Anisotropic Spin-Orbit Coupling ofd3andd8Solutes in Corundum

The $g$ shifts and the zero-field splittings of the ground states of the transition metal ions ${\mathrm{Cr}}^{3+}$ (a $d^3$ solute) and ${\mathrm{Ni}}^{2+}$ and ${\mathrm{Cu}}^{3+}$ ($d^8$ solutes) in corundum are studied within the expanded crystalline field framework, in which each $d$ electron is considered to itinerate between the transition metal ion and ligands. It is shown that two types of the distortion of the wave function by the trigonal field play very important roles in the production of the observed $g$ shifts and the zero-field splitting, if the transition metal ions are assumed not to move abnormally. One type is the distortion of the covalent $\pi$ bonds, and another is due to the configuration mixing effect by the trigonal field. The former operates favorably to explain the experimental facts in both cases of the $d^3$ and $d^8$ solutes, while the latter acts favorably in the case of the $d^3$ solutes and unfavorably in the case of the $d^8$ solutes.