A unified analysis of plasma-sheath transition in the Tonks–Langmuir model with warm ion source

Abstract

The paper presents a comprehensive kinetic theory of the famous Tonks–Langmuir model of a plane symmetric discharge, taking into account the thermal motion of ion source particles. The ion kinetics is governed by the ionization of neutrals at electron impacts. The plasma consisting of Boltzmann distributed electrons and singly charged ions is in contact with the absorbing negative wall. The derivations are performed in the frame of the “asymptotic two-scale” approximation, when the ionization mean-free path Li is much larger than the electron Debye length λD. In the limit (λD/Li)→0, the plasma-wall transition (PWT) layer can be split into two sublayers: a quasineutral presheath (PS) (with the scale-length Li) and the Debye sheath (DS) (with the scale λD). Such a subdivision of the PWT layer allows to investigate these sublayers separately and simplify the analysis of the influence of the ion source thermal motion (this has been neglected in the major part of publications up to now). The uniform description of the PWT layer as a single unit is complicated by the singular presheath and sheath structure and by a coupling with the eigenvalue problem originating from the plasma balance in the bounded system. The issue is clarified both analytically and numerically by construction of a matched asymptotic expressions. The equation and the length-scale governing the transition between neighboring PS and DS sublayers are derived. The eigenvalue problem combining the wall potential, the wall location, and the ionization mean-free path is discussed.