Why Paschen's law does not apply in low-pressure gas discharges with inhomogeneous fields

Low pressure (0.13<P<26.7 mbar) breakdown conditions were studied in a direct current point-plane gap discharge in air. Influences of the gas pressure P and geometrical factors (gap distance, electrode configuration with various tip and rod radii for the anode) on the breakdown voltage V s are presented. Curves of V s versus P were drawn in a Paschen-like manner, and a roughly 550 V minimum value V s(min) was obtained. This minimum always corresponds to the transition between the Townsend-like regime and the subnormal glow regime of the discharge, with a development in the lateral direction between the rod (anode) and the vessel walls. As the pressure increases, the discharge direction development is modified. For pressures higher than about 6.7 mbar, depending on the experimental conditions, an axial breakdown (between point and plane) is observed. This is highly dependent on the effective geometry of the discharge vessel (rod radius, rod-to-walls distance, point tip radius and point-plane gap) in the <3 cm inter-electrode gap distance considered. To explain this phenomenon, a simple mathematical model was established. It permits one, despite simplifying assumptions, to show why the V s(min) value is obtained with a lateral development of the discharge when at higher pressure breakdown it occurs axially. This model points out the relative role of the various parts of the anode (rod and tip curvature radii) in the breakdown potential as a function of pressure.