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 CaF₂:Tm²⁺, CaF₂:Ho²⁺, and MgO:Er³⁺ 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 CaF₂:Tm²⁺ and CaF₂:Ho²⁺ exhibit a small direct process, a T⁹ Raman process, and an e^-ΔkT resonance Raman process (CaF₂:Ho²⁺ only). MgO:Er³⁺ exhibits a large direct process, a complex Raman process with temperature dependence a sum of terms Tⁿ(n=5,,9), and the usual e^-ΔkT resonance Raman process. Calculated results are found to be quite sensitive to the value of <r²> employed. Reasonable agreement with the available experimental data is obtained.