Analytic relationships for characterizing electrical properties and electron density profiles in gas discharge columns

A previous paper presented numerically determined values of a dimensionless quantity which is useful for characterizing gas discharge columns with electron production and loss rates linear and quadratic in electron density. That quantity, denoted by S, is useful for calculating voltage-current characteristics and for indicating the shape of the electron density distribution, i.e., its degree of diffuseness or constriction. (In terms of the electron density ne, S is the integral of −∇2ne/ne over the column cross section.) Crawford has presented an analytic approximation for a limited range of S for the case of a circular column cross section. In this note Crawford’s expression is compared with values obtained numerically, and it is found to be a very good approximation over its limited range of applicability. In addition, modified versions of Crawford’s expression are provided for use with all possible relative magnitudes and combinations of signs of the linear and quadratic rates. Analytic relationships between the maximum electron density and average electron density are also provided, both for direct use and to facilitate use of the expressions for S.