Ionization Currents in Non-Uniform Electric Fields

It is demonstrated that calculation of the ionization current in a gaseous discharge by means of the classical Townsend equation $i=i_0{e}^{\int \alpha \mathrm{dx}}$ is likely to lead to large errors when the field distribution is not uniform. For a field approximately inversely proportional to distance from the cathode, the error was greater than 25 percent when the field intensity changed by more than 2.5 percent per mean free path of electrons just able to ionize. For fields of the type found at the cathode end of a glow discharge, therefore, the Townsend equation is seldom if ever applicable. A differential-difference equation for the electron current as a function of the electron energy and distance from the cathode was derived, and, by the use of semi-empirical functions where adequate data are not available, the ionization currents were calculated by step-by-step numerical methods for a restricted range of pressure and applied voltage. The results agree with measured currents within the range where the assumed functions apply. The method is much more laborious than integration of the Townsend equation, but it yields more information since the actual electron-energy distribution at each point of the discharge is found.