Theoretical explanation of spin-Hamiltonian parameters for the rhombic Mo5+ octahedral clusters in molybdenum phosphate glasses

The high-order perturbation formulas based on the two-mechanism model are used to calculate the spin-Hamiltonian parameters (g factors g_i and hyperfine structure constants A_i, where i = x, y, z) of the rhombic Mo⁵⁺ oxygen octahedral clusters in molybdenum phosphate glasses. These formulas consist of the crystal-field mechanism in the extensively applied crystal-field theory and of the charge-transfer mechanism (which is often neglected). In the calculations, only three adjustable parameters are applied and the six calculated spin-Hamiltonian parameters are reasonably coincident with the experimental values. The results are discussed.     

Unified calculations of spin-Hamiltonian parameters and optical band positions for Ni+ ion in AgGaSe2 crystal

The complete diagonalisation (of energy matrix) method based on the two-spin-orbit-parameter model is applied to unifiedly calculate the spin-Hamiltonian parameters (g factors g_//, g_⊥ and hyperfine structure constants A_//, A_⊥) and optical band positions for Ni⁺ ion in silver gallium selenide (AgGaSe₂) crystal. In the model, besides the contribution due to the spin-orbit parameter of central dⁿ ion (i.e., the one-spin-orbit-parameter model in the traditional crystal-field theory), that of ligand ions are taken into account. The calculated results are reasonably consistent with the experimental values. The local structure of Ni⁺ centre in AgGaSe₂ is estimated through the calculation. The complete diagonalisation method based on the one-spin-orbit-parameter model is also applied to calculate these electron paramagnetic resonance and optical data. It is found that although the calculated optical band positions are close to those based on the two-spin-orbit-parameter model and hence to the experimental values, the calculated spin-Hamiltonian parameters (in particular, the g factors) are in disagreement with the experimental values. The latter point is further confirmed from the calculations with the perturbation method. So, for the rational calculations of spin-Hamiltonian parameters of dⁿ clusters with ligand having large spin-orbit parameter, the contributions due to spin-orbit parameters of both the central dⁿ ion and ligand ion should be contained.     

Theoretical calculations of spin-Hamiltonian parameters and defect structures for Cu2+ in trigonally-distorted tetrahedral sites of ZnO and GaN crystals

The spin-Hamiltonian (SH) parameters (g factors g _//, g ⊥ and hyperfine structure constants ⁶³ A _//, ⁶³ A ⊥, ⁶⁵ A _//, ⁶⁵ A ⊥) for Cu²⁺ ions in the trigonally-distorted tetrahedral sites of ZnO and GaN crystals are calculated from a complete diagonalization (of energy matrix) method (CDM) based on a two spin-orbit parameter model for d ⁹ ions in trigonal symmetry. In the method, the Zeeman and hyperfine interaction terms are added to the Hamiltonian in the conventional CDM. The calculated results are in good agreement with the experimental values. The calculated SH parameters are also compared with those using the traditional diagonalization method or perturbation method only within the ² T ₂ term. It appears that, for exact calculations of SH parameters of d ⁹ ions in trigonal tetrahedral clusters in crystals, the present CDM is preferable to the traditional diagonalization method or perturbation method within the ² T ₂ term. The local structures of Cu²⁺ centers (which differ from the corresponding structure in the host crystal) in ZnO : Cu²⁺ and GaN : Cu²⁺ are obtained from the calculations. The results are discussed.     

Unified calculations of the optical band positions and spin-Hamiltonian parameters for V2+ ions in CdCl2 crystal

The eight optical spectral band positions and three spin-Hamiltonian parameters (g factors g_//, g_⊥ and zero-field splitting D) of V²⁺ ions in trigonal CdCl₂ crystal are calculated together from the complete diagonalisation (of energy matrix) method (CDM) based on the two-spin-orbit-parameter model (also called the cluster approach). In the model, differing from the usual one-spin-orbit-parameter model in the conventional crystal-field theory (where only the contribution to spin-Hamiltonian parameters due to the spin-orbit parameter of central dⁿ ion is considered), both the contributions from the spin-orbit parameter of central dⁿ ion and that of ligand ions are taken into account. The calculated results show reasonable agreement with the experimental values. The local lattice relaxation in the vicinity of V²⁺ ion due to the introduction of V²⁺ impurity is acquired from the calculations. The calculations of spin-Hamiltonian parameters from the CDM based on the one-spin-orbit-parameter and those from the perturbation theory method based on the two-spin-orbit-parameter model are also made for comparison. The results are discussed.     

