Electron energy distributions in uniform electric fields and the Townsend ionization coefficient

Abstract

The second-order differential equation which expresses the equilibrium condition of an electron swarm in a uniform electric field in a gas, the electrons suffering both elastic and inelastic collisions with the gas molecules, is solved by the Jeffreys or W.K.B. method of approximation. The distribution function F($\epsilon $) of electrons of energy $\epsilon $ is obtained immediately in a general form involving the elastic and inelastic collision cross-sections and without any restriction on the range of E/p (electric strength/gas pressure) save that introduced in the original differential equation. In almost all applications the approximation is likely to be of high accuracy, and easy to use. Several of the earlier derivations of F($\epsilon $) are obtained as special cases. Using the function F($\epsilon $) an attempt is made to relate the Townsend ionization coefficient $\alpha $ to the properties of the gas in a more general manner than hitherto, using realistic functions for the collision cross-section. It is finally expressed by the equation $\alpha $/p = A exp (-Bp/E) in which A and B are functions involving the properties of the gas and the ratio E/p. The important coefficient B is directly related to the form and magnitude of the total inelastic cross-section below the ionization potential and can be evaluated for a particular gas once the cross-section is known experimentally. The present theory shows clearly the influence of E/p on both A and B, a matter which has not been satisfactorily discussed previously. The theory is illustrated by calculations of F($\epsilon $) and $\alpha $/p for hydrogen over a range of E/p from 10 to 1000. The agreement between the calculated results and recent reliable observations of $\alpha $/p is surprisingly good considering the nature of the calculations and the wide range of E/p.