Relativistic Corrections to the Light-Scattering Spectrum of a Plasma

The light scattering spectrum of a plasma with drift-displaced Maxwellian velocity distribution is calculated to first order in $\frac{\overrightarrow{\mathrm{v}}}{c}$ ($c$ = 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 $\overrightarrow{\mathrm{v}}$, 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 (${\overrightarrow{\mathrm{k}}}^S-{\overrightarrow{\mathrm{k}}}^i$) of approximate magnitude $\left(\frac{5}{3}\right)v_e\left(\frac{v_e}{c}\right) sin\frac{1}{2}\theta$, where $v_e$ is the rms thermal electron velocity, $\theta$ the scattering angle, and ${\overrightarrow{\mathrm{k}}}^S\left({\overrightarrow{\mathrm{k}}}^i\right)$ the wave vector of the scattered (incident) radiation. In the coherent spectrum the relativistic corrections are found proportional to the quantity $\frac{\omega }{c}|{\overrightarrow{\mathrm{k}}}^S-{\overrightarrow{\mathrm{k}}}^i|$. Because of the proportionality to the frequency shift $\omega$, the corrections are negligible for the ion spectrum.