Fundamentals of Plasma Physics 2025, 31 - Uniform Plasma

Assume a small wave amplitude, but a large initial particle velocity, so that it moves substantially during one wave period. This is the case for significant thermal motion. If the motion is parallel to the B-field, such motion is significant wrt. ω/k_||. Thermal motion in perpendicular becomes an issue when the Larmor orbit becomes comparable to the wavelength, as the particle samples different phases as it traces out the orbit. Consider an electrostatic wave

At B → 0, this should recover the Bessel identity. At k_|| → 0, the dispersion relation reduces to definitions of the Larmor radius and the wavelength. This case is the Bernstein wave dispersion relation. It has an infinite number of roots. To choose only the electrons, drop the σ-index.

For electrons-only, distinguish between ω²_p << data-preserve-html-node="true" ω²_c, where the summed over fraction is negligible compared to unity except for ω² ~ O(n²ω²_ce), so only one term is near resonance. The resulting spectrum waves are slightly above the cyclotron harmonics. In the converse case, for large λ, the product of the exponential factor and modified Bessel function is unity, reducing to cyclotron harmonics. ω² = n²ω²_c as λ → ∞. For small λ, the lowest order n=1 term is independent of λ.