Transition from Townsend to glow discharge: Subcritical, mixed, or supercritical characteristics

The transition from Townsend to glow discharge is investigated numerically in one space dimension in full parameter space within the classical model: with electrons and positive ions drifting in the local electric field, impact ionization by electrons (alpha process), secondary electron emission from the cathode (gamma process) and space charge effects. We also perform a systematic analytical small current expansion about the Townsend limit up to third order in the current that fits our numerical data very well. Depending on the two determining parameters gamma and system size pd, the transition from Townsend to glow discharge can show the textbook subcritical behavior, but for smaller values of pd, we also find supercritical or some unexpected intermediate "mixed" behavior. Our work shows the same qualitative dependence of U = U(I,pd) for fixed gamma as the old experiments by Pokrovskaya-Soboleva and Klyarfeld. Furthermore, the analysis lays the basis for understanding the complex spatiotemporal patterns in short planar barrier discharge systems.