Equilibrium direction of magnetization under coherent rotation

The Stoner-Wohlfarth (SW) model describes coherent rotation of magnetization with uniaxial anisotropy. However, the traditional SW model can be analytically solved only for three specific directions, making it challenging to determine the direction of equilibrium magnetization in anisotropic ferromagnets under arbitrary fields. In this study, the uniaxial anisotropy energy in the SW model is extended to include a fourth-order term. By employing inverse function theory, an exact analytical solution for the equilibrium magnetization direction under any external field is derived. The impact of the fourth-order term on hysteresis loops is analyzed, showing increased coercivity for fields parallel to the easy axis and enhanced saturation fields for perpendicular ones. These results improve the understanding of static magnetism and provide a foundation for the quantitative analysis of magnetization scattering by electric and spin currents.