Magnetic interactions and microscopic spin Hamiltonian approaches for 3d3 ions at trigonal symmetry sites

The spin-Hamiltonian (SH) parameters (D, g//, and g⊥) for 4A2(3d3)-state ions at trigonal symmetry sites, taking into account the spin-spin (SS), the spin-other-orbit (SOO), the orbit-orbit (OO) magnetic interactions besides the well-known spin-orbit (SO) magnetic interaction, are studied in the intermediate-field coupling scheme using the CDM/MSH (Complete Diagonalization Method/ Microscopic Spin Hamiltonian) program recently developed. It is shown that the SH parameters arise from five microscopic mechanisms including SO coupling mechanism, SS coupling mechanism, SOO coupling mechanism, OO coupling mechanism, and SO-SS-SOO-OO combined coupling mechanism. The relative importance of the five (SO, SS, SOO, OO and combined SO-SS-SOO-OO) contributions to the SH parameters is investigated. It is shown that the SO coupling mechanism in these coupling mechanisms is the most important one. The effect of the OO coupling mechanism on the energy levels is appreciable whereas that on the SH parameters is negligible. The contribution from the SS coupling mechanism to the zero-field splitting (ZFS) parameter D is appreciable but is quite small to g-factors: g// and g⊥. In contrast, the contribution from the SOO coupling mechanism to the ZFS parameter D is quite small but is appreciable to g-factors. Two perturbation theory method approaches have been examined using CDM/MSH program. It is found that the analytical expressions developed by Macfarlane for D, g//, and g⊥ work well in most of the CF ranges considered whereas those developed by Zdansky for D do not work well in almost all the CF ranges considered. The illustrative evaluation is performed for typical laser material Cr3+: Al2O3. The good agreements between the theoretical values and the experimental finding are obtained. It is found that the percentage difference δD(=|DTotal(CDM)-DSO(CDM)|/|DTotal(CDM)|×100%) reaches 20.9% for laser material Cr3+: Al2O3. The investigation indicates that the contribution to the ZFS parameter D from the SS and SOO magnetic interactions should not be omitted.