Recombination on a discrete crystalline lattice

Up to now we have been considering defect diffusion in continuous approximation, despite the fact that crystalline lattice discreteness was explicitly taken into account defining the initial distribution for geminate pairs. Note, however, that such continuous diffusion approximation is valid only asymptotically when defects (particles) before recombination made large number of hops (see Kotomin and Doktorov [50]). This condition could be violated for recombination of very close defects which can happen in several hops. The lattice statement of the annihilation kinetics has been discussed in detail by Schroder et al. [3, 4, 83], Dederichs and Deutz [34]. Let us consider here just the most important points of this problem. 

Here w(n, t) is the probability to find an interstitial atom in a site n at time t, H n - rh) the hopping rate and p(n, t) the production rate. For the nearest neighbour hopping we have 

The boundary condition of instant annihilation when an interstitial atom finds itself in one of the unstable sites around vacancy reads w jl) = 0, jl G iy. (A shape and size of the instability region are discussed, e.g., by Schroder [3], Dzhumanov and Khabibullaev [16]). 

Now the effective recombination (annihilation) radius could be defined similarly to that in the continuous approximation 

In a model of nearest-neighbour hopping a(m) are nonzero on the surface sites of the recombination sphere only. The recombination probability sought for is 

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