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In this paper, we critically review the existing microscopic spin Hamiltonian (MSH) approaches, namely the complete diagonalization method (CDM) and the perturbation theory method (PTM), for 3d8(3d2) ions in a trigonal (C3v, D3, D3d) symmetry crystal field (CF). A new CDM is presented and a CFA/MSH computer package based on our crystal-field analysis (CFA) package for 3dN ions is developed for numerical calculations. Our method takes into account the contribution to the SH parameters (D, g∥ and g⊥) from all 45 CF states for 3d8(3d2) ions and is based on the complete diagonalization of the Hamiltonian including the electrostatic interactions, the CF terms (in the intermediate CF scheme) and the spin-orbit coupling. The CFA/MSH package enables us to study not only the CF energy levels and wavefunctions but also the SH parameters as functions of the CF parameters (B20, B40 and B43 or alternatively Dq, v and v') for 3d8(3d2) ions in trigonal symmetry. Extensive comparative studies of other MSH approaches are carried out using the CFA/MSH package. First, we check the accuracy of the approximate PTM based on the `quasi-fourth-order' perturbation formulae developed by Petrosyan and Mirzakhanyan (PM). The present investigations indicate that the PM formulae for the g-factors (g∥ and g⊥) indeed work well, especially for the cases of small v and v' and large Dq, whereas the PM formula for the zero-field splitting (ZFS) exhibits serious shortcomings. Earlier criticism of the PM approach by Zhou et al (Zhou K W, Zhao S B, Wu P F and Xie J K 1990 Phys. Status Solidi b 162 193) is then revisited. Second, we carry out an extensive comparison of the results of the present CFA/MSH package and those of other CDMs based on the strong- and weak-CF schemes. The CF energy levels and the SH parameters for 3d2 and 3d8 ions at C3v symmetry sites in several crystals are calculated and analysed. Our investigations reveal serious inconsistencies in the CDM results of Zhou et al and Li (Li Y 1995 J. Phys.: Condens. Matter 7 4075) based on the strong-CF scheme for Ni2+ ions in LiNbO3 crystals. The correctness of our CFA/MSH package is verified by comparing our results with the predictions of Ma et al (Ma D P, Ma N, Ma X D and Zhang H M 1998 J. Phys. Chem. Solids 59 1211, Ma D P, Ma X D, Chen J R and Liu Y Y 1997 Phys. Rev. B 56 1780) and Macfarlane (Macfarlane R M 1964 J. Chem. Phys. 40 373) for α-Al2O3 : V3+(3d2) and MgO : Ni2+(3d8). It appears that the two independent approaches show perfect agreement with our approach, unlike those of Zhou et al and Li, which turn out to be unreliable. Our results reveal that the contributions to the ZFS parameter from the higher excited states cannot be neglected; also, the ZFS parameter is very sensitive to slight changes of the crystal structure. Hence our CFA/MSH package, which takes into account the contributions to the ZFS parameter from the higher excited states, can provide reliable results and proves to be a useful tool for the studies of the effect of the l...

The microscopic origin of the spin Hamiltonian (SH) parameters for Ni 2+ (3d 8 ) ions in a trigonal type I symmetry (C 3v , D 3d , D 3 ) crystal field (CF) is studied. In addition to the spin-orbit (SO) interaction, we consider also the spin-spin (SS) and spin-other-orbit (SOO) interactions. The relative importance of the four (SO, SS, SOO, and combined SO-SS-SOO) contributions to the SH parameters is investigated using the CFA/MSH package and the complete diagonalization method (CDM). The SO mechanism is dominant for all CF parameter (CFP) ranges studied, except where the contributions D SO to the zero-field splitting (ZFS) parameter D change sign. For the trigonal CFP, v c ≈ 1200 cm −1 D due to the other three mechanisms exceeds D SO . Although |D SOO | is quite small, the combined |D SO−SOO | is appreciable. The SO-based perturbation theory (PT) works generally well for the g-factors: g and g ⊥ , while it fails for D in the vicinity of v c and for large |v | and v > 0. The high percentage discrepancy ratio δ D = 2020% for v c indicates unreliability of D SO (in PT). Applications to Ni 2+ ions at trigonal symmetry sites in LiNbO 3 , α-LiIO 3 , and Al 2 O 3 , are provided. The theoretical SH parameters are in good agreement with the experimental data. The low symmetry (C 3 ) effects induced by the angle ϕ are tentatively studied, but appear to be quite small.

Relationships involving the interelectronic repulsion parameters, F(k) (k = 2, 4, 6), the spin-orbit coupling constant, ζf, and J-mixing, with the (5)D0-(7)F0 energy, E, have been investigated for Eu(3+) using various approaches. First, the linear relationship between E and the (7)F1 splitting (or the second rank crystal field parameter) is shown to be applicable not only to glasses but also to solid-state crystalline systems with Eu(3+) site symmetry of C2, C2v, or lower. In these cases, the change in (5)D0-(7)F0 energy is mainly due to the J-mixing effect of (7)F(J) (J = 2, 4, 6: most notably J = 2) which depresses (7)F0, whereas the (5)D0 energy is relatively constant. The (5)D0-(7)F0 energy also depends upon certain energy parameters in the Hamiltonian, in particular, F(k) and ζf. Model calculations show that increase in F(4) or F(6) produces an increase in E, whereas increase in F(2) produces a decrease in E. An increase in ζf produces a decrease in E. These findings are rationalized. Most previous 4f(6) crystal field calculations have only considered the F and D terms of Eu(3+) so that the Slater parameters are not well-determined. More reliable energy level data sets and crystal field calculations for Eu(3+) with fluoride, oxide, or chloride ligands have been selected, and certain of these have been repeated since most previous calculations have errors in matrix elements. The fitted Slater parameters have been corrected for the effects of three-body Coulomb interactions. Some systems do not follow the ligand trend F ~ O > Cl for Slater and spin-orbit parameters. From the limited data available, the average values of the corrected Slater parameters are greater for fluoride compared with chloride ligands, but the differences are comparable with the standard deviations of the parameters. There is no clear nephelauxetic series for these three types of ligands, with respect to spin-orbit coupling. Previous correlations of E with various parameters are of limited value because the (5)D0-(7)F0 energy difference not only depends upon the F(k) and ζf parameters but in addition is sensitive to the importance of J-mixing for low symmetry systems.