Effect of surface anisotropy on the exchange resonance modes

Exchange resonance modes in small ferromagnetic spheres were first predicted for the case of a cylindrical symmetry of the magnetization configurations. It is shown here that when the constraint of this symmetry is removed, other modes are also possible, but their lowest resonance frequency is very nearly the same as that of one of the cylindrically symmetric modes. Therefore, new resonances may only be sought at higher frequencies than those observed in the most recent experiment. It is also shown that the dependence on the particle size observed in that experiment may be accounted for by assuming a rather strong surface anisotropy in the measured spheres. The original prediction of an $R^{\text{−2}}$ dependence, typical for the curling mode, may still be reached if either the particle size or the surface anisotropy is considerably reduced.