Temperature dependence of the Haven ratio inβ-alumina

A strong temperature dependence of the Haven ratio $H_R$, which has been found in $\beta$-alumina along with a strange discontinuity in the $H_R$ vs $T$ curve of Na—$\beta$-alumina at low temperature, is explained by a many-body diffusion theory utilizing the newly improved pair approximation of the path probability method of irreversible statistical mechanics based on a simple two-site model. The effect is due (1) to a strong temperature dependence of the "correlation effects" in both diffusion and conduction of ions as a result of changes with temperature in the population of conduction ions among different kinds of available lattice sites, and (2) to a difference in the statistical nature of tracer diffusion and ionic conduction. An apparent discontinuity in the $H_R$ vs $T$ curve observed for Na—$\beta$-alumina at low temperatures is connected to a percolation difficulty which occurs only for tracer diffusion near the stoichiometric composition because of the existence of preferred sites for conduction ions. The $H_R$ vs $T$ curves are very sensitive to the occupancy ratio of available sites of conduction ions (or the degree of nonstoichiometry), to the ratio of the site-occupancy energy difference $w$ between the preferred sites and the interstitial sites, and to the mutual interactions among conduction ions $\epsilon$. A remarkable difference in behavior between Na- and Ag—$\beta$-alumina is explained by the difference in the magnitude of $w$ for these two compounds.