Spin-lattice relaxation in an isotropic elastic continuum with spherical lattice waves and a cubic crystal field

Spin-lattice relaxation among hyperfine levels of the cubic systems Ca${\mathrm{F}}_2$:${\mathrm{Tm}}^{2+}$, Ca${\mathrm{F}}_2$:${\mathrm{Ho}}^{2+}$, and MgO:${\mathrm{Er}}^{3+}$ is studied theoretically from zero to high magnetic fields. Relaxation-rate expressions are derived employing a description of lattice dynamics in terms of normal modes of vibration transforming as spherical harmonics. Full use is made of symmetry considerations in formulating the ion-lattice interaction Hamiltonian and the computation of transition rates. Relaxation between ion levels not derived from a time-conjugate pair of electronic states is discussed in detail, with particular attention to the Raman process which may exhibit a complex temperature dependence in such cases. Relaxation rates are calculated from a crystal-field model and isotropic-elastic-continuum lattice dynamics. The systems Ca${\mathrm{F}}_2$:${\mathrm{Tm}}^{2+}$ and Ca${\mathrm{F}}_2$:${\mathrm{Ho}}^{2+}$ exhibit a small direct process, a $T^9$ Raman process, and an $e^{-\frac{\Delta }{\mathrm{kT}}}$ resonance Raman process (Ca${\mathrm{F}}_2$:${\mathrm{Ho}}^{2+}$ only). MgO:${\mathrm{Er}}^{3+}$ exhibits a large direct process, a complex Raman process with temperature dependence a sum of terms $T^n(n=5,\dots ,9)$, and the usual $e^{-\frac{\Delta }{\mathrm{kT}}}$ resonance Raman process. Calculated results are found to be quite sensitive to the value of $<r^2>$ employed. Reasonable agreement with the available experimental data is obtained.