν ∂Fρσ

µ0 Π µν (k) = 2 kα kβ χαµβν (k) = kα kβ

∂ 2 LI . ∂Fαµ ∂Fβν

(8.3.19)

The general expression for the linear response tensor is rather cumbersome. Using the expansion of the Lagrangian in powers of F0µν /Bc , the leading term and the next order correction to it give e2 4S(k 2 g µν − k µ k ν ) + 4kα F αµ kβ F βν + 7kα F αµ kβ F βν 45πBc2 2αc + (24S 2 + 13P 2 )(k 2 g µν − k µ k ν ) + 48S kα F αµ kβ F βν , 315πBc4 +26S kα F αµ kβ F βν − 26P kα F αµ kβ F βν + kα F αµ kβ F βν . (8.3.20)

µ0 Π µν (k) =

In a frame in which the the electric and magnetic fields are parallel and along the 3-axis and k is in the 1-3 plane, the leading terms in response tensor (8.3.20) have the form    2 −k⊥ − kz2 −ωk⊥ 0 −ωkz 2 2   e2  2(B 2 − E 2 ) −ωk⊥ −ω + kz 0 2 −k⊥ kz  µ0 Π µν =    2 0 0 −k 0 45πBc 2 2 −ωkz −k⊥ kz 0 −ω + k⊥  2 2  2 0 k⊥ kz EB −ωkz E kz E   0 0 0 0  +4  2 2  k⊥ kz EB 0 k⊥ B −ωk⊥ EB  −ωkz E 2 0 −ωk⊥ EB ω 2 E 2   kz2 B 2 0 −k⊥ kz EB −ωkz B 2   0 0 0 0  . +7  (8.3.21) 2 2  −k⊥ kz EB 0 k⊥ E ωk⊥ EB  −ωkz B 2 0 ωk⊥ EB ω2B2

It is of interest to note that the terms in the 3-tensor components of (8.3.21) include an electric response and magnetic response and a magneto-electric response, which is the component along f3µν , cf. (8.3.3). If the response is written in terms of the polarization, P , and the magnetization, M : P /ε0 =

340

8 Magnetized vacuum

χ(e) · E + χ(em) · B, µ0 M = χ(m) · B + χ(me) · E, where the χ are susceptibility 3-tensors. The terms proportional to ω 2 are included in χ(e) , the terms indpendent of ω 2 are included in χ(m) and the terms proportional to ω are included in χ(em) and χ(me) . It follows that the inclusion of a component of E along B introduces a magneto-electric response. In the absence of an electrostatic field (E = 0), (8.3.20) describes the response of a magnetized vacuum. In this case, comparison with (8.1.13) leads to the identifications 2 2 µν B f1 , kα F0αµ kβ F0βν = −k⊥

kα F0αµ kβ F0βν = −(ω 2 − kz2 )B 2 f2µν . (8.3.22) Then (8.3.20) is of the form (8.1.15), leading to the identifications 2αc B 2 µ0 Π0 (k) =− , 2 k 45πBc2

µ0 Π1 (k) 4αc B 2 , = 2 k⊥ 45πBc2

µ0 Π2 (k) 7αc B 2 = , 2 2 ω − kz 45πBc2 (8.3.23)

which is equivalent to (8.1.19). 8.3.4 Spontaneous pair creation An electrostatic field and an electromagnetic wrench are intrinsically unstable to decay into pairs. A simple physical model is for a static electric field, E, along the x axis, with potential Φ = −Ex. Consider a virtual pair at x = 0. Quantum mechanical tunneling implies a nonzero probability of the electron and positron appearing as real particles after tunneling a distance such the eΦ exceeds the rest energy m, that is, with a separation > 2m/eE. The probability of pair creation is identified by noting that probability amplitude for a system whose action integral is A is exp(iA), that is exp(iA/¯h) in ordinary units, so that the probability of the system remaining in its initial state decays as | exp(iA)|2 = exp(−2ImA). Provided that the probability of decayR is small, the probability of decay is 2ImA ≪ 1. The action is given by A = d4 x L(x). The probability of decay per unit time and per unit volume is identified as wSPC = 2ImL(x), (8.3.24) where SPC denotes spontaneous pair creation, with L(x) identified as the Heisenberg-Euler Lagrangian. The imaginary part of the Heisenberg-Euler Lagrangian, in the form (8.2.30), arises from the term in the integrand involving the function cosh(esX). On writing X = a + ib, one has (es)2 P

Re cosh(esX) cosh(eas) cos(ebs). = (eas)(ebs) Im cosh(esX) sinh(eas) sin(ebs)

(8.3.25)

The singularities of the function (8.3.25) occur where (ebs) cot(ebs) = ∞, which occurs at s = sn , with

8.3 Vacuum with a homogeneous electrostatic field

sn =

πn , eb

n = 1, 2, . . . .

341

(8.3.26)

The imaginary parts are determined by applying the Landau prescription to each of these singularities. This gives wSPC

∞ a X 1 e2 m2 = 2 ab exp −nπ coth nπ . 4π n eb b n=1

(8.3.27)

Further evaluation of (8.3.27) requires explicit values for a, b, with X = a+ ib. From the definition (8.2.18), one has X 2 = (B − iE)2 = B 2 − E 2 + 2iE · B. In the case of an electrostatic field, B = 0, one may choose a = 0, b = E. Then (8.3.27) gives [4] wSPC

∞ e2 E 2 X 1 m2 = . exp −nπ 4π 2 n=1 n2 eE

(8.3.28)

For an electromagnetic wrench, choosing the frame in which the electric and magnetic fields are parallel, one has a = B, b = E. Then (8.3.27) gives [?] wSPC

∞ B e2 EB X 1 m2 coth nπ . = exp −nπ 4π 2 n=1 n eE E

(8.3.29)

The probability wSPC is zero for a magnetostatic field, which does not decay, and nonzero for an electrostatic field or an electromagnetic wrench, which do decay. The probability (8.3.28) involves an exponential factor with exponent ∝ −Ec /E, with Ec = m2 /e the electric counterpart of the critical magnetic field Bc . In virtually all contexts one has E ≪ Ec and the derivation of (8.3.27) is valid only if this condition is satisfied. An exception where spontaneous pair creation is of interest and where this condition is not satisfied is for a model of bare strange stars (composed of u, d and s quarks) where there is a surface layer with an electric field ≫ Ec [?]; in this case the surface layer contains degenerate electrons with a sufficiently highly relativistic Fermi energy that strongly suppresses spontaneous pair creation through the Pauli exclusion principle. Although the Heisenberg-Euler Lagrangian allows one to determine the rate of decay of an electric field, it does not determine the spectrum of the pairs produced. For an electrostatic field, the spectrum of the pairs is an ill-defined concept because each electron and positron is subject to ongoing linear acceleration. For an electromagnetic wrench, the solutions of Dirac’s equation factorized into a part describing the motion in the t-z plane and a part describing the motion in the x-y plane, where E and B are assumed along the z axis. The solutions in the x-y plane may be described in terms of the Landau levels. Each pair produced must have the same eigenvalues as the vacuum, which requires that the electron and positron be in the same Landau

342

8 Magnetized vacuum

level with the same spin. However, the approach based on the HeisenbergEuler Lagrangian provides no information on the distribution of the pairs in these Landau levels. (Note that integer n in the sum in (8.3.29) is not the Landau level.)

8.4 Waves in superstrong fields

343

8.4 Waves in superstrong fields The magnetized vacuum is birefringent. For weak fields, B ≪ Bc , the properties of the two natural wave modes are well known, having been derived in the 1930s [?, ?]. These properties are rederived here and generalized to include the case where the magnetic field is not necessarily weak. The high-frequency limit (ω ≫ m) and the generalization to an electromagnetic wrench are also discussed. 8.4.1 Weak-field, weak-dispersion limit The well-know properties of waves in a birefringent vacuum are written down 2 in ??, specifically, the dispersion relations, in the forms k 2 = k± , |k|2 /ω 2 = 2 n± , are given by (??), (??), respectively, and the polarization vectors by (??). These results apply only for weak fields, B ≪ Bc , and low frequencies, ω ≪ m. The usual labelling of the modes is as the ⊥ and k modes, but unfortunately there are two different conventions. The more widely used convention is to define these in terms of the direction of the electric vector in the wave relative to the projection of the magnetostatic field on the plane orthogonal to k. The alternative convention, used notably [6], is to define the ⊥ and k modes in terms of the direction of the magnetic vector in the wave relative to the projection of the magnetostatic field on the plane orthogonal to k, and this interchanges the labelling ⊥↔k compared with the more widely used convention. The k and ⊥ modes were denoted the the 2- and 3-modes by [3]. 8.4.2 Wave properties in an arbitrarily strong field When the magnetostatic field is not weak, the properties of the wave modes are determined by the wave equation with the regularized vacuum response tensor given by the forms derived in §9.5. With the response tensor written in the form (8.1.15), the wave equation, Λµν (k)Aν (k) = 0 with Λµν (k) = k 2 g µν − k µ k ν + µ0 Π µν (k), gives 2 (k + µ0 Π0 (k))f0µν + µ0 Π1 (k)f1µν + µ0 Π2 (k)f2µν Aµ (k) = 0, (8.4.1)

with f0µν = g µν − k µ k ν /k 2 . The explicit forms (8.1.16), (8.1.17) for the in2 , variants in (8.4.1) imply that Π0 (k), Π1 (k), Π2 (k) are proportional to k 2 , k⊥ 2 2 ω − kz , respectively. The constants of proportionality depend implicitly on k, and a useful approximation is to evaluated these constants of proportionality in the weak-dispersion limit k 2 = 0. This involves writing µ0 Π0 (k) = χ0 k 2 ,

2 µ0 Π1 (k) = χ1 k⊥ ,

µ0 Π2 (k) = χ2 (ω 2 −kz2 ), (8.4.2)

and assuming that the χi are independent of k. The forms (8.4.2) are inserted into the wave equation (8.4.1), and the invariants are written in the form

344

8 Magnetized vacuum

2 k 2 = ω 2 (1 − n2), k⊥ = ω 2 n2 sin2 θ, ω 2 − kz2 = ω 2 (1 − n2 cos2 θ). This procedure allows one to solve for the wave properties without making the weak dispersion approximation. Choosing the temporal gauge, so that the polarization 4-vector, eµ ∝ µ A (k), reduces to eµ = [0, e], and choosing the same coordinate system such that the magnetic field is along the z-axis and k is in the x-z plane, the wave equation reduces to    ex a0d  0 b 0   ey  = 0, d0c ez

a = (1 − n2 cos2 θ) (1 + χ0 ),

c = (1 − n2 sin2 θ) (1 + χ0 ) + χ2 ,

b = (1 − n2 )(1 + χ0 ) + n2 sin2 θ χ1 , d = n2 sin θ cos θ (1 + χ0 ).

(8.4.3)

The dispersion equation is given by setting the determinant of the matrix of coefficients in (8.4.3) to zero, and this factors into the dispersion relations b = 0 and ac − d2 = 0. These apply to the ⊥ and k modes, respectively. The resulting wave dispersion relations are n2⊥ =

1 + χ0 , 1 + χ0 − χ1 sin2 θ

n2k =

1 + χ0 + χ2 , 1 + χ0 + χ2 cos2 θ

(8.4.4)

and the corresponding polarization 3-vectors are e⊥ = a,

ek =

(1 + χ0 + χ2 ) cos θ, 0, −(1 + χ0 ) sin θ

[(1 + χ0 + χ2 )2 cos2 θ + (1 + χ0 )2 sin2 θ]1/2

.

(8.4.5)

The ⊥-mode is strictly transverse, with e⊥ = a = (0, 1, 0). The k mode has a longitudinal component, which is exhibited by writing ek =

K2 κ + T2 t , (K22 + T22 )1/2

Kk = χ2 cos θ sin θ,

Tk = 1+χ0 +χ2 cos2 θ. (8.4.6)

8.4.3 Weak field limit In the weak-field limit the χi are small, of order B 2 /Bc2 . The power series expansions of the functions di in (8.1.17) then converge rapidly. On retaining only the leading order terms, the integrals, cf. (8.1.10), are elementary, and one finds, cf. (8.1.19), µ0 e2 B 2 µ0 e2 B 2 6 B4 12 B 4 χ0 = − , χ = , − − 1 90π 2 Bc2 7 Bc4 45π 2 Bc2 7 Bc4 7µ0 e2 B 2 26 B 4 χ2 = − . (8.4.7) 180π 2 Bc2 49 Bc4

8.4 Waves in superstrong fields

345

To lowest order in an expansion in the χi the dispersion relations (8.4.4) are independent of χ0 and reduce to n2⊥,k = 1 + χ1,2 sin2 θ. On inserting the values (8.4.7) into (8.4.4) the wave properties are reproduced. The next highest order terms in the expansion in B/Bc lead to the following corrections to the dispersion relations (??): 4αc B 2 12 B 4 sin2 θ, n2⊥ = 1 + − 45π Bc2 7 Bc4 7αc B 2 26 B 4 sin2 θ, (8.4.8) n2k = 1 + − 45π Bc2 49 Bc4 where correction terms of order α2c are neglected. As already noted, the polarization vector for the k mode has a longitudinal component of order B 2 /Bc2 determined by (??). 8.4.4 Strong field limit The strong field limit corresponds to B ≫ Bc . In this case (??) implies that Π0 (k) and Π1 (k) are small, being of order exp(−B/Bc ), and Π2 (k) is ap2 proximated by (8.1.20). For 4m2 ≪ ω 2 − kz2 , k⊥ ≪ 2eB (8.1.20) reduces to (8.1.35), giving χ2 ≈ (µ0 e2 /12π 2 )(B/Bc ), and then one has [7] CHECK ORIGINAL REFERENCE αc 3 ln(2B/Bc ) 2 sin2 θ, n⊥ = 1 + 1− 3π 2B/Bc  α B  1 + c − 1.9 sin2 θ for χ2 ≪ 1, 3π Bc n2k =   1 for χ2 ≫ 1. (8.4.9) cos2 θ Thus, in the limit B ≫ (3π/e2 )Bc = 2.5 × 1013 T, the modes become MHD modes with Alfv´en speed vA ≫ 1. The ⊥ and k modes are the counterparts of the magnetoacoustic and Alfv´en modes, respectively. 8.4.5 Dispersion in an electromagnetic wrench The inclusion of an electrostatic field with a component along the magnetostatic field an electromagnetic wrench) modifies both the dispersion relations and the polarization vectors compared with the case E · B = 0. In the weak-field, weak-dispersion limit, the response tensor (8.3.21), derived from the Heisenberg-Euler Langrangian, leads to the following wave properties given by (??). In this limit the polarization vectors are transverse, but unlike the case E = 0, cf. (??), they are neither perpendicular (to both B and k) nor parallel (to the projection of B on the plane orthogonal to k). For E 6= 0 the polarization vectors (??) are linear combinations of the

346

8 Magnetized vacuum

perpendicular and parallel polarizations. The polarization vectors are determined by the 3-vectors that arise in the problem, and these are the 3-vector components of kα F αµ and kα F αµ . For E 6= 0 these are given by (8.3.2) and it is apparent that they are not along the perpendicular and parallel directions, whereas they for E = 0 and also for B = 0, E 6= 0. The labelling of the modes as ⊥ and k is somewhat inappropriate for E/B 6= 0 because neither mode is strictly perpendicular or parallel. These labelling are retained here to make explicit the link to the modes in the limit E/B 6= 0 of most interest. In the more general case when the magnetic field is not necessarily weak, and the electric field is weak and satisfies E ≪ B, wave dispersion in an electromagnetic wrench may be treated by expanding the response tensor (8.3.4)–(8.3.7) in powers of E/B. In this case the labelling as the ⊥, k modes could be misleading: the ⊥ mode is polarized along the parallel (to E) direction and the k is polarized along the perpendicular direction in the limit B/E → 0. 8.4.6 High-frequency, weak-field limit The specific results derived so far for the wave dispersion apply for frequencies well below the pair-creation threshold, ω 2 sin2 θ ≪ 4m2 . The response tensor in the form (8.3.4)–(8.3.7) applies both below and above this threshold and so may be used to derive the wave properties at high frequencies. The dispersion at sufficiently high frequencies necessarily approaches the vacuum value k 2 = 0, and an appropriate simplifying assumption is to assume the weak-dispersion limit, when evaluating the response tensor. Denoting the values in the weak-dispersion limit by an asterisk, the φi that appear in (8.3.7) reduce to Z e4 1 χ∗i = − dη (1 − η 2 ) φ∗i 4π 0 Z ∞ (1 − η 2 )2 Π 3 e2 (E 2 + B 2 )ω 2 sin2 θ × dαα cos m2 α + 48 0 2 2 1 η η + φ∗1 = 1 − E2 + B2, 3 2 6 1 η2 η2 1 − η2 ∗ 2 + EB, (8.4.10) φ2 = E + 1− B2, φ∗3 = − 2 6 3 2 with η = β/α. The α-integral in (8.4.10) is performed in terms of the derivative, Gi′ (z), of a generalized Airy function Z 1 ∞ Gi (z) = dt sin(zt + 31 t3 ). (8.4.11) π 0 The integral in (8.4.10) then gives

8.4 Waves in superstrong fields

χ∗i =

e4 (2ξ 2 )2/3 m4

Z

1

dη

0

2

ξ=

347

φ∗i Gi′ ([2/ξ(1 − η 2 )]2/3 ), (1 − η 2 )1/3

ω(E + B 2 )1/2 sin θ . 2mBc

(8.4.12)

The limiting forms of Gi′ are Gi′ (z) = −

1 πz 2

for z ≫ 1,

Gi′ (z) =

Γ (2/3) 2π31/3

for z ≪ 1.

(8.4.13)

On using (8.4.13) to approximate (8.4.12), the upper limit applies for ξ ≪ 1 and leads to the wave properties (??), derived using the weak field limit of the Heisenberg-Euler Lagrangian. This implies that the wave properties (??) also apply for ω 2 sin2 θ ≪ 4m2 . The lower limit in (8.4.13) applies for ξ ≫ 1 and leads to n2⊥ = 1 − 2A

(E 2 + B 2 ) sin2 θ, Bc2

n2k = 1 − 3A

(E 2 + B 2 ) sin2 θ, (8.4.14) Bc2

with A = 3[Γ (2/3)]2 αc /7π 1/2 Γ (1/6)(2ξ 2 )2/3 . The polarization vectors are the same as in (??). According to (8.4.14) the refractive indices at high frequencies is less than unity, whereas they are greater than unity at low frequencies. For the case of a magnetostatic field, [7] noted that the sum rule Z ∞ dω [n⊥,k (ω) − 1] = 0, (8.4.15) 0

requires n⊥,k (ω) < 1 at high frequency, granted n⊥,k (ω) > 1 at low frequency. The noted that the refractive indices depend only ω only in the combination λ = (3/2)(ω/m)(B/Bc ), and that they increase monotonically with λ for λ < 1.2, decrease monotonically for 1.2λ ≤ 24, crossing unity at λ ∼ 24, and then increase towards the limit (8.4.14). The form (8.4.12) describes the real part of χ∗i , and the corresponding imaginary part describes damping due to one-photon absorption. The real and imaginary parts are related by the Kramers-Kronig relations. However, the imaginary part may be obtained by noting that the real and imaginary parts of the Airy functions satisfy the Kramers-Kronig relations. The imaginary part is obtained from (8.4.12) by the replacement Gi′ (z) → −iAi′ (z).

348

8 Magnetized vacuum

8.5 Mass operator for a magnetized electron The presence of a static electromagnetic field modifies the dispersion relation for an electron (or positron), from the form εn = (m2 +p2z +2neB)1/2 , through the dependence of radiative corrections that depend on the field. These modifications are usually expressed in terms of the mass operator, which is calculated in this section for a magnetized vacuum. The most familiar radiative correction is to the magnetic moment of the electron. 8.5.1 Qualitative consequences of g 6= 2 In the absence of radiative corrections the magnetic moment of the electron is µB = e/2m, corresponding to a gyromagnetic ratio g = 2. The radiative corrections to (g − 2)/2 = αc /2π have now been calculated to high order in αc and compared with experiment, leading to the claim that QED is the most accurately tested of all physical theories. The earliest calculations of the radiative correction to (g − 2)/2 predated the development of QED. In particular, Pauli [?] suggested that the effect of g 6= 2 could be treated by modifying the Dirac equation by the inclusion of an additional term: i/∂ + e/ A − m + 21 (g − 2)µB Fµν S µν Ψ (x) = 0. (8.5.1) The additional term in (8.5.1) leads to the Dirac Hamiltonian ˆ =α· p ˆ + eA + βm − β 21 (g − 2)µB Fµν S µν . H

(8.5.2)

The energy eigenvalues corresponding to the Hamiltonian (8.5.2) may be found to lowest order in (g − 2)/2 by operating twice with the Hamiltonian (8.5.2), keeping only the terms of first order in (g −2)/2 and requiring that the eigenvalue of the squared operator be E 2 . The terms of first order in (g − 2)/2 are proportional to the spin operator i h ˆ˜µν = 1 Hγ ˆ 0 Sˆµν + γ 0 Sˆµν H ˆ , (8.5.3) S 2m

ˆ identified as the Dirac Hamiltonian (8.5.2) for g = 2. For a magnetowith H ˆ˜µν is proporstatic field one has Fµν S µν = −σ · B, which implies that Fµν S tional to the component of the magnetic moment operator along the direction of B. The energy eigenvalues for g = 2 are E = ǫεn , and the generalization to g 6= 2 leads to [2] 1/2 E = ǫ [(m2 + 2neB)1/2 + s 12 (g − 2)µB B]2 + p2z ,

(8.5.4)

with s = ±1 the spin eigenvalues. A calculation using QED by Schwinger [4] gave (g − 2)/2 = αc /2π, which confirms that (8.5.4) is correct to first order in the fine structure constant αc .

8.5 Mass operator for a magnetized electron

349

An important qualitative point is that the magnetic moment operator appears naturally through (8.5.3), so that the spin eigenvalue in (8.5.4) is specifically the magnetic moment spin eigenvalue. It was pointed out by [2] that, for a magnetostatic field, the magnetic moment operator is the only spin operator that commutes with the Hamiltonian (8.5.2). The radiative correction causes all other spin operators to precess, so that their eigenvalues are not constants of the motion. For an electron at rest (pz = 0) in its ground state (n = 0, s = −1), (8.5.4) with (g − 2)/2 = αc /2π implies an energy eigenvalue E = m(1 − αc B/4πBc ). Early interest in pulsar physics led to suggestions that this energy could be zero, implying catastrophic spontaneous pair creation in a magnetic field with B/Bc > 4π/αc [?, ?, ?, ?]. However, this is incorrect [?], and the correction term is poorly approximated by αc B/4πBc for B > ∼ Bc . 8.5.2 Reduced mass operator The mass operator in a magnetized vacuum was calculated by Constantinescu [3]. However, inappropriate (Johnson-Lippmann) spin eigenfunctions were used in citeC72, rather than the appropriate (magnetic moment) spin eigenfunctions [11], so that the results are only directly relevant only for an electron at rest (ǫ = 1, pz = 0). A treatment using the appropriate states was given by Parle [11], using a method developed by Ritus [10]. The mass operator in the presence of a magnetic field does not have a simple momentum-space form. Its coordinate space form is the formally the same as in the absence of a magnetic field: −iM(x, x′ ) = −e2 γ µ G(x, x′ )γ ν Dµν (x − x′ ).

(8.5.5)

Following [11] it is convenient to introduce Z ′ ′ Mq′ q (E ′ , E) = d4 x d4 x′ ei(Et−E t ) Vg† (x, n, ǫpz )M(x, x′ )Vg′ (x′ , n′ , ǫ′ p′z ),

(8.5.6) where Vg (x, n, ǫpz ) is the diagonal matrix introduced in (5.5.1), and where q denotes quantum numbers n, ǫpz , g, and similarly for q ′ . An advantage of introducing Mq′ q (E ′ , E) is that it is diagonal in the sense that it is nonzero only for E ′ = E, q ′ = q. Thus one has M(E ′ , E)q′ q = M(E, n, ǫpz ) 2π∆(E − E ′ )

2π δ(ǫpz − ǫ′ p′z ) δnn′ δgg′ . (8.5.7) Lz

The quantity M(E, n, ǫpz ) is identified as a reduced mass operator. Let the reduced propagator (5.5.19) in the absence of particles be referred to as the bare reduced propagator. Using (5.5.19), the bare reduced propagator may be written in the form [Gn (E, Pz )]0 = (/ Πn − m)−1 ,

(8.5.8)

350

8 Magnetized vacuum

with E = P 0 . The reduced mass operator combines with the bare reduced propagator to give the exact reduced propagator in the same way as in the unmagnetized case, where the exact propagator, G, is given by G−1 = G−1 0 − M, with G0 = (/ P + m)−1 . The exact reduced propagator in the magnetized case becomes Gn (E, Pz ) = Π / n − m − M(E, n, Pz )]−1 . (8.5.9)

The energy eigenvalues correspond to poles of the exact propagator, and hence satisfy det Π / n − m − M(E, n, Pz ) = 0, (8.5.10) with E = P 0 , and where ”det” denotes the determinant.

