Study of the electron paramagnetic resonance spectra of Zn(antipyrine)2(NO3)2:VO

In this work, a full ligand-field energy matrix (10×10) diagonalization treatment for 3d¹ ions in tetragonal symmetry is developed on the basis of the two-s.o.-coupling-parameter model. Spin Hamiltonian parameters (g factors g_∥, g_⊥ and hyperfine structure constants A_∥, A_⊥) of the tetragonal V⁴⁺ center in Zn(antipyrine)₂(NO₃)₂ are calculated from the complete energy matrix diagonalization method and the perturbation theory method. The calculated results from both methods are not only close to each other but also in good agreement with the experimental values. Furthermore, the compressed defect structure of V⁴⁺ center 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.     

Studies of the optical spectrum band positions and spin-Hamiltonian parameters for Pa ion in ThX4 (X=Cl, Br) crystals

The optical spectrum band positions and spin-Hamiltonian (SH) parameters (g factors g_∥ and g_⊥ and hyperfine structure constants A_∥ and A_⊥) for 5f¹ ion Pa⁴⁺ at the tetragonal Th⁴⁺ site of ThX₄ (X=Cl, Br) crystals are calculated from a complete diagonalization (of energy matrix) method (CDM). In the CDM, the magnetic (or Zeeman) interaction and hyperfine interaction terms are added to the Hamiltonian in the conventional CDM and so the optical and EPR spectra data can be studied in a unified way. The calculated results are in reasonable agreement with experimental values. The possible misprints or small errors in the experimental g factors for Pa⁴⁺ in ThX₄ crystals are pointed out. The results are discussed.     

Theoretical Explanation of the Optical and Electron Paramagnetic Resonance Spectral Data for Trivalent Ytterbium Ions in β-Lead Fluoride

Six crystal-field energy levels and three electron paramagnetic resonance (EPR) data [g factor and hyperfine structure constants A(¹⁷¹Yb³⁺) and A(¹⁷³Yb³⁺)] for trivalent ytterbium (Yb³⁺) ions in the cubic β-lead fluoride (β-PbF₂) crystal are calculated by using a diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the Hamiltonian in the conventional diagonalization method, and so the optical and EPR data can be calculated in a unified way. The calculated results are in reasonable agreement with the experimental values, and the signs of constants A(¹⁷¹Yb³⁺) and A(¹⁷³Yb³⁺) are determined. The results are discussed.     

Theoretical investigation of ESR spectra and local structure for VO in Ba2Zn(HCOO)6(H2O)

The electron spin resonance spectra (characterized by g factors g_, g_⊥ and hyperfine structure constants A_ and A_⊥) of Ba₂Zn(HCOO)₆(H₂O) (BZFA): VO²⁺ crystal are calculated from high order perturbation formulas. The calculated results are in good agreement with the observed values. The local structure parameters of [VO(H₂O)₅]²⁺ clusters are also obtained from the calculation. The magnitudes of the metal-ligand distances parallel and perpendicular to the C₄-axis are, respectively, R_≈0.163 nm and R_⊥≈0.210 nm. It is shown that the local structure around the V⁴⁺ ion possesses a compressed tetragonal distortion along C₄-axis.     

Theoretical investigation of EPR and optical spectra of Mo(V) in [Mo6O19][N(C4H9)4]3 salt

The electron paramagnetic resonance (EPR) spectral data (the g factors and hyperfine structure constants) and d–d transition spectra for the tetragonal Mo⁵⁺ centre in [Mo₆O₁₉][N(C₄H₉)₄]₃ salt are theoretically investigated from the complete diagonalization method (CDM) for a 4d¹ ion in tetragonally compressed octahedron. The theoretical results are in good agreement with the experimental data. The dependency of the g factors of the ground state on the R_∥(MoO bond length) has been studied. It is shown that the g factors varied with the R_∥ approximately in a linear way.     

Theoretical explanation of g factors for Ti ions in LiNbO3 and LiTaO3 crystals

The EPR g factors g_// and g_⊥ for Ti³⁺ ions at the trigonal octahedral Li⁺ sites of LiNbO₃ and LiTaO₃ crystals are calculated from the third-order perturbation formulas of g factors for 3d¹ ion in trigonal symmetry. In the calculations, the crystal-field parameters are obtained from the structural data by using the superposition model. The calculated values are in reasonable agreement with the observed values. The results are discussed.     

