Ultimate recombination probability in the absence of an applied electric field

1 Ultimate recombination probability in the absence of an applied electric field 

Mozumder [315] presented an elegant and direct analysis of the longtime or ultimate recombination probability (the complement of which is the escape probability) for an ion-pair. The steady state arises after a time sufficiently long for all transient effects to have decayed away. Only the residual probability that an ion-pair still remains, p(r, t - °°) at some (large) distance of separation is of interest, and dp/dt - 0 as t - °°. Using eqn. (141) with the coulomb potential gives 

On taking the Laplace transform, and noting that the steady-state limit (f- °° where dp/dt 0) is equivalent to the limit s -+ 0, and using the initial condition (131), this reduces to the spherically symmetric 

Equation (147) describes the ultimate density distribution when ion-pairs were created with an initial separation of r0. Not only is it the long-time 

The escape probability is small unless ion-pairs were formed at separations of at least rc/2. In weakly polar or non-polar solvents, rc R and, to a very good approximation 

---