On the mechanism of the runaway of electrons in a gas: The upper branch of the self-sustained discharge ignition curve

Based on the results of simulation by the method of particles, it is shown that the Townsend mechanism of electron multiplication in a gas at a sufficiently large electrode spacing is valid at least up to such large values of $E/p$ at which relativistic electrons are generated. On the other hand, the phenomenon of electron runaway in a gas is determined by the electrode spacing, which must be either comparable with or smaller than the characteristic electron multiplication length, rather than the local criteria accepted presently. It is shown that, for a particular gas, the critical voltage across the electrodes at which the runaway electrons comprise a significant fraction is a universal function of the product of the electrode spacing by the gas pressure. This function also determines the condition of self-sustained discharge ignition. It not only incorporates the known Paschen curve but also additionally contains the upper branch, which describes the absence of a self-sustained discharge at a high voltage sufficiently rapidly supplied across the electrodes.