Plasma Density Fluctuations in a Magnetic Field

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 α and ε be the ratio of the Debye length and of an electron gyroradius, respectively, to the wavelength k^-1. A general formula is derived for the frequency spectrum of these fluctuations for arbitrary values of α, ε, and of the angle (φ-1/2π) 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 m/M is assumed.For large values of ε and α most of the intensity occurs at small frequencies: If sinφ(m/M)1/2ε, the main spectrum is continuous as in the absence of a magnetic field; if (m/M)1/2sinφ(m/M)1/2ε, it consists of lines with spacing about the ion gyrofrequency; if sinφ(m/M)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 φ, and the intensities of the various components are evaluated. For small φ, another spectral line is obtained at a "resonance" frequency intermediate between the electron and ion gyrofrequency.