Theory of Orbital Ordering . II

According to the two-band model in the previous theory, the spontaneous magnetic moment due to the orbital motion of electrons in solids points to a special direction given by the interband matrix element of the angular momentum of an electron around the lattice site to which it belongs. For the three-fold degenerate bands, the free energy of the system is shown to be independent of the direction of the magnetic moment. Also, the previous theory is extended into the case of an arbitrary electron number, taking into account the antiferro-ordering. The solution of the ordering equations in the ferroand antiferro-phases leads to the maximum magnitude of the magnetic moment to be expected for the degenerate pand d(t,g)-states, and the orbital phase diagram is obtained with the result that, for the electron number per atom between 1 and 2, the antiferro-phase rather than the ferro-phase is stable.