Impurity modes due to interstitials in CaF2

A Green-function formulation for the impurity modes due to interstitials in Ca${\mathrm{F}}_2$ is given. Two methods, one due to Brice and the other due to Maradudin et al., are used. The interstitials discussed are the ${\mathrm{F}}^{-}$ ion, the ${\mathrm{H}}^{-}$ ion, and atomic hydrogen in Ca${\mathrm{F}}_2$. Symmetry coordinates are worked out to block diagonalize the Green-function matrix $\mathit{g}$ and the perturbation matrix $\delta \mathit{l}$. Shell-model data fitted to the experimental phonon dispersion curves are employed to compute the necessary lattice Green functions. The force-constant changes are determined by studying the relaxations of the first and second neighbors of the interstitial. Good agreement between the calculated and experimental values of the localized modes due to ${\mathrm{H}}^{-}$ and atomic H interstitials is obtained. Resonant modes and their widths are computed for all the cases and the possibility of their experimental detection is discussed. The results of the alternate methods are compared.