Types of temperature dependence of single-ion magnetic anisotropy constants by general thermodynamic considerations

A general discussion of the temperature dependence of the magnetic anisotropy of single-ion origin is given based on the connection between the experimentally measured anisotropy constants and the theoretically more fundamental anisotropy coefficients. The cases of uniaxial and cubic crystal symmetry are considered in turn and the qualitative and quantitative differences between them are treated. General analytical arguments valid for at least a whole class of untrivial collective-excitation theories including the mean-field theory are implemented to describe exhaustively the types of temperature dependence of the anisotropy constants in the two-parameter phenomenological expression for the free energy. Anisotropy-flow diagrams in the plane (${\mathit{K}}_1$-${\mathit{K}}_2$) are given for both the uniaxial and the cubic cases. In both cases, three crossover-inducing wedges have been detected in the anisotropy phase diagram. A system which resides in one of these wedges at T=0 inevitably runs away to a phase with another easy-axis orientation at some spin-reorientation temperature ${\mathit{T}}_{\mathit{S}}$ when the temperature is increased. ${\mathit{T}}_{\mathit{S}}$ is determined for some representative cases. In the experimentally interesting case when a uniaxial system evolves from a tilted axis to an easy axis along the c axis or in an easy plane perpendicular to it, the temperature dependence of the cone angle θ(T) is given and a critical angle ${\mathrm{\theta }}_{\mathrm{cr}}$ resolving between the two possible crossover scenarios is determined. Prospective generalizations and applications are described for the implementation of the general procedures to the characterization of anisotropy in technologically important hard magnetic materials as typified by ${\mathrm{Nd}}_2$ ${\mathrm{Fe}}_{14}$B.