Propagation of Electromagnetic Waves in Plasmas

Abstract

Green's function techniques are used to treat the propagation of electromagnetic waves in uniform, weakly interacting plasmas near equilibrium in the absence of external magnetic fields. The frequency and the damping of electromagnetic waves in a medium are related to the local complex conductivity tensor, which is calculated by the diagrammatic techniques of modern field theory. Physical quantities are calculated in terms of a consistent many-particle perturbation expansion in powers of a (weak) coupling parameter. An open-diagram technique is introduced which simplifies the calculation of absorptive parts. For long-wavelength longitudinal waves (i.e., electron plasma oscillations) it is found that the main absorption mechanism in the electron-ion plasma is the two-particle collision process appropriately corrected for collective effects and not the one-particle (or Landau) damping process. Electron-ion collisions produce a damping effect which remains finite for long wavelengths. The effect of electron-electron collisions vanishes in this limit. The absorption of transverse radiation is also considered; calculations for the electron-ion plasma are in essential agreement with the recent work of Dawson and Oberman. The results for the absorptive part of the conductivity tensor for long-wavelength electromagnetic waves in a plasma where the phase velocity $\frac{\ensuremath{\omega}}{k}$ is much greater than the rms particle velocity is for the electron-ion plasma: $4\ensuremath{\pi} \mathrm{Im}{\ensuremath{\sigma}}_{\mathrm{ij}}(\mathbf{k},\ensuremath{\omega})=\frac{{\ensuremath{\Omega}}_{p}}{6\sqrt{2}{\ensuremath{\pi}}^{\frac{3}{2}}}\frac{\ensuremath{\Omega}_{p}^{}{}_{}{}^{2}}{{\ensuremath{\omega}}^{2}}\frac{k_{D}^{}{}_{}{}^{2}}{n}\mathrm{ln}\left(\frac{{C}_{a}(\ensuremath{\omega})}{\ensuremath{\beta}\ensuremath{\hbar}\ensuremath{\omega}}\right){\ensuremath{\delta}}_{\mathrm{ij}}$ where $\ensuremath{\Omega}_{p}^{}{}_{}{}^{2}=4\ensuremath{\pi}{e}^{2}n{m}^{\ensuremath{-}1}$, $k_{D}^{}{}_{}{}^{2}=4\ensuremath{\pi}{e}^{2}n\ensuremath{\beta}$, and $\ensuremath{\beta}={(\mathrm{kT})}^{\ensuremath{-}1}$. The effects of dynamic screening are entirely contained in definite integral ${C}_{a}(\ensuremath{\omega})$ which is numerically evaluated. The calculations are valid for temperatures and densities which satisfy the inequalities: ${(\frac{4\ensuremath{\pi}{e}^{2}n}{m})}^{\frac{3}{2}}{(\frac{\ensuremath{\hbar}}{\mathrm{kT}})}^{3}\ensuremath{\ll}{(\frac{4\ensuremath{\pi}{e}^{2}n}{\mathrm{kT}})}^{\frac{3}{2}}{n}^{\ensuremath{-}1}\ensuremath{\ll}{(\frac{4\ensuremath{\pi}{e}^{2}n}{m})}^{\frac{1}{2}}(\frac{\ensuremath{\hbar}}{\mathrm{kT}})\ensuremath{\ll}1.$ Reading from left to right these inequalities justify the use of Boltzmann statistics, the Born approximation, and the neglect of wave mechanical interference effects. The weak-coupling approximation is justified by ${(\frac{4\ensuremath{\pi}{e}^{2}n}{\mathrm{kT}})}^{\frac{3}{2}}{n}^{\ensuremath{-}1}\ensuremath{\ll}1$. These restrictions are satisfied, for example, if $T>{10}^{6}$ \ifmmode^\circ\else\textdegree\fi{}K and $n<{10}^{20}$ particles/${\mathrm{cm}}^{3}$. For these hot plasmas a natural short-wavelength cutoff appears at roughly the thermal de Broglie wavelength. Electrons and ions are found to produce comparable screening effects. To illustrate the application of these techniques to degenerate, low-temperature systems, the absorption process in a high-density electron gas is briefly considered.