Theoretical explanation of absorption spectra and ESR parameters of Cu in shattuckite

The optical absorption spectra and electronic spin resonance parameters (ESR g factors g_‖, g_⊥ and hyperfine structure constants A_‖, A_⊥) for Cu²⁺ in shattuckite crystal are calculated from the two spin–orbital coupling parameters model, high-order perturbation formulas and complete diagonalization (of energy matrix) method (CDM) of 3d⁹ ion in tetragonal symmetry. The calculated results are in good agreement with the observed values. Since the ESR parameters are sensitive to the local structure of a paramagnetic impurity center, the defect structure of Cu²⁺ center in shattuckite crystal is estimated. The results are discussed.     

Optical spectra and spin-Hamiltonian parameters of trivalent ytterbium in lead tungstate

By using crystal-field theory, the optical spectra and spin-Hamiltonian parameters (abbr. SH parameters, i.e. the anisotropic g factors g _∥, g _⊥, and hyperfine structure constants A _∥, A _⊥) of ¹⁷¹Yb^3 + and ¹⁷³Yb^3 + isotopes in the tetragonal PbWO₄ are calculated. The theoretical results agree well with the experimental values. The crystal-field parameters and the signs of the hyperfine structure constants for both ¹⁷¹Yb^3 + and ¹⁷³Yb^3 + isotopes are determined. The validities of the theoretical results are discussed.     

Theoretical examination of optical and EPR spectra for Cu ion in K2PdCl4 crystal

In this work, two d-d transition spectra and four EPR parameters g_∥, g_⊥, A_∥, A_⊥ of K₂PdCl₄/Cu²⁺ are uniformly interpreted based on Zhao's crystal-field model. The calculation result is in good agreement with the experiment findings. The ligand spin-orbit coupling is neglected on the calculation, which is consistent with the ab initio result by Hillier et al. [J. Am. Chem. Soc. 98 (1976) 95]     

Crystal field energy levels of the laser crystal Gd3Ga5O12: Nd

The crystal field energy levels of laser crystal Gd₃Ga₅O₁₂: Nd³⁺ are calculated using the diagonalization (of energy matrix) method. From the calculations, the 93 observed crystal field energy levels are explained reasonably and the root-mean-square (r.m.s.) deviation σ(≈25.6 cm⁻¹) and the scalar crystal-field strength parameter Nv (≈3847 cm⁻¹) are obtained. The results are discussed.     

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.     

Theoretical investigations of the local structure distortion and the spin Hamiltonian parameters of Cr ions at tetragonal charge-compensation defect CrF5O site in Cr: KMgF3 crystals

The local structure distortion and the spin Hamiltonian (SH) parameters, including the zero-field splitting (ZFS) parameter D and the Zeeman g-factors g_‖ and g_⊥, are theoretically investigated by means of complete diagonalization method (CDM) and the microscopic spin Hamiltonian theory for tetragonal charge compensation CrF₅O defect center in Cr³⁺:KMgF₃ crystals. The superposition model (SPM) calculations are carried out to provide the crystal field (CF) parameters. This investigation reveals that the replacement of O²⁻ for F⁻ and its induced lattice relaxation Δ₁(O²⁻) combined with an inward relaxation of the nearest five fluorine Δ₂(F⁻) give rise to a strong tetragonal crystal field, which in turn results in the large ZFS and large anisotropic g-factor Δg. The experimental SH parameters D and Δg can be reproduced well by assuming that O²⁻ moves towards the central ion Cr³⁺ by Δ₁(O²⁻)=0.172R₀ and the five F⁻ ions towards the central ion Cr³⁺ by Δ₂(F⁻)=0.022R₀. Our approach takes into account the spin-orbit (SO) interaction as well as the spin-spin (SS), spin-other-orbit (SOO), and orbit-orbit (OO) interactions omitted in previous studies. This shows that although the SO interaction is the most important one, the contributions to the SH parameters from other three magnetic interactions are appreciable and should not be omitted, especially for the ZFS parameter D.     

