How peculiar can the temperature variation of magnetic anisotropy be: limitations by a general theorem

The variation of anisotropy constants is studied within the three-constant approximation to the anisotropy free energy. The anisotropy-flow concept for tracing the evolution of the system in the anisotropy space of the system is introduced. Unexpected features of the variation of anisotropy are uncovered for realistic values of the intrinsic parameters as, for example, two zero points for the first anisotropy constants and, correspondingly, two successive reorientation transitions between phases with different easy axes of magnetization. The question of the ultimate possible peculiarity of variation of anisotropy and magnetostriction is addressed by formulating and proving a far-reaching theorem by virtue of which the superposition of p different functions with certain widely met properties may not have more than p-1 zeroes or internal extrema