The criterion for streamer development is determined by finding the level of ionization at which
significant field distortion from space charge in the gap arises. Taking all the electrons to be in
a spherical volume, with radius r, at the tip of the avalanche, the field is






where e is the electronic charge magnitude. The radius can be approximated by the diffusion radius




where D is the diffusion coefficient. The transit time of the electrons can be found from the
avalanche length x, and the drift velocity v[d][ ]or the electron mobility ke





The diffusion coefficient is found from



where Te is the electron temperature and κ is Boltzmann’s constant. Using



where V is the electron energy in volts, we have






Now let x[c] designate the distance at which Er becomes comparable with E. This yields Raether’s

avalanche-streamer criterion




where x[c] is in centimeters. In the uniform field the occurrence of a critical avalanche usually
develops into a spark. Thus we can substitute x[c][ ]= d