## 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$_\bot$ = 0 and those with g$_\bot \neq$ 0. The Brons-Van Vleck formula holds rigorously for the former case, and predicts a relaxation time inversely proportional to (H$^2$ + H$^2_{hyp.}$ + H$^2_{dip.}$)/(H$^2$ + H$^2_{hyp.}$ + $\frac{1}{2}H^2_{dip.}$), where H$^2_{hyp.}$ and H$^2_{dip.}$ are the mean square internal hyperfine and dipolar fields, respectively. For the case of g$_\bot \neq$ 0, a quite different dependence of relaxation time on field is found by the use of the G.V.H.S. formalism: 1/T$_1 \propto$ (H$^2$ + $\mu H^2_{hyp.}$ + $\frac{1}{2}\mu'H^2_{dip.}$)/(H$^2$ + H$^2_{hyp.}$ + $\frac{1}{2}H^2_{dip.}$). The terms $\mu$ and $\mu'$ are computed explicitly for the general case. The Van Vleck effective internal field approach predicts $\mu$ = 1, $\mu'$ = 2, but in general we find that $\mu$ < 1 and $\mu'$ 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 $\mu'$ will in general decrease with increasing temperature for Raman 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$_\bot$ = 0), but the feature of $\mu'$ > 2 for salts with g$_\bot \neq$ 0 seems to be characteristic of many magnetic salts.