The spin Hamiltonian

The orbital states of a multielectron 3d ion are represented by the states |L,M_L〉and the spin states by |S,M_S〉 . Since the orbital moments are quenched by the crystal field, the effects of the crystal field and spin-orbit coupling on the magnetic energy levels of an ion are conveniently represented by an effective spin Hamiltonian based on the 2S + 1 |M_S〉 sublevels of the ground state. The magnetic properties of the ion depend on the splitting of these levels in a magnetic field.

.......(1)
Following (1), the simplest spin Hamiltonian, used for sites with uniaxial symmetry, is

........(2)

An orthorhombic distortion adds a term E(S²_x− S²_y ). The next term in tetragonal symmetry is FS⁴_z . Cubic symmetry gives a term D_c(S⁴_x+ S⁴_y+ S⁴_z ). The splitting of the levels usually depends on the orientation of the applied field, which is reproduced by an anisotropic ˆg-tensor, so the Zeeman term is written as μ_B/¯h(B · ˆg · S) = (μ_B/¯h)Σ_i,j g_ijB_iS_j . The spin Hamiltonian approach is widely used in electron paramagnetic resonance, where the spectra are parameterized in terms of D, E, . . . , g||, g⊥. The ability to diagonalize large Hamiltonian matrices which include crystal-field, spin-orbit and Zeeman terms by computer has reduced the need for these approximation methods.