MeangValues of Rare-Earth Ions in Crystals

The mean $g$ value of a Kramers's doublet in a lattice with a symmetry axis is shown to have extrema of magnitude $\frac{2}{3}\Lambda \mathrm{}(J+1\mathrm{})\mathrm{}$ and $\frac{2}{3}\Lambda J$, where $\Lambda$ is the Landé splitting factor and only the lowest free-ion $J$ state is considered. The extrema are associated, respectively, with positive and negative values of the product $g_x{g}_y{g}_z$. The mathematical properties of the extrema are used to derive simple relationships between the parameters of a general wave function, and the results are applied to the anisotropic paramagnetic-resonance spectra of La${\mathrm{F}}_3$: ${\mathrm{Yb}}^{3+}$ and CaO: ${\mathrm{Er}}^{3+}$.