Low-temperature heat capacity of alkali-metal and silverβ-aluminas

The heat capacity of $\beta$-alumina containing Li, Na, K, Rb, and Ag as the diffusing ion is found to be a strong function of the ion at temperatures between 1.3 and 40 K. The excess heat capacity above a Debye contribution for the host lattice of Al and O ions is compatible with the low-frequency modes observed in the infrared resulting from the two degrees of freedom for the in-plane vibrations of the diffusing ions. Semiquantitative agreement is obtained using a model of Einstein oscillators with frequencies taken from the infrared results of Allen, Derosa, and Remeika and weights corresponding to the modes of the ions in doubly occupied unit cells being at lower frequencies than those of the singly occupied cells as calculated by Wang, Gaffari, and Choi. By analogy with amorphous materials deviations of the heat capacity at lower temperatures from $T^3$ behavior are attributed to tunneling modes. Assuming the mobile ions and the compensating defects to be distributed at random, a probability distribution of the difference in site energies between otherwise crystallographically equivalent sites is derived. The calculated tunneling specific heat, which is related to the probability of zero energy difference is in fair agreement with the measured values. The heat capacity results imply the existence of a distribution of both attempt frequencies and barrier heights for diffusion in $\beta$-alumina.