Antiferromagnetism and ferromagnetism in transitional-metal compounds

In any cubic field, or in the case of octahedral coordination by eight anions, the 3d-state of a transitional-metal cation splits into six states of t2g symmetry with orbital wave functions of the form 

Antiferromagnetic coupling will occur in oxides such as MnO, where there are five collinear spins on each Mn2+(3d5) ion, and the overlap from neighbouring ions is into states with opposite spin. On the other hand, ferromagnetic coupling can also occur if large overlap is possible into unoccupied states with the same spin as that on the ion under consideration. An example is 

An appropriate analysis is as follows. We set instead of (4) for the orbital on atom i 

For ferromagnetism to occur, this must be lower than (7). 

For later discussions it is important to consider what happens if one of the d-orbitals (eg or t2g) is not full (e.g. CoO with seven electrons per atom or TiCl3 with one). The d-orbital is now degenerate, and the orbital motion is no longer quenched. Consider the case of a transitional-metal ion in the d1 state. Suppose that the spins are parallel to the z-axis the spin-orbit coupling will separate the orbitals into functions of the forms 

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