Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 3 / a

For a partially filled $3d$ shell, electron–electron Coulomb repulsion and Hund's rules first favour particular total spin and orbital terms. The crystal field splitting is substantial because these orbitals overlap the ligand environment; it normally exceeds the spin-orbit coupling and often quenches the orbital moment. Competition between the splitting and the electron pairing energy determines high-spin versus low-spin filling. A common hierarchy is Coulomb term energies, then crystal-field and Hund-exchange scales, then spin–orbit and weak-field Zeeman energies; the relative sizes of crystal-field splitting and pairing are material-dependent. Hyperfine coupling, if the nucleus carries spin, adds a smaller splitting. An isolated ion has no inter-ion exchange ordering field.

For a partially filled $4f$ shell, the outer $5s$ and $5p$ shells shield the localized orbitals from the ligands. Coulomb interactions establish $L,S$ through Hund's rules, and spin-orbit coupling usually exceeds the much smaller crystal field splitting. First form a ground $J$ multiplet using $\mathbf{J}=\mathbf{L}+\mathbf{S}$, then split that multiplet by the crystal environment. The orbital contribution is therefore generally retained, and the g-factor is a multiplet Landé factor rather than an almost spin-only value. Weak external Zeeman and hyperfine interactions can subsequently resolve the remaining states. Exceptions with $J=0$ or unusual valence require separate treatment.