Temperature dependence of the Haven ratio in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>-alumina

Abstract

A strong temperature dependence of the Haven ratio ${H}_{R}$, which has been found in $\ensuremath{\beta}$-alumina along with a strange discontinuity in the ${H}_{R}$ vs $T$ curve of Na---$\ensuremath{\beta}$-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---$\ensuremath{\beta}$-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 $\ensuremath{\epsilon}$. A remarkable difference in behavior between Na- and Ag---$\ensuremath{\beta}$-alumina is explained by the difference in the magnitude of $w$ for these two compounds.