Theoretical studies of the spin-Hamiltonian parameters for the orthorhombic Pr4+ centers in Sr2CeO4 crystals

Theoretical studies of spin-Hamiltonian (SH) parameters associated with Pr⁴⁺ in Sr₂CeO₄ single crystals have been made by using the complete diagonalizing energy matrix method (CDM) for the 4f ¹ electronic configuration. The calculated results are in excellent agreement with the experimental data. The negative signs of the anisotropic g _i -factors and hyperfine structure constants A _i (where i = ∥ or ⊥) for the orthorhombic Pr⁴⁺ ion in Sr₂CeO₄ are suggested from the calculations. By comparing the results obtained by the CDM with the experimental data, one finds it is valid to interpret the SH parameters for 4f ¹ ions in crystals. The results are discussed.      

Theory studies of the local lattice distortions and the EPR parameters for Cu2+, Mn2+ and Fe3+ centres in ZnWO4

The local lattice distortions and the electron paramagnetic resonance (EPR) parameters (g factors, hyperfine structure constants and zero-field splittings) for Cu²⁺, Mn²⁺ and Fe³⁺ in ZnWO₄ are theoretically studied based on the perturbation calculations for rhombically elongated octahedral 3d⁹ and 3d⁵ complexes. The impurity centres on Zn²⁺ sites undergo the local elongations of 0.01, 0.002 and 0.013 Å along the C₂ axis and the planar bond angle variations of 8.1°, 8.0° and 8.6° for Cu²⁺, Mn²⁺ and Fe³⁺, respectively, due to the Jahn–Teller effect and size and charge mismatch. In contrast to the host Zn²⁺ site with obvious axial elongation (～0.31 Å) and perpendicular (angular) rhombic distortion, all the impurity centres demonstrate more regular octahedral due to the above local lattice distortions. The copper centre exhibits significant Jahn–Teller reductions for the spin-orbit coupling and orbital angular momentum interactions, characterised by the Jahn–Teller reduction factor J (≈0.29 ? 1). The calculated EPR parameters agree well with the experimental results. The local structures of the impurity centres are analysed in view of the corresponding lattice distortions.     

Theoretical investigation of the spin-Hamiltonian parameters and optical spectra for the doped Cu2+ in ZnCdO nanopowder

The spin-Hamiltonian (SH) parameters (g factors g_||, g_⊥ and hyperfine structure constants A_||, A_⊥) and d–d transitions for ZnCdO:Cu²⁺ are calculated based on the perturbation formulas for a 3d⁹ ion in tetragonally elongated octahedra. Good agreement between the calculated results (four SH parameters and three optical absorption bands) and the experimental results can be obtained. Since the SH parameters are sensitive to the local structure of a paramagnetic impurity center, the tetragonal distortion (characterized by the relative elongation ratio ρ ≈ 3.5% along the C₄ axis) of the impurity center due to the Jahn–Teller effect is also acquired from the calculations. The negative and positive signs of hyperfine structure constants A_|| and A_⊥ for ZnCdO:Cu²⁺, respectively, are also suggested in the discussions.     

Theoretical investigations of the EPR parameters and local structure for Cu2+ in BeO

The electron paramagnetic resonance (EPR) parameters (the anisotropic g factors, the hyperfine structure parameters and the quadrupole coupling constant Q) and local structure for Cu²⁺ in BeO are theoretically investigated from the perturbation formulas of these parameters for a 3d⁹ ion under trigonally distorted tetrahedra. The ligand orbital and spin-orbit coupling contributions are included in the basis of the cluster approach, in view of the strong covalency of the [CuO₄]^6? cluster. From the calculations, the impurity Cu²⁺ is suggested not to occupy exactly the ideal Be²⁺ site but to suffer a slight inward displacement (≈0.024 Å) toward the ligand triangle along the C₃ axis. The theoretical EPR parameters show good agreement with the experimental data.     

