Unusual temperature dependence of the magnetic anisotropy constant in barium ferrite BaFe$_{12}$O$_{19}$

We report magnetic hysteresis loops in a wide temperature range (4-700 K) for silica-coated barium ferrite BaFe$_{12}$O$_{19}$ nanoparticles. The saturation magnetization $M_{s}$ and the first magnetic anisotropy constant $K$ are determined simultaneously from the magnetic hysteresis loop using the law of approach to saturation. It is remarkable that $K$ is linearly proportional to $M_{s}$ and varies precisely with temperature as $K(T) = K(0)[1-(T/T_{C})^{1.58}]$ in the whole temperature range below the Curie temperature $T_{C}$ (740 K). The unusual temperature dependence of the anisotropy constant and its linear relation with the saturation magnetization in BaFe$_{12}$O$_{19}$ are not predicted from the existing theoretical models.