Contribution of the Orbit-Lattice Interaction to the Splitting of theS728Ground State

A calculation of the static contribution of the orbit-lattice interaction to the ground-state splitting of rare-earth $S$-state ions is made, and the effects of mixing due to spin-orbit coupling are treated explicitly. Both the zero-point and the temperature-dependent phonon contributions are determined, and it is shown that, for ${\mathrm{Gd}}^{3+}$ in Ca${\mathrm{F}}_2$, the splitting produced by these terms (a) is approximately three orders of magnitude smaller than the measured value, (b) has the opposite sign to the observed splitting, and (c) possesses (by itself) an incorrect temperature behavior. Consequently, in agreement with our initial calculations, which did not include spin-orbit coupling corrections, we conclude that the static orbit-lattice interaction is not responsible for the ground-state splitting of $S$-state ions.