Phonon resonances associated with a vacancy in CaF2

A Green-function formulation for calculating the phonon resonances due to an anion vacancy in Ca${\mathrm{F}}_2$ is given. From a study of the relaxations of the first and second neighbors of an anion vacancy in Ca${\mathrm{F}}_2$, we have computed the force-constant changes, felt by these first and second neighbors due to the anion vacancy. These force-constant changes along with the lattice Green function of Ca${\mathrm{F}}_2$ (computed from shell-model data fitted to the experimental phonon dispersion curves) are employed to estimate the potential resonant modes in this system. Symmetry coordinates are used to block diagonalize the $g$ and $\delta l$ matrices and hence to study the resonant modes occurring under the different individual irreducible representations. Resonant modes are predicted under the $A$, $F_1$, and $F_2$ representations. The possibility of detecting some of these modes experimentally is discussed.