Theory of the Magnetic Anisotropy in KMnF3

A theoretical calculation is made of the magnetic anisotropy in the cubic perovskite structure of KMn${\mathrm{F}}_3$ at room temperature and in its distorted structures at lower temperatures. These distortions are of two types: first, a small tetragonal distortion of the entire crystal; and then, below the antiferromagnetic Néel point, a distortion of the octahedron of fluorine atoms surrounding each manganese. The cubic anisotropy is obtained from a general spin-wave calculation of the zero-point dipole-dipole energy in a cubic antiferromagnet. The result is found to be the same as that for the ferromagnetic case. The anisotropy from the tetragonal distortion is obtained from the change in the classical Lorentz factors. In calculating the effect of the fluorine distortion, a generalization is introduced of Kondo's method for obtaining the anisotropic effective spin Hamiltonian produced by overlap and electron transfer between an ${\mathrm{Mn}}^{++}$ ion and its non-magnetic neighbors. In its present form the method permits the ready calculation of this anisotropy for any symmetry and number of neighbors. Comparison with the microwave resonance and torque measurements of Portis, Teaney, and Heeger, reveals the last effect to be the most important and confirms the form of the spin Hamiltonian found here and its approximate magnitude.