Analytic model of the hollow cathode effect

An analytic model of the hollow cathode effect (HCE) is suggested. The model is based on an approximate kinetic treatment of fast electrons which are responsible for ionization processes in the cathode cavity. The difference between exponential electron multiplication in the cathode sheath and linear multiplication in the plasma is shown to be responsible for the HCE. The HCE takes place when the fast electrons from one side of the cathode sheath are capable of reaching the opposite side of the sheath and produce a substantial fraction of the total ionization in the sheath. Simple analytic solutions of the non-local Boltzmann equation in a spatially inhomogeneous electric field are obtained for two different cases which correspond to the onset of the HCE and to the well developed oscillatory motion of electrons. The main approximations which lead to analytical solutions are a continuous slowing down approximation and assumption of predominantly forward electron scattering. The calculated electron distribution functions are in good agreement with Monte Carlo (MC) numerical simulations. The electron multiplication coefficient is obtained and used for a quantitative description of the HCE.