Onsagers Theory of Geminate-Ion Recombination

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as 

Under the joint influence of the external field F and the field of the geminate 

Similarly the outer solution r = rg with a sink at infinity n( ) = 0) is given by 

Demanding continuity of n at r = rg, using the Kirchhoff relation I — l = 1, and identifying (for unit input current) as the escape probability 0, we get 

Equation (9.2) shows the effect of the reaction radius rl on the escape probability, which, remarkably, is free of the diffusion coefficient. Normally rc r which reduces Eq. (9.2) to the celebrated Onsager formula 0 = exp(- r /rg) as given by Eq. (9.1). 

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