Exact Solution of the Collisionless Plasma-Sheath Equation

TLDR

The plasma‑sheath problem in low‑pressure plane discharges is solved exactly without arbitrary plasma–sheath separation. The authors numerically solve the collisionless plasma‑sheath equation for α between 10⁻³ and 10⁻¹ under three ion‑generation models: uniform, proportional to electron density, and proportional to its square. For large α the transition from quasi‑neutral plasma to thick sheath is smooth, whereas for small α the conventional abrupt transition is confirmed, and the study delivers accurate potential profiles and evaluates ion current density, wall potential, space‑charge density, mean ion energy, and sheath thickness across all cases.

Abstract

The plasma-sheath problem for the low-pressure discharge in plane geometry is treated exactly, that is, with no arbitrary division into plasma and sheath regions. Numerical solutions are presented for various values of the parameter α, which is of the order of the ratio of the Debye length to the discharge width for 10−3 ≤ α ≤ 10−1; and for three assumptions regarding the ion generation rate, namely generation uniform, proportional to electron density, and proportional to the square of electron density. For the higher values of α, corresponding to weak laboratory discharges, there is a smooth transition from a quasi-neutral plasma region to a thick sheath. At the smaller values of α, the conventional model of a quasi-neutral plasma region passing rather abruptly into a narrow sheath region is substantiated. In all cases, accurate values for the potential profile throughout the plasma and sheath regions are given and compared with the separate asymptotic plasma and sheath solutions for α = 0. The ion current density, wall potential, space-charge density, mean ion energy, and sheath thickness are discussed.