Relativistic Corrections to the Light-Scattering Spectrum of a Plasma

Abstract

The light scattering spectrum of a plasma with drift-displaced Maxwellian velocity distribution is calculated to first order in $\frac{\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}{c}$ ($c$ = velocity of light). The scattered intensity is obtained from the nonrelativistic Vlasov equation, and from the Lienard-Wiechert potentials produced by electrons which move with velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$, and are accelerated by the incident electric and magnetic radiation fields. For the incoherent spectrum, the relativistic effects are found equivalent to an apparent drift-velocity component parallel to the scattering vector (${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{S}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{i}$) of approximate magnitude $(\frac{5}{3}){v}_{e}(\frac{{v}_{e}}{c}) sin\frac{1}{2}\ensuremath{\theta}$, where ${v}_{e}$ is the rms thermal electron velocity, $\ensuremath{\theta}$ the scattering angle, and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{S}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{i})$ the wave vector of the scattered (incident) radiation. In the coherent spectrum the relativistic corrections are found proportional to the quantity $\frac{\ensuremath{\omega}}{c}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{S}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{i}|$. Because of the proportionality to the frequency shift $\ensuremath{\omega}$, the corrections are negligible for the ion spectrum.