Theoretical investigations of the spin Hamiltonian parameters and local tetragonal distortions for Cu2+ in crystalline and amorphous TeO2 and GeO2

The spin Hamiltonian parameters (i.e., anisotropic g factors and hyperfine structure constants) and local tetragonal distortions for Cu²⁺ in crystalline and amorphous TeO₂ and GeO₂ are theoretically investigated using the high-order perturbation formulas of these parameters for a tetragonally elongated octahedral 3d⁹ cluster. The impurity Cu²⁺ occupying the octahedral sites are found to experience the relative tetragonal elongation ratios of about 11.4% and 9.5% for crystalline TeO₂ and GeO₂ and 10.8% and 6.6% for amorphous TeO₂ and GeO₂, respectively, along the C₄ axis due to the Jahn–Teller effect. This reveals the larger tetragonal elongation distortions for the Cu²⁺ centres in crystalline than amorphous systems (especially TeO₂). The theoretical spin Hamiltonian parameters show good agreement with the experimental data. The results are discussed.     

Theoretical investigation of the optical spectra and g factors for Cu2+ in dioptase

The optical spectra of Cu²⁺ in dioptase are calculated using crystal-field theory. Good agreement between measured and calculated energy values is obtained under D _4h point-symmetry approximation. The electron paramagnetic resonance g factors, g _// and g _⊥, are also investigated from high-order perturbation formulae. The local structure of Cu²⁺ in dioptase is obtained using these formulae. Theoretical results are in perfect agreement with experimental findings.     

EPR g factors and tetragonal distortion for the isoelectronic Ni and Cu centers in the CuGaSe2 crystal

The EPR g factors, g_|| and g_⊥, for the isoelectronic 3d⁹ ions Ni⁺ and Cu²⁺ at the tetragonal Cu⁺ site of the CuGaSe₂ crystal are calculated from the high-order perturbation formulas based on a two-spin-orbit-parameter model. In the model, both the contributions to g factors from the spin-orbit parameter of central 3d⁹ ion and that of ligand ion are contained. The calculated results appear to be consistent with the experimental values. The tetragonal distortions (characterized by θ−θ₀, where θ is the angle between the metal-ligand bond and C₄ axis, and θ₀≈54.74° is the same angle in cubic symmetry) of Ni⁺ and Cu²⁺ centers, which are different from the corresponding angle in the host CuGaSe₂ crystal and from impurity to impurity, are obtained from the calculations. The difference of the sign of g_||−g_⊥ between the isoelectronic Ni⁺ and Cu²⁺ centers is found to be due to the different tetragonal distortions of both centers in the CuGaSe₂ crystal.     

Theoretical calculations of the optical band positions and spin-Hamiltonian parameters for Cr ions in Y2Ti2O7 crystal

The optical band positions and spin-Hamiltonian parameters (g factors g_g_? and zero-field splitting D) for the trigonal Cr³⁺ centers in Y₂Ti₂O₇ crystal are calculated from the complete diagonalization (of energy matrix) method based on the two-spin-orbit-parameter model. In the calculations, the contributions to spectral data from both the spin-orbit parameter of central dⁿ ion and that of ligand ion are considered and the crystal field parameters used are estimated from the superposition model. The calculated results are in reasonable agreement with the experimental values. The defect structures of Cr³⁺ center is suggested.     

Studies of the optical spectra and spin-Hamiltonian parameters for the trivalent ytterbium ions in lithium yttrium fluoride crystals

In this paper, the crystal field (CF) levels and spin-Hamiltonian (SH) parameters (g factors g _∥ and g _⊥ and hyperfine structure constants A _∥ and A _⊥) of the rare-earth ion Yb³⁺ in lithium yttrium fluoride crystals are calculated under D _2d point symmetry assumption. Two main methods are used in the calculation to study the SH parameters: one is the perturbation theory method and the other is the complete diagonalization (energy matrix) method (CDM). Comparing the calculated results with the experimental data, we can see that the CDM is more effective to calculate the SH parameters. In addition, the CF J-mixing of all excited-state multiplets into the ground-state multiplet ²F_7/2 is considered. The validity of the calculated results is discussed.     