8.5.3 Unrenormalized form of reduced mass operator Convenient forms for the propagators in (8.5.5) are the form (5.3.12) for the electron propagator, and the form Z µ0 g µν ∞ dµ exp(− 21 iµx2 ), Dµν (x) = − (8.5.11) 8π 2 0 for the photon propagator, which is derived using the identity Z ∞ Z i 1 1 d4 k e−ikx = = − dµ exp(− 12 iµx2 ), (2π)4 k 2 + i0 4π 2 x2 − i0 8π 2 0

(8.5.12)

The mass operator, M(P ), is a Dirac matrix, and hence it may be represented in terms of the basis 1, γ µ , S µν , γ µ γ 5 , γ 5 . These terms can appear only combined with an invariant, a 4-vector, an anti-symmetric tensor, a pseudo 4-vector and a pseudo invariant, respectively. The Maxwell tensor, F µν , is an anti-symmetric tensor, and it is convenient to consider only inertial frames in which it corresponds to a magnetostatic field along the z axis. The theory must be invariant under Lorentz transformations along the z axis, and this implies that the only possible ways in which the components of γ µ can appear are in the combinations pn γ 2 , Eγ 0 − Pz γ 3 , and Sµν F µν , which is proportional to Bσz . The only possible combination that involves γ µ γ 5 is (Eγ 0 −Pz γ 3 )Bσz . The mass operator in a magnetostatic field must be a sum of terms of these allowed forms: αc M= mI1 + pn γ 2 I2 + (Eγ 0 − Pz γ 3 )I3 + mσz I4 + (Eγ 0 − Pz γ 3 )σz I5 , 2π (8.5.13) where the Ii = Ii (E, n, Pz ), i = 1–5 are invariants. Explicit forms for the invariants are given in terms of the variables, cf. (5.3.12) and (8.5.11) L= such that one has

eB B = , m2 Bc

w=

m2 , 2λ

u=

µ , µ+λ

(8.5.14)

8.5 Mass operator for a magnetized electron

351

∞ 1 E 2 − ε2n (Pz ) Ii (E, n, Pz ) = L dw du exp −iw u − (1 − u2 ) m2 0 0 n a − ib 2iLw(1−u) 1 Ki , e × 2 a + b2 a + ib Z

Z

a = u tan(Lw) + Lw(1 − u),

b = Lw(1 − u) tan(Lw),

(8.5.15)

with the Ki , i = 1–5, given by K1 = 2a,

K2 = b/ cos(Lw) sin(Lw), K4 = 2ib,

K3 = (1 − u)[a − b tan(Lw)],

K5 = i(1 − u)[b + a tan(Lw)].

(8.5.16)

The mass operator contains singular terms that must be removed through a regularization procedure. The introduction of a magnetic field introduces no intrinsically new singularities, and hence the singular terms in the mass operator for B 6= 0 must be the same as the singular terms that are present in the limit B → 0. The limit B → 0 corresponds to L → 0 in (8.5.15). The convergence of the integrals in this limit was considered by [3].The integral (8.5.15) is divergent for i = 1–3. These divergent terms are those in the mass operator for an unmagnetized electron, and they are removed by the conventional renormalization procedure. The regularization of the integrals (8.5.15) involves subtracting the divergent terms in the limit L → 0, so that the remaining quantities are finite. These regularized terms, which are denoted by Ii′ in the following discussion, necessarily satisfy Ii′ → 0 for L → 0. The terms I4′ , I5 approach zero ∝ L for L → 0, and I1′ , I2′ , I3′ approach zero ∝ L2 for L → 0. 8.5.4 Energy states in a magnetic field Using the regularized form (8.5.13) of the mass operator it is straightforward to evaluate the determinant in (8.5.10). Then (8.5.13) may be written in the form [11] 2 2 ′2 2 ′ 2 ′ 2 E −Pz2 −p′2 E −Pz2 −p′2 n −(m −∆) −4pn ∆ = 0, (8.5.17) n −(m +∆)

where the following quantities are introduced: f1 − f4 f1 − f4 f1 + f4 f1 + f4 ′ 1 1 m = 2m , ∆ = 2m , + − 1 + f5 1 − f5 1 + f5 1 − f5 p′n = pn

f2 , (1 − f52 )1/2

f1,2 =

′ 1 + αc I1,2 , 1 + αc I3′

f4,5 =

′ αc I4,5 . 1 + αc I3′

(8.5.18)

The solutions of (8.5.17) for the energy eigenvalues are 1/2 2 1/2 , E = ǫ (m′2 + p′2 + s∆ + Pz2 n)

(8.5.19)

352

8 Magnetized vacuum

where s = ±1 is the spin eigenvalue corresponding to the magnetic moment operator. Further evaluation of the energy eigenvalues (8.5.19) requires explicit evaluation of the invariants Ii′ , i = 1–5, and only a few special cases seem to have been considered in detail: the weak-field limit and the correction to the ground state energy for arbitrary B. 8.5.5 Weak-field correction to the magnetic moment In the weak-field limit, L = B/Bc ≪ 1, an expansion to first order in B/Bc gives Z 2 Z ∞ αc B αc . (8.5.20) ∆ ≈ im L du u(1 − u) dw e−iuw = m 2π 4π Bc 0 0 When this value of ∆ is inserted in (8.5.19), the result corresponds to Schwinger’s [4] correction to the magnetic moment of the electron, µ = (e/2m)(1 + αc /2π). This correction is included in Pauli’s [?] modified form (8.5.1) of the Dirac equation with g − 2 = αc /2π. The corrected energy eigenvalues are then given by (8.5.4). 8.5.6 Correction to the ground state energy For the ground state, n = 0, s = −1 the correction given by inserting the weak-field approximation (8.5.20) into (8.5.19) is such as to decrease the energy eigenvalue. Jancovici [?] calculated the ground state energy in the limit B/Bc ≫ 1 and found ( ) 2 αc 2B 3 ln E0 = m + m −C − + ··· , (8.5.21) 4π Bc 2 where C = 0.577 . . . is Euler’s constant. Note that the sign correction changes, for the ground state, from negative in (8.5.20) for B/Bc ≪ 1 to positive in (8.5.21) for B/Bc ≫ 1. 8.5.7 Transitions between split levels In the absence of radiative corrections the levels n > 0 as doubly degenerate, and according to (8.5.19) this degeneracy is broken by the radiative correction. A spin-flip transition, from s = 1 to s = −1 at fixed n is then possible, with emission of a photon with energy ω ≈ 2∆ in the frame Pz = 0. The rate of such transitions for ∆ ≪ m is given by [11] 3 2 pn B α6c wspinflip = m , (8.5.22) 8(2π)5 Bc ε2n

where a photon with frequency ω ≈ 2∆ is emitted. This transition rate has the same dependence on B as the transition n → n − 1 without spin flip, but for n = 1 is smaller by a factor ∼ 1015 .

References

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

J. Toll, I.A. Batalin, A.E. Shabad 1971, ZETP 60, 894; JETP 33, 483 D.H. Constantinescu (1972) Nucl. Phys. B36, 121 D.B. Melrose, R.J. Stoneham (1976) Nuovo Cim. 32A, 435 D.B. Melrose, R.J. Stoneham 1977) J. Phys. A 10, 1211 A.E. Shabad (1975) Ann, Phys. 90, 166 W.-Y. Tsai (1974) PRD 10, 2699 P. Bakshi, R.A Cover, G., Kalman (1976) PRD 14, 2532 J. Schwinger 1951, PR 82 , 664 A. Minguzzi 1956, Nuovo Cim. 7, 501 S.L. Adler (1971) Ann. Phys. 67, 599 W.-Y. Tsai, T. Erber (1974) PRD 10 492

353

9 Response of magnetized systems

Exact QPD expressions for the linear, quadratic and cubic response tensors for a relativistic magnetized quantum plasma follow directly from the amplitude for the bubble, triangle and box diagrams, respectively. Compared with their classical counterparts, discussed in §2.2, the QPD expressions include four quantum effects: degeneracy, the quantization of the Landau levels, the spin, the quantum recoil and dispersion associated with one-photon pair creation. Degeneracy can be included in a classical theory by interpreting the distribution function, for example as a Fermi-Dirac distribution. The quantization of the Landau levels is particularly important in strong magnetic fields, with the energy separation between levels determined by the parameter m(B/Bc ). The Landau levels n > 0 are doubly degenerate and the wave functions for the two states depend on the choice of the spin operator, which appears in the vertex function, introduced in §5.4. If the occupation number does not depend on the spin, the electron gas is said to be ‘spin-independent’. The response tensor for a spin-independent electron gas is obtained by summing over the spin states, either directly or indirectly using the Ritus method §5.5. The exact expressions are too cumbersome for most purposes, and one needs to consider various approximate and limiting cases. Important limiting cases include the one-dimensional limit, when all the electrons are in the Landau 2 level n = 0, the limit x = k⊥ /2eB ≪ 1, in which expansion of the J-functions in power series in x converges rapidly, and the completely degenerate limit, in which the occupation numbers are either unity or zero. General expressions for the linear response tensor are written down and discussed in §9.1. The sum over spins is performed explicitly for both spinindependent and spin-dependent electron gas in §9.2. Limiting cases of the linear response tensor are discussed in §9.3. The response tensor is evaluated for a relativistic thermal distribution in §9.4, and for a completely degenerate electron gas in §9.5. Wave dispersion in relativistic quantum magnetized plasma is discussed in §9.6. Nonlinear response tensors for a relativistic magnetized quantum plasma are written down in §9.7. Photon splitting is discussed in §9.8.

356

9 Response of magnetized systems

9.1 Response of a magnetized electron gas General expressions for the linear response tensor for a magnetized Dirac electron gas are written down and discussed in this section. The tensor may be separated into hermitian and antihermitian parts, which describe the reactive and dissipative parts of the response, respectively. The tensor may also be separated into nongyrotropic and gyrotropic parts, which are even and odd, respectively, in the interchange of electrons and positrons. 9.1.1 Calculation of the response tensor There are several alternative methods for calculating the response of a relativistic quantum electron gas in the absence of a magnetic field, and these may be adapted to apply in a magnetic field. Three methods are discussed in §8.4 of volume 1: the forward-scattering method, the Wigner-matrix method and the density-matrix approach. From a formal quantum mechanical viewpoint, these three methods may be interpreted in terms of different pictures, where a picture involves the choice of how to include time dependence. The forward-scattering method is based on the S-matrix approach, also called the Greens function approach, which involves adopting the interaction picture. In the interaction picture the wave functions evolve due to the interaction Hamiltonian, and the operators evolve due to the unperturbed Hamiltonian. The Wigner function in nonrelativistic quantum mechanics is defined in terms of the outer product of the Schr¨odinger wave function with itself. The evolution is in the Schr¨odinger picture, in which all the time dependence is in the wave functions. The Wigner matrix in the relativistic case is a generalization of the nonrelativistic Wigner function; it is a 4 × 4 Dirac matrix defined in terms of the outer product of the Dirac wave function and its adjoint. The evolution of the Wigner matrix is determined by the Dirac equation, which corresponds to choosing the Schr¨odinger picture. The density matrix is defined as an operator constructed from the state function (a vector in a Hilbert space, usually written as a ket) and its adjoint (a bra). The density-matrix method is based on the Heisenberg picture, in which all the time dependence is in the operators. Expressions for the response tensor for a nonrelativistic magnetized quantum electron gas were derived in the 1960s using both the density-matrix method [42, 43] and the Wigner-function method [?]. (An explicit expression for the resulting nonrelativistic form for the response tensor is written down at the end of this section.) The S-matrix approach is adopted here, but it should be emphasized that the three methods are formally equivalent, and the choice of method has no effect on the result. In the density-matrix approach, the unperturbed density matrix describes the background distribution of particles, and is equivalent to the occupation number. The occupation number arises from a statistical average of the number operator, which is the outer product of creation and annihilation operators. The Wigner matrix is defined by analogy with the Wigner function

9.1 Response of a magnetized electron gas q

k

(a) k

q’

k

(c)

(b)

q

k

q’

q

357

q’ k

q

k

q’

Fig. 9.1. Cuts in the upper and lower electron lines in the Feynman bubble diagram (a) produce the forward scattering diagrams (b) and (c), respectively.

in nonrelativistic theory. A conventional definition is Z f (x, p) = d3 y eip·y hψ(x − 12 y)ψ ∗ (x + 12 y)i,

(9.1.1)

where the angular brackets denote a statistical average, and where ψ(x) is the Schr¨odinger wave function. The Schr¨odinger equation is used to derive an equation for the evolution for the Wigner function f (x, p). The Wigner-matrix approach has the advantage of being closely analogous to the classical Vlasov approach, but the disadvantage of involving a relatively unfamiliar form for the unperturbed distribution. The Wigner function (9.1.1) for a nonrelativistic magnetized thermal distribution involves a sum over generalized Laguerre polynomials [?]. The generalizations of both the Wigner-matrix method [?, ?] and the density-matrix method [?] have been used to derive the response tensor in the relativistic magnetized case. Several different forms of the S-matrix or Greens approach have been used to calculate general forms for the response tensor [refs]. The particular form used here is based on the amplitudes for closed loop diagrams and statistically averaged propagators [refs]. In the S-matrix approach, the linear response tensor for a relativistic quantum electron gas in a magnetic field is derived from the Feynman amplitude for the bubble diagram. As discussed in §8.1 of volume 1, the amplitude for the bubble diagram gives the vacuum polarization tensor. By replacing the electron propagators in the bubble diagram by their statistical average over the electron gas, the calculation of the vacuum polarization tensor also includes the response tensor for the electron gas. A related method, that is also discussed here, is based on forward Compton scattering. This corresponds to Compton scattering in the case where the initial and final photons are the same and the initial and final electron states are the same. A complication in the magnetized case, compared with the unmagnetized case, is that the electron propagator in a magnetic field does not have a simple momentum space representation. Two alternative methods are discussed in §5.5 for overcoming this difficulty: the vertex formalism and the Ritus method. The derivation of the linear response tenor (9.1.6) using the vertex formalism starts from the Feynman amplitude for the bubble diagram Fig. 9.1a. The closed loop corresponds to two propagators between two vertices. Applying the rules for the vertex method given in §5.6 for writing down the Feynman

358

9 Response of magnetized systems

′ µ amplitude to the bubble diagram, Fig. 9.1a, there is a factor ie Γqǫ′ qǫ (k) as ǫǫ′ ν sociated with the vertex on the left, and a factor ie Γqq associated ′ (−k) with the vertex on the right. There is a factor iG¯qǫ (E) associated with each (statistically averaged reduced) propagator, given by (5.6.14); if the energy is E in the propagator with the arrow pointing to the right in Fig. 9.1a, it is E ′ = E − ω in the propagator with the arrow pointing to the left in Fig. 9.1a. The loop energy E is undetermined, and one integrates over it. Only the two terms that arise from the resonant part of one propagator and the nonresonant (principal value) part of the other are retained. Taking the resonant part is equivalent to replacing the virtual intermediate state with a real state, which can be represented by cutting the relevant electron line. Cutting the bubble diagram, Fig. 9.1a leads to the diagrams Fig. 9.1b,c, which correspond to forward scattering. Conservation of parallel momentum is included through the factor (2π/eB)2πδ(ǫ′ p′z − ǫpz + kz ), which appears here through (5.6.11). The sums over the states q, q ′ includes integrals over (eB/2π)dpz /2π, (eB/2π)dp′z /2π, which include the undetermined parallel momentum. On carrying out the integral over E one obtains an explicit form for Π µν (k). For a spin-independent electron gas, this procedure combined with the Ritus methods results in an expression for Π µν (k) that has some analogies to the corresponding result in the unmagnetized case, which is given in equation (8.1.2) of volume 1. Before statistically averaging, the result for the unmagnetiized case is Z d4 P Tr [γ µ G(P )γ ν G(P − k)], (9.1.2) Π µν (k) = ie2 (2π)4

with G(P ) = (/ P + m)/(P 2 − m2 + i0), P µ = ǫpµ . The analogous result implied by the Ritus method for the magnetized case is, after statistically averaging, X ie3 B Z dE Z dpz Z dp′ z µν Π (k) = 2πδ(ǫ′ p′z − ǫpz + kz ) 2π 2π 2π 2π ′ ′ ǫ ,ǫ,n ,n ν ′ ¯ ǫn (E, pz )Jnn ¯ǫ × Tr Jnµ′ n (k⊥ ) G (9.1.3) ′ (−k⊥ ) Gn′ (E − ω, pz ) , with Gnǫ (E, pz ) given by (5.6.15), which gives, 1 P/nǫ + m ǫ ǫ ¯ ℘ E − ǫε − iǫ[1 − 2nn (pz )]πδ(E − ǫεn ) , Gn (E, pz ) = 2ǫεn n

(9.1.4)

with P/nǫ = γ 0 ǫεn − γ 2 pn − γ 3 ǫpz . The vertex function is replaced by the Dirac matrix, (5.5.26), viz. µ ′ n Jnµ′ n (k⊥ ) = (−ie−iψ )n −n Jnn−1 ′ −n (x) P+ + Jn′ −n (x) P− γk µ iψ n + − ie−iψ Jnn−1 Jn′ −n−1 (x) P+ γ⊥ , (9.1.5) ′ −n+1 (x) P− + ie

The derivation in terms of forward Compton scattering leads to the form (9.1.9). The two terms ′inside the µνsquare ′ brackets µνin (9.1.9) arise from the scattering amplitudes Mqǫ′ qǫ (k, k′ ) 1 , Mqǫ′ qǫ (k, k′ ) 2 in (7.1.2). Forward scattering corresponds to k ′ = k and q ′ = q, but a more careful relabeling of the

9.1 Response of a magnetized electron gas

359

µν electron states is required. In M (k, k′ ) 1 , the appropriate relabeling is ′ µν q → q ′ , q ′ → q ′ , q ′′ → q, and in Mqǫ′ qǫ (k, k′ ) 2 , the appropriate relabeling is q → q, q ′ → q, q ′′ → q ′ . With this relabeling, the denominators become ±(ω − ǫεq + ǫ′ εq′ ), respectively. The denominators are reinterpreted as propagators, with integrals over dE/2π times 2πδ(E − ǫεq ) and dE ′ /2π times 2πδ(E ′ − ǫ′ εq′ ), respectively, with E ′ = E − ω. These propagators are replaced by their statistical averages, and the calculation proceeds as above.

ǫ′ ǫ q′ q

9.1.2 Explicit forms for Π µν (k) The general expression for the linear response tensor, derived from the amplitude for the bubble diagram using the vertex formalism, is Z Z dp′z dpz e3 B X 2πδ(ǫ′ p′z − ǫpz + kz ) Π µν (k) = − 2π 2π 2π ′ ′ ǫ,q,ǫ ,q 1 ′ 2 (ǫ

′

ν − ǫ) + ǫnǫq − ǫ′ nǫq′ ǫ′ ǫ µ ǫǫ′ × Γq′ q (k) Γqq′ (−k) , (9.1.6) ′ ′ ω − ǫεq + ǫ εq ǫǫ′ ν ǫ′ ǫ ∗ν with Γqq = Γq′ q (k) . The resonance condition is to be imposed by ′ (−k) making the replacement ω → ω + i0 in the denominator. The response tensor (9.1.6) includes the response of the magnetized vacuum, which corresponds to the term 21 (ǫ′ − ǫ) in the numerator. The resulting expression for the vacuum response diverges and a finite result for the vacuum response tensor is derived from it by a regularization procedure, as discussed in §9.5. The vacuum contribution is omitted in the following discussion. To illustrate the intepretation of the sums in (9.1.6), consider the contribution from the terms with occupation number nǫq . This includes electrons (ǫ = 1) and positrons (ǫ = −1) in the state q. For electrons, the contribution involves the electron absorbing and re-emitting (or emitting and re-absorbing) an identical wave quantum, with the electron returning to the state q. The sum over q is over all the electrons for which n+ q is nonzero, and the sum over ǫ includes the contribution of positrons for which n− q is nonzero. Between the aborption and re-emission the electron is in a virtual state with quantum numbers q ′ , and the sum over q ′ is over all possible virtual states. The sum over ǫ′ includes these virtual electron states (ǫ′ = 1) and also virtual positron states (ǫ′ = −1). The quantum numbers are labeled collectively by q = n, s, pz and q ′ = ′ ′ ′ n , s , pz , and the dependence of the occupation number on q is shown explicitly by writing nǫq → nǫns (pz ). The energy eigenstates, εq → εn (pz ) = [m2 + p2z + 2neB]1/2 , are independent of the spin, and it is convenient to use the notation εn (pz ) → εn , εn′ (p′z ) → ε′n′ . (9.1.7) Explicit expressions for the vertex function in (9.1.6) depend on the choice of spin operator. As discussed in §5.4, and the choice of the magnetic momentum operator is the most appropriate, leading to the form (5.4.18), viz.

