Effect of Electron Pressure on Plasma Electron Oscillations

A general equation for electron motion in a plasma is developed which includes a term arising from electron gas pressure. The resulting expression is $\frac{{\partial }^2\xi }{\partial t^2}+\left(\frac{4\pi n{e}^2}{m}\right)\xi =\left(\frac{\mathrm{kT}}{m}\right){\nabla }^2\xi ,$ where $\xi$ is electron displacement, $n$ electron density, and $T$ electron gas temperature. From this it is found that the possible frequencies of free vibration form a series given by $f_i={(\frac{\mathrm{kT}}{\lambda _i^{}{}_{}{}^{2}m}+\frac{n{e}^2}{\pi m})}^{\frac{1}{2}}$. The lower limit corresponds to the Tonks-Langmuir value ${\left(\frac{n{e}^2}{\pi m}\right)}^{\frac{1}{2}}$, while the other frequencies depend upon the possible standing waves which may exist. The theory explains the observed variation of frequency with electron gas temperature.