Plasma Density Fluctuations in a Magnetic Field

Abstract

Sinusoidal electron charge density fluctuations with propagation vector k are considered for a fully ionized gas in complete thermodynamic equilibrium in a constant magnetic field. Let $\ensuremath{\alpha}$ and $\ensuremath{\epsilon}$ be the ratio of the Debye length and of an electron gyroradius, respectively, to the wavelength ${k}^{\ensuremath{-}1}$. A general formula is derived for the frequency spectrum of these fluctuations for arbitrary values of $\ensuremath{\alpha}$, $\ensuremath{\epsilon}$, and of the angle ($\ensuremath{\varphi}\ensuremath{-}\frac{1}{2}\ensuremath{\pi}$) between k and the magnetic field. The dispersion relation implied by this expression has been obtained previously by Gross and by Bernstein, but the method of derivation is different. A very small electron-ion mass ratio $\frac{m}{M}$ is assumed.For large values of $\ensuremath{\epsilon}$ and $\ensuremath{\alpha}$ most of the intensity occurs at small frequencies: If $sin\ensuremath{\varphi}\ensuremath{\gg}{(\frac{m}{M})}^{\frac{1}{2}}\ensuremath{\epsilon}$, the main spectrum is continuous as in the absence of a magnetic field; if ${(\frac{m}{M})}^{\frac{1}{2}}\ensuremath{\ll}sin\ensuremath{\varphi}\ensuremath{\ll}{(\frac{m}{M})}^{\frac{1}{2}}\ensuremath{\epsilon}$, it consists of lines with spacing about the ion gyrofrequency; if $sin\ensuremath{\varphi}\ensuremath{\ll}{(\frac{m}{M})}^{\frac{1}{2}}$, it consists mainly of a line at zero frequency. Weaker spectral lines are obtained which correspond to plasma oscillations, the existence of "frequency gaps" is confirmed for small angles $\ensuremath{\varphi}$, and the intensities of the various components are evaluated. For small $\ensuremath{\varphi}$, another spectral line is obtained at a "resonance" frequency intermediate between the electron and ion gyrofrequency.