Theory of the width of the ferromagnetic resonance line of europium iron garnet

Abstract

A successful theory of the width of the ferromagnetic resonance line of Eu iron garnet must explain its variation with temperature, operating frequency and direction. It is found that this goal can be achieved by using the so-called ‘ longitudinal ’ theory in which the width is caused by the fact that the population redistribution among the components of J = 1 lags behind the values appropriate to instantaneous equilibrium with the r.f. field. Agreement with experiment requires that the relaxation processes responsible for the redistribution be primarily of the ‘ inter ’ rather than ‘in tra’ type, i.e. associated with transitions between the states J = 0 and J = 1 of the Eu 3+ ion, rather than being internal to J = 1. The mathematical theory is closely related to that of an earlier paper by two of the authors on the static (i.e. free energy) anisotropy of Eu i.g. in that it involves the simultaneous consideration of the crystalline field and anisotropic exchange. Numerical values of the relevant constants are employed which are consistent with those in the earlier paper. Anisotropic exchange is less important relative to the crystalline potential than it was there, but nevertheless improves the agreement with experiment. The present problem requires in addition a dynamic parameter connected with the relaxation rate. The necessary relaxation time is short and has to be about the same for all components of J = 1. The latter requirement is shown to preclude the relaxation mechanism being that of phonon modulation of exchange interaction. The final part of the paper is concerned with the equations of motion relating to the relaxation process. Their solution shows that rigorously there is more than one relaxation time, but the theory based on a single such time is found to be a good approximation in some cases.