The generalized atomic hopping problem—occupancy correlation functions

Abstract

We develop a formalism for calculating the occupancy correlation functions associated with atomic hopping processes where atoms at certain lattice sites can hop to nearby vacant lattice sites. Although the problem is trivial if ${\Gamma }_{\alpha \beta }={\Gamma }_{\beta \alpha }$ where ${\Gamma }_{\alpha \beta }$ is the hopping rate from site $\alpha$ to site $\beta$, it becomes a true many-body problem if ${\Gamma }_{\alpha \beta }\ne {\Gamma }_{\beta \alpha }$. The formalism yields a diagrammatic expansion of the moments of the correlation functions. Certain classes of moment diagrams can then be approximately summed to yield the correlation functions themselves. Applications to the hopping of interstitial hydrogen in metals as well as the usual atomic migration are discussed.