Analytical model of atmospheric pressure, helium/trace gas radio-frequency capacitive Penning discharges

Abstract

Atmospheric and near-atmospheric pressure, helium/trace gas radio-frequency capacitive discharges have wide applications. An analytic equilibrium solution is developed based on a homogeneous, current-driven discharge model that includes sheath and electron multiplication effects and contains two electron populations. A simplified chemistry is used with four unknown densities: hot electrons, warm electrons, positive ions and metastables. The dominant electron–ion pair production is Penning ionization, and the dominant ion losses are to the walls. The equilibrium particle balances are used to determine a single ionization balance equation for the warm electron temperature, which is solved, both approximately within the α- and γ-modes, and exactly by conventional root-finding techniques. All other discharge parameters are found, the extinction and α-γ transitions are determined, and a similarity law is given, in which the equilibrium for a short gap at high pressure can be rescaled to a longer gap at lower pressure. Within the α-mode, we find the scaling of the discharge parameters with current density, frequency, gas density and gap width. The analytic results are compared to hybrid and particle-in-cell (PIC) results for He/0.1%N2, and to hybrid results for He/0.1%H2O. For nitrogen, a full reaction set is used for the hybrid calculations and a simplified reaction set for the PIC simulations. For the chemically complex water trace gas, a set of 209 reactions among 43 species is used. The analytic results are found to be in reasonably good agreement with the more elaborate hybrid and PIC calculations.