Theory of Direct Exchange in Ferromagnetism

An extensive investigation is presented on the role of direct exchange as the mechanism responsible for ferromagnetism. The direct exchange integral $J$ which arises in Heisenberg's theory of ferromagnetism and which has been a subject of considerable speculation and controversy (particularly concerning its sign behavior as a function of internuclear separation) is considered for several cases for which (as Löwdin has shown) $J$ is rigorously defined. (1) A pair of atoms with a single electron per atom (the hydrogenic case). $J$ is calculated for the unrealistic but historically interesting case of hydrogen $3d$ functions and the computationally more difficult case of the exchange between $3d$ orbitals for the iron series elements. The fact that the iron series $3d$ orbitals are not eigenfunctions of the free atom (hydrogenic) Hamiltonian is shown to profoundly affect the results. Calculations for all pairs of $3d$ orbitals show that $J$ is sensitive to the angular dependence of the wave functions (and the precise radial shape as well). (2) A single hole in otherwise closed shells (such as a pair of iron series atoms in the $3d^9$ configuration). The effect on $J$ of "clothing" the atoms with the remaining electrons is discussed first with regard to the effect of the core electrons on the one-electron potentials and secondly with respect to the effect of the overlap of the core electrons. (From an analysis of these terms it is suggested that the paired "$4s$" conduction electrons of the metal can play an important role in "direct exchange," quite aside from a Zener type of effect.) We find that the direct exchange parameter $J$ is large and negative for the two-electron case [case (1)] and negative, but smaller, for the "clothed" $3d^9$ case [case (2)], whereas for ferromagnetism it should be positive. From this one may conclude that either the direct exchange mechanism is not the dominant source of the ferromagnetism of the transition metals or that the direct exchange model is an inappropriate description of their magnetic behavior. Finally, a more exact model of direct exchange is discussed, as are some of the problems inherent in carrying it out.