Linear longitudinal oscillations in collisionless plasma diodes with thin sheaths. Part I. Method

A method is proposed for treating linear longitudinal perturbations in one-dimensional collisionless plasma diodes with a uniform plasma region and thin electrode sheaths. The method is comprehensive in that it allows for very general equilibrium, initial, boundary, and external-circuit conditions. Upon Laplace-transforming the Vlasov and Poisson equations in both space and time, appropriate evaluation of all pertinent relations leads to a set of 2+2nσ(nσ is the number of particle species) coupled integral equations in x and v for the following quantities (which are the time Laplace transforms of the respective physical perturbations): j̃e(ω) (external-circuit current density), Ẽ(x,ω) (electrostatic field), f̃σl(v>0,ω), and f̃σr(v<0,ω) (velocity distribution functions of the plasma-bound particles at the left- and right-hand plasma boundaries, respectively), where σ is the species index. The formal solution of these integral equations and the inverse Laplace transformation (ω→t) are discussed in general terms. In particular, it is shown that the intrinsic eigenfrequencies are given by the zeros of the coefficient determinant of the integral equations. A comparison with previous treatments is given, and it is concluded that extensions of the method proposed to more general systems should be feasible.