Calculation of the phonon spectra ofβ-alumina

The phonon spectra and the density of states for $\beta$-alumina ($1.3M_2\mathrm{O}·11{\mathrm{Al}}_2{\mathrm{O}}_3$) have been computed for the first time assuming that: (a) only the dynamics of the monovalent metal ions $M^{+}$ need be treated, the other ions are approximated by an elastic potential derived from a recent calculation of Wang, Gaffari, and Choi; (b) the screened Coulomb interaction among $M^{+}$ ions in a single layer is taken into full account, but different layers are assumed to be noninteracting; and (c) overall charge neutrality is maintained by a uniform, negatively-charged background. Two ideal lattice structures are considered: (i) the stoichiometric structure, and (ii) an extended superlattice structure. Allowance for nonideal structures is made, using a self-consistent perturbation method. The observed infrared reflection and Raman spectra are in close correspondence with the computed vibrational density of states, and features in these optical spectra are thus identified and ascribed to specific ionic normal modes. The specific heat is also calculated and in good agreement with high-temperature data.