Zero-field splitting of the $4f^7$ state: An electrostatic theory for ${\mathrm{Gd}}^{3+}$-$M^{+}$ complexes in Ca${\mathrm{F}}_2$

From the known bulk properties of Ca${\mathrm{F}}_2$ crystals and the theoretical parameters of the impurity ions ${\mathrm{Gd}}^{3+}$ and $M^{+}$ ($M=\mathrm{L}\mathrm{i},\mathrm{N}\mathrm{a},\mathrm{ }\mathrm{\dots }\mathrm{ }\mathrm{Cs}$) describing the Coulomb, polarization, and the repulsive interactions with neighboring ions in the crystal lattice, we have calculated the ionic positions associated with minimum total potential energy as a function of the $M^{+}$ radius. These results have been employed to evaluate the behavior of the crystal-field parameters $c_1^0$, $B_2^0$, and $B_2^2$, which has been investigated extensively with various experimental techniques in earlier papers from this laboratory. It is concluded from the results given in this paper that with the electrostatic model employed here the crystal-field parameters mentioned above can be described quite well. For the parameter $B_2^2$ it is suggested that it is necessary to allow rather large numbers of ions neighboring the ${\mathrm{Gd}}^{3+}$-$M^{+}$ complex to relax towards new equilibrium positions.