## Abstract

A general approach to spin-lattice relaxation is given for salts to which a crystalline field theory is appropriate. In particular, the theory of Elliott & Stevens for the interaction of a rare-earth ion with static ionic surroundings is generalized phenomenologically to represent the interaction of the rare-earth ion with the lattice vibrational modes. Evaluation of the spin-lattice interaction in terms of a few constants is possible. One- and two-phonon processes are investigated and the relaxation times for non-Kramers and Kramers salts computed. For the one-phonon (or direct) process the non-Kramers salts exhibit the typical behaviour T$_1 \propto$ H$^{-2}$T$^{-1}$, and the Kramers salts T$_1 \propto$ H$^{-4}$T$^{-1}$. It is shown that, for a given Zeeman splitting of the ground doublet, the latter may exhibit an enormous anisotropy with respect to the direction of the external field, approximately proportional to the anisotropy of the temperature-independent part of the susceptibility. Application of the general theory is made to two salts, holmium and dysprosium ethyl sulphate; the former a non-Kramers, the latter a Kramers salt. It is shown that the dysprosium salt would be expected to show a relaxation time in the direct process region which will vary as sin$^{-2} \theta$ cos$^{-2} \theta$ H$^{-4}$T$^{-1}$, where $\theta$ is the angle the external magnetic field makes with the crystallographic symmetry axis. For two-phonon processes, the additional distinction of whether the Debye energy (K$\theta_D$) is less than or greater than the crystalline field splitting $\Delta$ between the ground state and the first excited state must be made. Non-Kramers salts to which the former condition apply (K$\theta_D$ < $\Delta$) are shown to possess two-phonon relaxation processes of the usual Raman type. The relaxation time is proportional to T$^{-7}$ and is independent of magnetic field. When K$\theta_D$ > $\Delta$, there is present in addition a term arising from a resonance process, analogous to the resonance radiation effect in gases. Phonons of energy $\sim \Delta$ are absorbed and emitted by the spin system preferentially because of a phonon resonance with the crystalline field splitting of the spin states. As normally KT is much less than $\Delta$, this leads to a relaxation time proportional to exp ($\Delta$/KT). This process will dominate the Raman process except at very high and low temperatures. It is shown to be significant right down to the liquid-helium range by comparison with the relaxation rate due to direct processes. Kramers salts, when K$\theta_D$ < $\Delta$, owing to a cancellation in the rate equation, exhibit a Raman relaxation time proportional to T$^{-9}$ and independent of field. This `Van Vleck cancellation' is shown to be a consequence of time reversal symmetry. When K$\theta_D$ > $\Delta$, the resonance process is also present, the relaxation time again being proportional to exp ($\Delta$/KT). The resonance process is now shown to be dominant down to 1 or 2$^\circ$K for many rare-earth salts. Experimental verification is found for the resonance relaxation process in the rare-earth ethyl sulphates. In general, it is expected that this mechanism will be significant for any magnetic salt in which K$\theta_D$ > $\Delta$.