The far-infrared conductivity of Na, Ag, Rb, and K $\beta$-alumina is measured from 10 to 200 ${\mathrm{cm}}^{-1\mathrm{}}$. A broad band of absorption is observed whose detailed structure depends on growth conditions. The frequencies observed agree quantitatively with model calculations by Wang, Gaffari, and Choi (WGC) for the vibrations of the mobile ions. The ionic conductivity of Na, Ag, Rb, and K $\beta$-alumina obtained from a melt, a ${\mathrm{Bi}}_2$${\mathrm{O}}_3$ flux, and a Co-doped ${\mathrm{Bi}}_2$${\mathrm{O}}_3$ flux is also measured. Invariably the melt-grown material which has the lowest mobile-ion content has the higher conductivity, the pure flux-grown material the lowest, while the Co-doped material lies between the two. This result points to the crucial but subtle role played by the concentration of mobile ions, compensating defect, and the interaction between the two. Despite the variation of the observed conductivities with growth conditions, convincing correlations can be made between the vibrational properties, the model calculation of WGC, and the observed conductivities, that suggest that the basic jump mechanism is the interstitialcy and it is quantitatively described by the WGC model. It is suggested that the observed variation in conductivity are due to the inhibition of this mechanism by short-range order among the mobile ions and compensating defects.

• Received 29 July 1977

DOI:https://doi.org/10.1103/PhysRevB.17.4031