Spin-lattice relaxation in rare-earth salts

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 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-2θ cos-2θH-4T-1, where θ 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 (KθD) is less than or greater than the crystalline field splitting Δ between the ground state and the first excited state must be made. Non-Kramers salts to which the former condition apply (KθD Δ) 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 KθD Δ, there is present in addition a term arising from a resonance process, analogous to the resonance radiation effect in gases. Phonons of energy ~ Δ 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 Δ, this leads to a relaxation time proportional to exp (Δ/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 KθD Δ, 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 KθD Δ, the resonance process is also present, the relaxation time again being proportional to exp (Δ/KT). The resonance process is now shown to be dominant down to 1 or 2 °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 KθD Δ.