Extended Tonks–Langmuir-type model with non-Boltzmann-distributed electrons and cold ion sources

Abstract

AbstractA general formalism for calculating the potential distribution Φ(z) in the quasineutral region of a new class of plane Tonks–Langmuir (TL)-type bounded-plasma-system (BPS) models differing from the well-known ‘classical’ TL model (Tonks, L. and Langmuir, I. 1929 A general theory of the plasma of an arc. Phys. Rev. 34, 876) by allowing for arbitrary (but still cold) ion sources and arbitrary electron distributions is developed. With individual particles usually undergoing microscopic collision/sink/source (CSS) events, extensive use is made here of the basic kinetic-theory concept of ‘CSS-free trajectories’ (i.e., the characteristics of the kinetic equation). Two types of electron populations, occupying the ‘type-t’ and ‘type-p’ domains of electron phase space, are distinguished. By definition, the type-t and type-p domains are made up of phase points lying on type-t (‘trapped’) CSS-free trajectories (not intersecting the walls and closing on themselves) and type-p (‘passing’) ones (starting at one of the walls and ending at the other). This work being the first step, it is assumed that ε ≡ λD/l → 0+ (where λD and l are a typical Debye length and a typical ionization length respectively) so that the system exhibits a finite quasineutral ‘plasma’ region and two infinitesimally thin ‘sheath’ regions associated with the ‘sheath-edge singularities’ | dΦ/dz|z→±zs → ∞. The potential in the plasma region is required to satisfy a plasma equation (quasineutrality condition) of the form ni {Φ} =ne (Φ), where the electron density ne (Φ) is given and the ion density ni {Φ} is expressed in terms of trajectory integrals of the ion kinetic equation, with the ions produced by electron-impact ionization of cold neutrals. While previous TL-type models were characterized by electrons diffusing under the influence of frequent collisions with the neutral background particles and approximated by Maxwellian (Riemann, K.-U. 2006 Plasma-sheath transition in the kinetic Tonks–Langmuir model. Phys. Plasmas 13, 063508) or bi-Maxwellian (Godyak, V. A. et al. 1995 Tonks–Langmuir problem for a bi-Maxwellian plasma. IEEE Trans. Plasma Sci. 23, 728) electron velocity distribution functions (VDFs), which satisfy the zero-CSS-term (Vlasov) kinetic equation and imply zero electron currents, we here propose a more general class of electron VDFs allowing, in an approximate manner, for non-zero CSS terms and finite electron currents inside the plasma region. The sheath-edge and floating-wall potentials are calculated by balancing the ion and electron current densities at sheath-edge singularities. In a first detailed application, the type-t and type-p electron VDFs are assumed to be ‘inner’ and ‘outer’ cut-off Maxwellians respectively, with different amplitudes and ‘formal’ temperatures, implying the perfectly CSS-free limit. For the special case of equal type-t and type-p electron VDF amplitudes and formal temperatures, the classical Boltzmann distribution for electrons is formally retrieved. Special cases with other amplitude and formal-temperature ratios show significant deviations from the classical case.