Transition from Townsend to glow discharge: subcritical, mixed or supercritical

The full parameter space of the transition from Townsend to glow discharge is investigated numerically in one space dimension in 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 total current that fits our numerical data very well. Depending on $\gamma$ 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 intermediate ``mixed'' behavior. The analysis in particular lays the basis for understanding the complex spatio-temporal patterns in planar barrier discharge systems.