High frequency permeability of the composite with ferromagnetic spherical shells

The paper studies high-frequency permeability of the composite materials consisting of hollow ferromagnetic particles embedded into the non-magnetic media. We model the ferromagnetic particles in composite by spherical shell: the thickness of the ferromagnetic region $d$ compared to the particles' diameter $D$ can vary in a wide range, from $d\ll D$ to $d\sim D$. We assume that the magnetization distribution in such a particle is non-uniform, but forms a vortex-like structure: the magnetization is twisted in some plain outside two vortex cores placed at the poles of the particle. The high-frequency permeability of such a composite material has been studied in the limit of non-interacting particles. We study the dependence of the permeability on the ratio $d/D$. It was shown, in particular, that in the limit $d/D\ll1$ the frequency dependence of the particle's susceptibility is quite similar to that for the thin film. At the same time, the magnetization oscillations in the ac field are non-homogeneous.