Defect model and spin-Hamiltonian parameters for the tetragonal Nd center in the tetragonal phase of SrTiO3 crystal

The spin-Hamiltonian parameters (g factors g_∥, g_⊥ and hyperfine structure constants ¹⁴³A_∥, ¹⁴³A_⊥, ¹⁴⁵A_∥ and ¹⁴⁵A_⊥) of the tetragonal Nd³⁺ center in the low-temperature (T≈4.2 K) tetragonal phase of SrTiO₃ are calculated from a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are attached to the conventional Hamiltonian and a 52×52 energy matrix concerning the ground term ⁴H_J (J=9/2, 11/2, 13/2, 15/2) is constructed. The Nd³⁺ center is attributed to Nd³⁺ occupying the 12-fold coordinated Sr²⁺ site in SrTiO₃. Differing from the defect model assumed in the previous paper that the tetragonal distortion of this Nd³⁺ center is due to the association of one interstitial oxygen ion at a nearest neighborhood of Nd³⁺ and the Nd³⁺ displacement Δz along C₄ axis, we suggest that it is due to the distortion of SrTiO₃ lattice in the tetragonal phase. The calculated g factors g_∥ and g_⊥ show good agreement with the experimental values, suggesting that our defect model of Nd³⁺ center in SrTiO₃ is reasonable. The experimental hyperfine structure constants were not reported and so our calculated results remain to be checked by EPR experiment.     

Theoretical Calculations of g Factors and Hyperfine Structure Constants for Co2+ inCd2+(I) and Cd2+(II) Sites of CsCdCl3 Crystal

The g factors g‖,g⊥ and hyperfine structure constants A‖,A⊥ for two trigonal Co2+ centers (i.e.,Co2+ in Cd2+ (I) and Cd2+ (II) sites) in CsCdCl3:Co2+ crystals are calculated from the high-order perturbation formulas based on the cluster approach.In the calculation,the contributionsfrom covalency effect and configuration interaction effect are considered and the parameters related to both effects are obtained from the optical spectrum and the structure data of the studied system.The results are in good agreement with the observed values.     

Investigations of the spin-Hamiltonian parameters for Er3+ ions in AlN crystal

The spin-Hamiltonian parameters (g factor g _//, g _⊥ and hyperfine structure constants A _//, A _⊥) for Er³⁺ ion at the trigonal Al³⁺ site of AlN crystal are calculated by diagonalising the 52 × 52 energy matrix. The matrix are related to the ground mutiplet ⁴I_15/2 and the first to third excited multiplets ⁴I_13/2, ⁴I_11/2 and ⁴I_9/2 for 4f¹¹ ions in trigonal crystal field under an external magnetic field. The crystal-field parameters used in the matrix are obtained from the superposition model and the local lattice relaxation due to the substitution of Er³⁺ for Al³⁺ is considered. The calculated spin-Hamiltonian parameters are in reasonable agreement with the experimental values and the signs of hyperfine structure constants are suggested. The results are discussed.     

Characterization of tetravalent vanadium functional centres in metal oxides derived from a spin-Hamiltonian analysis

The electronic and structural characterization of vanadium functional centres in metal oxides by means of electron paramagnetic resonance (EPR) defines an important topic in solid-state research. In that respect, transfer of determined spin-Hamiltonian parameters into electronic and structural information often imposes a ‘bottleneck’ to interpret the obtained EPR spectra. Using two semi-empirical models, EPR spin-Hamiltonian parameters for tetravalent vanadium can be analysed first to distinguish between either V⁴⁺ and vanadyl VO²⁺-centres and second to determine the location of the corresponding centre either in the ‘bulk’ or at the surface region of metal oxide particles.