Electron density distributions in gas discharge columns with electron production and loss rates linear and quadratic in electron density

Previously, a simple yet exact characteristic equation was derived for steady-state longitudinally uniform positive columns in which the electron density ne is given by the continuity equation D∇2ne+νne +kn2e =0 with the coefficients independent of position and with ne =0 at the boundaries, which can be of arbitrary cross-sectional shape. The effective coefficients ν and k may be positive or negative and can each represent the net effect of various electron production and loss processes. The general characteristic equation contains a dimensionless quantity S (the integral of −∇2ne/ne over the cross section) which provides information about the shape of the electron density distribution. The S integral has been evaluated numerically for circular and rectangular column cross sections. For the case of a circular cross section the variation of the density profile and S have been determined for all possible relative magnitudes and combinations of signs of ν and k. The results provide a generalized parameter map which indicates steady-state operating conditions for a large class of discharge columns. S varies from large positive values for diffuse columns (for k<0, i.e., with electron loss by volume recombination) to large negative values for constricted columns (for ν<0, i.e., with loss by attachment). For ν<0, the results indicate that no steady-state distributions exist with attachment-dominated loss at the location of maximum ne.