Lattice distortions and local compressibility around trivalent rare-earth impurities in fluorites

We have calculated the lattice distortions around trivalent rare-earth dilute impurities, occupying substitutionally metal sites in fluorites. Explicit results are given for the equilibrium positions of the nearest fluorine ligands, $R$, the induced electric dipole moments, and the local hydrostatic strains for $M{\mathrm{F}}_2$ ($M=\mathrm{C}\mathrm{d},\mathrm{C}\mathrm{a},\mathrm{S}\mathrm{r},\mathrm{P}\mathrm{b},\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{B}\mathrm{a}$). These results are used to study the impurity-ligand distance dependence of the fourth-order cubic-crystal-field parameter, $b_4$, for ${\mathrm{Gd}}^{3+}$ and the isoelectronic ion ${\mathrm{Eu}}^{2+}$. Comparison is made with the change of $b_4$ with hydrostatic stress using the calculated local compressibility of the lattice. A consistent description of the experimental data is obtained assuming $b_4\propto R^{-m}$ with $m\sim 10$.