Vacancy-assisted tracer diffusion in a multicomponent kinetic alloy consisting of $x^{\lambda }N$ atoms with hopping rate $J^{\lambda }$ (where $\lambda \equiv A,B,C$, etc.) and $\upsilon N$ vacancies (where $\upsilon =1-\Sigma {\lambda }^{}{x}^{\lambda }$) distributed randomly over a regular $d$-dimensional (where $d\ge 2$) hypercubic, or close-packed, lattice of $N$ sites is analyzed through a self-consistent renormalization of a recent theory of Tahir-Kheli and Elliott combined with a generalization of concepts introduced by Manning. The result for the tracer-diffusion correlation factor is the following: $f^{\mathrm{tr}}=\frac{H^{\prime \prime }\left(\mathrm{tr}\right)}{\left[H^{\prime \prime }\right(\mathrm{tr})+2\mathrm{}{J}^0]}$, where $J^0$ is the tracer-hopping rate, $H^{\prime \prime }\left(\mathrm{tr}\right)$ is a generalized effective vacancy escape frequency, $H^{\prime \prime }\left(\mathrm{tr}\right)=\left[\frac{M}{(1-\upsilon )}\right[J^0\upsilon f^{\mathrm{tr}}+J^{\mathrm{eff}}]$, where $J^{\mathrm{eff}}$ is an effective hopping rate of the background atoms averaged with a weighting factor proportional to $x^{\lambda }$ and $f^{\lambda }$, i.e., $J^{\mathrm{eff}}=\frac{\Sigma {\lambda }^{}\left(J^{\lambda }{x}^{\lambda }{f}^{\lambda }\right)}{\Sigma {\lambda }^{}\left(x^{\lambda }{f}^{\lambda }\right)}$ and $M=-\frac{(1+〈cos\theta 〉)}{〈cos\theta 〉}$. For a single-component alloy, with particle concentration $x$, $J^{\lambda }=J$, and vacancy concentration $\upsilon =1-x$ our theory provides an excellent overall description of the correlation factor as long as $\frac{J}{J^0}\gtrsim z^{-2\mathrm{}}$. Indeed, even for $J\to 0$, the calculated results agree with the Monte Carlo estimates, except in the immediate vicinity of the percolation threshold, ${\upsilon }_p$, which is located self-consistently to an accuracy of the order $\frac{1}{z}$.

• Received 4 April 1983

DOI:https://doi.org/10.1103/PhysRevB.28.3049