Abstract: The local structure and the spin Hamiltonian parameters g factors g// and gL and the hyperfine structure constants A// and AL for the tetragonal Ni+ center (i.e., [NiF4]3- cluster) in CaF2 are theoretically studied from the high order perturbation formulas of these parameters for a 3d9 ion in tetragonal symmetry (elongated octahedron). Based on the studies, the impurity Ni+ is found to locate at the distance of about 0.35Å from the nearest fluorine plane. The theoretical spin Hamiltonian parameters show good agreement with the experimental data.
Introduction
Ni-doped fluorite-type crystals have been extensively investigated because of the paramagnetic defects arising from X-ray irradiation (Studzinski et al., 1984; Alonso et al., 1983; Casas-González et al., 1980). These impurity centers usually exhibit tetragonal symmetry, as observed by Electron Paramagnetic Resonance (EPR) and Electron Nuclear Double Resonance (ENDOR) techniques (Alonso et al., 1980, 1983; Studzinski et al., 1984). For example, the spin Hamiltonian parameters g factors g// and g⊥ and the hyperfine structure constants A// and A⊥ for the tetragonal Ni+ center in CaF2 produced by X-ray irradiation were measured (Casas-González et al., 1980). The off-center displacement and the anisotropic g factors for the tetragonal Ni+ center in CaF2 were investigated by using the simple perturbation formulas of the g factors for a 3d9 ion in tetragonally elongated octahedra (Casas-González et al., 1980). It was found that the impurity Ni+ may not occupy exactly the host Ca2+ site but suffer so large displacement along [100] (or C4) axis that it is only 0.37Å away from the nearest fluorine plane (Casas-González et al., 1980). In their theoretical treatment, however, the third-order perturbation terms were not completely involved and the fourth-order ones were neglected as well. Meanwhile, the energy denominators in the formulas of the g factors were not correlated with the local structure around the impurity center, but taken from the values of Ni+ doped LiF and NaF (Hayes and Wilkens, 1964). In addition, the hyperfine structure constants were not interpreted, either. In order to explain the g factors and the hyperfine structure constants for the tetragonal center in CaF2:Ni+ and to investigate the local structure of the impurity Ni+ more exactly, high order perturbation formulas of these parameters are applied and the related energy differences are quantitatively determined from the local structure of this center. The results are discussed.
Theory and Calculations
In pure fluorite, Ca2+ is coordinated to eight oxygen ions forming
an ideal cube. When irradiated by X-ray, Ni-doped CaF2 can exhibit
Ni+ center of tetragonal symmetry by Ni+ occupying the
host Ca2+ site with additional off-center shift along one of [100]
axis (Casas-González et al., 1980). This may be due to the smaller
size and the less charge of the impurity Ni+ than those of the host
Ca2+, which is likely to make the Ni+ unstable on the
host Ca2+ site and tend to suffer a significant axial displacement.
As a result, the impurity Ni+ would be much close to one of the fluorine
plane in the cube and the near square planar [NiF4]3-
cluster (i.e., the Ni+ has a small distance ΔZ from the plane)
is formed. The other four ligands are much farther from the impurity and their
influence may be ignored for simplicity. This tetragonal center can be regarded
as an elongated octahedron, with its local structure characterized by the distance
ΔZ.
For a Ni+(3d9) ion in tetragonally elongated octahedra, the lower 2E irreducible representation may be separated into two orbital singlets 2B1 (|x2-y2 >) and 2A1(| z2 >), with the former lying lowest, while the upper 2T2 representation would split into an orbital singlet 2B2(|xy>) and a doublet 2E(|xz>, |yz>) (Abragam and Bleaney, 1970). It is noted that in the treatments of the previous work (Casas-González et al., 1980), the notations 2B1 and 2B2 are interchanged, due to a rotation of the frame of axes. The perturbation formulas of the spin Hamiltonian parameters of the 2B1 ground state for a 3d9 ion in tetragonal symmetry can be expressed as follows (Wei et al., 2005):
| (1) |
where gs(≈2.0023) is the spin-only value. k is the orbital reduction factor. κ is the core polarization constant. ζd and P are, respectively, the spin-orbit coupling coefficient and the dipolar hyperfine structure parameter of the 3d9 ion in crystals. They can be written in terms of the corresponding free-ion values, i.e., ζd ≈k ζd0 and P ≈k P0. E1 and E2 are the energy separations between the excited 2B2g and 2Eg and the ground 2B1g states (Wei et al., 2005):
E1= 10Dq, E2 = 10Dq
-3Ds+5Dt |
(2) |
Here Dq is the cubic field parameter and Ds and Dt the tetragonal field parameters. From the superposition model and the geometrical relationship of the [NiF4]3- cluster, the tetragonal field parameters can be expressed as
| (3) |
where t2 and t4 are the power-law exponents, we take
t2 ≈3 and t4 ≈5 here. }2
| Table 1: | The spin Hamiltonian parameters for the tetragonal Ni+ center in CaF2 |
| aCalculations based on the simple perturbation formulas (Casas-Gonzáles, 1980) and the energy denominators for LiF and NaF:Ni+ (Hayes and Wilkens, 1964) in the previous work. bCalculations based on the high order perturbation formulas and the local structure (i.e., the distance ΔZ of Eq. 4) in this study cCasas-Gonzáles, 1980 | |
In the studied system, ζd0 are about 629 cm-1 for the free Ni+ ion (Griffith, 1964). The orbital reduction factor k (≈ 0.68) is adopted here. Substituting these parameters into the formulas of the g factors and fitting the theoretical values to the experimental data, we have
ΔZ ≈ 0.35Å |
(4) |
The corresponding calculated values are shown in Table 1. The energy levels in Eq. 1 are also obtained from Eq. 2 and 3, i.e., E1 ≈4000 cm-1 and E2 ≈8138 cm-1.
