Theory of atomic diffusion in cubic crystals with impurities based on the correlated-walks theory

Abstract

A walker is allowed to move on the simple cubic lattice with the following rules: If it should arrive at any site, it may move in the same direction as that of the previous step with probability $\alpha$, turn at right angles with probability $\gamma$, reverse with probability $\beta$, or remain with probability $\sigma$, normalized such that $\alpha +4\mathrm{}\gamma +\beta +\sigma =1\mathrm{}$. If it were at rest, it may move in any direction with the same probability $\mu$ or remain with probability ${\sigma }^{\prime }$, normalized such that $6\mu +{\sigma }^{\prime }=1\mathrm{}$. An exact expression for the mean-square displacement $〈r^2〉$ after $N$ units of time is derived. From this expression, the diffusion coefficient $D$ is obtained as follows: $D=\left(\frac{1}{6}\right)(1+\alpha -\beta ){(1-\alpha +\beta )}^{-1\mathrm{}}\times {[1+\frac{\sigma }{(1-{\sigma }^{\prime })}]}^{-1\mathrm{}}{a}_0^2{\tau }^{-1\mathrm{}}$, where $a_0$ is the step length and $\tau$ the unit of time. Similar results are obtained for the face-centered and body-centered cubic lattices. These results are used to discuss the atomic diffusion in cubic crystals with impurities, which act as traps. Comparison with previous experimental and theoretical results is made and discussed.