Electron Spin-Lattice Relaxation of a Paramagnetic Mass Defect

In this paper, we study the influence of the resonance modes associated with a heavy paramagnetic-impurity ion in a simple-cubic host lattice on the electron spin-lattice relaxation time of the impurity. The relaxation rate is expressed in terms of the phonon propagator for a harmonic lattice containing a mass defect, and is evaluated at low temperatures using a long-wavelength approximation and a Debye spectrum for the density of states. We find that although the direct process is unchanged by the presence of the mass defect, the Raman process is drastically altered. A term having a $T^{11}$ temperature dependence appears in addition to the usual $T^7$ Raman term; the $T^{11}$ term predominates when ${\omega }_R\ll {\omega }_D$, ${\omega }_R$ being the frequency of the resonance mode and ${\omega }_D$ the Debye frequency. Furthermore, if ${\omega }_R\ll {\omega }_D$, a term having an $\exp (-\frac{\hslash {\omega }_R}{\mathrm{kT}})$ temperature dependence for $\mathrm{kT}<\mathrm{}\hslash {\omega }_R$ will take over from the $T^{11}$ term when $10kT>\mathrm{}\hslash {\omega }_R$. The enhancement of the Raman relaxation of ${\mathrm{Cr}}^{3+}$ impurities in MgO observed by Castle, Feldman, and Klemens can be attributed to the excess mass of the chromium ion, and thus provides experimental support for our theory.