Cold Electron Runaway Below the Friction Curve

Cold electron runaway means that free electrons in gases are accelerated by electric fields from eV energies to energies above tens of keV where they can be accelerated further. To run away, the electrons need to overcome a barrier at intermediate energies where they can lose much energy in collisions. When they have reached the runaway regime, they can produce high-energy radiation by bremsstrahlung that can be detected as (terrestrial) gamma ray flashes. When can thermal electrons from active discharges like streamers and leaders reach the runaway regime? The deterministic approach to this question is based on an energy-dependent electron friction that has to be overcome by electric acceleration. Taking the stochastic nature of the electron molecule collisions into account, we find (1) that the classical friction curve in the energy regime up to 1keV does not characterize the mean electron energy, but rather, it seems to approximate the upper limit of the electron energy distribution and (2) that electrons can tunnel through the barrier when the field is close to 3MV/m, below the so-called cold runaway threshold (or critical field) of approximately 26MV/m in air at standard temperature and pressure. (3) This is only true in the bulk perspective where the electron liberation and attachment in a given electric field is taken into account in the continuously refreshing electron ensemble. In a flux simulation that follows individual electrons as long as they are free, electron attachment reduces electron runaway very strongly in air, differently to what we observe in nitrogen.