Relativistic limitations on runaway electrons

The non-relativistic theory of a plasma in an electric field E predicts that there will always be runaway electrons, although their number will be exponentially small for fields less than the Dreicer field ED. However, when E/ED ∼ kT/mec2, the ratio of the electron thermal energy to the rest mass energy, relativistic effects become important. After comparing earlier non-relativistic calculations we extend the approach of Kruskal and Bernstein to take account of relativistic effects and also to investigate the influence of impurities. It is found that below the critical electric field ER = ED (kT/mec2) absolutely no runaways are generated. In addition, the number of runaway electrons produced by electric fields in excess of ER is calculated and we find significant modifications to the non-relativistic estimates when (ED/E)2 (kT/mec2) > 1.