360

9 Response of magnetized systems ′

′

ǫ′ ǫ(+)

[Γqǫ′ qǫ (k)]µ = (ieiψ )n−n Aq′ q ǫ′ ǫ(−)

−Aq′ q ǫ′ ǫ(±)

Aq′ q

(0)

Jq′ q (k), ǫ′ ǫ(−)

(+)

Jq′ q (k), −iAq′ q ′

= a′ǫ′ s′ aǫs ± a′−ǫ′ s′ a−ǫs ,

′ (−) (0) Jq′ q (k), Bqǫ′ ǫq Jq′ q (k) ,

Bqǫ′ ǫq = a′ǫ′ s′ a−ǫs + a′−ǫ′ s′ aǫs ,

(0)

(±)

′ ′ n Jq′ q (k) = b′s′ bs Jnn−1 ′ −n (x) + s sb−s′ b−s Jn′ −n (x),

′ iψ n Jn′ −n−1 (x), Jq′ q (k) = s′ b′−s′ bs e−iψ Jnn−1 ′ −n+1 (x) ± sbs′ b−s e 1/2 1/2 ε0n + sm εn ± ε0n a± = P± , bs = , 2εn 2ε0n

(9.1.8)

with P± defined by (5.2.13). The response tensor (9.1.6) may be written in a variety of in alternative forms. One form involves only nǫq in the numerator. This form is obtained by rewriting the term proportional to ǫ′ nq′ through the relabeling ǫ′ , q ′ → ǫ, q and ǫ, q → ǫ′′ , q ′′ . Neglecting the vacuum response, (9.1.6) becomes ′ µ ′ ∗ν Z e3 B X dpz ǫ X Γqǫ′ qǫ (k) Γqǫ′ qǫ (k) µν ǫnq Π (k) = − 2π ǫ,q 2π ω − ǫεq + ǫ′ εq′ ǫ′ ,q′ ǫǫ′′ µ ǫǫ′′ ∗ν X Γqq Γqq′′ (k) ′′ (k) − , (9.1.9) ′′ ω − ǫ εq′′ + ǫεq ′′ ′′ ǫ ,q

where ǫ′ p′z = ǫpz − kz , ǫ′′ p′′z = ǫpz + kz are implicit. There are two sets of sums in (9.1.9), over ǫ, q and ǫ′ , q ′ (or ǫ′′ , q ′′ in the second sum inside the square brackets). These sums are over only the discrete quantum numbers, q = n, s and q ′ = n′ , s′ , with the integral over the continuous quantum number pz included explicitly, and with ǫ′ p′z = ǫpz − kz (and ǫ′′ p′′z = ǫpz +kz ) implicit. The sum over ǫ, q is weighted by the occupation number, nǫq , which can be rewritten as nǫns (pz ) to make the dependence on the quantum numbers explicit. The sum over ǫ′ , q ′ is interpreted as a sum over virtual states. A particle with quantum numbers ǫ, n, s, pz contributes to the response through transitions to all intermediate states ǫ′ , n′ , s′ , p′z returning to ǫ, n, s, pz . Note that this includes transitions in which an initial electron becomes a virtual positron before returning to its initial state. This effect has no counterpart in nonrelativistic quantum mechanics or in classical theory. Although such virtual transitions do not contribute to the dispersion in the classical limit, they cannot be ignored entirely when taking the classical limit of (9.1.9): they can contribute to the non-dispersive part. 9.1.3 Antihermitian part of the response tensor The form (9.1.6) of the response tensor is convenient for identifying its antihermitian part. The anithermitian part describes the resonant contributions,

9.1 Response of a magnetized electron gas

361

which correspond to transitions in which the intermediate state is real rather than virtual. There are resonant contributions from each of the four denominators in (9.1.9). Using the Plemelj formula (2.1.17), one finds Z ′ dpz 1 ′ e3 B X [ 2 (ǫ − ǫ) + ǫnǫq − ǫ′ nǫq′ ] Π Aµν (k) = iπ 2π 2π ǫ,q,ǫ′ ,q′ ǫ′ ǫ µ ǫǫ′ ν × Γq′ q (k) Γqq′ (−k) δ(ω − ǫεq + ǫ′ εq′ ), (9.1.10) The sum over ǫ, ǫ′ involves four terms. The term ǫ = ǫ′ = 1 is associated with gyromagnetic absorption by electrons. In this case the primed state is identified as the electron state before absorption, and the unprimed state as the electron state after absorption. The term ǫ = ǫ′ = −1 is associated with gyromagnetic absorption by positrons; the role of the initial and final states is opposite to that for electrons, with the unprimed state now the initial state. The terms ǫ 6= ǫ′ are associated with dispersion due to one-photon pair creation. One-photon pair creation exists in the vacuum, and the presence of electrons or positrons suppresses the vacuum contribution. Specifically, for − ǫ = 1, ǫ′ = −1 the factor in square brackets is −[1 − n+ q − nq′ ], and it implies that one-photon pair creation is suppressed when the state into which the electron or the positron would be created is occupied, in accord with the Pauli exclusion principle. The integral in (9.1.11) is performed over the δ-function using (6.1.32)– (6.1.34), giving Π Aµν (k) = iπ

e3 B 4π 2

X

′

[ 12 (ǫ′ − ǫ) + ǫnǫn (pz± ) − ǫ′ nǫn′ (p′z± )]

ǫ,q,ǫ′ ,q′ ,±

×

ǫ′ ǫ µ ǫǫ′ ν εn± ε′n′ ± Γq′ q (k) ± Γqq′ (−k) ± , 2 − kz ) gnn′

(ω 2

(9.1.11)

with the resonant values given by (6.1.17)–(6.1.19). Specifically, one has ǫpz± = kz fnn′ ± ωgnn′ ,

ǫεn± = ωfnn′ ± kz gnn′ ,

(ε0n )2 − (ε0n′ )2 + ω 2 − kz2 , 2(ω 2 − kz2 ) 2 ω − kz2 − (ε0n − ε0n′ )2 ω 2 − kz2 − (ε0n + ε0n′ )2 , = 4(ω 2 − kz2 )2 fnn′ =

2 gnn ′

(9.1.12)

with (ε0n )2 = m2 + p2n , and with ǫ′ p′z± = ǫpz± − kz , ǫ′ ε′± = ǫε± − ω. The ± subscripts of the vertex functions in (9.1.11) indicate that they are to be evaluated at these resonant values. 9.1.4 Kramers-Kronig relations The Kramers-Kronig relations follow from the requirement that the response be causal, which implies

362

9 Response of magnetized systems

Π µν (ω, k) = i

Z

∞

−∞

dω ′ Π µν (ω ′ , k) . 2π ω − ω ′ + i0

(9.1.13)

In particular, (9.1.13) implies that the hermitian part of the response tensor is related to the antihermitian part by a Hilbert transform, Z ∞ dω ′ i µν Π Hµν (ω, k) = ΠNR (ω, k) − ℘ Π Aµν (ω ′ , k), (9.1.14) ′ π −∞ ω − ω µν where ΠNR (ω, k) is an undetermined part whose Hilbert transform is zero. µν The usefulness of the result (9.1.14) is limited by the fact that ΠNR (ω, k) is undetermined by the method.

9.1.5 Gyrotropic and nongyrotropic parts The Onsager relations, cf. §1.4 of volume 1, allow one to separate the response tensor into nongyrotropic and gyrotropic parts. In terms of the components of the response tensor in a magnetic field, the Onsager relations imply Π 00 (ω, −k) |−B = Π 00 (ω, k) |B ,

Π 0i (ω, −k) |−B = −Π i0 (ω, k) |B ,

Π ij (ω, −k) |−B = Π ji (ω, k) |B .

(9.1.15)

The gyrotropic terms change sign under k, B → −k, −B, whereas the nongyrotropic terms do not. With B along the 3-axis, and with k defining the vector eµ1 = [0, k⊥ /k⊥ ] and with eµ2 = f µν e1ν , the gyrotropic terms are the 02, 12- and 23-components, and their negatives, the 20-, 21- and 32-components, respectively. The response tensor in the forms (9.1.6) and (9.1.9) satisfies the Onsager relations, as may be shown using the properties of the vertex function. An explicit form for the vertex function for the magnetic-moment eigenstates is given by (9.1.8): the change k → −k implies e±iψ → −e±iψ and kz → −kz , which preserves ǫ′ p′z = ǫpz − kz . The change n ↔ n′ causes n′ −1 n′ n−n′ n n−n′ n a change Jn−n Jn′ −n (x), Jn−n Jn−n−1 (x), ′ (x) = (−) ′ +1 (x) = −(−) ′ ′ n−1 n n−n Jn−n′ −1 (x) = −(−) Jn−n+1 (x). With these changes, one finds that only the 2-component of the vertex function (9.1.8) changes sign. This confirms that the Onsager relations in the form (9.1.15) are satisfied by (9.1.6) and (9.1.9) with (9.1.8). The nongyrotropic and gyrotropic parts are symmetric and antisymmetric, respectively, under the interchange of electrons and positrons. It follows that, on writing − nsum = n+ q q + nq ,

− ndiff = n+ q q − nq ,

+ ǫndiff nǫq = 12 [nsum q q ], (9.1.16)

the nongyrotropic and gyrotropic parts are proportional to nsum and ndiff q q , respectively.

9.2 Explicit 4-tensor form for Π µν (k)

363

9.2 Explicit 4-tensor form for Π µν (k) The general form (9.1.6) involves sums over real and virtual electron and positron states, ǫ, ǫ′ = ±1, and over spin states s, s′ = ±1. In this section an alternative general form is derived by performing these sums explicitly. After discussing spin dependence, this general expression for the response tensor is written down in terms of nǫns (pz ). The steps involved in performing the sums are then summarized. 9.2.1 Spin-dependent occupation number In general, the occupation number, nǫq → nǫns (pz ), is different for the two spin states, s = ±1. By introducing the sum and difference, n ¯ ǫn (pz ) = 12 [nǫn+ (pz ) + nǫn− (pz )],

∆nǫn (pz ) = 21 [nǫn+ (pz ) − nǫn− (pz )], (9.2.1) one can make the dependence on s explicit by writing nǫns (pz ) = n ¯ ǫn (pz ) + s∆nǫn (pz ).

(9.2.2)

A ‘spin-independent’ electron gas means ∆nǫn (pz ) = 0. On inserting (9.2.2) into (9.1.6), the dependence on s, s′ becomes explicit; the sums are performed explicitly in §9.2. The spin operator for the vertex function (9.1.8) is the component of the magnetic moment operator along the magnetic field. The magnetic moment eigenvalues are 12 gµB s, where g ≈ 2 is the gyromagnetic ratio, and (SI units) µB = e¯h/2me = 2.74 × 10−24 J T−1

(9.2.3)

is the Bohr magnetion. Hence a spin-dependent electron gas has a non-zero magnetization, M , directed along the magnetic field field. The contributions of the electrons and positrons in the nth Landau level to the number density and magnetization are given by ǫ X Z dpz eB 2¯ nǫn (pz ) nn . (9.2.4) = Mnǫ 2π 2π gµB ∆nǫn (pz ) n

Thus the spin-independent part of the response tensor is proportional to nǫn and the spin-dependent part is proportional to Mnǫ . 9.2.2 Explicit form for given nǫns (pz ) The form resulting from the lengthy calculation summarized below is X ǫ µν [Πns ] (k), (9.2.5) Π µν (k) = ǫ,n,s

364

9 Response of magnetized systems

where the contribution from given ǫ, n, s is Z dpz nǫns (pz ) e3 B X ǫ µν [Nnǫ ′ ns (pk , k)]µν . (9.2.6) [Πns ] (k) = 2 − ε′2 ] 2π 2π ε [(ω − ǫε ) ′ n n n ′ n

The dependence on ǫ, s of the 4-tensor in the integrand of (9.2.6) can be made explicit by writing [Nnǫ ′ ns (pk , k)]µν = 12 {[Nn′ n (ǫpk , k)]µν + ǫ[Gn′ n (ǫpk , k)]µν }

+ 12 s{[∆Nn′ n (pk , k)]µν + ǫ[∆Gn′ n (pk , k)]µν }.

(9.2.7)

The combination of non-gyrotropic 4-tensors is [Nn′ n (ǫpk , k)]µν + s[∆Nn′ n (ǫpk , k)]µν = 2 [2pµk pνk − ǫ(pµk kkν + kkµ pνk ) + (pk)k gkµν ][(Jnn−1 ′ −n ) (1 + s

m m ) + (Jnn′ −n )2 (1 − s 0 )] 0 εn εn

µν n−1 2 n 2 2 n 2 −p2n [gkµν [(Jnn−1 ′ −n ) + (Jn′ −n ) ] + g⊥ [(Jn′ −n+1 ) + (Jn′ −n−1 ) ]] m m µν 2 +ǫ(pk)k g⊥ [(Jnn−1 ) + (Jnn′ −n−1 )2 (1 − s 0 )] ′ −n+1 ) (1 + s ε0n εn µ ν µ ν n−1 n n +2pn′ pn [gkµν Jnn−1 ′ −n Jn′ −n + (e1 e1 − e2 e2 )Jn′ −n+1 Jn′ −n−1 ]

n−1 n n +pn (kkµ eν1 + kkν eµ1 )[Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 ] m m n −ǫ(pµk eν1 + pνk eµ1 ){pn [Jnn−1 ) + Jnn′ −n Jnn−1 )] ′ −n Jn′ −n−1 (1 + s ′ −n+1 (1 − s ε0n ε0n m m n−1 ) + Jnn′ −n Jnn′ −n−1 (1 − s 0 )]}, (9.2.8) +pn′ [Jnn−1 ′ −n Jn′ −n+1 (1 + s ε0n εn

and the combination of gyrotropic 4-tensors is 2 n 2 [Gn′ n (ǫpk , k)]µν + s[∆Gn′ n (ǫpk , k)]µν = i{p2n f µν [(Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) ] m m 2 −ǫ(pk)k f µν [(Jnn−1 ) − (Jnn′ −n−1 )2 (1 − s 0 )] ′ −n+1 ) (1 + s 0 εn εn n−1 n n +pn (kkµ eν2 − kkν eµ2 )[Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 ] m m n −ǫ(pµk eν2 − pνk eµ2 ){pn [Jnn−1 ) − Jnn′ −n Jnn−1 )] ′ −n Jn′ −n−1 (1 + s ′ −n+1 (1 − s ε0n ε0n m m n−1 ) − Jnn′ −n Jnn′ −n−1 (1 − s 0 )]}. (9.2.9) −pn′ [Jnn−1 ′ −n Jn′ −n+1 (1 + s ε0n εn

Various approximate and special forms of (9.2.5) with (9.2.6)–(9.2.9) are discussed in the next section. In the remainder of this section, the derivation of (9.2.5)–(9.2.9) is summarized. 9.2.3 Sums over s′ , s After writing the occupation numbers in (9.1.6) using (9.1.16) and (9.2.2), the dependence on s, s′ becomes explicit. The sums over s′ , s lead to three 4-tensors:

9.2 Explicit 4-tensor form for Π µν (k)



ǫ′ ǫ n′ n ǫ′ ǫ n′ n ′ǫ′ ǫ n′ n

C 1  D  2ǫ′ ǫε′n′ εn D ′



(ǫ′ p′k , ǫpk ) 1 ′  X   ǫ′ ǫ (ǫ′ p′k , ǫpk )  = s [Γq′ q (k)]µ [Γqǫ′ qǫ (k)]∗ν . s′ .s s′ (ǫ′ p′k , ǫpk ) 



365

(9.2.10)

The tensor Cnǫ ′ǫn (ǫ′ p′k , ǫpk ) applies to the spin-independent part, and the other two tensors contribute only to the spin-dependent part. In the following, only ′ the derivation of Cnǫ ′ǫn (ǫ′ p′k , ǫpk ) is discussed explicitly. The response tensor for a spin-independent electron gas becomes µν ′ ′ 3 X Z dpz ′ n (P , Pk ) ǫ C n e B k nǫ (pz ) ′ Π µν (k) = − 2π 2π n 2εn′ εn (ω − ǫεn + ǫ′ ε′n′ ) ′ ′ ǫ,n;ǫ ,n µν Z ǫ Cn′ n (Pk′ , Pk ) dp′z ǫ′ ′ . (9.2.11) − n ′ (p ) 2π n z 2ε′n′ εn (ω − ǫεn + ǫ′ ε′n′ ) with P ′µ = [ǫ′ ε′n′ , 0, 0, ǫpz − kz ] in the first integral and P µ = [ǫεn , 0, 0, ǫ′ p′z + kz ] in the second integral. The sums over ǫ′ , ǫ in the two terms, respectively, in (9.2.11) give µν µν X ǫ′ Cn′ n (Pk′ , Pk ) Cn′ n (Pk − kk , Pk ) , =− 2ε′n′ εn (ω − ǫεn + ǫ′ ε′n′ ) εn [(ω − ǫεn )2 − ε′2 n′ ] ǫ′ µν µν X ǫ Cn′ n (Pk′ , Pk ) Cn′ n (Pk′ , Pk′ + kk ) = ′ . (9.2.12) 2ε′n′ εn (ω − ǫεn + ǫ′ ε′n′ ) εn′ [(ω + ǫ′ ε′n′ )2 − ε2n ] ǫ The second integral in (9.2.12) reduces to a form similar to the first integral by making the changes p′z → pz , n′ ↔ n, ǫ′ → −ǫ. 9.2.4 Separation into nongyrotropic and gyrotropic parts The tensors that appear in (9.2.12) separate into nongyrotropic and gyrotropic parts: µν µν µν Cn′ n (Pk − kk , Pk ) = Nn′ n (Pk , k) + ǫ Gn′ n (Pk , k) , µν µν µν = Nn′ n (Pk , k) − ǫ Gn′ n (Pk , k) . (9.2.13) Cnn′ (−Pk , −Pk + kk ) The nongyrotropic part is given by µν 2 n 2 Nn′ n (Pk , k) = (Pk − kk )µ Pkν + (Pk − kk )ν Pkµ (Jnn−1 ′ −n ) + (Jn′ −n ) 2 n 2 −[(P 2 )k − (P k)k − m2 ]gkµν (Jnn−1 ′ −n ) + (Jn′ −n ) µν n−1 (Jn′ −n+1 )2 + (Jnn′ −n−1 )2 −[(P 2 )k − (P k)k − m2 ]g⊥ µ ν µ ν n−1 n n +2pn′ pn gkµν Jnn−1 ′ −n Jn′ −n + e1 e1 − e2 e2 Jn′ −n+1 Jn′ −n−1 n−1 n n −pn (Pk − kk )µ eν1 + (Pk − kk )ν eµ1 Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 n−1 n n (9.2.14) −pn′ Pkµ eν1 + Pkν eµ1 ] Jnn−1 ′ −n Jn′ −n+1 + Jn′ −n Jn′ −n−1 ,

366

9 Response of magnetized systems

and the gyrotropic part is given by µν 2 n 2 ǫ Gn′ n (Pk , k) = i [(P 2 )k − (P k)k − m2 ] f µν (Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) n−1 n n −pn (Pk − k)µk eν2 + (Pk − kk )ν eµ2 Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 , n−1 n n , (9.2.15) +pn′ Pkµ eν2 − Pkν eµ2 Jnn−1 ′ −n Jn′ −n+1 − Jn′ −n Jn′ −n−1

µ with eµ1 = k⊥ /k⊥ , eµ2 = f µν k⊥ν /k⊥ , and where the argument of the J2 functions is x = k⊥ /2eB. The expressions (9.2.25), (9.2.15) may be rewritten in a variety of different specific forms using identities satisfied by the J-functions, given in §A.1. For example, one can eliminate the explicit dependence on pn′ using the identities n pn′ Jnn′ −n (x) = pn Jnn−1 ′ −n (x) + k⊥ Jn′ −n−1 (x), n−1 n pn′ Jnn−1 ′ −n (x) = pn Jn′ −n (x) + k⊥ Jn′ −n+1 (x).

(9.2.16)

9.2.5 Sum over electron and positron states To perform the sum over ǫ, it is convenient to write the occupation number in terms of parts that are symmetric and antisymmetric under the interchange of electrons and positrons, as in (9.1.16). The resulting expression for the response tensor for a spin-independent electron gas is µν µν Z + ndiff e3 B X dpz nsum n (pz ) Nn′ n (ǫpk , k) n (pz ) Gn′ n (ǫpk , k) µν Π (k) = − 2π 2π 2εn [ǫ(pk)k − (pk)nn′ ] ′ ǫ,n,n

(9.2.17)

where the resonant value ǫ(pk)k = (pk)nn′ , with (pk)nn′ = (k 2 )k fnn′ = 12 (ω 2 − kz2 + p2n − p2n′ ),

(9.2.18)

corresponds to the ± solutions (6.1.19), (6.1.20), with fnn′ is defined by (6.1.17). It is straightforward to perform the sum over ǫ in (9.2.18), giving Z e3 B X dpz 1 Π µν (k) = − 2 − (pk)2 ] 2π 2π ε [(pk) n nn′ k n,n′ µν µν × nsum n (pz ) (pk)nn′ Nn′ n (pk , k) + + (pk)k Nn′ n (pk , k) − µν µν , (9.2.19) +ndiff n (pz ) (pk)k Gn′ n (pk , k) + + (pk)nn′ Gn′ n (pk , k) −

µν µν where Nn′ n (pk , k) ± , Gn′ n (pk , k) ± are the parts that are even and odd, respectively, under ǫ → −ǫ. The form (9.2.19) is useful when considering the nonquantum limit. An alternative form is more convenient for evaluating the integral over pz .

9.2 Explicit 4-tensor form for Π µν (k)

367

9.2.6 Rationalized resonant denominator An alternative form for the response tensor is obtained by rationalizing the denominator. The sums over ǫ, ǫ′ imply that the integrand in (9.1.6) contains four denominators, ω − ǫεn + ǫ′ ε′n′ with ǫ, ǫ′ = ±1. By multiplying each such denominator by the other three factors, the common denominator becomes D(ω, εn , ε′n′ ), defined by (6.1.14) as the product of all four factors. The existing denominator in (9.2.18) arises from (ω − ǫεn )2 − ε′2 n′ = −2[ǫεn ω − ǫpz kz − (pk)nn′ ], and D(ω, εn , ε′n′ ) is obtained by multiplying the numerator and de2 nominator of (9.2.18) by (ω + ǫεn )2 − ε′2 n′ = 2[ǫεn ω + ǫpz kz + (k )k fnn′ ]. The ′ major advantage of this procedure is that D(ω, εn , εn′ ) is a rational function, being a quadratic function of pz : 1 1 , =− ′ 2 2 D(ω, εn , εn′ ) 4(ω − kz )(pz − pz+ )(pz − pz− )

(9.2.20)

or of p′z , given by pz , pz± → p′z , p′z± in (9.2.20). It is convenient to rationalize in two steps, by writing 1 ω + ǫεn − ǫ′ ε′n′

=

ω − ǫεn − ǫ′ ε′n′ . (ω − ǫεn )2 − ε′2 n′

(9.2.21)

The form (9.2.20) of the denominator follows by multiplying both numerator 2 2 and denominator of (9.4.19) by (ω + ǫεn )2 − ε′2 n′ = ω − kz + 2ǫ(ωεn + kz pz ) + ′ 2(n − n )eB. The response tensor in the form (9.2.18) becomes X Z dpz ǫ(εn ω + pz kz ) + 1 (k 2 )k fnn′ e3 B 2 8π(ω 2 − kz2 ) 2π ε (p − p )(p − pz− ) n z z+ z ǫ,n,n′ sum µν µν × nn (pz ) Nn′ n (ǫpk , k) + ndiff , n (pz ) Gn′ n (ǫpk , k)

Π µν (k) =

(9.2.22)

with pz± implicit functions of n′ , n through (9.1.12). The response tensor in the form (9.2.22) is the starting point for application to specific electron distributions. The integral over pz in is expressed in terms of two plasma dispersion functions in §9.4 for an arbitrary occupation number. The form (9.2.22) may also be used to derive the vacuum response tensor. By inspection of (9.1.6), the response tensor for a medium transforms into that for the vacuum by replacing all occupation numbers by − 12 . This corresponds 1 d to nsum n (pz ) → − 2 , nn (pz ) → 0 in (9.2.22). The resulting expression for the unregularized vacuum response tensor is Π µν (k) = −

X Z dpz ǫ(εn ω + pz kz ) + 1 (k 2 )k fnn′ µν e3 B 2 Nn′ n (ǫpk , k) . 2 16π(k )k 2π εn (pz − pz+ )(pz − pz− ) ′ ǫ,n,n

(9.2.23)

368

9 Response of magnetized systems

9.2.7 Spin-dependent electron gas The generalization to an spin-dependent electron gas, with ∆nǫn (pz ) 6= 0 in (9.2.2), involves evaluating the tensors D, D′ in (9.2.10). The resulting expression for Π µν (k) becomes Z ǫ(εn ω + pz kz ) + (ω 2 − kz2 )fnn′ e3 B X dpz µν Π (k) = − 2π 2π 2εn (ω 2 − kz2 )(ǫpz − pz+ )(ǫpz − pz− ) ′ ǫ,n ,n

µν µν ×{¯ nsum + ǫ¯ ndiff n (pz )[Nn′ n (ǫpk , k)] n (pz )[Gn′ n (ǫpk , k)]

µν µν +∆nsum + ǫ∆ndiff n (pz )[∆Nn′ n (ǫpk , k)] n (pz )[∆Gn′ n (ǫpk , k)] }, (9.2.24)

where the resonant denominator can be rewritten using (9.4.19). The four tensors in the integrand in (9.2.24) are [Nn′ n (ǫpk , k)]µν = 12 {[Cn′ n (Pk − kk , Pk )]µν + [Cnn′ (−Pk , −Pk + kk )]µν }, [Gn′ n (ǫpk , k)]µν = 12 {[Cn′ n (Pk − kk , Pk )]µν − [Cnn′ (−Pk , −Pk + kk )]µν },

′ µν [∆Nn′ n (ǫpk , k)]µν = 21 {[Dn′ n (Pk − kk , Pk )]µν + [Dnn }, ′ (−Pk , −Pk + kk )]

′ µν }. [∆Gn′ n (ǫpk , k)]µν = 12 {[Dn′ n (Pk − kk , Pk )]µν − [Dnn ′ (−Pk , −Pk + kk )] (9.2.25)

Explicit evaluation of the first two of (9.2.25) gives, for the nongyrotropic part, [Nn′ n (ǫpk , k)]µν = [2pµk pνk − ǫ(pµk kkν + kkµ pνk )

2 n 2 −[p2n − ǫ(pk)k ]gkµν ][(Jnn−1 ′ −n ) + (Jn′ −n ) ]

µν 2 2 n −[p2n − ǫ(pk)k ]g⊥ [(Jnn−1 ′ −n+1 ) + (Jn′ −n−1 ) ]

µ ν µ ν n−1 n n +2pn′ pn [gkµν Jnn−1 ′ −n Jn′ −n + (e1 e1 − e2 e2 )Jn′ −n+1 Jn′ −n−1 ] n−1 n n +pn (kkµ eν1 + kkν eµ1 )[Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 ]

n−1 n n −(ǫpµk eν1 + ǫpνk eµ1 ){pn [Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 ]

n−1 n n +pn′ [Jnn−1 ′ −n Jn′ −n+1 + Jn′ −n Jn′ −n−1 ]},

(9.2.26)

and, for the gyrotropic part, 2 n 2 [Gn′ n (ǫpk , k)]µν = i{[p2n − ǫ(pk)k ]f µν [(Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) ]

n−1 n n −(ǫpµk eν2 − ǫpνk eµ2 ){pn [Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 ] n−1 n n −pn′ [Jnn−1 ′ −n Jn′ −n+1 − Jn′ −n Jn′ −n−1 ]}

n−1 n n +pn (kkµ eν2 − kkν eµ2 )[Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 ]}. (9.2.27)

The final two in (9.2.25) describe the spin-dependent parts in (9.2.24), and their explicit evaluation gives, for the non-gyrotropic part,

9.2 Explicit 4-tensor form for Π µν (k)

[∆Nn′ n (ǫpk , k)]µν =

369

m {[2pµk pνk − ǫ(pµk kkν + kkµ pνk ) ε0n 2 n 2 +ǫ(pk)k gkµν ][(Jnn−1 ′ −n ) − (Jn′ −n ) ]

µν 2 n 2 +ǫ(pk)k g⊥ [(Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) ]

n−1 n n −(ǫpµk eν1 + ǫpνk eµ1 ){pn [Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 ]

n−1 n n +pn′ [Jnn−1 ′ −n Jn′ −n+1 − Jn′ −n Jn′ −n−1 ]},

(9.2.28)

and, for the gyrotropic part, [∆Gn′ n (ǫpk , k)]µν = i

m 2 n 2 {−ǫ(pk)k f µν [(Jnn−1 ′ −n+1 ) + (Jn′ −n−1 ) ] ε0n

n−1 n n −(ǫpµk eν2 − ǫpνk eµ2 ){pn [Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 ]

n−1 n n −pn′ [Jnn−1 ′ −n Jn′ −n+1 + Jn′ −n Jn′ −n−1 ]}}.

