On the Anisotropy of Cubic Ferromagnetic Crystals

It is shown that the dependence of the intensity of magnetization on direction in cubic single crystals probably results from the interplay between orbital valence and spin-orbit interaction. Because of the spin-orbit coupling, the spin vectors responsible for the ferromagnetism feel slightly the anisotropic electrostatic forces which connect the orbital angular momenta of different atoms and whose bonding effect is called orbital valence. In consequence there is apparent dipole-dipole coupling between the spins of different atoms with a constant of proportionality about fifty times larger than results from pure magnetic forces between spins. The same mechanism also gives rise to apparent quadrupole-quadrupole coupling, but it is shown very generally that the latter is possible only if the spin quantum numbers of the atoms are greater than ½. Although dipole-dipole coupling is well known not to contribute to cubic anisotropy when the elementary magnets are all parallel, there is an appreciable contribution in the second approximation of perturbation theory in which complete parallelism is not assumed. The perturbation calculations can be carried through for both the dipole or quadrupole models, provided a Weiss molecular field is used to portray exchange interaction. Both lead to a constant $K_1$ of about the observed order of magnitude in the expression $F_0+K_1({{\alpha }_1}^2{{\alpha }_2}^2+{{\alpha }_2}^2{{\alpha }_3}^2+{{\alpha }_3}^2{{\alpha }_1}^2)+K_2{{\alpha }_1}^2{{\alpha }_2}^2{{\alpha }_3}^2$ for the free energy. The temperature variation of $K_1$ is given correctly in so far as $K_1$ vanishes much more rapidly than the intensity of magnetization near the Curie point, but the calculations are not sufficiently refined to give quantitative details of the temperature dependence, such as, for example, the different behavior of iron and nickel. The empirical values of $K_2$ seem somewhat larger than one would expect provisionally from dimensional considerations, but higher order calculations are needed before this point can be definitely settled. Our model is admittedly somewhat phenomenological, but in our opinion comes closer to physical reality than others in the literature, which are criticized. An explanation is given of why the so-called lattice sums in magnetostriction have larger magnitude, and sometimes different sign, than computed for pure magnetic coupling between the spins. In the final section a brief discussion is included on the anisotropy of hexagonal crystals.