Ionic recombination in an ambient gas I. Extension of quasi-equilibrium statistical method into nonlinear region

In the past the quasi-equilibrium statistical method has been successfully applied in the calculation of the ionic recombination coefficient α in an ambient gas of sufficiently low density for a to be directly proportional to the density. As originally formulated the method contained nothing analogous to the nonlinearity predicted by the simple theory of Thomson (1924) and it was presumed that this was an inherent limitation. However the method has now been extended, nonlinearity being introduced by treating ion-pairs having inward radial motion separately from those having outward radial motion. This gives rise to coupled integro-differential equations which must be solved numerically. A certain power series expansion for a is obtained which bears a striking similarity to an expansion which may be derived from the closed formula of Thomson.