(9.2.29)

Rearrangement of (9.2.24)–(9.2.29) leads to the form (9.2.5)–(9.2.9).

370

9 Response of magnetized systems

9.3 Special and limiting cases of the response tensor In this section, general expressions for the linear response tensor derived in §9.1 are applied to various special and limiting cases. Special cases include parallel propagation and a one-dimensional distribution, in which all the electrons (and positrons) are in their lowest Landau level. Two limiting cases are the nonrelativistic and the nonquantum limits. Major simplification occurs in 2 the limit x = k⊥ /2eB ≪ 1, when the power series expansion of the J-functions converges rapidly. 9.3.1 Response tensor for parallel propagation A major simplification to the response tensor occurs in the limit of parallel 2 propagation, k⊥ = 0, due to the argument, x = k⊥ /2eB, being zero. The n n J-functions reduce to J0 (0) = 1, Jν (0) = 0 for ν 6= 0. The sums over n, n′ give zero except for n′ − n = 0, ±1. For a spin-independent electron gas, for parallel propagation the response tensor in the form (9.1.6) summed over s, s′ reduces to ǫ′ ǫ ′ µν Z ǫ ′ ǫ′ ′ e3 B X dpz [ǫn( pz ) − ǫ nn′ (pz )] Cn′ n (Pk , Pk ) k µν Π (k) = − , (9.3.1) 2π ′ ′ 2π 2ǫ′ ǫε′n′ εn (ω − ǫεn + ǫ′ εn′ ) ǫ ,ǫ,n ,n

′ µν with ǫ′ p′z = ǫpz − kz implicit, and where πnǫ ′ǫn k denotes the tensor ǫ′ ǫ ′ µν Cn′ n Pk , Pk k , given by (9.2.13)–(9.2.15) in general, for k⊥ = 0. There are nonzero components only for n′ = n, n ± 1, and these gives µν ǫ′ ǫ ′ Cn′ n (Pk , Pk ) k = Pk′µ Pkν + Pkµ Pk′ν − (P ′ P )k − m2 − p2n gkµν gn δn′ ,n µν (δn′ +1,n + δn′ −1,n ) − if µν (δn′ +1,n − δn′ −1,n ) , (9.3.2) − (P ′ P )k − m2 g⊥

with g0 = 1, gn = 2 for n ≥ 1, and with Pk′µ = [ε′n′ , 0, 0, p′z ], Pkµ = [εn , 0, 0, pz ]. Inserting (9.3.2) into (9.3.1) gives Z dpz e3 B X µν Π (k) = − 2π ′ 2π ǫ ,ǫ,n

′

gn [ǫ′ nǫn (pz ) − ǫnǫn (p′z )] ′µ ν Pk Pk + Pkµ Pk′ν − (P ′ P )k − m2 − p2n gkµν ′ ′ ′ 2εn εn (ω − ǫεn + ǫ εn ) ′ X [ǫ′ nǫn (pz ) − ǫnǫn±1 µν (p′z )] ′ 2 µν (P P )k − m (g⊥ ± if ) , (9.3.3) − 2ε′n±1 εn (ω − ǫεn + ǫ′ ε′n±1 ) ±

with ε′n = εn (p′z ). An alternative form for the response tensor, with a rationalized denominator, follows by setting k⊥ = 0 in (9.4.9):

9.3 Special and limiting cases of the response tensor

371

X Z dpz ǫ(εn ω + pz kz ) + (n′ − n)eB + (k 2 )k fnn′ e3 B Π µν (k) = 4(ω 2 − kz2 ) 2π 2π εn (pz − pz+ )(pz − pz− ) ǫ,n,n′ sum µν µν × nn (pz ) Nn′ n (ǫpk , k) k + ǫndiff . (9.3.4) n (pz ) Gn′ n (ǫpk , k) k

µν Nn′ n (ǫpk , k) k = 2Pkµ Pkν − Pkµ kkν − Pkν kkµ + (P k)k − p2n gkµν gn δn′ ,n µν (δn′ +1,n + δn′ −1,n ) + (P k)k − p2n g⊥ µν µν 2 Gn′ n (ǫpk , k) k = −i (P k)k − pn f⊥ (δn′ +1,n − δn′ −1,n ). (9.3.5)

Only n′ = n, n ± 1 contribute in (9.5.3). In the case of parallel propagation there are only three independent components of the response tensor. Specifically, in a coordinate system with B and k along the 3-axis, the only nonzero space components are Π 11 = Π 22 , Π 12 = −Π 21 , Π 33 . The only nonzero space components in the numerator in (9.5.3) are  ′  11   Nnǫ ′ǫn (ǫpk , k) k [p2n − ǫ(εn ω − pz kz )](δn′ +1,n + δn′ −1,n )  ′   12   ǫǫ     Gn′ n (ǫpk , k) k  =  − i[p2n − ǫ(εn ω − pz kz )](δn′ +1,n − δn′ −1,n ) .  ′    33 Nnǫ ′ǫn (ǫpk , k) k [2p2z + p2n − ǫ(εn ω + pz kz )]gn δn′ ,n

(9.3.6)

9.3.2 One-dimensional electron gas In a superstrong magnetic field the time scale for electrons to radiate away their perpendicular energy is so short that all the electrons (and positrons) can plausibly be assumed to be in the ground Landau state, n = 0. This is referred to as a one-dimensional electron gas. Three equivalent forms for the response tensor for a one-dimensional electron gas are written down here. The first form follows by making the one-dimensional to the response tensor (9.1.6), giving Z Z dp′z dpz e3 B X 2πδ(ǫ′ p′z − ǫpz + kz ) Π µν (k) = − 2π 2π 2π ǫ,ǫ′ ,n′ ′ µν ′ µν ′ ǫnǫ0 (pz ) πnǫ ′ǫ0 (k) − ǫ′ nǫ0 (p′z ) π0ǫ′ǫn (k) . (9.3.7) × ω − ǫε0 (pz ) + ǫ′ εn′ (p′z ) ′ µν The components of πnǫ ′ǫ0 (k) follow by setting n = 0 in (5.4.24). The anithermitian part of the tensor (9.3.7) follows by making the one-dimensional assumption in (9.1.11). The second form for the response tensor for a one-dimensional electron gas follows by setting n = 0 in (9.2.18). This gives

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9 Response of magnetized systems

Z e3 B X dpz 1 2π 2π 2ε [ǫ(pk) − (pk)0n′ ] 0 k ǫ,n′ sum µν µν × n0 (pz ) Nn′ 0 (ǫpk , k) + ǫndiff , 0 (pz ) Gn′ 0 (ǫpk , k)

Π µν (k) = −

(9.3.8)

with pµk = [ε0 (pz ), 0, 0, pz ], (pk)0n′ = 12 (k 2 )k − n′ eB, (k 2 )k = (ω 2 − kz2 ), and where (9.2.25), (9.2.15) reduce to µν µ ν Nn′ 0 (ǫpk , k) = 2pk pk − ǫpµk kkν − ǫpνk kkµ + ǫ(pk)k gkµν (Jn0′ )2 µν µ ν (Jn0′ −1 )2 , (9.3.9) +ǫ(pk)k g⊥ − ǫ pµk k⊥ + pνk k⊥ µν µ ν (Jn0′ −1 )2 , (9.3.10) Gn′ 0 (ǫpk , k) = iǫ (pk)k f µν − pµk kG − pνk kG

µ µ = k⊥ eµ2 , and where the identify pn′ Jn0′ (x) = respectively, with k⊥ = k⊥ eµ1 , kG 0 k⊥ Jn′ −1 (x) is used. The third form is obtained by performing the sum over ǫ in (9.3.10), which is equivalent to making the one-dimensional assumption in (9.2.22). This gives µν Z Bn′ (pk , k) e3 B X dpz 1 µν Π (k) = − , (9.3.11) 2π ′ 2π ε0 (pk)2k − [ 12 (k 2 )k − n′ eB]2 n

with the numerator rewritten using 2(pk)0n′ = (k 2 )k − p2n′ and pn′ Jn0′ = k⊥ Jn0′ −1 so that it gives

µν = nsum (pz ) (k 2 )k pµk pνk − (pk)k (pµk kkν + pνk kkµ ) + (pk)2k gkµν (Jn0′ )2 Bn′ (pk , k) 0 µ µν 2 µ ν ν − k⊥ pk pk (Jn0′ −1 )2 − (pk)k pµk k⊥ + pνk k⊥ +nsum (pz ) (pk)2k g⊥ 0 1 2 µ 0 µν ν +indiff (Jn′ −1 )2 − pµk kG − pνk kG 0 (pz ) [ 2 (k )k − eB] (pk)k f µ 2 ν −k⊥ (pk)k f µν − pµk kG − pνk kG (Jn0′ −2 )2 . (9.3.12)

The expressions (9.3.7)–(9.3.11) are exact for a one-dimensional distribution, and simplified forms require additional simplifying assumptions. 9.3.3 Nonquantum limit: one-dimensional case

The classical expression for the response tensor for a one-dimensional electron gas is written down in (??), and in an alternative form in (??). The form (??) is reproduced by the nonquantum limit of (9.3.11) with (9.3.12). To facilitate the comparison, (??) is first rewritten in the notation used in this section, giving Z (pz ) µν e3 B dpz nsum 0 Π µν (k) = − A (pk /m, k), (9.3.13) 2π 2π ε0 with pµk /m = uµ and

9.3 Special and limiting cases of the response tensor

kkµ uν

+ uµ kkν

373

(k 2 )k uµ uν 1 µν + (ku)2 g⊥ ku (ku)2 (ku)2 − Ω02 µ ν µ ν ν ν 2 µ ν u − uµ kG −ku(k⊥ u + uµ k⊥ ) − k⊥ u u + iηΩ0 ku f µν + kG , (9.3.14)

Aµν (u, k) = gkµν −

+

sum with Ω0 = eB/m, η = −ndiff (pz ). 0 (pz )/n0 The nonquantum limit corresponds formally to h ¯ → 0, and to take this limit explicitly one needs to revert to ordinary units by restoring h ¯ . A potential complication is that the nonquantum limit also corresponds to the Landau level, n, becoming arbitrarily large, so that h ¯ n tends to a finite limit, p2⊥ /2eB, but this is not relevant for a one-dimensional distribution, which has n = 0 or p⊥ = 0. In the reduction of the relativistic quantum expression (9.3.11) with (9.3.12) to the classical expression (9.3.13) with (9.3.14) it suffices to note that h ¯ appears in two ways. First, it appears in the arguments of the 2 J-functions, which is x = h ¯ k⊥ /2eB, and second in the dimensionless ratio k/p, which becomes ¯hk/p in ordinary units. Setting x = 0, one has J00 (0) = 1 with Jn0′ (0) = 0 for n′ 6= 0, so that only terms n′ = 0, 1, 2 remain in the sum in (9.3.11) with (9.3.12). Retaining only the lowest order powers in k/p in the numerator and denominator corresponds to neglecting 12 (k 2 )k and the final term proportional to (Jn0′ −1 )2 . This corresponds to neglecting the quantum recoil. With these approximations, on writing pµk = muµ and eB = mΩ0 , (9.3.11) with (9.3.12) reproduces (9.3.13) with (9.3.14).

9.3.4 Nonquantum limit: Vlasov form In the general case the occupation numbers, n± n (pz ), are nonzero for arbitrary n, and the nonquantum limit corresponds to h ¯ → 0, n → ∞, h ¯ n → p2⊥ /2eB. It is of interest to derive the nonquantum limit in two different forms. These reproduce two different classical forms in §2.2: the Vlasov form (2.2.28) and the forward-scattering form (2.2.12). The Vlasov form follows naturally by taking the nonquantum limit of (9.1.6). A preliminary step in the reduction (9.1.6) to (2.2.28) involves a subtlety. There is no dispersion associated with pair creation in classical theory, and this suggests that one should neglect the terms with ǫ′ 6= ǫ in (9.1.6) and retain only the dispersion associated with gyromagnetic absorption by electrons (ǫ = ǫ′ = 1) and positrons (ǫ = ǫ′ = −1). The preliminary step is to concentrate on the contribution of electrons, with the positrons known to contribute with the same sign for the nongytropic part and with the opposite sign for the gytropic part. The subtlety is that the terms with ǫ′ 6= ǫ do need to be retained to reproduce (2.2.28) exactly. There are four important steps in the reduction (9.1.6) to (2.2.28), two of which are the same as in the reduction in §6.1.3 of the probability (6.1.5) for gyromagnetic emission to its nonquantum counterpart (6.1.6). One step is to replace the vertex functions by their nonquantum limits (6.1.12): Γ µ → (ieiψ )a U µ (a, k)/γ, with a = n−n′ remaining finite in the limit n, n′ → ∞. With only the contribution of the electrons retained explicitly, the sign of the charge in U µ (n − n′ , k) is η = −1. Another step is to replace the

374

9 Response of magnetized systems

resonant denominator by it nonquantum counterpart, as in the replacement of the resonance condition (6.1.7) by its nonquantum counterpart (ku)k −aΩ0 . A third step is to expand the difference between the quantumnumbers in (9.1.6) ˆ introduced in (6.1.8). One has using the differential operator D. h + ′ 1 ˆ2 ˆ n+ 1 − D + (p ) = n (p − k ) = D + · · · n+ (9.3.15) ′ z z n−a z n (pz ), n 2

and retain only up to first order. The final step is to replace n+ n (pz ) by its classical counterpart. A further step is to replace n by p2⊥ /2eB, the sum over n by the integral over dp⊥ p⊥ /eB, the sum over n′ by a sum over a, and to + rewrite n+ n (pz ) as n (p). Then one has ∂ ∂ + ′ + k n+ (p ) − n n+ (p), (p ) = −(ω − aΩ/γ − k v ) + ω − k v ] z z z z z z n n′ z ∂p⊥ ∂pz (9.3.16) The results reproduce the term involving the sum over harmonic number in (2.2.28) which, when rewritten in the notation used in this section, becomes Z ∂ ∂ d3 p µ ν 1 µ ν µν 2 − b b uz Π (k) = 2e uk uk (2π)3 u⊥ ∂p⊥ ∂pz ∞ µ ∗ν X ∂ U (a, k)U (a, k) (ku)k ∂ + kz n+ (p), (9.3.17) − (ku) − aΩ u ∂p ∂p 0 ⊥ ⊥ z k a=−∞ The terms in (9.3.17) that do not involve the sum over harmonics, a, arise in two ways. One is from the term proportional to ω − aΩ/γ − kz vz in (9.3.16), which factor cancels with the resonant denominator, allowing the sum over a to be performed using the sum rules for Bessel functions. The other is from the nonquantum limit of the terms, with ǫ 6= ǫ′ in (9.1.6), associated with pair creation. 9.3.5 Nonquantum limit: forward-scattering form The nonquantum limit of the form (9.2.18) for the response tensor reproduces the classical form (2.2.12), derived using the forward-scattering method. After modification in notation to be consistent with that used in the quantum treatment, the classical form (2.2.12) becomes X Z d3 p n(p) Π µν (k) = −e2 (2π)3 ε a gkµν Ja2 −

kkµ U ∗ν (a, k) + kkν U µ (a, k) (ku)k − aΩ0

Ja +

[(ku)k − aΩ0 ]2 g µν J 2 + [(ku)k = aΩ0 ]2 − a2 Ω02 ⊥ a

m2 (k 2 )k U µ (a, k)U ∗ν (a, k) [(ku)k − aΩ0 ]2

9.3 Special and limiting cases of the response tensor µ ∗ν k⊥ U (a, k)

375

ν µ + k⊥ U (a, k) m2 (k 2 )⊥ U µ (a, k)U ∗ν (a, k) Ja + (ku)k − aΩ0 [(ku)k − aΩ0 ]2 µ ∗ν ν µ iηΩ0 [(ku)k − aΩ0 ] kG U (a, k) − kG U (a, k) µν 2 Ja , (9.3.18) f Ja + + [(ku)k − aΩ0 ]2 − Ω02 (ku)k − aΩ0

−

with Ja = Ja (k⊥ u⊥ /Ω0 ) and with η = −ndiff (p)/nsum (p) ranging between η = −1 for a pure electron gas to η = 1 for a pure positron gas. (CHECK SIGN OF η) The reduction of (9.2.18) to this limit involves neglecting the quantum recoil term, and taking the limits h ¯ → 0, n → ∞, h ¯ n → p2⊥ /2eB. Only the contribution of the electrons needs to be considered explicitly; the positrons contribute with the same sign to the nongyrotropic part and with the opposite sign to the gyrotropic part. On neglecting the quantum recoil term, 12 (ω 2 − kz2 ), the resonant denominator reproduces its classical form, (ku)k − aΩ0 for electrons, ǫ = 1. TBC (9.2.18) gives Z ¯n′ n (pk , kk ) µν + nd (pz ) G ¯ n′ n (pk , kk ) µν ¯ n (pz ) N e3 B X dpz n n µν Π (k) = − 2π 2π εn (pk)k − ǫ(n − n′ )eB ′ n,n

(9.3.19) with (9.2.25) and (9.2.15) giving 2 n 2 ¯n′ n (pk , kk ) µν = − p2 − m2 g µν (J n−1 N k n′ −n+1 ) + (Jn′ −n−1 ) ⊥ 2 n 2 + 2pµk pνk − p2k − m2 gkµν (Jnn−1 ′ −n ) + (Jn′ −n ) µ ν µ ν n−1 n n +2pn′ pn gkµν Jnn−1 ′ −n Jn′ −n + e1 e1 − e2 e2 Jn′ −n+1 Jn′ −n−1 n−1 n n (9.3.20) +pn (kkµ eν1 + kkν eµ1 Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 , µν 2 2 n 2 ¯ n′ n (pk , kk ) G = i pk − m2 f µν (Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) µ ν n−1 n n (9.3.21) +pn kk e2 − kkν eµ2 Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 ,

respectively.

9.3.6 First quantum corrections The quantum corrections to the classical limits, (9.3.17) and (9.3.18), of the response tensor are of three different types: the quantum recoil, spindependence, and corrections of order B/Bc . These three types of correction require different treatments. It is straightforward to include the quantum recoil, simply by retaining the term 12 (k 2 )k = 12 (ω 2 − kz2 ) in the resonant denominator in the derivations of nonquantum limits (9.3.17) and (9.3.18). The ˆ 2 in the expansion quantum recoil arises in (9.1.6) from the terms of order D

376

9 Response of magnetized systems

(6.1.8) of the resonant denominator. The recoil term, 21 (ω 2 − kz2 ), appears directly in the resonant denominator in (9.2.18). Modifications to the nonquantum limits (9.3.17) and (9.3.18) to include the quantum recoil are obtained simply by retaining these corrections when otherwise taking the limit h ¯ → 0, n → ∞. Spin dependence is described by occupation numbers nǫn,s (pz ) that are different for s = ±1. The response tensor for a spin-dependent electron gas may be separated into a spin-averaged part and a spin-dependent part. Only the spin-averaged part is nonzero in the nonquantum limit, and it is given by the classical forms discussed above with nǫn (pz ), replaced by 12 [nǫn,+ (pz ) + nǫn,− (pz )]. The spin-dependent part is intrinsically quantum mechanical, in the sense that it vanishes in the limit h ¯ → 0, and it is proportional to nǫn,+ (pz ) − ǫ nn,− (pz ). Physically, this part may be attributed to dispersion associated with absorption involving a spin flip. The derivation of the relevant correction to the response tensor is lengthy, and the result is relatively cumbersome. When spin-dependent effects are of interest, it is more practicable to start from the exact expression (9.1.6), and to evaluate the vertex function for s′ = −s to find the spin-dependent part before making other approximations. Quantum corrections arise from the expansion of the J-functions in two different ways:. For large-n, Jνn is expanded in powers of 1/n, as given by (6.1.9). Only the leading term is retained in the nonquantum limit, and the first order term gives a quantum correction, with 1/n identified as h ¯ 2eB/p2⊥ . Another term of the same order arises from the fact that the vertex function involves both Jνn−1 and Jνn , which has the same limit for n → ∞, but which differ when the next order term is retained. The next order term is given by (6.1.10). These corrections become large for small n, in which case it can be simpler to use the exact expressions for the J-functions, rather than to include extra corrections to the large-n limit. 9.3.7 Long wavelength limit The long-wavelength limit, |k| → 0, is of interest historically in that it was the first limit case to be derived using the relativistic quantum theory [19]. In this limit, (9.1.6) gives e3 B Π (k) = Π (k) = − 2 4π 11

22

Z

∞

−∞

dpz

∞ X

n=0

(εn+1 − εn )[¯ nn+1 (pz ) − n ¯ n (pz )] m2 + p2z 1− ω 2 − (εn+1 − εn )2 εn+1 εn m2 + p2z (εn+1 + εn )[¯ nn+1 (pz ) + n ¯ n (pz )] , 1+ + ω 2 − (εn+1 + εn )2 εn+1 εn Z ∞ X ie3 B ∞ 12 21 d d Π (k) = −Π (k) = [nn+1 (pz ) − nn (pz )] dpz 4π 2 −∞ n=0

9.3 Special and limiting cases of the response tensor

377

ω m2 + p2z m2 + p2z ω + , 1− 1+ ω 2 − (εn+1 − εn )2 εn+1 εn ω 2 − (εn+1 + εn )2 εn+1 εn Z ∞ X εn n ¯ n (pz ) e3 B ∞ p2z dpz gn 2 Π 33 (k) = − 2 1 − , (9.3.22) π ω − 4ε2n ε2n −∞ n=0 with the remaining components equal to zero. (A sign error in the 12-term in [19] is corrected in (9.3.22).) The degeneracy factor in (9.3.22) is g0 = 1, gn = 2 for n ≥ 1. For a completely degenerate electron gas in the one-dimensional limit, when all the electrons are in their ground state, n = 0, the nonzero components of the response tensor (9.3.22) give [19] 2 ζF − χ2+ m2 − χ2− ζF e3 B σ √ ln Π 11 (k) = Π 22 (k) = , + ln 2π 2 m ζF2 − χ2− m2 − χ2+ 2 σ 2 − m2 e3 B Π 12 (k) = −Π 21 (k) = −i 2 2π σ (ζF − χ+ )(ζF + χ− ) (m + χ+ )(m − χ− ) × √ , ln (ζF + χ+ )(ζF − χ− ) (m − χ+ )(m + χ− ) 2 σ 2 − m2 ζF + ω/2 e3 Bm2 p Π 33 (k) = arctan p 2 2 2 π ω m − ω /4 m2 − ω 2 /4 s s s m + ω/2 ζF − ω/2 m − ω/2 − arctan , − arctan + arctan m − ω/2 m2 − ω 2 /4 m + ω/2 ζF = µF + m + [µF (m + 2µF )]1/2 , σ=

eB ω − , ω 2

χ± = σ ± (σ 2 − m2 )1/2 .

