Correlation factors for atomic diffusion in nondilute multicomponent alloys with arbitrary vacancy concentration

Vacancy-assisted tracer diffusion in a multicomponent kinetic alloy consisting of x^λN atoms with hopping rate J^λ (where λ≡A,B,C, etc.) and υN vacancies (where υ=1-Σλ^x^λ) distributed randomly over a regular d-dimensional (where d≥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: fᵗʳ=H^''(tr)[H^''(tr)+2J⁰], where J⁰ is the tracer-hopping rate, H^''(tr) is a generalized effective vacancy escape frequency, H^''(tr)=[M(1-υ)[J⁰υfᵗʳ+Jᵉᶠᶠ], where Jᵉᶠᶠ is an effective hopping rate of the background atoms averaged with a weighting factor proportional to x^λ and f^λ, i.e., ${J}ᵉᶠᶠ={{Σ}{{λ}}^{}({J}^{{λ}}{x}^{{λ}}{f}^{{λ}})}{{Σ}{{λ}}^{}({x}^{{λ}}{f}^{{λ}})}$ and $M={-}{(1+〈cos{θ}〉)}{〈cos{θ}〉}$. For a single-component alloy, with particle concentration $x$, ${J}^{{λ}}=J$, and vacancy concentration ${υ}=1{-}x$ our theory provides an excellent overall description of the correlation factor as long as ${J}{{J}⁰}{}{z}^{{-}2}$. Indeed, even for $J{→}0$, the calculated results agree with the Monte Carlo estimates, except in the immediate vicinity of the percolation threshold, ${{υ}}ₚ$, which is located self-consistently to an accuracy of the order $1/z$.