Electron Spin-Lattice Relaxation by the Two-Phonon Process

In this paper, we study electron-spin-lattice relaxation via the two-phonon process. We employ a simple model of a non-Kramers ion which allows the effect of the perturbation produced by changes in the force constants and mass which are associated with the impurity to be considered. We study in detail the case in which a localized resonance in the lattice is produced with a frequency in the acoustical continuum. At temperatures $T$ large compared to the Debye temperature, the relaxation rate is found to be proportional to $T^2$, but with a coefficient which differs from that obtained when the above-mentioned perturbations are neglected. When the resonance in the lattice is sharp and well defined, we find a term in the relaxation rate which has an exponential temperature dependence of the form $\exp (-\frac{{\Theta }_L}{T})$ when $T\ll {\Theta }_L$. Numerical estimates indicate that this term may dominate the relaxation rate at temperatures in the upper end of the liquid-helium range. We also find a term proportional to $T^7$ when $T\ll {\Theta }_L$, but the coefficient is modified by the presence of the perturbations resulting from the force constant and mass changes.