Spin-lattice relaxation in rare-earth salts: field dependence of the two-phonon process

Abstract

The field dependence of the two phonon process is computed taking into account hyperfine and dipolar fields. The general Gorter-Van Vleck-Hebel-Slichter (G.V.H.S.) formalism is adapted to the rare-earth ions, the effective Hamiltonian techniques of the previous paper being used. It is found that a distinction is necessary between salts with g ⊥ = 0 and those with g ⊥ ≠ 0. The Brons-Van Vleck formula holds rigorously for the former case, and predicts a relaxation time inversely proportional to τ − 1 = 0.15 × 10 11 + 6 × 10 7 T 2 s − 1 where H hyp 2 and H dip 2 are the mean square internal hyperfine and dipolar fields, respectively. For the case of g ⊥ ≠ 0, a quite different dependence of relaxation time on field is found by the use of th e G.V.H.S. formalism : 1 / T 1 ∝ ( H 2 + μ H hyp 2 + 1 2 μ ′ H dip 2 ) / ( H 2 + H hyp 2 + 1 2 H dip 2 ) The terms μ and μ' are computed explicitly for the general case. The Van Vleck effective internal field approach predicts μ = 1, μ' = 2, but in general we find that μ < 1 and μ' may be greater than 2. This arises from the failure of the effective field approximation and from processes neglected by Van Vleck in his treatment; specifically, the effects of spin correlation and of off-diagonal terms in the dipolar Hamiltonian which ‘help’ the oscillating electric field to flip spins. The term μ' will in general decrease with increasing temperature for Hainan processes, and be temperature-independent for resonance processes. It should increase with concentration for both. Agreement with experiment is not very good for the case of dysprosium ethyl sulphate (g ⊥ = 0), but the feature of μ' > 2 for salts with g ⊥ ≠ 0 seems to be characteristic of many magnetic salts.