The diffusion length for gas discharge columns with electron production and loss rates linear and quadratic in electron density

Abstract

Steady-state positive column calculations can sometimes be simplified by writing the electron diffusion rate in terms of an effective length, thereby replacing a Laplacian with an algebraic quantity. When the electron continuity equation is purely linear in electron density, the characteristic diffusion length is useful for this purpose, since it can be used to relate the electron diffusion loss rate to the dimensions of a plasma and to calculate the electron temperature or operating electric field. If additional, nonlinear processes are introduced, the characteristic diffusion length is no longer appropriate for representing the diffusion rate. This paper presents a simple yet exact definition of a generalized diffusion length for calculations of the spatially averaged electron density in plasmas with electron production and loss rates linear and quadratic in electron density. This generalized effective diffusion length properly accounts for the average effect of electron diffusion for all possible signs and relative magnitudes of the linear and quadratic rates. Some properties of the effective diffusion length differ significantly from those of the characteristic diffusion length, which represents a special limiting case. Problems with other definitions of an effective diffusion length are discussed.