A characteristic class of quantities in nonequilibrium thermodynamics and a statistical justification of the local equilibrium approximation

A theory of bandwidth anisotropy in metallic ferromagnets developed previously is specialised to the case of 5d electrons in a hexagonal close-packed lattice. This theory is combined with a model for 5d electrons in the heavy rare earth metals to give a new theory for the low temperature values of the magnetic anisotropy coefficientsκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy, Ho, Er and Tm. In this theory the magnetic anisotropy is due to a combination of (i) crystal fields acting on 5d and 4f electrons and (ii) bandwidth anisotropy associated with a dependence of 5d bandwidths on magnetization direction. After use is made of empirical upper limits on the eighth order magnetic anisotropy in Gd, there remain four partially adjustable parameters of importance in the theory. These can be chosen to give a good fit to the six observed values forκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy and Ho. Crystal fields corresponding to negative point charges are seen by 5d electrons, but because of 4f – 5d interactions effective fields of larger magnitude and opposite sign act on 4f shells. Bandwidth anisotropy gives a significant contribution toκ ₄ ⁰ of opposite sign to that due to crystal fields, and dominates the latter in Tb and Er.