Theory of Normal Cathode Fall in Glow Discharges

Abstract

Basis of theory of cathode fall.---From the principle of minimum energy dissipation it is shown that the actual potential distribution in the fall space must be that one which is most favorable to ionization, subject to limitations imposed by Poisson's equation. With the aid of this principle, taking as known quantities the normal cathode fall ${V}_{n}$ and Townsend's ionization constants $N$ and ${V}_{0}$, expressions are derived for the potential distribution, the thickness of the fall space ${d}_{n}$, the distribution of ionization ${n}_{x}$ and the current density ${j}_{n}$ which agree acceptably with experimental values.Results of theory.---The most important results are given by the equations $p{d}_{n}=0.85\frac{{V}_{n}}{N{V}_{0}}(1+\frac{{V}_{0}}{0.97{V}_{n}}+\ensuremath{\cdots})$ $\frac{{j}_{n}}{{p}^{2}}=1.85{(10)}^{\ensuremath{-}4}{\left(\frac{L}{M}\right)}^{\frac{1}{2}}\frac{{(\mathrm{kN})}^{\frac{5}{2}}{{V}_{n}}^{\frac{3}{2}}}{(1\ensuremath{-}\frac{{n}_{0}}{{n}_{d}})},$ in which $L$, $M$ and $p$ are molecular mean free path, molecular weight of positive ions and gas pressure.Explanation of normal and abnormal cathode falls.---It appears that the normal cathode fall is found when the potential is distributed in the manner most favorable to ionization, without restriction by space charge considerations. When the current density exceeds the normal value, space charge considerations require a less favorable potential distribution and hence a larger cathode fall,---which is the abnormal case.