Oscillations of a Nonuniform Plasma

The oscillations of a plasma which is confined by an rf field are investigated. The confining potential is approximated as ψ(x) = ½mω02x2. Longitudinal plasma oscillations in the x direction are determined from the self-consistent Boltzmann transport equation without the collision term (Vlasov equation). This equation is linearized about equilibrium velocity distribution f0 = exp [−(ψ + ½mv2)/κT]. By expanding the electric field in Hermite polynomials, it is possible to reduce exactly the resulting integro-differential equation to an infinite system of linear equations for the expansion coefficients. The resonant frequencies are the roots of the determinant of the system. The frequency spectrum so obtained is quite unlike those obtained for Sturm-Liouville problems. This spectrum contains the integral multiples of ω0 as limit points. As e2n/mω02 → 0, the resonant frequencies coalesce into these limit points, each of these frequencies μω0 (μ = integer) being infinitely degenerate. Since all frequencies are real, the oscillations are not damped. The resonant frequencies are determined approximately as functions of e2n/mω02 as the roots of principal sub-determinants of finite order N. This procedure converges rapidly with increasing N.