Diffusion tensor for atomic migration in crystals

The diffusion of holes or interstitials in a crystal is treated by several types of asymptotic analyses, of which the starting point is the Fokker-Planck equation. First, we show that the standard Smoluchowski equation holds, for large as well as intermediate or small values of the interatomic potential Phi, as an approximation of the Fokker-Planck equation. Next, we derive from the Smoluchowski equation a ''homogenized'' Smoluchowski equation, which contains only constant coefficients. Finally, we expand these coefficients for large Phi and obtain explicit formulas for the diffusion coefficients in terms of Phi and other physical quantities