Electron Density Fluctuations in a Plasma
We consider the spatial Fourier transform ${\ensuremath{\rho}}_{\mathrm{ke}}$ for wave vector k of the charge distribution of the electrons in a plasma with particle density $n$, electron and ion temperatures $T$ and ${T}_{i}$ and Debye length $D$. We assume the absence of a magnetic field, neglect collisions and assume $n{D}^{3}\ensuremath{\gg}1$. The statistical average of ${|{\ensuremath{\rho}}_{\mathrm{ke}}|}^{2}$ is calculated as a function of $\ensuremath{\alpha}=\frac{1}{\mathrm{kD}}$ assuming complete thermodynamic equilibrium; that component of ${|{\ensuremath{\rho}}_{\mathrm{ke}}|}^{2}$ which keeps in phase with the ion charge density fluctuations is also calculated.The frequency spectrum of the time-varying function ${\ensuremath{\rho}}_{\mathrm{ke}}$ is obtained at thermal equilibrium and simplified, assuming the ion mass to be much larger than the electron mass, for general values of $\ensuremath{\alpha}$ and $\frac{T}{{T}_{i}}$. For small $\ensuremath{\alpha}$ the main component of the spectrum has the characteristic Doppler broadening shape corresponding to the electron's thermal velocity. For large $\ensuremath{\alpha}$ we have a component with narrow width corresponding roughly to the ion velocity Doppler spread and very narrow side bands at plus and minus the frequency of electrostatic plasma oscillations.