A simple yet general expression characterizes the electrical properties of a steady-state positive column of arbitrary cross-sectional shape with electron production and loss rates linear or quadratic in electron density in addition to electron loss by diffusion. This equation relates effective ionization and diffusion coefficients (or applied electric field), the total number of electrons per unit column length (or electron current), the area of the column cross section, and the cross-sectional shape, which is represented by a single dimensionless number. The derivation and some implications of this equation are described. For the limiting case of a linear production rate only (e.g., one-step electron-impact ionization), the effective ionization coefficient depends on the cross-sectional area but is independent of the number of electrons per unit length. However, in the limit of a quadratic production rate only (e.g., two-step electron-impact ionization), the dependence is reversed; the effective coefficient varies with the number of electrons per unit length but is independent of the size of the cross section.
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Gerald L. Rogoff (1979) studied this question.