Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Applied Mathematics


Mikko Karttunen


The dependence of the magnetic hardness on the microstructure of magnetic solids is investigated, using a field theoretical approach, called the Magnetic Phase Field Crystal model. We constructed the free energy by extending the Phase Field Crystal (PFC) formalism and including terms to incorporate the ferromagnetic phase transition and the anisotropic magneto-elastic effects, i.e., the magnetostriction effect. Using this model we performed both analytical calculations and numerical simulations to study the coupling between the magnetic and elastic properties in ferromagnetic solids. By analytically minimizing the free energy, we calculated the equilibrium phases of the system to be liquid, non-magnetic solid and magnetic solid. We also studied the anisotropic manegostriction effect using the infinitesimal strain theory. Finally we calculated the hysteresis loop of a single crystal by minimizing the material's magnetic free energy. These analytical calculations gave us an insight into the properties of the model. We then used numerical simulations to solve the dynamical equations of motion and to track the evolution of the density and magnetization fields. Using simulations we confirmed the analytically calculated phase diagram and the hysteresis loop of a single crystal. We also performed simulations to address the effect of the grain size on the magnetic hardness. In these simulations we computed the coercivity of the system for different grain sizes and showed that the results are in agreement with the experimental data on magnetic nanocrystalline alloys. This is a quite interesting result which enables us to comprehend the mechanism of the formation and growth of the domains in the presence of the grains and the mutual effects of the elastic and magnetic properties. Finally we studied the effect of the coercivity on the grain boundary angle and showed that the coercivity decreases with increasing the grain boundary angle. The importance of such studies lies in today's need for more efficient electronic devices such as transformers and magnetic recording devices. The PFC formalism used in this research, although being a coarse-grained free energy, can resolve the atomic structure and symmetries of the solid and therefore many natural properties of the solid that are associated with the symmetry and periodicity, spontaneously emerge in this formalism. This includes elastic and plastic deformations, differently oriented grains and grain boundaries in polycrystals and formation and diffusion of the defects. This features makes this method ideal for the subject of this research.