Random Walker in a Cubic Crystalline Structure

Let us now consider a random walker in a three-dimensional cubic lattice. The atom will jump between sites of the normal lattice for a substitutional diffuser, and from interstitial to interstitial site for an interstitial diffuser. In the present case, the Einstein-Smoluchovskii equation for the diffusion coefficient in three dimensions which is a generalization of Equation 5.36, that is, 

In the first approximation, we ignore the correlation effects that are contained in the double summation, that is, since the steps are not correlated, consequently 

In the general case, correlation effects are present during the diffusion process. This fact implies that 

In the case of diffusion in a cubic lattice, it is possible to only consider nearest-neighbor jumps of equal lengths, /, into, z, adjacent sites. Besides, the time between jumps is, x, then t=Nv, in addition, the jump frequency is 

Consequently, it is possible to define the correlation factor as follows [4,7]  

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