Energy balance of the bulk, Maxwellian electrons in spatially inhomogeneous negative-glow plasmas

The energy balance of the Maxwellian (bulk) electrons is analyzed in spatially inhomogeneous negative-glow plasmas (NGP). The purpose is to give a comprehensive model which enables the electron temperature in the NGP to be predicted. Since the bulk of the electron distribution function (EDF) in the NGP is Maxwellian, the rates of many important plasma processes (e.g., ambipolar diffusion, recombination, stepwise processes), as well as the plasma potential, are controlled by the electron temperature. Knowledge of the electron temperature is thus of particular importance for such types of plasma. In order to calculate the EDF in the elastic energy range (slow electrons), a spatially inhomogeneous kinetic equation is employed, in which the electron-electron collision integral is fully incorporated. Owing to the complicated (nonlinear integro-differential) form of the electron-electron collision integral, the direct numerical solution of the full kinetic equation represents a difficult task. An efficient way to render the problem tractable consists in breaking the slow electrons up into two distinct groups, namely, the Maxwellian (trapped) and superthermal (untrapped) electrons. The parameters of the Maxwellian EDF can be found from the particle- and energy-balance equations. The superthermal EDF can be found from a reduced kinetic equation. The separation of the electron population into two groups allowed us to obtain an energy-balance equation for the Maxwellian (cold, trapped) electrons, which properly accounts for the most important physical mechanisms, such as heating due to Coulomb collisions with the superthermal (hot, untrapped) electrons. It is shown that the problem of finding the electron temperature in a weakly collisional NGP can be described correctly only at the kinetic level. In this situation, the use of the fluid approximation, in which the electron ensemble is treated in terms of its density and mean energy, results in a physically incorrect energy-balance equation. Furthermore, it is demonstrated that the “nonlocal” effects may be critical for the problem of finding the EDF in general, and the electron temperature in particular, so that the “local” (kinetic) models may also produce erroneous results. The principal terms in the energy-balance equation are identified, and this equation is simplified to allow a ready solution and implementation into a plasma code. The validity of the proposed model for predicting the electron temperature was confirmed by numerical calculations of the EDF from the full kinetic equation. The results of the paper can be applied to the NGP generated in direct-current glow discharges with planar or hollow cathodes, as well as to negative-glow-like plasmas, such as beam-generated and afterglow (decaying) plasmas.