Instability criterion for plasma oscillations in a magnetic field

The dispersion equation of Cauchy type for longitudinal plasma oscillations in a magnetic field is exactly derived on the basis of Vlasov's collision-free kinetic equation for arbitrary velocity distribution of electron ∫_-∞^∞ Φ(k, ν)/ν-ζ dν = k²/ω_p², where ζ = ω/k_II, Φ(k, ν) = 2π/Nk_II [k_II Σ_ℓ=0^∞ A_2ℓ(ν) (k_⊥/Ω)^2ℓ + k_⊥Σ_ℓ=0^∞ A_2ℓ+1(ν)(k_⊥/Ω)^2ℓ+1], A_2ℓ(ν) = 1/(ℓ!)²2^2ℓ Σ_r=0^2ℓ (-)^r (_r^2ℓ) M_2ℓ+1¹ (ν + ℓ-r/k_II Ω), A_2ℓ+1 (ν) = 1/(ℓ!)²2^2ℓ+1 Σ_r=0^2ℓ (-)^r (_r^2ℓ+1) [M_2ℓ+1 (ν + ℓ-r/k_II Ω) + M_2ℓ+1 (ν + ℓ-r+1/k_II Ω)], M_2ℓ+1(u) =∫₀^∞w^2ℓ+1 f₀ (w, u)dw, M_2ℓ+1′(u) = ∂/∂u M_2ℓ+1 (u) N and Ω are the total density and the gyration frequency of electron. A_2ℓ+1, A_2ℓ are determined by the higher order moments of the component of velocity distribution function which is perpendicular to the magnetic field. The motion of the ion is easily taken into account by replacing Φ intoΦ₊+Φ_-. It is found that the terms of lowest order in Φ for k_⊥/Ω is the predominant factor for the drift instability and that higher terms of more than second order contribute to the instability due to anisotropic velocity distributions. The criterion and detailed conditions for plasma instabilities are tabulated through the knowledge of potential theory. This criterion is applied to several interesting distribution functions. (author)