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 MF₂ ($M=Cd,{{0ex}{0ex}}Ca,{{0ex}{0ex}}Sr,{{0ex}{0ex}}Pb,{{0ex}{0ex}}and{{0ex}{0ex}}Ba$). These results are used to study the impurity-ligand distance dependence of the fourth-order cubic-crystal-field parameter, ${b}₄$, for ${Gd}³⁺$ and the isoelectronic ion ${Eu}²⁺$. 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 ${b}₄{∝}{R}^{{-}m}$ with $m{~}10$.