Generating functionology
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In this paper a rigorous probabilistic two-probability-parameter model of a diffusion barrier is investigated that describes comprehensively reflection, absorption, and segregation phenomena at a diffusion barrier. As a special case, a rigorous analysis of counting paths for 1D random walk in the presence of a reflecting barrier is presented. This paper defines and makes distinction between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. A combinatorial formula is derived showing that in the presence of a totally reflecting barrier (at mb=0) the probability of a particle departing from position m=2j and arriving at position m=2k on the positive axis after N= 2M steps is given by W2j(2k,N) = [C(2M, M-j +k)-C(2M, M-j-k-l)]/[C(2M,M)+ 2 Σi=0j C(2M,M + i)], where C(n,m) denotes the binomial coefficient. This formula enables easy computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N = 2M as 1/(M + 1) an_d that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as √4M. Thus, the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.