Modelling of the hollow cathode plasma for thw pre-breakdown phase of a pseudospark discharge

Abstract

The ionization growth during the pre-breakdown phase of a pseudospark discharge is numerically simulated within a hollow cathode region. A swarm model based on the local-field approximation is applied. The continuity equations for electrons and ions are solved simultaneously with Poisson's equation as a function of time in a two-dimensional axially symmetric geometry. The distortion of the electric field due to the hollow-cathode geometry and due to space-charge effects is incorporated. The numerics of the coding is outlined, and problems, like boundary conditions, and solutions adopted for this type of geometry are described. The validity and the robustness of the numerical scheme are discussed. The results of detailed tests proof that errors related to the limited physical model and to the numerics are tolerable. According to the simulations, the plasma density inside the hollow-cathode region at first grows exponentially in time during several 100 ns, mainly because of electron impact ionization and photon emission releasing secondary electrons at the surfaces by photoeffect. The electrons follow the electric field lines as a nearly incompressible fluid, resulting in a focusing towards the axis combined with an increase in velocity. Then they are extracted quickly into the main discharge gap, leaving the much slower ions behind nearly stationary. Consequently a positive space charge builds up, which has its maximum on axis at the cathode hole. During a transition phase of several 10 ns this charge starts to shield off the electric field from the main gap and enhances it inside the hollow cathode, causing increased ionization locally. Further the force related to the positive charge attracts the electrons by means of the modified electric field towards its maximum, thereby compressing the electron fluid. During several ns this leads to an overexponential growth in electron density in this vicinity, triggering a similar growth in the positive-ion density. The result is the onset of the pseudospark as an extremely high-current-density electron beam with a very small diameter leaving the hollow cathode on axis. Such electron beams can be extracted experimentally in pseudospark discharges through a hole in the anode. The pre-breakdown phase in a hollow cathode geometry was also studied experimentally by means of high-speed streak photography. It is shown that the light-emitting region begins in time from a point on axis close to the hole, corresponding to the predictions of the simulation.