Numerical results on two-body and (two + three)-body lattice dynamics of CaF2, SrF2, and BaF2

Abstract

Utilizing semiempirical two-body and (two + three)-body force fields, respectively, two distinct sets of shell-model calculations are reported on the crystal dynamics of CaF2, SrF2, and BaF2. The central force computations cover longwave properties, dispersion curves, specific heat, and thermal expansion. Also included are numerical results for mean square amplitudes as calculated conventionally and, in addition, by a new computational device which circumvents the need for diagonalizing the dynamical matrix in uninformative regions of k space. On comparing theory and measurement it appears, as expected, that the two-body potentials cannot alone account quantitatively for the experimental findings. The discrepancies, which decrease in the order CaF2, SrF3, and BaF2, probably do not prevent the models from being suitable for various orientational lattice dynamics calculations, say theoretical defect work. The calculations predict different branch pairings at the intersection of symmetry elements in the Brillouin zone for the three alkaline-earth fluorides, the CaF2 realization of compatibility requirements being confirmed by available neutron results. In the BaF2 case the predicted, but as yet unmeasured, dispersion relations exhibit a complete separation of optical and acoustic branches; the latter bearing a strong resemblance to those typical of monoatomic fcc lattices. The numerical work based on (two + three)-body force fields is limited to exploratory calculations concerning the longwave optical properties. Although indicating that Löwdin forces originating in both cation-anion as well as anion-anion overlap may be important, static Heitler-London values for overlap charge approximated by point charges are not suitable for rectifying the two-body shell models. However, in anticipation of a refined theory, present and previous formulas concerning three-body contributions may be useful when treated as adjustable adjunctions to the central force equations of motion.