(9.3.23)

The form (9.3.23) is derived only for σ > m, ω < 2m. In the nonrelativistic limit, µF ≪ m, and at low frequencies, ω 2 ≪ mΩe , (9.3.23) reduces to the cold plasma dielectric tensor. The approximate forms (9.3.22) and (9.3.23) are of limited usefulness due to the assumption kz → 0 being overly restrictive. 9.3.8 Response tensor for a nonrelativistic quantum gas Before discussing the relativistic case in detail, it is useful to note the form of the response tensor for a nonrelativistic magnetized quantum electron gas. There are no positrons in a nonrelativistic electron gas. In the early calculations of the response tensor, the electron spin was not taken into account explicitly. When the spin is ignored, the nonrelativistic energy eigenvalues consists of the kinetic energy of the parallel motion, with the circular motion perpendicular to the magnetic field quantized as a simple harmonic oscillator.

378

9 Response of magnetized systems

This gives energy eigenvalues (SI units) εn (pz ) = mc2 + ∆εn (pz ), with the kinetic energy given by ∆εn (pz ) =

p2z + (n + 12 )¯ hΩe . 2m

(9.3.24)

However, the spin is taken into account in a degeneracy factor, with the ground state, n = 0, nondegenerale and the excited Landau levels doubly degenerate, described by a degeneracy factor gn = 1 for n = 0, gn = 2 for n ≥ 1. The electron density is then given by (SI units) ne =

Z ∞ X eB dpz nn (pz ), 2π¯ h 2π¯ h n=0

(9.3.25)

where nn (pz ) is the occupation number. It might be remarked that in the Wigner-function method, the background distribution is described by the Wigner function, which contains the same information as the occupation number, but which has a much more cumbersome form; the specific form in the nonrelativistic case was derived by [?]. The dielectric tensor is K i j (k) = δji + Π i j (k)/εω 2 , with the contravariant component Π ij (k) numerically equal to minus the mixed component Π i j (k). The specific expression for the dielectric tensor given in Ref. [36], when rewritten in the notation used here, gives K i j (k) = δji +

Π i j (k) , ε0 ω 2

Π i j (k) = −

e 2 ne i δ j − τ i j (k) , m

(9.3.26)

with ne given by (9.3.25). The contravariant components τ ij (k), which are numerically equal to minus the mixed components τ i j (k), are given by (SI units) Z dpz [V i (k)]nn′ [V j (k)]∗nn′ m X eB [gn nn (pz ) − gn′ nn′ (p′z )] , τ ij (k) = − ′ (pz ) ne 2π¯ h 2π¯ h ω − ω nn ′ n,n

(9.3.27) with p′z = pz − ¯ hkz , and with the resonant frequency in the denominator given by (9.3.28) hωnn′ (pz ) = ∆εn (pz ) − ∆εn′ (p′z ). ¯

In the coordinate system in which the magnetic field is along the 3-axis and k is in the 1-3 plane, the components of [ti (k)]nn′ are TBC n+1 V 1 (k)]nn′ = pn Jnn−1 ′ −n+1 − pn+1 Jn′ −n−1 , (2)

[V 2 (k)]nn′ = (eB)1/2 Inn′ , (3)

[V 3 (k)]nn′ = pz Inn′ ,

(9.3.29)

9.3 Special and limiting cases of the response tensor

379

with (1)

(9.3.30)

Inn′ =

(2)

(9.3.31)

(3) Inn′

(9.3.32)

Inn′ =

=

380

9 Response of magnetized systems

9.4 Magnetized thermal distribution The response tensor for a thermal distribution of particles plays a central role in theory of dispersion in plasmas. In relativistic quantum theory, a thermal distribution of electrons is a Fermi-Dirac distribution. Specific forms for the response tensor are derived in this section and the next section in two limiting cases of a Fermi-Dirac distribution: the nondegenerate and completely degenerate limits, respectively. The response tensor in the form (9.2.22) may be evaluated explicitly in these two limits. Before considering a thermal distribution explicitly, it is helpful to establish that for an arbitrary occupation number, the response tensor (9.2.22) may be evaluated in terms of two plasma dispersion functions. It is these two plasma dispersion functions that can be evaluated explicitly in the nondegenerate and completely degenerate limits. 9.4.1 Reduction of integrals The response tensor in the form (9.2.22) involves a pz -integral with a numerator that depends on pz through the occupation number and powers of εn and pz , and with a denominator εn (pz − pz+ )(pz − pz+ ). An important step in evaluating the integral is to show that it can be expressed in terms of two plasma dispersion functions, each defined for a given occupation number. Using ε2n = (ε0n )2 + p2z , all the relevant integrals may be expressed in terms of the following two dimensionless integrals: Z Z nǫ (pz ) dpz nǫn (pz ) ǫ ǫ I1± (ω, kz ) = dpz n , I2± (ω, kz ) = ε0n . (9.4.1) pz − pz± εn pz − pz± It is straightforward to prove the identities Z ǫ X pnz± I1± (ω, kz ) pnz nǫn (pz ) dpz ± , = (pz − pz+ )(pz − pz− ) 2ωgnn′ ± Z ǫ X pnz± I2± (ω, kz ) dpz pnz nǫn (pz ) 0 ± . εn = εn (pz − pz+ )(pz − pz− ) 2ωg nn′ ± The specific integrals that appear in (9.2.22) are of the form Z nǫ (pz ) ǫ(εn ω + pz kz ) + (pk)nn′ a b dpz n ε p εn (pz − pz+ )(pz − pz− ) n z X ǫ ǫ ǫ ± ε0n I1± (ω, kz ) + εn± I2± (ω, kz ) εan± pbz± , = 2gnn′ ε0n ± where a, b ≥ 0 are integers.

(9.4.2)

(9.4.3)

9.4 Magnetized thermal distribution

381

9.4.2 Plasma dispersion functions It is useful to define two plasma dispersion functions by Z 1 nǫ (pz ) ǫ dt n , J± (ω, kz ) = t − tz± −1

(9.4.4)

and the two further functions with t± → −1/t± in (9.4.4). The variable t, with −1 ≤ t ≤ 1, is defined by pz 2t = , ε0n 1 − t2

εn 1 + t2 = . ε0n 1 − t2

(9.4.5)

The parameters t± are defined to satisfy 2t± pz± = , ε0n 1 − t2±

1 + t2± εn± = , ε0n 1 − t2±

t± =

εn± − ε0n , pz±

(9.4.6)

where t = t± , −1/t± are the two solutions of pz = pz± . One finds that the combination of plasma dispersion functions in (9.4.3) can be expressed in form Z dpz ǫ ǫ ǫ ǫ n (pz ) + 2ε0n J± (ω, kz ), (9.4.7) (ω, kz ) + εn± I2± (ω, kz ) = ε0n ε0n I1± εn n with the functions defined by t± → −1/t± in (9.4.4) canceling in (9.4.7). The non-dispersive term (9.4.7) is proportional to the proper number density in the level n: Z eB dpz ε0n ǫ nǫnpr = n (pz ) = n ¯ npr + ǫndnpr , (9.4.8) 2π 2π εn n where the final expression involves the separation into the sum and difference, respectively, of the electron and positron contributions. In terms of the plasma dispersion functions (9.4.4) the response tensor (9.2.22) becomes X e2 nǫnpr constant termµν Π µν (k) = ǫ,n

+

X

ǫ,n,n′ ,±

µν µν d J¯± (ω, kz ) Nn′ n (pk± , k) + ǫJ± (ω, kz ) Gn′ n (pk± , k) e B . 8π 2 (ω 2 − kz2 ) 3

(9.4.9)

with pµk± = [εn± , 0, 0, pz±], and with ǫ d J± (ω, kz ) = J¯± (ω, kz ) + ǫJ± (ω, kz ).

(9.4.10)

382

9 Response of magnetized systems

9.4.3 TBA A subtle feature is the absence of an explicit dependence of (9.2.22) on a specific frame, such as the rest frame, associated with the plasma. There is an implicit dependence on a special frame through the dependence of the occupation number on pz . Introducing 4-velocity, u ˜µ of the rest frame, which µ allows one to introduce the 4-vector b along the magnetic field, as in the choice (1.1.26) of basis 4-vectors, one interprets ε, pz as the invariants pk u ˜, −pk b, respectively. Similarly, one interprets ω, kz as kk u˜, −kk b, respectively. With this interpretation, the rest frame is introduced by writing pµk = εn u ˜ µ + pz b µ ,

kkµ = ω u ˜µ + kz bµ

(9.4.11)

in the explicit forms (9.2.25), (9.2.15) for the tensors that appear in the integrand of (9.2.22). 9.4.4 Fermi-Dirac distribution for magnetized electrons For a Fermi gas, a relativistic thermal distribution of electrons is a Fermi-Dirac distribution, with occupation number n± (ε) =

1 , e(ε−µ± )/T − 1

(9.4.12)

where T is the temperature (in energy units), and where µ± are the chemical potentials for the electrons and positrons, respectively. The rest energy, m, is excluded in the chemical potential here. In thermal equilibrium the chemical potentials satisfy µ+ + µ− = 0. Suppose that the number density, ne , of free electrons is specified. In equilibrium the electron gas consists of this fixed number of free electrons plus a number of pairs that is determined by the conditions for thermal equilibrium. The fixed number ne requires Z ∞ eB X dpz + gn [n (pz ) − n− n (pz )] = ne , 2π n=0 2π n

(9.4.13)

where g0 = 1, gn = 2 for n ≥ 1, is the degeneracy factor. In the nondegenerate limit, the chemical potential µ+ − m is large and negative. The unit term in the denominator in (9.4.12) is then negligible, and the resulting distribution reduces to a sum of one-dimensional J¨ uttner distributions. In this case it is convenient to incorporate the chemical potential into a normalization constant, A± , by writing " 1/2 # B p2z ± , (9.4.14) nn (pz ) = A± exp −ρ 1 + 2 + 2n m Bc with A± determined in terms of the number densities, n± , by

9.4 Magnetized thermal distribution ∞ X

eB n± gn = A± 2π n=0

Z

∞

−∞

dpz −ρεn /m e , 2π

383

(9.4.15)

with εn /m = [1+p2z /m2 +2nB/Bc )1/2 . By changing the variable of integration to χ, with sinh χ = pz /m(1 + 2nB/Bc )1/2 , the integral in (9.4.15) reduces to an integral representation of a Macdonald function: Z (x/2)ν Γ ( 12 ) ∞ Kν (x) = dχ sinh2ν χ e−x cosh χ , (9.4.16) Γ (ν + 12 ) 0 The number densities in a nondegenerate, relativistic, thermal electron gas are given by ∞ X n± eBε0n K1 (ρε0n /m), (9.4.17) = gn A± 2π 2 n=0

with ε0n /m = (1 + 2nB/Bc )1/2 . The result (9.4.17) simplifies in the nonrelativistic limit when the asymptotic expansion Kν (x) = (π/2x)1/2 e−x iapplies, provided that one also has B ≪ Bc , such that the approximation (1 + 2nB/Bc )1/2 ≈ 1 + nB/Bc applies. The sum over n in (9.4.17) gives ∞ X

gn e−ρnB/Bc =

n=0

1 . tanh(ρB/2Bc )

(9.4.18)

In the ultrarelativistic limit, ρ ≪ 1, the Macdonald functions in (9.4.17) may be approximated by the leading term in the power series expansion, K1 (x) ≈ 1/x. The approximation K1 (x) ≈ 1/x is justified only for ρ(2nB/Bc )1/2 ≪ 1. For ρ ≪ 1 and ρ(2B/Bc )1/2 ≫ 1 nearly all the particles are in the ground state, n = 0, so that the electron gas is essentially one dimensional. 9.4.5 Plasma dispersion functions The response tensor (9.1.6) involves a pz -integral over a resonant denominator, ω − ǫεq + ǫ′ εq′ , and such integrals are evaluated in terms of appropriately defined plasma dispersion functions. In the unmagnetized case, it is shown in §9.4.3 of volume 1 that the relativistic quantum expression for the response tensor for a J¨ uttner distribution may be evaluated in terms of a relativistic plasma dispersion function T (z, ρ), whose properties are summarized in §4.4.3 of volume 1. The magnetized case may be treated in a similar manner. The first step in such a treatment involves writing the resonant denominator and the integral over pz in an appropriate form. It is convenient to start from the form (9.2.22) for the response function, in which the resonant denominator is rationalized to D(ω, εn , ε′n′ ), given by (9.2.20), specifically, D(ω, εn , ε′n′ ) = −4(ω 2 − kz2 )(pz − pz+ )(pz − pz− ), with pz± implicitly dependent on n, n′ . The response tensor involves a sum over

384

9 Response of magnetized systems

n, n′ , and the objective is to evaluate the pz integral for each n, n′ . For given n the pz -integral may be rewritten in terms of a t-integral: 2t pz = , 0 εn 1 − t2

εn 1 + t2 = , ε0n 1 − t2

dpz = 2ε0n dt

1 + t2 . (1 − t2 )2

(9.4.19)

For given n, n′ , pz± may be written in terms of analogous parameters, t± : ǫpz± = kz fnn′ ± ωgnn′ = ǫε0n

2t± , 1 − t2±

(9.4.20)

The denominator (9.2.20) becomes 4(ω 2 − kz2 )(ε0n )2 t+ t− −1 (t − t+ )(t − t−1 + )(t − t− )(t − t− ). (1 + t2+ )(1 + t2− ) (9.4.21) The product of four factors in the denominator may be expressed as a sum of four terms, each involving a single denominator, 1/(t − t+ ), 1/(t − t−1 + ), 1/(t − −1 t− ), 1/(t − t− ). For the J¨ uttner distribution, the resulting integrals may be expressed in terms of the relativistic plasma dispersion function Z 1 1 + t′2 dt′ , (9.4.22) I(t, ρ) = exp −ρ ′ 1 − t′2 −1 t − t D(ω, εn , ε′n′ ) = −

This function is evaluated in §9.4.3 of volume 1 in terms of the relativistic plasma dispersion function Z 1 dv ′ exp(−ργ ′ ), (9.4.23) T (v, ρ) = ′ −1 v − v with γ ′ = (1 − v ′2 )−1/2 . The properties of T (v, ρ) are summarized in §4.4.3 of volume 1. The denominators in (9.4.21) appear in pairs, 1/(t−t± ), 1/(t−t−1 ± ), and a pair of such in t-integrals of the form (9.4.22) gives T ′ (v, ρ) + 2γ 2 ρK1 (ρ) , ργ 3 v (9.4.24) with T ′ (v, ρ) = ∂T (v, ρ)/∂v. The relativistic plasma dispersion functions that appear are T (v± , ρ), T ′ (v± , ρ) with I(t, ρ) + I(1/t, ρ) = T (v, ρ),

I(t, ρ) − I(1/t, ρ) = −

v± =

2t± kz fnn′ ± ωgnn′ , = 2 1 + t± ωfnn′ ± kz gnn′

with fnn′ , gnn′ defined by (6.1.17), (6.1.18), respectively.

(9.4.25)

9.4 Magnetized thermal distribution

385

9.4.6 Evaluation of specific integrals The specific pz -integrals (or p′z -integrals) that appear in the response tensor in the form (9.2.22) involve not only also fac the denominator µν (9.4.21), butµν ′ ′ tors (ω + ǫεn )2 − ε′2 times either N (ǫp , k) or G (ǫp , k) in the nn nn k k n′ numerator. These factors can be written as functions of pz and εn ; they do not involve ε′n′ . Evaluation of integrals involving additional powers of pz in the integrand is straightforward, and additional power of εn are treated by differentiating with respect to ρ, and using v

∂T (v, ρ) (1 − v 2 ) ′ = 2K1 (ρ) + T (v, ρ). ∂ρ ρ

TO BE COMPLETED 9.4.7 Parallel propagation: nondegenerate limit TBA 9.4.8 Complete response tensor: nondegenerate limit TBA 9.4.9 Antihermitian part: nondegenerate limit TBA

(9.4.26)

386

9 Response of magnetized systems

9.5 Completely degenerate electron gas The linear response tensor for a completely degenerate electron gas is relevant solid state plasmas, and an expression for the response tensor was derived by Lindhard [?] in the nonrelativistic case, and generalized by Jancovici [13] to the relativistic case. The generalization to include a magnetic field in the relativistic case TBC 9.5.1 Completely degenerate limit The degenerate limit corresponds to T → 0 in the Fermi-Dirac distribution (9.4.12). A degenerate distribution in the absence of a magnetic field corresponds to all the states being filled for ε < εF , and empty for ε > εF , where the Fermi energy, εF , is equal to the chemical potential. Similarly, fo a degenerate distribution in the presence of a magnetic field, the Fermi energy is equal to the chemical potential. The occupation number at each Landau level are filled up to the Fermi energy. This corresponds to all the states being filled for |pz | < pnF , where pnF = (ε2F − m2 − 2neB)1/2 .

(9.5.1)

The Landau levels or n < nF , where n = nF is the maximum n for which pnF , as defined by (9.5.1) is real. The states and are empty for |pz | > pnF and for n > nF . For nF < 1 only the ground state, n = 0. All positron states are empty in a completely degenerate electron gas. The number density of electrons is Z nF nF X X eB pnF dpz eB (9.5.2) gn gn 2 (ε2F − m2 − 2neB)1/2 , ne = = 2π −pnF 2π 2π 0 0 with g0 = 1, gn = 2 for n ≥ 1. One may regard (9.5.2) as determining nF for given ne , εF , B.

Π

µν

X Z dpz ǫ(pk)k + (n′ − n)eB + 1 (k 2 )k e3 B 2 2 2 (k) = 4(ω − kz ) 2π 2π εn (pz − pz+ )(pz − pz− ) ′ ǫ,n,n µν µν × n ¯ n (pz ) Nn′ n (ǫpk , k) + ǫndn (pz ) Gn′ n (ǫpk , k) .

µν µν µν Nn′ n (Pk , k) = Nn′ n (Pk , k) + + Nn′ n (Pk , k) − , µν 2 n 2 Nn′ n (Pk , k) + = 2Pkµ Pkν − p2n gkµν (Jnn−1 ′ −n ) + (Jn′ −n ) µν n−1 (Jn′ −n+1 )2 + (Jnn′ −n−1 )2 −p2n g⊥

(9.5.3)

9.5 Completely degenerate electron gas

387

µ ν µ ν n−1 n n +2pn′ pn gkµν Jnn−1 ′ −n Jn′ −n + e1 e1 − e2 e2 Jn′ −n+1 Jn′ −n−1 n−1 n n +pn kkµ eν1 + kkν eµ1 Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 , µν 2 n 2 Nn′ n (Pk , k) − = − Pkµ kkν + Pkν kkµ − (P k)k gkµν (Jnn−1 ′ −n ) + (Jn′ −n ) µν n−1 (Jn′ −n+1 )2 + (Jnn′ −n−1 )2 +(P k)k g⊥ n−1 n n −pn Pkµ eν1 + Pkν eµ1 Jnn−1 ′ −n Jn′ −n−1 + Jn′ −n Jn′ −n+1 n−1 n n −pn′ Pkµ eν1 + Pkν eµ1 ] Jnn−1 (9.5.4) ′ −n Jn′ −n+1 + Jn′ −n Jn′ −n−1 , µν µν µν Gn′ n (Pk , k) = Gn′ n (Pk , k) + + Gn′ n (Pk , k) − , µν 2 n 2 Gn′ n (Pk , k) + = i p2n f µν (Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) n−1 n n +pn kkµ eν2 − kkν eµ2 Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 , µν 2 n 2 Gn′ n (Pk , k) − = i − (P k)k f µν (Jnn−1 ′ −n+1 ) − (Jn′ −n−1 ) n−1 n n −pn Pkµ eν2 − Pkν eµ2 Jnn−1 ′ −n Jn′ −n−1 − Jn′ −n Jn′ −n+1 n−1 n n , (9.5.5) +pn′ Pkµ eν2 − Pkν eµ2 Jnn−1 ′ −n Jn′ −n+1 − Jn′ −n Jn′ −n−1

9.5.2 Logarithmic plasma dispersion functions For a completely degenerate distribution, the counterpart of a plasma dispersion function (for a thermal distribution) is play by logarithmic functions. The resonant integrals in (??) may be reduced to two forms: Z pnF Z pnF 1 ε0 1 dpz I1± = dpz n , I2± = , (9.5.6) (pz − pz± ) εn (pz − pz± ) −pnF −pnF with ε0n = (m2 + 2neB)1/2 . Combinations of integrals I1± lead to logarithmic functions with arguments Λ1 = Λ2 =

2 (pnF − p+ )(pnF + p− ) (pnF − ωgnn′ )2 − kz2 fnn ′ = 2 , (pnF + p+ )(pnF − p− ) (pnF + ωgnn′ )2 − kz2 fnn ′

2 (pnF − kz fnn′ )2 − ω 2 gnn (pnF − p+ )(pnF − p− ) ′ = 2 , 2 2 (pnF + p+ )(pnF + p− ) (pnF + kz fnn′ ) − ω gnn ′

(9.5.7)

where the explicit forms (6.1.19) for p± are used. The integrals I2± may be evaluated by writing pz 2t = , ε0n 1 − t2

εn 1 + t2 = , 0 εn 1 − t2

and the two solutions of each of pz = pz± as

dpz 2dt = , εn 1 − t2

(9.5.8)

388

9 Response of magnetized systems

"

2

#1/2

Λ3 = =

+1

ε0 − n , pz+

"

2

#1/2

1 ε0n , t4 = − , pz− t3 (9.5.9) respectively. Combinations of integrals I2± lead to logarithmic functions with arguments of three types, with two given by (9.5.7), and the third by TO BE CHECKED

t1 =

ε0n pz+

1 t2 = − , t1

t3 =

ε0n pz−

+1

−

(tnF − t1 )(tnF + t2 )(tnF + t3 )(tnF − t4 ) (tnF + t1 )(tnF − t2 )(tnF − t3 )(tnF + t4 )

[(εF kz − pnF ω)fnn′ + (ωεF − pnF kz )gnn′ ][(εF kz + pnF ω)fnn′ − (ωεF + pnF kz )gnn′ ] , (9.5.10) [(εF kz + pnF ω)fnn′ + (ωεF + pnF kz )gnn′ ][(εF kz − pnF ω)fnn′ − (ωεF − pnF kz )gnn′ ]

where t = tnF corresponds to pz = pnF . The three logarithmic functions (9.5.7), (9.5.10) appear in the case of parallel propagation [21, ?].; it follows that no new plasma dispersion functions appear in the generalization from k⊥ = 0 to k⊥ 6= 0. 9.5.3 Complete response tensor: degenerate limit 9.5.4 Parallel propagation: degenerate limit TBA 9.5.5 Antihermitian part: degenerate limit TBA

9.6 Wave dispersion in relativistic quantum magnetized plasma

389

9.6 Wave dispersion in relativistic quantum magnetized plasma Each generalization of the response tensor to include additional physical effects leads to modifications of the existing wave modes and the introduction of additional wave modes. Including relativistic quantum effects in a magnetized electron gas leads to modifications (to the wave modes of a relativistic, magnetized, nonquantum electron gas) due to the quantum recoil, degeneracy, and spin effects. Additional wave modes appear, associated with dispersive effects due to the discreteness of the Landau levels and with pair creation. These additional modes are referred to as gyromagnetic absorption modes, and pair modes, respectively. The discussion in this section emphasizes the appearance of these additional modes. For this purpose it suffices to consider the special case of parallel propagation. TBC 9.6.1 Dispersion relations: parallel propagation TBA 9.6.2 Gyromagnetic absorption modes TBA 9.6.3 Pair modes TBA

