Ionization waves in electron-beam-assisted, shielded capillary discharge

A theory of propagation of cathode-directed ionization waves during the early stages of an electrical breakdown in a shielded, low-pressure capillary is developed. The discharge process occurs due to the ionization of the low-density gas in the capillary by an electron beam that is emanating from a hollow cathode. Due to the strong electric field in the capillary the electrons are in the fast acceleration regime. Consequently, the full momentum equation for the electrons is employed, rather than the electron drift velocity approach. The smallness of the ratio of the capillary radius to the characteristic length of the electric potential variation in the axial direction allows the construction of a quasi-one-dimensional model. The latter retains the important two-dimensional nature of the electron flow as well as the electrodynamic boundary conditions at the capillary wall and the conducting shield and results in a set of one-dimensional, time-dependent partial differential equations for the on-axis distributions of the physical quantities. It is shown that those equations admit self-similar solutions that represent ionization waves propagating with constant velocities. The resulting set of ordinary differential equations is solved numerically for various initial conditions representing a nonperturbed steady state ahead of the ionization front and the resulting features of the ionization waves are investigated and discussed. The obtained solutions describe both ionization growth and virtual anode propagation and represent fast ionization waves in plasma waveguides, for which the maximum value of the mean electron velocity is much higher than the wave velocity. The space-charge distribution associated with the ionization waves is found in the form of plasma oscillations with a continuously increasing frequency and a solitary envelope. The calculated wave velocity increases with the gas pressure and this tendency is in agreement with corresponding experimental observations.