Ionization waves in shielded capillary discharges

A theory of propagation of ionization waves during the early stages of an electrical breakdown in a shielded, low-pressure capillary is presented and its results are compared to the results of recent experiments that were purposefully conducted to detect such waves. Due to the high values of the ratio of the electric field to the atomic concentration in the discharge tube, neither the drift velocity approximation nor the use of the Townsend ionization coefficient is valid. Instead, the full momentum equation for the electrons is employed, as well as experimentally measured values of the cross section for ionization by electron impact. A quasi one-dimensional model is obtained by assuming some appropriate radial profiles for the physical variables. Those profiles, together with the appropriate boundary conditions, provide a set of time dependent one-dimensional equations for the on-axis values of the physical variables. In particular, a wave-guide-like expression is assumed for the electric potential whose wave-guide parameter is a function of time and the axial coordinate. Numerical solutions of the quasi one-dimensional equations have been obtained for cathode-directed ionization waves that are associated with virtual anode propagation as well as for anode-directed waves that are associated with virtual cathode propagation. In the former case the cathode is grounded while the anode is positively biased while in the latter case the anode is grounded and the cathode is negatively biased. The numerical solutions indicate that the cathode-directed waves may propagate with much lower velocity than its anode-directed counterpart, which agree with the experimentally observed properties of the various waves. Properties of the two types of ionization waves, their dependence on the capillary initial pressure and geometry, and possible explanations for the difference in the propagation velocities will be discussed.