Kinetic model of an inverted sheath in a bounded plasma system

A one-dimensional, kinetic model of inverted sheath formation in a plasma system bounded by two infinitely large planar electrodes (the source and the collector) has been developed for the first time. It is assumed that ions and electrons are injected into the system from the source with half-Maxwellian distributions, and emitted electrons are also injected from the collector with a half-Maxwellian distribution. It is assumed that the potential increases monotonically from the source to the collector. Consequently, the distribution functions of ions, electrons, and emitted electrons anywhere in the system can be written as functions of the potential. Zero and first moments of the distribution functions give particle densities and fluxes. From these, the floating condition for the collector is derived and the Poisson equation is written. The first integrals of the Poisson equation give the conditions for the electric field at the source and at the collector. The model consists of five basic equations: (1) collector floating condition, (2) neutrality condition at the inflection point of the potential, (3) source electric field condition, (4) collector electric field condition, and (5) Poisson equation. The model contains nine parameters. Five of them are plasma parameters: (1) ion mass μ, (2) ion temperature τ, (3) ion source strength α, (4) temperature of emitted electrons σ, and (5) emission coefficient ε. Then there are two potentials, (1) floating potential of the collector $ΨC$ and potential at the inflection point $ΨP$ and (2) electric fields, (1) electric field at the collector η_C and (2) electric field at the source η_S. If five of them are selected, the other four can be found from the system of equations (1)–(4). Numerical solutions of the Poisson equation give axial profiles of the potential, electric field, and space charge density. The model can be used for parametric analysis of the inverted sheath formation. Usually μ, τ, α, ε, and σ are selected and then $ΨC, ΨP$⁠, η_C, and η_S are found from the system of equations (1)–(4). This means that the particle densities are selected independently, but the potentials and electric fields are then calculated in a self-consistent way with the selected parameters.