390

9 Response of magnetized systems

9.7 Nonlinear response tensors As with the linear response, the nonlinear responses are of interest for both an electron gas and for the magnetized vacuum. The method used in §5.1 to calculate the response of an electron gas generalizes in a straightforward manner to the hierarchy of nonlinear responses, as defined by the weak turbulence expansion (1.4.4). This method also leads to explicit expressions for the nonlinear response tensors for the magnetized vacuum. The alternative exact method used in §?? to derive the linear response of the magnetized vacuum also generalizes straightforwardly to the nonlinear responses. However, the resulting expressions for the quadratic response tensor are very cumbersome, and simplifications need to be made to derive useful approximations. The approximate method based on the use of the Heisenberg-Euler Lagrangian, which is used in §?? to derive a relatively simple expression for the linear response tensor in the weak field low-frequency limit, also leads to relatively simple, approximate expressions for the quadratic and cubic nonlinear response tensors. Another approximation for the linear response tensor that generalizes in a straightforward way, applies in the strong field limit and involves assuming that the only virtual particle states that contribute significantly correspond to the lowest Landau orbitals. 9.7.1 Closed loop diagrams In QPD, the (n + 1)th rank response tensor is derived from the amplitude for a closed loop diagram with n+1 sides, with n = 1, n = 2, n = 3 corresponding to the linear, quadratic and cubic responses, respectively. The Feynman amplitude for an n-sided electron loop follows from the general nth order amplitude with n contractions over the electron wavefunctions. The relevant S-matrix component in coordinate space is given by Z Z Z (−e)n Sˆ(n) = − d4 x1 d4 x2 . . . d4 xn γ ν1 G(x1 −x2 )γ ν2 . . . γ νn G(xn −x1 ) n! (9.7.1) × : Aˆν1 (x1 ) Aˆν2 (x2 ) . . . Aˆνn (xn ) : . In the unmagnetized case one rewrites (9.7.1) in momentum space simply by expressing the propagators in terms of their Fourier transforms and then carrying out the space-time integrals, which all give δ-functions. The n δfunctions ensure conservation of 4-momentum at the n vertices, and one can carry out n − 1 of the 4-momentum integrals over these δ-functions, leaving a single δ-function, which expressed conservation of the external 4-momentum, plus the integral over the undetermined loop 4-momentum. In the magnetized case this procedure is not possible because the Fourier transform of the propagator does not exist. For an electron gas, this difficulty is overcome by writing (9.7.1) in terms of vertex functions whose Fourier transform does exist. The energy and the parallel momentum are conserved at each vertex, and

9.7 Nonlinear response tensors

391

are treated in the same manner as any of the components of 4-momentum in the unmagnetized case. The the propagators in (9.7.1) are interpreted in terms of the form (??), which may be written as Z X dE −iE(t−t′ ) e Ψqǫ (x)Ψ ,ǫq (x′ ) G(x, x′ ) = 2π ǫq 1 × Π (9.7.2) − iǫ(1 − 2nǫq ) πδ(E − ǫεq ) . E − ǫεq The wavefunctions in (9.7.1) with a common x-dependence occur in pairs and each such pair is expressed in terms of a vertex function using the inverse of the definition (??), Z 3 d K iK·x ǫ′ ǫ ǫ′ µ ǫ Ψ q′ (x)γ Ψq (x) = e [γq′ q (K)]µ . (9.7.3) (2π)3 There are n such vertex functions, with arguments K 1 , . . . , K n say. The 4potentials in (9.7.1) are expressed in terms of their Fourier transforms, Z d4 ki −iki xi ˆ ˆ (9.7.4) Aνi (ki ). e Aνi (xi ) = (2π)4 The n space integrals in (9.7.1) are then trivial, giving n δ-functions of the form (2π)3 δ 3 (K i + ki ), and the n integrals over the K i are then trivial. The n time integrals in (9.7.1) also give n δ functions; n − 1 of the E-integrals are performed over these over these n δ-function, leaving a single E-integral over an undetermined loop energy, and a single δ-function expressing conservation of the external energy, ω1 + ω2 + · · · ωn = 0. Thus (9.7.1) reduces to Z (−i)n ˆ¯ ˆ¯ )A(x ˆ/ 1 )ψ(x ˆ ˆ 1) : ˆ Sˆ(n) = d4 xn . . . d4 x1 : ψ(x n / n )ψ(xn ) . . . ψ(x1 )A(x n! Z 4 (ie)n d kn d4 k1 νn ···ν1 = (−k1 , . . . , −kn ) Aˆνn (kn ) . . . Aˆν1 (k1 ), · · · L n! (2π)4 (2π)4 X Lνn ···ν1 (k1 , . . . , kn ) = γqǫ11qǫnn (kn ) . . . γqǫ22qǫ11 (k1 ) 2πδ(ω1 + ω2 + · · · ωn ) q1 ,...,qn

×

Z

dE 2π

n Y

r=1

1 ǫ − iǫr (1 − 2nqr ) πδ(Er − ǫr εqr ) . Π Er − ǫr εqr

(9.7.5)

The loop energy E is an undetermined additive constant in each of the Er , whose difference are determined by energy conservation at each vertex, Er − Er−1 + ωr = 0. The n factors in square brackets in (9.7.5) have both real (principal value) parts and imaginary (resonant of δ-function) parts. The only physical contributions are those corresponding to the imaginary part of one factor times

392

9 Response of magnetized systems

the real parts of all the others. (The term with all principal value parts gives zero, and terms with more than one resonant part are nonphysical due to the use of the Feynman propagator.) Thus the final integral in (9.7.5) is to be interpreted according to Z n dE Y 1 − iǫr (1 − 2nǫqr ) πδ(Er − ǫr εqr ) Π 2π r=1 Er − ǫr εqr Z n dE X −iǫi (1 − 2nǫqi ) πδ(Ei − ǫi εqi ) Q → . (9.7.6) n 2π i=1 r6=i [Er − ǫr εqr ]

This procedure leads only to the nondissipative part of the response tensor, and the causal condition needs to be imposed separately to obtain the dissipative part. The causal condition in included by the Landau prescription of giving the frequencies in Er −Er−1 +ωr = 0 an infinitesimal positive imaginary part. ′ The specific form of the vertex functions [γqǫ′ ǫq (k)]µ in (9.7.5) depends on the choice of gauge. The gauge-dependent factors can be included in a fac′ ′ tor dǫq′ǫq (k) multiplying a gauge-independent vertex function [Γqǫ′ qǫ (k)]µ . It is straightforward to carry out the sums or integrals over the gauge-dependent quantum numbers. Specifically for the Landau gauge, this involves writing ′

′

′

[γqǫ′ ǫq (k)]µ = dǫq′ǫq (k) [Γqǫ′ qǫ (k)]µ , ′ ′

′

dǫq′ǫq (k) =

eikx (ǫpy +ǫ py )/2eB 2πδ(ǫpy − ǫ′ p′ y − ky ) 2πδ(ǫpz − ǫ′ p′ z − kz ). V (eB)1/2 (9.7.7)

There are n integrals over the p1y . . . pny and n integrals over the p1z . . . pnz , and n − 1 of each of these are performed over the δ-functions in the gaugedependent factors The remaining δ-functions imply the y- and z-components of k1 + k2 + · · · + kn = 0, the remaining py integral is performed over the py -dependence in the phase factor arising from the phase in and it gives eB times the δ-functions expressing the x-component of k1 + k2 + · · · + kn = 0; a single pz integral remains. In this way, one finds   X (ki × kj )z  Lνn ···ν1 (k1 , . . . , kn ) = (2π)4 δ 4 (k1 + · · · + kn ) exp − 2eB j
where the sums over the qs denotes the remaining quantum numbers ǫ, n, s. The n pz -components are related by (pz )r − (pz )r−1 + (kz )r = 0, with the integral over pz over the undetermined additive constant in each of the (pz )r .

9.7 Nonlinear response tensors

393

9.7.2 General expression for the nth order response tensor The final step in the formal identification of the nonlinear response tensors involves the identification of the interaction energy in terms of S-matrix elements with the interaction energy implied by the nonlinear response. This corresponds to the identification −iΠ ν0 ν1 ...νn (k0 , k1 , . . . , kn )(2π)4 δ 4 (k1 + · · · + kn ) (ie)n+1 X 1...n νn′ ···ν1′ ν0 = P1′ ...n′ L (kn′ , . . . , k1′ , k0 ), n!

(9.7.9)

P

where the sum is over all permutations of 1′ . . . n′ amongst 1 . . . n. Applying (9.7.9) to the linear case n = 1 gives the expression (9.1.6). The notation in (9.7.9) corresponds to a linear response tensor Π µν (k0 , k1 ) with two argument that satisfy k0 + k1 = 0. The notation for the linear response tensor used elsewhere is related to the notation used in (9.7.9) by Π µν (k) = Π µν (−k, k). 9.7.3 Quadratic and cubic response tensors The general expression (9.7.9) leads to the following form for the quadratic nonliear response tensor, n = 2: µνρ Π µνρ (k1 , k2 , k3 ) = αµνρ 1 (k1 , k2 , k3 ) + α1 (k1 , k3 , k2 ),   Z 3 X e B dp (k × k ) z i j z  αµνρ exp − 1 (k1 , k2 , k3 ) = − 4π 2π 2eB j
2

+ +

3

1 2 ǫ2 (1

(ω2 + ǫ2 εq2

− 2nǫq22 ) − ǫ1 εq1 )(ω3 + ǫ3 εq3 − ǫ2 εq2 ) 1 2 ǫ3 (1

(ω3 + ǫ3 εq3

− 2nǫq33 ) − ǫ2 εq2 )(ω1 + ǫ1 εq1 − ǫ3 εq3 )

×[Γqǫ33qǫ11 (−k1 )]µ [Γqǫ11qǫ22 (−k2 )]ν [Γqǫ22qǫ33 (−k3 )]ρ .

(9.7.10)

The response of the medium arises from the terms proportional to the occupation numbers. For the linear response the sum and the difference of the contribution of the electrons and positrons appear in the nongyrotropic and gyrotropic parts of the response tensor, respectively, where nongyrotropic terms may be defined as those that do not vanish in the limit B → 0. For the quadratic response the situation is reversed: the difference between the contribution of the electrons and positrons appears in the nongyrotropic terms,

394

9 Response of magnetized systems

which give zero for a pure pair plasma with identical distributions of electrons and positrons. The gyrotropic terms, which vanish ∝ B for B → 0, contain the sum of the contributions of the electrons and positrons. The terms in (9.7.10) that are independent of the occupation numbers, nq , correspond to the quadratic response of the vacuum. Unlike the linear response, the quadratic response contains no divergent terms and so does not need to be regularized. The cubic response tensor follows from (9.7.9) with n = 3: X µν ν ν P22′ 33′ 44′ α1 2′ 3′ 4′ (k1 , k2′ , k3′ , k4′ ), Π µν2 ν3 ν4 (k1 , k2 , k3 , k4 ) = P

3

e B = 12π

dpz 2π   4 1 ǫr X X X (ki × kj )z 2 ǫr (1 − 2nqr )   × exp − 2eB dr dr+1 dr+2 q ,q ,q ,q r=1 j
(k1 , k2 , k3 , k4 ) αµνρσ 1

Z

1

2

3

4

×[Γqǫ44qǫ11 (−k1 )]µ [Γqǫ11qǫ22 (−k2 )]ν [Γqǫ22qǫ33 (−k3 )]ρ [Γqǫ33qǫ44 (−k4 )]σ , dr = ωr + ǫr εqr − ǫr−1 εqr−1 ),

(9.7.11)

with r+4 ≡ r. The response of the medium arises from the terms proportional to the occupation numbers; as for the linear response, the contributions of the electons and positrons add for the nongyrotropic terms. The terms in (9.7.11) that are independent of the occupation numbers, nq , correspond to the cubic response of the vacuum. As remarked in connection with the cubic response tensor for B → 0, the divergent parts are removed by discarding the part that does not satisfy the charge-continuity and gaugeinvariance conditions. This may be achieved by projecting (9.7.11) onto the set of basis vectors (8.1.13). 9.7.4 Sum over spin states Provided that the occupation numbers of the particles do not depend on the spin (or that one is uninterested in any such dependence and wishes to average over it), one can sum over the spins in the expressions (9.7.10) and (9.7.11). The procedure is analogous to that used for the linear response tensor. Specifically, in place of (??), for the quadratice response one writes X µνρ ǫ1 ǫ2 ǫ3 = [Γqǫ33qǫ11 (k1 )]µ [Γqǫ11qǫ22 (k2 )]ν [Γqǫ22qǫ33 (k3 )]ρ πq1 q2 q3 (k1 , k2 , k3 ) s1 ,s2 ,s3

Π / n +m µ Π / n +m µ Π / n +m = Tr Gµ (n3 , n1 , k1 ) 1 G (n1 , n2 , k2 ) 2 G (n2 , n3 , k3 ) 3 2ǫ1 εn1 2ǫ2 εn2 2ǫ3 εn3 (9.7.12)

with explicit forms for the matrices implies by (??) and (??). Explicit evaluation of (9.7.12) is cumbersome, involving lengthy expressions for the 43 = 64

9.7 Nonlinear response tensors

395

different components of the tensor. As for the linear response, one can separate into nongyrotropic and gyrotropic terms, which may be defined as terms that are even and odd under reversal in the sense of the arrow, in the bubble, triangle and square diagram for the linear, quadratic and cubic responses respectively. The result (9.7.12) generalizes to, for an arbitrary number of vertex functions, X µn n−1 ǫn = Tr Gµ (n3 , n1 , k1 ) (k )] [Γqǫnnqǫ11 (k1 )]µ1 [Γqǫ11qǫ22 (k2 )]µ2 . . . [Γqǫn−1 n qn s1 ,...,sn

Π / n1 + m µ Π /n +m Π /n +m , G (n1 , n2 , k2 ) 2 . . . Gµ (nn−1 , nn , kn ) n 2ǫ1 εn1 2ǫ2 εn2 2ǫn εnn (9.7.13)

with n = 4 being relevant to the cubic response. In the evaluation of the quadratic response tensor for a magnetized vacuum using the form (9.7.10), the sum over spins is performed using (9.7.12), and the sum over the signs ǫi is then straightforward. The full result is cumbersome to write down; relevant particular combinations of components were written down by [8, 9], in connection with what is called the S-matrix treatment of photon splitting in vacuo. Algebraically simpler forms for the nonlinear response tensors are available, as discussed below. However, the forms (9.7.10), (9.7.11) are the most convenient forms in situations where the sums over the principal quantum numbers, n1 , . . ., converge rapidly, which is the case when the arguments of the generalized Laguerre polynomials are sum. This 2 conditions requires ki⊥ /2eB ≪ 1 for all wavenumbers, ki . For photon splitting this requires ωi sin θi ≪ m(2B/Bc )1/2 for all three photons. Thus the form (9.7.10) is convenient for treating photon splitting in vacuo for sufficiently low frequencies on in sufficiently strong fields, B > ∼ Bc for frequencies below the 2m. In other case the alternative form for pair creation threshold, ωi sin θi < ∼ the quadratic response tensor of the vacuum is more convenient. 9.7.5 Alternative form for the quadratic response tensor The expressions included in (9.7.10), (9.7.11) and (9.7.12), (9.7.13) for, respectively, the quadratic and cubic responses of the magnetized vacuum, are generalizations of the form (8.1.21) of the linear response tensor for the vacuum. These forms involve a sums over the simple harmonic quantum numbers, that label the Landau orbitals, and integrals over pz . The alternative form for the linear response tensor, given by (8.1.15) with (??), is derived using the form for the propagator, and it involves integrals over parameters α and β of an integrand that contains only elementary functions of these variables. There are analogous alternative forms for the quadratic and cubic response tensors of the vacuum The quadratic response tensor for a magnetized vacuum is given by, in coordinate space,

396

9 Response of magnetized systems µρν ′ ′′ ′′ ′ [αµνρ 1 (x, x , x ) + α1 (x, x , x )] , µ ′ ′′ 3 ′ ν ′ ′′ ρ ′′ αµνρ 1 (x, x , x ) = ie Tr γ G(x, x )γ G(x , x )γ G(x , x) .

Π µνρ (x, x′ , x′′ ) =

1 2

(9.7.14)

On inserting the specific form for the propagator, (9.7.14) gives ′

′′

αβ

′ ′′ 3 ie(x−x )α (x −x)β F αµνρ 1 (x, x , x ) = ie e ×Tr γ µ ∆(x, x′ )γ ν ∆(x′ , x′′ )γ ρ ∆(x′′ , x) .

(9.7.15)

There is freedom to locate the space-time origin, and this may be used to write (9.7.15) in terms only of the differences between the space-time points. Specifically, if one chooses the origin at x = 0, then (9.7.15) simplifies to Z Z Z e3 (eB)3 −iex′α x′′β F αβ ∞ ds ∞ ds′ ∞ ds′′ ′ ′′ e αµνρ (x , x ) = −i 1 (16π 2 )3 s 0 s′ 0 s′′ 0 2

′

′′

e−im (s+s +s ) D1µνρ (x′ , x′′ ), sin(eBs) sin(eBs′ ) sin(eBs′′ ) eBx′2 i eB(x′ − x′′ )2⊥ ⊥ × exp − + 4 tan(eBs) tan(eBs′ ) ′2 x′′2 xk (x′ − x′′ )2k eBx′′2 k ⊥ , + + + + tan(eBs′′ ) s s′ s′′

(9.7.16)

(γx′ )k eB(γx′ )⊥ µ = Tr γ exp(−iΣeBs) m − − 2s 2 sin(eBs) ′ ′′ (γ(x − x ))k eB(γ(x′ − x′′ ))⊥ ′ ν + ×γ exp(−iΣeBs ) m + 2s′ 2 sin(eBs′ ) ′′ ′′ (γx )k eB(γx )⊥ ρ ′ ×γ exp(−iΣeBs ) m + , (9.7.17) + 2s′′ 2 sin(eBs′′ )

D1µνρ (x′ , x′′ )

The quadratic response tensor in momentum space then follows from µρν ′ ′′ ′′ ′ Π µνρ (−k, k ′ , k ′′ ) = αµνρ 1 (−k, k , k ) + α1 (−k, k , k ), Z ′ ′ ′′ ′′ ′ ′′ ′ ′′ (−k, k , k ) = d4 x d4 x′ e−ik x −ik x αµνρ αµνρ 1 (x , x ), 1

(9.7.18)

with the dependence on k = k ′ + k ′′ includes so that the symmetries of the response tensor can be exhibited. The This form of the quadratic response tensor for the vacuum was evaluated explicitly by [10], based on earlier work [11], to find Z ∞ Z ∞ Z ∞ e4 B ′ ′′ ′ ds′′ D1µνρ (−k, k ′ , k ′′ ) αµνρ (−k, k , k ) = i ds ds 1 16π 2 0 0 0 ′ 2 2 ′ ′′ ss kk + ss′′ kk′2 + ss′ kk′′2 e−im (s+s +s ) × exp i (s + s′ + s′′ ) sin[eB(s + s′ + s′′ )] s + s′ + s′′

9.7 Nonlinear response tensors

(a)

397

(b)

Fig. 9.2. a) The box diagram, and (b) the hexagon diagram, whose Feynman amplides lead to the terms in (9.7.23) proportional to F and F 3 , respectively. There are additional contributions from diagrams in which the order of the vertices is changed relative to the diagrams illustrated.

2 ′2 ′′2 C(s)S(s′ )S(s′′ )k⊥ + S(s)C(s′ )S(s′′ )k⊥ + S(s)S(s′ )C(s′′ )k⊥ × exp i eB sin[eB(s + s′ + s′′ )] S(s)S(s′ )S(s′′ ) kα kβ′ f αβ × exp 2i , eB sin[eB(s + s′ + s′′ )] S(s) = sin(eBs),

C(s) = cos(eBs),

f αβ = F αβ /B,

(9.7.19)

The explicit expression for Z 1 ′ ′′ ′′ ′ ′ ′ ′′ ′′ µνρ ′ ′′ D1 (−k, k , k ) = d4 x d4 x′ e−ik x −ik x e− 2 ieB(x y −x y ) D1 (x′ , x′′ ),

(9.7.20) is very cumbersome, and is not written down here. To exhibit its symmetries it is helpful to write k → −k0 , k ′ → k1 , k ′′ → k2 , so that one has k0 +k1 +k2 = 0. It is also helpful to make the dependence on s, s′ , s′′ explicit and to write s → s0 , s′ → s1 , s′′ → s2 . Then one has D1µνρ (k0 , k1 , k2 ; s0 , s1 , s2 ) = D1νρµ (k1 , k2 , k0 ; s1 , s2 , s0 ).

(9.7.21)

However, even when all the symmetries are taken into account, the resulting expression remains very cumbersome. Fortunately, for most purposes the lowfrequency response suffices, and this may be derived much more simply by using the Heisenberg-Euler Lagrangian. 9.7.6 Quadratic response tensor For the quadratic (n = 2) response tensor this (8.3.13) Π µνρ (k0 , k1 , k2 ) ==

i α β γ ∂ 3 LI . k0 k1 k2 2 ∂F α µ ∂F β ν ∂F γ ρ

(9.7.22)

398

9 Response of magnetized systems

Inserting the explicit form for the Heisenberg-Euler Lagrangian and carrying out the indicated operations gives µ0 Π µνρ (k0 , k1 , k2 ) =

iαc 4 k0 k1 g µν − k1µ k0ν ) k2γ F γρ 2 90πBc

+7k0α k1β ǫµανβ k2γ F γρ + perm. iαc + − 16k0α F αµ k1β F βν + 26k0α F αµ k1β F βν k2γ F γρ 4 315πBc − k0 k1 g µν − k1µ k0ν ) 48S k2γ F γρ − 26P k2γ F γρ (9.7.23) +26k0α k1β ǫµανβ S k2γ F γρ + P k2γ F γρ + perm. ,

where “+perm.” indicates two additional sets of terms obtained from those written, one by making the interchanges k0 , µ ↔ k2 , ρ and the other by making the interchanges k1 , ν ↔ k2 , ρ. The two terms in (9.7.23) are proportional to the first power, F , and the third power, F 3 , of the external field. These arise from the Feynman amplitudes for the box and hexagon diagrams, respectively, as illustrated in Figure 6.4. For the box diagram one of the external lines is associated with the static electromagnetic field, and the other three vertices with the three waves involved; for the hexagon diagram, three of the vertices are associated with the static electromagnetic field, leading to the dependences as F and F 3 , respectively. 9.7.7 Cubic response tensor The same procedure applied to the cubic response leads to a leading term that is independent of the external field:

αc 4 k0 k1 g µν − k1µ k0ν ) k2 k3 g ρσ − k3ρ k2σ ) 2 270πBc +7k0α k1β ǫµανβ k2α k3β ǫρασβ + perm. , (9.7.24)

µ0 Π µνρσ (k0 , k1 , k2 , k2 ) =

where “+perm.” corresponds to two other pairs of terms obtained from the pair written by the interchanges k1 , ν ↔ k2 , ρ and k1 , ν ↔ k3 , σ.

9.8 Photon splitting

399

9.8 Photon splitting The theory for photon splitting is formally identical to the semiclassical theory of three-wave interactions discussed in §5.5. However, ‘photon splitting’ is usually restricted to the specific case (a) in which the quadratic nonlinear response is that of the magnetized vacuum, and (b) in which a single photon decays into two photons, both of which involve intrinsically quantum mechanical effects. 9.8.1 Kinematics of photon splitting Photon splitting in a vacuum with no external field is forbidden: the matrix element for the process is zero, due to the quadratic response tensor being zero. In the presence of an external magnetic field the quadratic response tensor is nonzero, allowing the matrix element for three-wave coupling to be nonzero. The birefringence of the magnetized implies six possible three-wave interactions involving the two modes. To see this, let a specific splitting be denoted by i → j , l, with i, j, l = 1, 2, labelling the modes; there are eight possibilities, but the two pairs i → j , l, i → l , j for j 6= l are equivalent, leaving only six different possible splittings. In the simplest (weak-field, weakdispersion) approximation only one of these (1 → 2 , 2 or ⊥→k , k) is allowed, with the others being excluded either because the matrix element is zero or because they are kinematically forbidden. More generally, the other splittings are possible in principle and the conditions under which they might occur need to be considered in detail. 9.8.2 Three-wave matching Conservation of 4-momentum in a photon splitting process, denoted γ → γ ′ , γ ′′ , requires that the three-wave matching condition, k = k ′ + k ′′ , be satisfied. This condition is ω = ω ′ + ω ′′ ,

k = k′ + k′′ .