Investigations of the optical spectra and spin-Hamiltonian parameters for the rhombic Yb centers in garnets

Two complete diagonalization (of energy matrix) methods (CDM-I and CDM-II) are used to calculate the six optical spectral band positions and nine spin-Hamiltonian (SH) parameters (g factors g_i and hyperfine structure constants ¹⁷¹A_i, ¹⁷³A_i, where i=x, y, z) for Yb³⁺ ions in the rhombic dodecahedral sites of garnets Y₃Al₅O₁₂, Lu₃Al₅O₁₂ and Y₃Ga₅O₁₂. In CDM-I, the Hamiltonian concerning energy matrix does not contain the Zeeman and hyperfine interaction terms, whereas in CDM-II, it does. So, in CDM-I, the SH parameters are obtained by first-order perturbation method or the equivalence between SH and Zeeman term (or hyperfine interaction term) and in CDM-II, the SH parameters and optical spectral band positions are calculated together. The results obtained from both methods are not only close to each other, but also in reasonable agreement with the observed values. So the second-order perturbation formulas of SH parameters developed recently are incorrect and unnecessary.     

An interpretation of the g factors for the tetragonally-compressed Cr centers in YVO4 and YPO4 crystals

The g factors g_∥ and g_⊥ for the tetragonally-compressed (CrO₄)³⁻ clusters in YMO₄ (M=V, P) crystals are calculated from the high-order perturbation formulas based on the two-mechanism model for the compressed d¹ tetrahedra with the ground state |d_z²〉. From the calculated values and by considering a small admixture of the first excited state |d_x²−y²〉 to the ground state |d_z²〉 due to the vibrational motion of ligands (which leads a twinkling compressed tetrahedron to become an elongated one), the observed g_∥ and g_⊥ for Cr⁵⁺ centers in YMO₄ crystals are explained reasonably. The difficulty of the large deviations of g_∥ from g_e (≈2.0023) in the two systems is therefore removed and the above dynamic effect may be the cause which results in the large deviation of g_∥ from g_e for some (CrO₄)³⁻ clusters in crystals.     

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.     

Investigation of the local structure of Cu2+ ions doped in alkali lead tetraborate glasses by their electron paramagnetic resonance and optical spectra

The local structure of the Cu²⁺ centers in alkali lead tetraborate glasses was theoretically studied based on the optical spectra data and high-order perturbation formulas of the spin Hamiltonian parameters (electron paramagnetic resonance g factors g_∥, g_⊥ and hyperfine structure constants A_∥, A_⊥) for a 3d⁹ ion in a tetragonally elongated octahedron. In these formulas, the relative axial elongation of the ligand O^2? octahedron around the Cu²⁺ due to the Jahn–Teller effect is taken into account by considering the contributions to the g factors from the tetragonal distortion which is characterized by the tetragonal crystal-field parameters Ds and Dt. From the calculations, the ligand O^2? octahedral around Cu²⁺ is determined to suffer about 19.2% relative elongation along the C₄ axis of the alkali lead tetraborate glass system, and a negative sign for A_∥ and a positive sign for A_⊥ for these Cu²⁺ centers are suggested in the discussion.     

Investigations of the spin-Hamiltonian parameters and defect structures for Gd ions in YMO4 (M=V, P, As) crystals

A 56×56 energy matrix containing the ground multiplet ⁸S_7/2 and the excited multiplets ⁶L_7/2 (where L=P, D, F, G, H, I) for 4f⁷ ion Gd³⁺ at a tetragonal crystal field and under an external magnetic field is constructed. By diagonalizing the energy matrix, the spin-Hamiltonian parameters (g factors g_‖, g_⊥ and zero-field splittings b₂⁰, b₄⁰, b₄⁴, b₆⁰, b₆⁴) for Gd³⁺ ion at the tetragonal Y³⁺ site of YMO₄ (M=V, P, As) crystals are calculated. The calculated results are in reasonable agreement with the experimental values. The defect structures of Gd³⁺ centers in YMO₄ crystals are estimated from the calculation. The results are discussed.