Plasma Oscillations in a Static Magnetic Field

A theory of the small-amplitude oscillations of an ionized gas in a static magnetic field is developed, including the effects of temperature motions. The Boltzmann equation is solved for this problem, and exact expressions are obtained for the distribution function and dispersion relation. A general feature of the dispersion relation is the existence of gaps in the spectrum at frequencies which are approximately multiples of ωc=eH/mc. The magnitude of the gap depends on the temperature of the gas, being proportional to it for long wavelengths. This leads to the prediction of selective reflection of waves impinging on a plasma with frequency in the forbidden range.For ckωₚ, ωc the waves split into approximately longitudinal plasma waves and transverse waves. Detailed analysis is made of the plasma waves for ωc small and ωc large. At long wavelengths the frequency is ω²ωₚ^_²+ωc^_²+β(κTm)k², where β depends on ωₚ and ωc. For waves near the Debye length the waves are heavily damped.Two simplified treatments of plasma oscillations based on transport equations are compared with the above treatment. Expressions of the form ω²ωₚ^_²+ωc^_²+β(κTm)k² are obtained where the factor β is independent of ωc and ωₚ. In addition, the transport treatments fail to predict the heavy damping near the Debye length and the existence of gaps in the frequency spectrum.