(9.8.1)

A specific photon splitting is allowed only if the conditions (9.8.1) can be satisfied. A specific splitting is kinematically forbidden if the dispersion relations do not allow (9.8.1) to be satified for that specific splitting. A useful approximation is the weak-dispersion limit in which the refractive indices are approximated by unity. In this case of vacuum dispersion the threewave matching conditions (9.8.1) are satisfied and they require that the three photons be collinear. To show this, note when one has k 2 = 0, k ′2 = 0, k ′′2 = 0, the condition k 2 = (k ′ + k ′′ )2 requires k ′ k ′′ = 0, that is, ω ′ ω ′′ = k′ · k′′ . Then ω ′ = |k′ |, ω ′′ = |k′′ | requires that to satisfy ω ′ ω ′′ = k′ · k′′ the angle between k′ , k′′ must be zero. A similar consideration of k ′′2 = (k − k ′ )2 implies that the angle between k, k′ is π. It follows that the three photons are collinear.

400

9 Response of magnetized systems

In treating photon splitting in a magnetized vacuum the weak-dispersion approximation implies that the three photons are collinear. One is free to make a Lorentz transformation such that the initial photon is propagating perpendicular to the magnetic field. an exception is when the initial photon is propagating along B, but then photon splitting is forbidden.) Specifically, for a photon propagating in an arbitrary direction a boost along B can be made such that in the new frame the photon is propagating perpendicular to B. Then the two final photons are propagating perpendicular to B in the same direction as the initial photon. The perpendicular wavenumbers, k⊥ = ω sin θ, ′ ′′ k⊥ = ω ′ sin θ′ , k⊥ = ω ′′ sin θ′′ , and the magnetic field B) are unchanged as a result of such a boost. In the laboratory frame, one has sin θ = sin θ′ = sin θ′′ from collinearity. Hence, one choosing the frame in which all three photons are propagating perpendicular to B to evaluate the matrix elements, the matrix element is expressed in terms of the variables in the laboratory frame simply ′ by interpreting the wavenumbers according to k⊥ = ω sin θ, k⊥ = ω ′ sin θ, ′′ ′′ k⊥ = ω sin θ. 9.8.3 Probability for photon splitting The probability for photon splitting is formally identical to the probability defined by which reproduces the semiclassical result wMP Q (−k, k ′ , k ′′ ) = 4µ30

RM (k)RP (k ′ )RQ (k ′′ ) ωM (k)ωP (k ′ )ωQ (k ′′ )

′′ ×|Π MP Q (−k, k ′ , k ′′ )|2 (2π)4 δ 4 (kM − kP′ − kQ ), ν ′ ρ ′′ (2) ′ ′′ Π MP Q (−k, k ′ , k ′′ ) = e∗µ µνρ (−kM , kP , kQ ). M (k)eP (k )eQ (k )Π

(9.8.2)

where M , P , Q labels the modes. The probability (9.8.2) applies when all three wave modes are different; if the modes P , Q are the same, the factor 4 is replaced by 2. In the following the factor 4 is replaced by 2, and it is then implicit that one is to regard the two possibilities i → j , l, i → l , j as different for j 6= l, and one is to sum over them to take account of the factor of 2 omitted by this convention. In photon splitting, the modes are identified as the modes 1 and 2 discussed in §??, and the quadratic response tensor, Π (2) µνρ (k0 , k1 , k2 ), is identified as that of the magnetized vacuum, with k0 = −k, k1 = k ′ , k2 = k ′′ . For photon splitting i → j , l it is convenient to rewrite (9.8.2) as wijl (−k, k ′ , k ′′ ) = µ0

|M ijl (−k, k ′ , k ′′ )|2 (2π)4 δ 4 (k − k ′ − k ′′ ), 4ωω ′ ω ′′

ν ρ (2) ′ ′′ M ijl (−k, k ′ , k ′′ ) = e∗µ µνρ (−k, k , k ), i ej el µ0 Π

(9.8.3)

where the ratio of electric to total energy for each of the modes is well approximated by 1/2 here. The quantity M ijl (−k, k ′ , k ′′ ) is referred to as the matrix element for the process i → j , l.

9.8 Photon splitting

401

The assumption that the three photons are collinear implies that the directions defined by both of 4-vectors (8.3.2), viz. B1µ = kα F αµ , B2µ = kα F αµ , that appear in the quadratic response tensor, are the same for all three photons. Moreover, these directions are the same as the directions of the polarization vectors, eµ1 , eµ2 , respectively. These directions are orthogonal, and hence one has (e1 )µ B2µ = 0 = (e2 )µ B1µ . Furthermore, when the dispersion is neglected, then the invariants kk ′ , kk ′′ , k ′ k ′′ all vanish for collinear photons. These properties lead to the matrix elements for three of the possible six splittings being zero in the weak-field, weak-dispersion limit. 9.8.4 Photon splitting in the weak-field limit In the weak-field limit, the quadratic response tensorhas the form (9.7.23) derived from the Heisenberg-Euler Lagrangian. Consider the terms (9.7.23) that arise from the box diagram in Figure 6.4, that is, the terms linear in the strength of the external field. The terms shown explicitly are 4 k0 k1 g µν − µ ν k1 k0 k2γ F γρ and 7k0α k1β ǫµανβ k2γ F γρ . We have k0 → −k, k1 → k ′ , k2 → k ′′ here. The projection of the polarization vectors onto these terms gives zero. Term by term, this arises as follows. The term proportional to g µν gives zero due to kk ′ = 0; the term proportional to k µ k ′ν gives zero because the waves are transverse; and the projection onto kα kβ′ ǫµανβ vanishes because k, k ′ are parallel. Hence, the lowest order contibution comes from the hexagon diagram, which are the terms proportional to the cube of the external magnetic field in (9.7.23). The matrix elements for the six possible splittings, involve three classes of terms arising from the hexagon diagram in (9.7.23). The second class contains are factor k0 k1 g µν − k1µ k0ν whose projection onto the polarization vectors vanishes, as discussed above. The third class contains a factor k0α k1β ǫµανβ which vanishes for collinear, dispersionless waves. Hence, we need consider only the contribution to the matrix element arising from the first class of terms. On taking the permutations into account there are contributions from a completely symmetric term −48k0α F αµ k1β F βν k2γ F γρ ∝ B1µ B1ν B1ρ , and contributions from −26k0α F αµ k1β F βν k2γ F γρ ∝ B2µ B2ν B1ρ and two similar terms obtained the cyclic permutations of (0, µ), (1, ν), (2, ρ). The vanishing of the projections (e1 )µ B2µ = 0 = (e2 )µ B1µ implies that the first of these term gives a nonzero contribution only for the splitting 1 → 1 , 1, and the second set of terms gives a nonzero contribution only for the splittings involving one photon in mode 1 and two photons in mode 2, that is, for 1 → 2 , 2 (⊥→k , k) and 2 → 1 , 2 (k→⊥ , k). The matrix elements for the other three possible splittings are zero in this approximation. Hence, for a magentized vacuum in the weak-field limit, one has CHECK SIGNS M ijl = tijl

iαc ωω ′ ω ′′ sin3 θ, πBc

t⊥⊥⊥ =

48 B 3 , 315 Bc3

t⊥k k = tk⊥k =

26 B 3 , 315 Bc3 (9.8.4)

402

9 Response of magnetized systems

with tk⊥⊥ = t⊥⊥k =k k k = 0, and where the arguments (−k, k ′ , k ′′ ) are omitted for simplicity in writing. 9.8.5 CP invariance Adler [6] gave an argument based on the invariance of QED under the parity, P, and charge-conjugation, C, transformations to explain why half the possible splitting are forbidden, cf. (9.8.4). Adler’s argument, applied to the more general case of an arbitrary static electromagnetic field, is as follows. The invariance of QED under C implies that only even powers of the electromagnetic field appear, and hence that the theory is invariant under B → −B, E → −E. The P transformation reverses the signs of vectors but not of axial vectors, implying the QED is invariant under k → −k, B → B, E → −E. Combining these, CP invariance requires that the theory be unchanged by the replacements CP-transformation:

k → −k,

B → −B,

E → E.

(9.8.5)

Applying the CP transformation to the wave modes (8.3.15), one finds e1 → +e1 , e2 → −e2 , so that mode 1 (mode ⊥) has eigenvalue ηCP = +1 and mode 2 (mode k) has eigenvalue ηCP = −1 under this transformation. Inclusion of the small longitudinal component of the polarization of mode 2 does not change the argument. In photon splitting, CP must be conserved. Hence, the selection rule is satisfied if and only if the number of photons of mode 2 is even. Thus the allowed processes are 1 → 1 , 1, 1 → 2 , 2, 2 → 1 , 2, and the processes 1 → 1 , 2, 2 → 1 , 1, 2 → 2 , 2 are forbidden by CP invariance. In terms of the more conventional labelling of the modes, 1 →⊥, 2 →k, the allowed decays are 2 → 1,2 1 → 2,2 1 → 1,1 . (9.8.6) , , allowed decays: k→⊥ , k ⊥→k , k ⊥→⊥ , ⊥ However, this argument does not imply that the other splittings are identically zero in all cases: in the presence of a plasma the effect of the plasma on either the wave dispersion or the nonlinear response of the vacuum breaks the C symmetry. 9.8.6 Decay rates in the weak-field approximation The allowed splittings imply a nonzero absorption coefficient or decay rate for the initial photons. In natural units the absorption coefficient and the decay rate are equivalent, with the absorption coefficient being the probability per unit time that the incident photon splits, and the decay rate being the probability per unit path length that the incident photon splits. Let this rate be denoted Rijl for the splitting i → j , l.

9.8 Photon splitting

403

The decay rates for the processes allowed according to (9.8.4) or (9.8.6) are given by integrating the relevant probability over the phase space of the final photons: Z 3 ′ 3 ′′ d k d k wijl (−k, k ′ , k ′′ ). (9.8.7) Rijl = (2π)3 (2π)3

On noting the dependence of the probability (9.8.3) with (9.8.4) on k, k ′ , k ′′ , the integral that needs to be evaluated is Z 3 ′ 3 ′′ d k d k J(ω) = ωω ′ ω ′′ (2π)4 δ 4 (k − k ′ − k ′′ ) (9.8.8) (2π)3 (2π)3 It is straightforward to evaluate I in the weak-dispersion limit, k 2 = 0, k ′2 = 0, k ′′2 = 0. After performing the k′′ -integral over δ 3 (k − k′ − k′′ ) and using the dispersion relations to write |k| = ω, |k′ | = ω ′ , |k′′ | = ω ′′ , one has Z 1 Z ω dω ′ ω ′2 d cos χ ωω ′ ω ′′ δ(ω − ω ′ − ω ′′ ) (9.8.9) J(ω) = 2π 1 0

with ω ′′ = (ω 2 + ω ′2 − 2ωω ′ cos χ)1/2 , where χ is the angle between k, k′ . The cos χ-integral is performed over the remaining δ-function, and the ω ′ -integral is then elementary, giving ω5 J(ω) = . (9.8.10) 60π However, as discussed below, the neglect of dispersion is justified only if the process is kinematically allowed, and for the processes that are kinematically forbidden (8.1.17) is replaced by J(ω) = 0. The resulting decay rates for photon splitting are 6 ω 5 2 B α3c m Rijl = Nijl sin2 θ, 60π 2 Bc m N⊥⊥⊥ =

48 , 315

N ⊥k k = N k⊥k =

26 , 315

(9.8.11)

with N k⊥⊥ = N ⊥⊥k = N k k k = 0, and where Bc2 = m2 /e, αc = µ0 e2 /4π are used. 9.8.7 Kinematic restrictions The weak-dispersion approximation needs to be complemented with a detailed consideration of the implications of the actual dispersion. In the absence of dispersion, the three photons are collinear, and hence one has cos χ = 1 in (9.8.9). The inclusion of the small differences of the refractive indices from unity changes the angles between the three photons slightly. If this change tends to reduce cos χ then the process is allowed and (8.1.17) is a good approximation to the integral J(ω). However, if this change tends to increase

404

9 Response of magnetized systems

cos χ, then cos χ becomes nonphysical, and the three-wave matching cannot be satisfied. Such splittings are kinematically forbidden and in such cases (8.1.17) is to be replaced by J(ω) = 0. In the weak-field low-frequency limit, for the two modes one has k = ω(1 + δn1,2 ) with δn2 = (7/4)δn1 > δn1 (δnk > δn⊥ ), where (8.3.15) is used. Splitting is forbidden for |k| > |k′ | + |k′′ |, which is the case for the 2 → 1 , 1, 2 → 1 , 2, where the numbers refer to the modes. For the splittings 1 → 1 , 1 and 2 → 2 , 2 the ω → 0 limits of δn1,2 and inadequate to determine whether three-wave matching is possible. Adler [6] considered the frequency dependence of the refractive indices and argued that in an expansion ω/m ≪ 1 the lowest order nonzero contribution is ∝ ω 2 and is positive for both modes. When this dependence is taken into account, one finds |k| − |k′ | + |k′′ | ∝ n′′ (0)[ω ′3 + ω ′′3 − (ω ′ + ω ′′ )3 < 0, with n′′ (ω) = d2 n(ω)/dω 2 , which applies for both 1 → 1, 1 and 2 → 2, 2. It follows that both the splitting 1 → 1, 1 and 2 → 2, 2 are kinematically forbidden in the low frequency regime. At higher frequencies the refractive indices have a maximum as a function of frequency, and this kinematic argument that these splitting are forbidden is no longer valid. 9.8.8 Low-frequencies and arbitrary field strengths The matrix elements (9.8.4) are valid at low-frequencies and for weak fields. Retaining the low-frequency approximation, one may relax the weak-field approximation by deriving the them from the full Heisenberg-Euler Lagrangian, rather that using the weak-field expansion in the form (9.7.23). The nonzero matrix elements are [6]: Z ∞ iαc ωω ′ ω ′′ ds −m2 s ijl M ijl = sin3 θ e m , πB s 0 1 u2 u cosh u 1 u cosh u 3 + + + + m⊥⊥⊥ = − 4u 6 sinh u 4 6 sinh2 u 2 sinh3 u 3 1 1 3 cosh u 3u2 2 ⊥k k k⊥k + −u m =m = − . (9.8.12) 2 4u sinh u 4 2 sinh3 u sinh u with u = eBs. In the weak-field limit (9.8.12) reduces to (9.8.4), and in the strong-field limit, B ≫ Bc , (9.7.12) gives iαc ωω ′ ω ′′ sin3 θ, 3πB

iαc ωω ′ ω ′′ sin3 θ, 6πBc (9.8.13) The approximations (9.8.13) are relatively straightforward to derive by starting from the expression (9.7.11) . . .. It follows that in the strong-field limit, if they are not kinematically forbidden, ⊥→k , k and k→⊥ , k are the dominant the splittings. M ⊥⊥⊥ = −

M ⊥k k = M k⊥k = −

9.8 Photon splitting

405

9.8.9 S-matrix approach An alternative treatment of photon splitting [8] was based on the S-matrix approach. Formally, this approach is equivalent to treating the splitting using (9.8.3) with the quadratic response tensor identified as the vacuum contributions to the form (9.7.10) with (9.7.11). Thus, the S-matrix elements involve sums over quantum numbers. The sum over the spins is straightforward, and the integral over pz . although tedious, can be performed using elementary methods. The sum over n remain, and its evaluation contains some subtleties [9]. This alternative approach leads to relatively cumbersome expressions in general, but can be simpler to use for B > ∼ Bc when the sum over n converges rapidly. The S-matrix contains terms of with odd powers of the frequencies, and only the terms cubic in the frequencies (proportional to ωω ′ ω ′′ ) is considered here. After summing over the spins and integrating over pz , these terms are of the form ∞ X 4π 2 (4παc )3/2 (ωω ′ ω ′′ )1/2 ′ ′′ ′ ′′ δ(k − k − k )δ(ω − ω − ω ) T (n), x x x (2V )3/2 n=0 (9.8.14) where the photons are collinear along the x axis. The T (n) do not depend on the frequencies, and the remaining factors do not depend on B. Explicit expressions for the T (n) → Tijl (n) for the two independent splittings, ijl =⊥kk, ijl =⊥⊥⊥, are relatively cumbersome: 2n 2B(n + 2) 2(n + 1) 1 h 8B 2 n 2B(n − 1) + 2 − + 2 T⊥kk (n) = − 4 + 2 2B 3ε0,n 3ε0,n ε0,n 3ε20,n+1 ε0,n+1 h i ε2 1 7 7 0,n+1 +3(2n + 1) − 2n3 + n2 12 + + n 11 + +1+ ln B B 2B ε20,n h n2 1 n 2 1 i ε20,n+2 i + n3 + ; (9.8.15) 7+ + 7+ +1+ ln 2 B 2 B 2B ε20,n

Sf i =

T⊥⊥⊥ (n) = +

1 n 6B(n + 1)(n + 2)2 4B 2 (n + 1)2 − + 2 2B ε0,n+2 ε40,n+1

B(n + 1)(12n2 + 24n + 9) 4Bn2 (B + 2) Bn(−6n2 + 2n + 3) − + ε20,n+1 ε40,n ε20,n 1h 24 6 i ε20,n+1 −12n − 3 − 23n3 + 58n2 + n 73 − + 38 − ln 4 B B ε20,n 1h 6 6 i ε20,n+2 − 11n2 + n 27 + + 16 + ln 2 2 B B ε0,n+1 h i ε2 3 2 1 7n 3 37n 3 0,n+2 + + 10 + ln + + n 25 + 2 2 2 2 B B ε0,n

406

9 Response of magnetized systems

2 1

-1

0

2

4

6

Fig. 9.3. The rates of splitting of a perpedicular photon into two perpendicular photons (light circles) and into two parallel photons (dark circles) are plotted on a log-log scale (tick marks represent powers of 10) as functions of B/Bc .

+

o i ε2 1 h 3n3 33n 0,n+3 , + 9n2 + + 9 ln 2 2 2 ε20,n

(9.8.16)

with ε0,n = m(1 + 2nB/Bc )1/2 . The decay rate is Rijl =

1 α3c m ω 5 4 60π 2 m

∞ X 2 Tijl (n) .

(9.8.17)

n=0

In practice, the sum over n must be terminated as some nmax . The EulerMaclaurin summation formula is used to include the contributions from n > nmax : ∞ X

n=0

T (n) = T (0) + · · · + T (i − 1) + 12 T (i) +

Z

i

∞

dn T (n) −

1 ′′′ 1 ′ T (i) + T (i) + · · · . 12 720

(9.8.18)

where primes denote derivatives. The choices of i and of the number of derivatives of T (i) to be retained are made by trial and error, such that increasing the numbers does not lead to siginificant improvement beyond some predetermined accuracy. The sum over n converges increasingly rapidly with increasing B, allowing the sum to be truncated at small values for B > ∼ Bc . As shown in Figure 6.5, for the splitting of photons in mode 1 (⊥), the weak-field dominance of the channel ⊥→⊥ , ⊥ compared to the channel ⊥→k , k for B ≪ Bc , revereses for B > ∼ Bc , and the relative rate for ⊥→⊥ , ⊥ becomes negligible for B ≫ Bc . Moreover, [12] found that only ⊥→k , k splitting is allowed for B ≫ Bc . 9.8.10 Photon splitting in an electromagnetic wrench A general expression for photon splitting in an electromagnetic requires a general expression for the quadratic response tensor, which is given formally

References

407

by (9.7.14) with the propagator identified as that for an electromagnetic wrenchThe general expressions are very cumbersome and are not written down here. To evaluate the matrix elements in the low-frequency limit, the approximations in which the photons are collinear and propagating along the x axis imply that one requires only the components of the quadratic response tensor with µ, ν, ρ = y, z provided that the longitudinal component of the polarization of mode 2 is neglected. Due to the symmetry properties, only the four components µνρ = yyy, zzz, yzz, zyy need be shown explicitly. These are of the form Z ∞ iαc e3 EBωω ′ ω ′′ ds −m2 s ijl M ijl = sin3 θ e m , π s 0 3 cot w 3w2 cos w cosh u cos w − + myyy = s2 2 sinh4 u 2su(u2 + w2 ) sin w sinh3 u sin w sinh2 u 3w2 cos w 3w(u2 − w2 ) cosh u 1 3ws(u2 + w2 ) − + , + 2 − (u + w2 )2 4u sin2 w sinh u 2 sin w sinh2 u 4 sin2 w sinh2 u (9.8.19)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20.

Melrose and Stoneham (1976) Melrose and Stoneham 1977) Shabad (1975) J. Schwinger 1951, PR 82 , 664 A. Minguzzi 1956, Nuovo Cim. 7, 501 S.L. Adler (1971) Ann. Phys. 67, 599 W.-Y. Tsai, T. Erber (1974) PRD 10 492 Mentzel, Berg & Wunner Phys. Rev. D 50, 1125 (1994) Weise, Baring & Melrose (1998) Stoneham (1979) 00 Bogoliubov and Shirkov (1959) Baier, V.N ., Milstein, A.I., and Shaisultanov, R.Zh. Sov. Phys. JETP 63, 665 (1996) bibitemT61 Tsytovich V N 1961 Sov. Phys.-JETP 13 1249 Jancovici B 1962 Nuovo Cimento 25 428 Hayes L M and Melrose D B 1984 Aust. J. Phys. 37 615 & 639 Kowalenko V, Frankel N E and Hines K C 1985 Phys. Rep. 126 109 Itoh N.et al. 1992 Astrophys. J. 395 622 Braaten E and Segel D 1983 Phys. Rev. D48 1478 Melrose D B 2008 Quantum plasmadynamics Unmagnetized plasmas, Springer, New York Svetozarova G N and Tsytovich V N 1962 Izv. Vuzov. Radiofiz. 5 658 Melrose D B 1974 Plasma. Phys. 16 845

408 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

9 Response of magnetized systems Cover R A, Kalman G and Bakshi P 1979 Phys. Rev. D 20 3015 P´erez Rojas H and Shabad A E 1979 Ann. Phys. 121 432 P´erez Rojas H and Shabad A E 1982 Ann. Phys. 138 1 Delsante A E and Frankel N E 1980 Ann. Phys. 125 135 Melrose D B and Parle A J 1983 Aust. J. Phys. 36 755 & 799 Pulsifer P and Kalman G 1992 Phys. Rev. A 45, 5820 Shabad A Ye 1991 Polarization of the vacuum and a quantum relativistic gas in a magnetic field, Nova Science Publ., New York Sokolov A A and Ternov I M 1968 Synchrotron radiation, Akademie, Berlin Weise J I 2008 Phys. Rev. E 78 046408 Hardy S J and Thoma M K 2000 Phys. Rev. D 63 025014 Baring M G and Harding A K 2007 Astrophys. Space Sci. 308 109 Ritus V I 1970 Sov. Phys. JETP Lett. 12 289 Abramowitz M and Stegun I A 1965 Handbook of Mathematical Functions, Dover, New York Godfrey B B, Newberger B S and Taggart K A 1975 IEEE Trans. Plasma Sci. PS-3 60 Padden W E P 1992 Aust. J. Phys. 45 165 Canuto V and Ventura J 1972 Astrophys. Space Sci. 18 104 Gonthier P L, et al. 2000 Astrophys. J. 540 907 Cutkovsky R E 1960 J. Math. Phys. 1 429 Melrose D B 1997 J. Plasma Phys. 57 479 D.C. Kelly 1964 Phys. Rev. 134, A641–A649 M.P. Greene, H.J. Lee, J.J. Quinn, S. Rodriguez 1969 Phys. Rev. 177, 1019– 1036 P.S. Zyryanov, V.P. Kalashnikov 1962 Sov. Phys. JETP 14, 799 J.J. Quinn, S. Rodriguez 1962 Phys. Rev. 128, 2487

A Special functions

A.1 Bessel functions and J -functions A.1.1 Ordinary Bessel functions The expansion in Bessel functions is based on the generating function e

iz sin φ

=

∞ X

einφ Jn (z).

(A.1.1)

n=−∞

The recursion formulas n Jn−1 (z) + Jn+1 (z) = 2 Jn (z), z

(A.1.2)

Jn−1 (z) − Jn+1 (z) = 2Jn′ (z).

(A.1.3)

Sum rules for Bessel functions: ∞ X

Js2 (z) = 1,

s=−∞

∞ X

sJs2 (z) = 0,

s=−∞ ∞ X

∞ X

Js (z)Js′ (z) = 0,

s=−∞ ∞ X

s2 Js2 (z) = 12 z 2 ,

Js′2 (z) = 21 .

