Then the exchange and magnetostatic energies of the vortices have been compared. The analytical values for the energies have been obtained with the formulas given by Usov et al. [113,114]. The finite element results are in good agreement with the analytical results and it is shown in Tab. 9.3, that the energy of the equilibrium magnetization distribution, which has been found with the FE model, is indeed smaller than that of the rigid vortex model.
In the first row of Tab. 9.3 the analytical results for the rigid vortex model are given, which have been obtained using Ref. [114]. The second row gives the result, which is obtained by the FE program, if the magnetization distribution is initialized with the rigid vortex model. The third row shows how the results improve, if a finite element mesh with smaller mesh size is used (cf. Fig. 9.5), and the fourth row gives the deviation from the analytical solution. In the fifth row the magnetostatic energy, which is almost solely caused by the vortex core, is given for a FE model, where the core has been meshed with a very fine mesh and the rest of the nanodot has been omitted. Finally, the sixth row gives the results after relaxing the magnetization into equilibrium, and the seventh row indicates the deviation from the analytical result. Thus, the rigid vortex model is a good approximation, but the FE calculation shows, that there is a slightly different magnetization distribution, which has a lower total energy.
