Magnetic Interaction and Resultant Anisotropy in Unstrained Ferromagnetic Crystals

The mutual potential energy of two equal and parallel magnets is derived for several especially symmetrical distributions of magnetic moment in each. If R is the distance between magnet centers the first three terms in order of importance depend upon R^-3, R^-5 and R^-7. These may be called "dipole," "quadrupole" and "sextupole" terms, respectively. Magnets previously considered as representing the atoms in a ferromagnetic crystal are special cases of one of the general cases here treated. Sextupole terms have not heretofore been included. The distributions of magnetic moment now dealt with permit a closer correspondence between the model magnet in iron, cobalt, and nickel and the probable distribution in these atoms of the electrons responsible for ferromagnetism.In order to compute ferromagnetic anisotropy---differences in potential energy for differences in direction of magnetization in a crystal---sums of zonal harmonics over the points occupied by atoms are also needed. Such sums for second and fourth-order harmonics have previously been reported for the crystal structures of interest. Sums of sixth-order harmonics are now given.Within the space assigned in current pictures of the atom to "ferromagnetic" electrons we find ample room for distributions of magnetic moment, agreeing in total amount with observed saturation values, to account in general for reported anisotropies in iron, cobalt, nickel and their cubic alloys. These distributions seem to depend principally upon the maximum number of codirected electron spins. Reported changes of anisotropy with rise of temperature are such as would result from relative rotations of atomic axes out of exact parallelism.The effects of magnetostrictive strain, here omitted from consideration, may explain some discrepancies in the sign or magnitude of the sextupole terms. In hexagonal cobalt it must be assumed that the molecular field itself is anisotropic.