In the formulas of the hyperfine structure constants, the free-ion value P0 ≈-140x10-4 cm-1 for Ni+ can be obtained from those for isoelectronic 3d9 ions (McGarvey, 1967) by extrapolation. By fitting the calculated hyperfine structure constants to the observed values, the core polarization constant is determined:
κ ≈ 0.62 |
(5) |
The corresponding hyperfine structure constants are also shown in Table 1. For comparisons, the theoretical results of the g factors based on the simple perturbation formulas and the energy separations (E1 ≈4900 cm-1 and E2 ≈8000 cm-1) taken from Ni+ doped LiF and NaF (Hayes and Wilkens, 1964) are collected in Table 1.
Discussion
From Table 1, it can be found that the theoretical spin Hamiltonian parameters based on the high order perturbation formulas and the distance ΔZ (≈ 0.35Å) in this work are in good agreement with the observed values. The calculated results of present studies are also better than those in the previous work (Casas-González et al., 1980).
The distance (≈ 0.35Å) of the impurity Ni+ from the fluorine plane based on the analyses of the spin Hamiltonian parameters in this work is consistent with that (≈ 0.37Å) based on the simple perturbation formulas of the g factors (Casas-González et al., 1980). The above result is also supported by the recent density functional theory (DFT) studies on the same system, which yields the distance of about 0.33Å (Aramburu et al., 2002). Therefore, the local structure of the impurity Ni+ center in CaF2 obtained in this study can be regarded as reasonable. In fact, when the host Ca2+ is replaced by the smaller and less charged Ni+, the impurity may be unstable at the host Ca2+ site and then suffer a large off-center displacement due to the size and/or charge mismatching substitution. As a result, the Ni+ is very close to one fluorine plane and this center can be conveniently described as [NiF4]3- cluster.
The theoretical g factors obtained in this study are slightly better than those in the previous studies (Casas-González et al., 1980). This means that the high order perturbation formulas of the spin Hamiltonian parameters can be regarded as more valid than the simple ones. Meanwhile, the energy denominators (E1 ≈4000 cm-1 and E2 ≈8138 cm-1) obtained from the local structure of the impurity center in present studies are somewhat different from those (E1 ≈4900 cm-1 and E2 ≈8000 cm-1) for Ni+ in LiF and NaF (Hayes and Wilkens, 1964). Further, the calculated hyperfine structure constants are in good agreement with the experimental data. Therefore, the spin Hamiltonian parameters and the related parameters used here can be regarded as reasonable.
There are some errors in the above calculations. First, approximation of the theoretical model can lead to some errors in the final results. Second, the displacements of the four nearest and four next nearest F- ions are not considered in the analyses. In fact, these fluorine ions may shift slightly towards the center of the cube due to the large off-center displacement of the impurity Ni+. For the sake of simplicity and reduction of number of the adjustable parameters, the errors arising from neglecting of the above ligand displacements may be taken as absorbed in the distance ΔZ and the orbital reduction factor k in the calculations. Therefore, the ΔZ (≈ 0.35Å) obtained in this study can be tentatively regarded as the effective distance between the impurity and its nearest ligand plane. Finally, the contributions from the ligand orbitals and spin-orbit coupling coefficient are also ignored here. Fortunately, these contributions are expected to be unimportant and negligible due to the small magnitude of the spin-orbit coupling interaction for the ligand F- compared with that of the impurity Ni+.
Conclusions
The local structure and the spin Hamiltonian parameters for the tetragonal Ni+ center in CaF:Ni+ are theoretically investigated from the high order perturbation formulas of these parameters in this study. It is found that the impurity Ni+ locates at the distance of about 0.35Å from the nearest fluorine plane, i.e., the [NiF4]3- cluster is expected.
Acknowledgment
This study was supported by the Youth Foundation of Science and Technology of UESTC under grant No. JX04022. The authors are grateful to Mr. Ze-Xin Feng for his helpful computations.