Conditions for electron runaway under leader breakdown of long gaps
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Conditions for Electron Runaway under Leader Breakdown of Long Gaps K. N. Ul’yanov Lenin All-Russia Electrotechnical Institute, Krasnokazarmennaya ul. 12, Moscow, 111250 Russia Received April 12, 2007; in final form, July 30, 2007
Abstract—An original hydrodynamic model in which inelastic collisions in the equations of motion and energy balance play a decisive role is developed and applied to simulate electron avalanches in strong electric fields. The mean energy and drift velocity of electrons, as well as the ionization coefficient and electric field in a wide range of mean electron energies, are determined for helium and xenon. A criterion is derived for the runaway of the average electron in discharges with ionization multiplication. It is shown that runaway can take place at any value of E/p, provided that the momentum mean free path exceeds the gap length. The voltage corresponding to electron runaway is found for helium, xenon, and air as a function of the electric field, the electron mean energy, and the parameter pd. Conditions for the formation of a precursor in electronegative gases are analyzed. It is shown that the presence of a precursor with a high electric conductance is necessary for the formation of a new leader step. The voltage and time ranges corresponding to efficient electron runaway and X-ray generation during leader breakdown in air are determined. PACS numbers: 52.80.Vp DOI: 10.1134/S1063780X08040107
1. INTRODUCTION An approximation based on the Townsend coefficients is widely used in the theory of streamer and glow discharges. The first Townsend coefficient α(E/p) describes gas ionization by electrons. At a high electron collision frequency, the electron density in a uniform electric field depends on the coordinate as n = n(0)exp(αx), where α(E/p) = νi /V, with V and νi being the electron drift velocity and ionization frequency, respectively. The problem is to determine α(E/p) and V(E/p). The dependence V(E/p) can be found from the equation of motion, in which the cross sections are averaged using the electron velocity distribution function f. The function f is found either analytically or numerically. This, however, is a rather complicated task if the electron energy ε(E/p) is less than the threshold energies for inelastic collisions and the gas atoms are ionized and excited by fast electrons from the tail of the distribution function f. The problem is significantly simplified in a strong electric field, when ε exceeds the threshold energies. In this case, ionization and excitation are mainly caused by the bulk electrons and the collision frequencies can be calculated in the “average electron” model, assuming that the collision frequencies correspond to the average electron energy ε(E/p). Such an approach was used in [1] to find a criterion for electron runaway in a strong electric field. In this paper, the approach developed in [1] is employed to describe electron motion in strong electric fields in the ionization multiplication mode for helium, xenon, and air, which ha
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