The hopping recombination

In this Section we describe briefly the principal effects arising during replacement of the continuous diffusion, equation (4.1.23), by its more general analog 

Here the diffusion operator DA. is replaced by the operator L describing motion as stochastic hops in the continuous coordinate space. Let (p r) be the distribution function of hop lengths normalized as 

Defect motion at a large relative distance could always be treated as diffusion with the coefficient D =, where r is the mean (waiting) time between successive hops. Thus we arrive at 

In view of the spherical symmetry of the problem, this expression could be simplified and reduced to the one-dimensional integral 

The quasi-steady-state hopping recombination rate K oo) = Kq is related to the coefficient i eff via equation (4.2.14) as in the diffusion-controlled case. As in equation (4.2.15), this i eff is defined by the asymptotics of the solution, Y r,oo) = y r), as r - cxd. It is important, however, that / eff cannot generally be treated as the effective recombination radius. It holds provided that the hop length is much smaller than the distinctive scale ro of tunnelling recombination 

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