Extended cold-source Tonks-Langmuir-type model with non-Boltzmann-distributed electrons

摘要

A general formalism for calculating the potential distribution <& (z) in the quasiheUtral region of a Tonks-Langmuir (TL) -type model allowing for arbitrary cold ion source's and arbitrary electron distributions is presented. The kinetic concept of "cbllision/sink/source (CSS) -free electron trajectories (characteristics)" is extensively used Two types of electron populations are distinguished: the "type-t" ones (populating the "trapped" characteristics, which do not intersect the walls and close on themselves) and the ''type-p" ones (populating the "passing" ones, which start out at one of the walls1 and end at the other). The. potential "in-the plasma region satisfies a- "plasma equation'' of the form n~i {Φ}=n~e (Φ), with the electron density n~e (Φ) given and the ion density n~i>{Φ} expressed in terms of trajectory integrals of, the+ion kinetic equation. While previous TL-type models, including the "classical" TL model i[l], were approximated by Maxwellian [2] or bi-Maxwellian [3] electron velocity distribution functions (VPFs), which imply zero CSS terms ("Vlasovian electrons") and electron currents, we here propose a more general class of electron VPFs allowing for non-zero CSS terms ("non-Vlasovian electrons") and electron currents inside the plasma region. The sheath-edge and floating-wall potentials are calculated by balancing the ion and electron current densities at the sheath-edge singularities. In a first detailed application, Vlasovian electrons are assumed for which the type-t and type-p VPFs are "inner" and "outer" cut-off Maxwellians, respectively, with different amplitudes and "formal" temperatures. For the special case of equal amplitudes and formal temperatures, the classical Boltzmann electron distribution is formally retrieved. Special cases with other amplitude and formal-temperature ratios show significant deviations from the classical Maxwellian case. This work is a first attempt at introducing electron VDFs different. from Maxwellian or bi-Maxwellian VPFs, leading to the conclusion that substantial efforts'will be required in the future to arrive at more realistic electron VDFs. The water-bag distribution function for kinetic modeling