Anisotropy of the Spin-Lattice Relaxation for the Kramers Doublets in Cubic Crystals. II. Analysis of Specific Cases

Using the expressions derived earlier for the anisotropy of spin-lattice relaxation for Kramers doublets in cubic crystals, numerical estimates have been made of the anisotropies for ${\mathrm{Co}}^{2+}$ in MgO and ${\mathrm{Yb}}^{3+}$ in Ca${\mathrm{F}}_2$ crystals. The results show that the ${\Gamma }_{3g}$ mode of the complex formed by the nearest neighbors around the ${\mathrm{Co}}^{2+}$ ion contribute most significantly to the relaxation rate at low temperature, and the anisotropy varies as {$1-(l^4+m^4+n^4)$}, where $l$, $m$, $n$ are the direction cosines of the Zeeman field about the crystal axes. On the other hand, for ${\mathrm{Yb}}^{3+}$ in Ca${\mathrm{F}}_2$, one of the two ${\Gamma }_{5g}$ modes is most effective for relaxation processes, and this causes an anisotropy which is given by ($l^4+m^4+n^4$). The implications of these results for other paramagnetic crystals are discussed.