(A.1.4)

s=−∞

s=−∞

A.1.2 Modified Bessel functions ez cos φ =

∞ X

Is (z)e±isφ ,

(A.1.5)

s=−∞

then gives what is effectively an expansion in gyroharmonics. These functions satisfy the differential equation 1 ′ s2 ′′ Is (z) + Is (z) − 1 + 2 Is (z) = 0, (A.1.6) z z

410

A Special functions

and the recursion relations Is−1 (z) − Is+1 (z) = 2(s/z)Is (z),

Is−1 (z) + Is+1 (z) = 2Is′ (z).

(A.1.7)

A.1.3 Macdonald functions Recursion relations Kν−1 (z) − Kν+1 (z) = −2(ν/z)Kν (z),

Kν−1 (z) + Kν+1 (z) = −2Kν′ (z).

An integral representation of the Macdonald function Kν (x) is Z (x/2)ν Γ ( 12 ) ∞ Kν (x) = dχ sinh2ν χ e−x cosh χ Γ (ν + 12 ) 0

(A.1.8)

(A.1.9)

The Gamma function satisfies Γ (x + 1) = xΓ (x),

Γ (1) = 1,

Γ ( 21 ) = π 1/2 .

(A.1.10)

The integral (A.1.9) also applies when ν is negative, and then K−ν (x) = Kν (x) implies Z (x/2)−ν Γ (ν + 12 ) cos πν ∞ e−x cosh χ Kν (x) = , (A.1.11) dχ 1 Γ(2) sinh2ν χ 0 where the identity Γ ( 21 + ν)Γ ( 12 − ν) =

π cos πν

(A.1.12)

is used. A.1.4 Properties of Kν (x) The following are some standard properties of the Macdonald functions, Kν (x) [1, 2, 3]. The Macdonald functions Kν (x) are modified Bessel functions of order ν. They satisfy the differential equation 1 d ν2 d2 K (x) + K (x) − 1 + Kν (x) = 0, (A.1.13) ν ν dx2 x dx x2 and the recursion relations ν Kν−1 (x) − Kν+1 (x) = −2 Kν (x), x d Kν−1 (x) + Kν+1 (x) = −2 Kν (x). dx

(A.1.14) (A.1.15)

A.1 Bessel functions and J-functions

411

One also has K−ν (x) = Kν (x). The recursion relations imply 1 d ±ν x Kν (x) = −x±ν−1 Kν∓1 (x). x dx

(A.1.16)

The expansion of Kν (x) for small x is

Kν (x) ≈ 2ν−1 Γ (ν) x−ν .

(A.1.17)

The asymptotic expansion for large x is π 1/2 4ν 2 − 1 (4ν 2 − 1)(4ν 2 − 9) −x Kν (x) = e 1+ + · · · . (A.1.18) + 2x 8x 128x2 Another class of functions related to Kn (x) K0 , denoted by Kin (ρ). One has  K0 (ρ)    R∞ ρ dx Kin−1 (x) Kin (ρ) = |n|    Ki (ρ) = (−)n d  K0 (ρ) n dρ|n|

are the multiple integrals of for n = 0, for n > 0,

(A.1.19)

for n < 0.

These functions satisfy the recurrence relation

rKir+1 (ρ) = −ρKir (ρ) + (r − 1)Kir−1 (ρ) + ρKir−2 (ρ), and they have the integral representation Z ∞ e−ρ cosh χ Kin (ρ) = dχ . coshn χ 0

(A.1.20)

(A.1.21)

Note that (A.1.20) allows one to write an arbitrary Kin (ρ) in terms of any other plus a combination of Macdonald functions. It is conventional to choose Ki2 (ρ) as the only one to appear explicitly. For example, (A.1.20) implies Ki1 (ρ) = −Ki2 (ρ)/ρ + K1 (ρ). Expansions of Kin (x) give !  1 n+1 1 n−1 Γ ( Γ ( )Γ (n + ) )Γ ( )  2 2 2 2   1− x + ··· for x ≪ 1, n 2  2Γ (n + 1) Γ ( ) 2 Kin (x) =   π 1/2 −x 1 + 4n   e + ··· for x ≫ 1. 1− 2x 8x (A.1.22) A.1.5 Airy functions Z 1 ∞ Ai (z) = dt cos(zt + 31 t3 ), π 0

1 Gi (z) = π

Z

0

∞

dt sin(zt + 13 t3 ). (A.1.23)

412

A Special functions

The approximations available for Gi (z) are for large and small z. The leading terms in the asymptotic expansion for z ≫ 1 are [4] 1 1 1 1 2 ′ − 2 + ··· , Gi (z) ∼ Gi (z) ∼ + 4 + ··· , π z z π z Z z 2 1 2C + ln 3 (A.1.24) − 3 + ··· , dz ′ Gi (z ′ ) ∼ ln z + π 3 3z where C = 0.577 · · · is Euler’s constant. The expansion for z ≪ 1 gives 1 31/3 32/3 z2 Gi (z) = Γ (4/3) + Γ (5/3) z − + ··· , π 2 4 2 Gi (0) = 0.205,

Gi′ (0) = 0.149.

(A.1.25)

Rothman [4] found that the asymptotic expansion is accurate for z > ∼ 8 and tabulated the functions for lower z. Z ∞ π 2 (1 − 4µ2 ) , (A.1.26) dξ ξ 2 Kµ2 (ξ) = 32 cos πµ 0 A.1.6 J-functions Definition The J-functions used here are defined by 1/2 n! n ν n+ν Jν (x) = (−) J−ν (x) = e−x/2 xν/2 Lνn (x). (n + ν)!

(A.1.27)

with Lνn (x) the generalized Laguerre polynomial, defined by Lνn (x) =

ex x−ν dn −x n+ν (e x ). n! dxn

(A.1.28)

Sokolov and Ternov function The function defined by Sokolov and Ternov [5, 6] is related to (A.1.27) by ′

n In,n′ (x) = Jn−n ′ (x).

(A.1.29)

Recursion formulae The J-functions satisfy recursion relations n−1 x1/2 Jν+1 (x) = (n + ν)1/2 Jνn−1 (x) − n1/2 Jνn (x),

(A.1.30)

n x1/2 Jν−1 (x) = −n1/2 Jνn−1 (x) + (n + ν)1/2 Jνn (x),

(A.1.31)

A.1 Bessel functions and J-functions

413

and also n−1 n νJνn−1 (x) = x1/2 n1/2 Jν+1 (x) + (n + ν)1/2 Jν−1 (x) , n−1 n νJνn (x) = x1/2 (n + ν)1/2 Jν+1 (x) + n1/2 Jν−1 (x) .

(A.1.32) (A.1.33)

A further pair of relations that is similar to the recursion relations for Bessel functions is n n (x + ν)Jνn (x) = [x(n + ν)]1/2 Jν−1 (x) + [x(n + ν + 1)]1/2 Jν+1 (x),

(A.1.34)

d n n n J (x) = [x(n + ν)]1/2 Jν−1 (x) − [x(n + ν + 1)]1/2 Jν+1 (x). dx ν

(A.1.35)

2x

Sum rules The sum rules ∞ X

′

n′ =0 ∞ X

′

′′

n n nn Jn−n , ′ (x)Jn′′ −n′ (x) = δ

(A.1.36)

′

n 2 (n′ − n)[Jn−n ′ (x)] = x,

(A.1.37)

n′ =0

were derived by Sokolov and Ternov [5]. Orthogonality relation Z

0

∞

′

′

dx Jνn (x)Jνn (x) = δ nn .

(A.1.38)

Integral identities Z

∞

1/2

dxx

0

Z

0

∞

[Jνn (x)]2

1/2

= (n + ν + 1)

n + 12 1+ 4(n + ν + 1)

dx x [Jνn (x)]2 = 2n + ν + 23 ,

,

(A.1.39) (A.1.40)

Particular values ν+1 Jν0 (x) = (−)ν J−ν (x) =

xν/2 e−x/2 (ν!)1/2

(A.1.41)

ν+1 Jν1 (x) = (−)ν J−ν (x) =

xν/2 e−x/2 (ν + 1 − x), ((ν + 1)!)1/2

(A.1.42)

414

A Special functions ν+2 Jν2 (x) = (−)ν J−ν (x) =

xν/2 e−x/2 (2!(ν + 2)!)1/2

×[(ν + 1)(ν + 2) − 2(ν + 2)x + x2 ], ν+3 Jν3 (x) = (−)ν J−ν (x) =

(A.1.43)

xν/2 e−x/2 [(ν + 1)(ν + 2)(ν + 3) (3!(ν + 3)!)1/2

−3(ν + 2)(ν + 3)x + 3(ν + 3)x2 − x3 ],

(A.1.44)

Approximations For x ≪ 1, the J-functions may be approximated by the leading term in their expansion in powers of x: Jnn′ −n (x)

=

n′ ! n!

1/2

′

x(n −n)/2 , (n′ − n)!

(A.1.45)

which applies for n′ ≥ n. The limit x → 0 gives J0n (0) = 1,

Jνn (0) = 0

for ν 6= 0.

(A.1.46)

The expansion of the J-functions in terms of Bessel functions, Jνn

z2 4n

=

(n + ν)! n!nν

b0 = 1, b1 = − 21 (ν (a + 1)ba+1 = − 12 (ν +

1/2 X ∞

ba

+ 1),

b2 = 18 (ν + 1)(ν + 2),

a=0

z a Jν+a (z), 2n

1)ba + (ν + a)ba−1 − 41 nba−2 , 1 4

converges rapidly for sufficiently large n,

(A.1.47)

A.2 Plasma dispersion functions

415

A.2 Plasma dispersion functions A.2.1 Relativistic thermal function T (z, ρ) The function T (z, ρ), defined by (2.3.25), has alternative integral representations: Z ∞ z + tanh χ dχ sinh χ e−ρ cosh χ ln T (z, ρ) = −ρ z − tanh χ 0 Z ∞ −ρ cosh χ e = 2z dχ 2 ) cosh2 χ − 1 (1 − z 0 Z z K1 (ρR) 2ρ dζ =− . (A.2.1) 1 − z2 R The first form follows directly from (2.3.25) with the variable χ defined by (??), and the second form is related to it by a partial integration. The third form is the real part of a Trubnikov function, cf. (??) and (A.2.12) below. The function T (z, ρ) satisfies a set of partial differential equations that includes (9.4.26) and (??). The full set is [7]: (1 − z 2 )

∂2 T (z, ρ) = 2zK0(ρ) + T (z, ρ), ∂ρ2

z(1 − z 2 )3 T ′′ (z, ρ) − (1 − z 2 )2 (1 + 2z 2 ) T ′ (z, ρ) − ρ2 z 3 T (z, ρ)

= 2z 2 ρ2 K0 (ρ) + 2(1 − z 2 )ρK1 (ρ),

z

∂ (1 − z 2 ) ′ T (z, ρ) = 2K1 (ρ) + T (z, ρ), ∂ρ ρ

(A.2.2)

(A.2.3) (A.2.4)

with T ′ (z, ρ) = ∂T (z, ρ)/∂z, T ′′ (z, ρ) = ∂ 2 T (z, ρ)/∂z 2. A.2.2 Trubnikov functions Another class of relativistic plasma dispersion functions appear in Trubnikov’s form for the response functions, cf. (??), (??). A general class of Trubnikov functions is defined by writing Z ∞ Kν r(ξ) , (A.2.5) tnν (z, ρ) = (k˜ u)n+1 dξ ξ n rν (ξ) 0 with the argument of the Macdonald functions given by (??), and where the power of k˜ u is included so that the integral is dimensionless. Expressions relating these functions to T (z, ρ), T ′ (z, ρ) follow from recursion formulas that allow them all to be generated from the simplest of them, once this is expressed in terms of T (z, ρ), T ′ (z, ρ). Recursion formulas are obtained as follows. First, using the identity (??) with f (ξ) = (k˜ u ξ)n , one obtains

416

A Special functions

  Kν (ρ) 2 2 for n = 0, z iρz n+1 n ρν t (z, ρ) + tν+1 (z, ρ) = 1 − z 2 ν+1 1 − z 2  n−1 ntν (z, ρ) for n > 0.

(A.2.6)

Next, differentiate (A.2.5), using the identity (A.1.16), to obtain n tn+1 ν+1 (z, ρ) = −iρ tν+1 (z, ρ) − i

∂tnν (z, ρ) . ∂ρ

These two equations can be combined to give  2 2  Kν (ρ) n for n = 0, 1 − z ∂tν (z, ρ) iz ρν tnν+1 (z, ρ) = − + ρ ∂ρ ρ  n−1 n tν (z, ρ) for n > 0.

(A.2.7)

(A.2.8)

A further identity follows by differentiating (A.2.5) with respect to z: 3 tn+2 ν+1 (z, ρ) = z

∂tnν (z, ρ) . ∂z

(A.2.9)

A convenient starting point for generating all the functions follows by considering the integral Z ∞ ∂T (z, ρ) e−ρ cosh χ =− dχ ∂ρ sinh χ − z cosh χ −∞ Z ∞ Z ∞ =i dξ dχ e−ρ cosh χ+i(sinh χ−z cosh χ)ξ , (A.2.10) 0

−∞

where (??) is used. The final integral is evaluated using Trubnikov’s method, leading to the identity t00 (z, ρ) =

i (1 − z 2 ) ′ iz ∂T (z, ρ) = 2K1 (ρ) + T (z, ρ) , 2 ∂ρ 2 ρ

(A.2.11)

where (A.2.4) is used. Then (A.2.8) with n = 0 allows one to construct all the functions t0ν (z, ρ) for n = 0 and ν > 0. In particular, one finds t01 (z, ρ) = −

iz T (z, ρ), 2ρ

(A.2.12)

which gives the final expression for T (z, ρ) in (A.2.1). The functions with n > 0 are generated from those with n = 0 by using (A.2.6). The specific functions that appear in Trubnikov’s form (??), (??) for the response tensors are t02 (z, ρ) − t23 (z, ρ)/z 2, t02 (z, ρ) and using the foregoing results to evaluate these, one may demonstrate the equivalence of Trubnikov’s forms (??), (??) for αL (k), αT (k) and the forms of (??), (??) in terms of T (z, ρ), T ′ (z, ρ).

A.2 Plasma dispersion functions

417

A.2.3 Shkarofsky and Dnestrovskii functions The generalized Shkarofsky functions are defined by (??) for real q, integer r > 0 and complex z, a with Im (z − a) > 0 by Z ∞ (it)r at2 Fq,r (z, a) = −i dt exp izt − (1 − it)q 1 − it 0 Z ∞ r a (it) exp i(z − a)t + = −ie−a . (A.2.13) dt (1 − it)q 1 − it 0 The definition is extended to Im (z −a) < 0 by analytic continuation. Generalized Dnestrovskii functions are defined by (??), viz. Fq,r (z) = Fq,r (z, 0). The usual Shkarofsky functions, (??), and Dnestrovskii functions, (2.4.30), are the special cases (??), viz. Fq (z, a) = Fq,0 (z, a), Fq (z) = Fq,0 (z), respectively. The Shkarofsky functions and the Dnestrovskii functions are related by an expansion in modified Bessel functions, cf. (A.1.5), applied to (??) with r = 0, which gives ∞ X e−2a Is (2a) Fq−s (z). (A.2.14) Fq (z, a) = s=−∞

The expansion (A.2.14) is used to find corrections to the case of perpendicular propagation through an expansion in a = kk2 c2 ρ/ω 2 . A.2.4 Recursion relations and differential equations Recursion relations satisfied by the Shkarofsky functions are aFq−2 (z, a) = 1 + (a − z)Fq (z, a) − qFq+1 (z, a),

(A.2.15)

= Fq (z, a) − Fq−1 (z, a),

(A.2.16)

Fq′ (z, a)

Fq′′ (z, a) = Fq (z, a) − 2Fq−1 (z, a) + Fq−2 (z, a),

(A.2.17)

where a prime denotes a derivative with respect to z. Eliminating Fq−1 (z, a) and Fq−2 (z, a) between these gives a second order differential equation satisfied by the Shkarofsky functions: (a−z)Fq′′ (z, a)−[2(a−z)−q−2]Fq′ (z, a)−(z +q−2)Fq (z, a)+1 = 0. (A.2.18) Recursion relations for the Dnestrovskii functions follow from (A.2.15) and (A.2.16) for a = 0: (q − 1)Fq (z) = 1 − zFq−1 (z),

Fq′ (z) = Fq (z) − Fq−1 (z).

(A.2.19) (A.2.20)

Eliminating Fq−1 (z) between these gives a first order differential equation satisfied by the Dnestrovskii functions:

418

A Special functions

zFq′ (z) = (z + q − 1)Fq (z) − 1.

(A.2.21)

The function Fq (z) also satisfies (A.2.18) with a = 0. Equation (A.2.20) integrates to give Z ∞ Fq (z) = z q−1 ez Γ (1 − q, z), Γ (q, z) = dζ ζ q−1 e−ζ , (A.2.22) z

where Γ (q, z) is the incomplete gamma function. A.2.5 Limiting cases The expansion of the Dnestrovskii functions for small arguments z follows from (A.2.22) and the relevant expansion of the incomplete gamma functions: Fq (z) = z q−1 ez Γ (1 − q) − =z

q−1 z

∞ X j

e Γ (1 − q) − e

z

z j Γ (1 − q) Γ (j + q − 1)j!

∞ X (−z)j Γ (1 − q) j

Γ (j + 2 − q)

.

(A.2.23)

For real, positive z there is an expansion in generalized Laguerre polynomials: Fq (z) =

(1−q) ∞ X (z) Lj j=0

j+1

.

(A.2.24)

For large argument, |z| ≫ 1, the limit Fq (z) ∼

∞ X

(−1)j z −1−j Γ (q + j)

(A.2.25)

j=0

applies for arg (z) < 3π/2. A.2.6 Half-integer q In evaluating (2.4.22) in terms of Shkarofsky functions, the function and its derivative with q = 5/2 appear. The expansion (2.4.34) then leads to Dnestrovskii functions with half-integer q. For q a positive half-integer, the Dnestrovskii functions are expressible in terms of the plasma dispersion function of Fried and Conte (1961), (??), viz. Z(z) = π −1/2

Z

∞

−∞

2

dt

2 φ(z) e−t =− + iπ 1/2 e−z , t−z z

where φ(u) is defined by (??). The relevant form is

(A.2.26)

A.3 Dirac algebra

419

q−3/2

Γ (q)Fq (z) =

X j=0

(−z)j Γ (q − 1 − j) + π 1/2 (−z)q−3/2 [iz 1/2 ez Z(iz 1/2 )]. (A.2.27)

Expansions for small and large arguments are  ∞ X    (−z)j Γ (q − 1 − j) − iπ(−z)q−1 ez   j=0 Γ (q)Fq (z) = ∞ X    Γ (q + j)(−z)−1−j − iσπ(−z)q−1 ez  − j=0

for |z|2 ≪ 1, for |z| ≫ 1,

(A.2.28) with σ = 0 for arg z < π, σ = 1 for arg z = π and σ = 2 for π < arg z < 2π.

A.3 Dirac algebra In this section some results associated with the properties of Dirac matrices are summarized. A.3.1 Definitions The Dirac matrices are defined to satisfy γ µ γ ν + γ ν γ µ = 2g µν ,

(A.3.29)

where the unit Dirac matrix is implicit on the right hand side. The Dirac Hamitonian is ˆ =α·p ˆ + βm, H

α = γ 0 γ,

β = γ0,

(A.3.30)

The requirement that the Dirac Hamiltonian be self-adjoint implies (γ µ )† = γ 0 γ µ γ 0 .

(A.3.31)

A.3.2 Standard representation The specific choice for the Dirac matrices used here is referred to as the standard representation. It corresponds to     1 0 0 0 0 0 0 1 0 1 0 0  0 0 1 0   γ0 =  γ1 =   0 0 −1 0  ,  0 −1 0 0  , 0 0 0 −1 −1 0 0 0     0 0 1 0 0 0 0 −i  0 0 0 −1   0 0 i 0   (A.3.32) γ3 =  γ2 =   −1 0 0 0  .  0 i 0 0, 0 1 0 0 −i 0 0 0

420

A Special functions

A convenient way of writing these and other 4× 4 matrices is in terms of block matrices. Let 0 and 1 be the null and unit 2 × 2 matrices. One writes 0 1 σ 0 , , ρx = Σ= 1 0 0 σ 1 0 0 −i1 , (A.3.33) , ρz = ρy = 0 −1 i1 0 where the 2 × 2 matrices 0 1 0 −i σx = , σy = , 1 0 i 0

σz =

1 0 0 −1

,

(A.3.34)

are the usual Pauli matrices. In this representation one has γ µ = [ρz , iρy Σ],

α = ρx σ,

β = ρz .

(A.3.35)

A.3.3 Dirac matrices σ µν and γ 5 Two additional Dirac matrices that play an important role in the theory are σ µν = 12 [γ µ , γ ν ],

(A.3.36)

which plays the role of a spin angular momentum, and γ 5 = −iγ 0 γ 1 γ 2 γ 3 ,

(A.3.37)

which satisfies the relations γ µ γ 5 + γ 5 γ µ = 0,

(γ 5 )2 = 1,

(γ 5 )† = γ 5 .

(A.3.38)

One also has γ µ γ ν γ ρ γ σ γ 5 = −iǫµνρσ .

(A.3.39)

5

In the standard representation one has γ = −ρx . The spin 4-tensor σ µν , defined by (A.3.36), has components   0 αx αy αz  −αx 0 −iσz iσy  . (A.3.40) σ µν =   −αy iσz 0 −iσx  −αz −iσy iσx 0

Different definitions of σ µν and γ 5 are used in the literature. The choices made here are those made in [?]. In particular, note that the many authors choose γ 5 with the opposite sign, and that this affects the sign of the projection operators for neutrinos, cf. (??).

References

421

A.3.4 Basic set of Dirac matrices There are sixteen independent 4 × 4 matrices and for the Dirac matrices it is sometimes convenient to choose a set of 16 basis vectors. A specific choice of 16 independent matrices is the set h i γ A = 1, γ µ , iσ µν , iγ µ γ 5 , γ 5 . (A.3.41)

This choice involves a scalar and a pseudo scalar (1, γ 5 ), a 4-vector and a pseudo 4-vector (γ µ , iγ µ γ 5 ) and an antisymmetric second rank 4-tensor (σ µν ). These have 1, 1, 4, 4, and 6 components, respectively. This set is chosen such that the analogous set, γA with indices down, γA = [1, γµ , iσµν , iγµ γ 5 , γ 5 ] satisfy A γ A γA = 1 (no sum), γ A γB = δB . (A.3.42) The expansion of an arbitrary Dirac matrix, O say, in this basis gives X (A.3.43) cA γ A , cA = 14 Tr [γA O]. O= A

A.3.5 Traces of products of γ-matrices The traces of products of γ-matrices are important in detailed calculations in QED. Consider T α1 α2 ...αn = Tr γ α1 γ α2 . . . γ αn . (A.3.44)

The trace of γ µ is zero, as are the traces of σ µν , γ µ γ 5 and γ 5 . The trace of a product of an odd number of γ-matrices is also zero: T α1 α2 ...αn = 0 for n odd. The trace of a product of two γ-matrices is nonzero. This trace is evaluated as follows. First the invariance of the trace of a product of matrices under cyclic permutations of the matrices implies T µν = T νµ . The trace of (A.3.29) implies T µν = 4g µν , where the factor of 4 arising from the trace of the unit 4 × 4 matrix. Using the invariance of the trace under cyclic permutations and (A.3.29) allows one to evaluate the traces (A.3.44) for all even n. One finds i h (A.3.45) T µν = 4g µν , T µνρσ = 4 g µν g ρσ − g µρ g νσ + g µσ g νρ ,

i h T µνρσαβ = 4 g µν T ρσαβ − g µρ T νσαβ + g µσ T νραβ − g µα T νρσα + g µβ T νρσα , (A.3.46) and so on.

References 1. Watson (1944)

422 2. 3. 4. 5. 6. 7.

A Special functions Gradshteyn and Ryzhik (1965) Abramowitz and Stegun (1965) Rothman (1954) Sokolov and Ternov (1968) Sokolov and Ternov (1986) Godfrey et al.(1975)

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