Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 4 / a

For a free transition-metal ion, degeneracy among different orbital states can support an orbital contribution to its magnetic moment. In a crystal, the electric fields and bonding of the surrounding ligands split those states. If the orbital ground state becomes nondegenerate and is separated from excited orbital states, its wavefunction can usually be chosen real in zero field. Since the orbital angular momentum operator is imaginary in a real-orbital basis, its expectation value then vanishes.

This suppression is orbital quenching. The remaining leading magnetic moment is the spin contribution, approximately $g{\mu }_B\mathbf{S}$ with $g\simeq 2$. Spin-orbit coupling admixes excited orbital states and produces residual anisotropy and shifts in the g-factor; quenching does not mean that every orbital matrix element or every higher-order orbital response vanishes. A degenerate ground orbital multiplet need not be quenched until a further distortion removes its degeneracy.