Analytical model of a longitudinal hollow cathode discharge

This paper presents a simple analytical model of a longitudinal hollow cathode discharge used in metal vapour lasers. The model describes the principle relations between the voltage, current, plasma density and axial structure of the discharge. Contrary to standard dc discharges, this discharge does not require electron multiplication in the cathode fall (CF) to produce ions, but rather to satisfy the electron energy balance. A self-sustainment condition is obtained from the energy balance per electron-ion pair. From this, it follows that there is a maximum voltage at which the CF thickness tends to zero and the current density tends asymptotically to infinity. The discharge develops axial non-uniformity and an axial electric field in order to evacuate the created electrons to the anode, such that the characteristic time for transport losses is the same for electrons as for ions. The axial profiles of the current density, plasma density and potential are obtained from the electron continuity equation. It is shown that additional energy absorption from the axial field, similar to electron heating in dc positive columns, modifies the self-sustainment condition and thus leads to a shift in the voltage-current characteristic, depending on the cathode length.