Kinetic theory of a spherical probe in a stationary collision-dominated isothermal plasma

Abstract

Constitutive equations for the charge carrier fluxes in a stationary collision-dominated isothermal plasma surrounding a spherical electrode are derived, from the Boltzmann equation, by a moment method using a two-stream Maxwellian distribution of the type introduced by Lees. In contrast to previous treatments, the constraint of uniform temperature is imposed and its importance emphasized. This condition, together with particle conservation, is shown to necessitate the use of a four-parameter Lees distribution and leads to generalized hydrodynamic equations. These are shown to reduce to conventional flux equations in terms of diffusion and mobility components only when the fields are spatially slowly varying. In addition, boundary conditions are derived by the same formalism. They are formulated for typical physical processes which may be operative at a real probe surface. As an example, the effect of the nonlinearity on the pure diffusion problem is analyzed in detail in the Appendix. A semiquantitative discussion of the current-voltage characteristics for negative probe potentials is also presented. It is shown that the charge carrier distribution around the probe conforms to the ’’free-fall’’ model only at low probe voltages. Otherwise, the model proposed by Su and Lam holds. The domain covered by the present analysis is determined by the following conditions on the probe radius rp, mean free path l, and Debye length λ: (i) l3≪λ2rp, (ii) (e ‖φp ‖l/kTrp) <1 for l<λ, and (iii) (e ‖φp ‖l/kTrp) ≪λ2/l2 for l≳λ, where φp is the probe voltage. Analytic approximations to the current-voltage characteristics in both these regimes are presented.