We describe an accurate periodic boundary condition (PBC) called the symmetric PBC in the calculation of the magnetostatie interaction field in the finite-differentiation-method fast-Fourier-transform (FDM-FFT) micromagneties. The micromagnetic cells in the regular mesh used by the FDM-FFT method are finite-sized elements, but not geometrical points. Therefore, the key PBC operations for FDM-FFT methods are splitting and relocating the micromagnetic cell surfaces to stay symmetrically inside the box of half-total sizes with respect to the origin. The properties of the demagnetizing matrix of the split micromagnetic cells are discussed, and the sum rules of demagnetizing matrix are fulfilled by the symmetric PBC.
Microwave permeability spectra of single Co nanotube under equilibrium state have been studied by micromagnetics simulation.More than four obvious resonance peaks have been found(11.72,24.20,33.18 and 39.55 GHz).Such large resonance frequency cannot be found in other traditional magnetic materials.The configurations of magnetic moments along the nanotube have been simulated.The results show that the top end of nanotube has a"flow-out"pattern of magnetic moments configuration.The bottom end has a"flow-in"pattern of magnetic moments configuration.The magnetic moments within the main body of nanotube are aligned perfectly along the length of nanotube.The magnitude of natural resonance peak is strongly related to the volume fraction of a zone,which has the same orientation of magnetic moments.Large microwave permeability values have been found for single nanotube.The generalized Snoek’s law has been used to validate the micromagnetics simulations in this paper.