1. Introduction

Mixed-alkali effect in glasses has received much attention with respect to its technological and theoretical interests in describing the physics and chemistry of glasses which are of the potential applications in the development of new tunable solid-state lasers, solar-energy converters, and fiber-optic communication devices ^1-3. It is well known that the transition metal ions are conveniently applied as probes to provide useful information about the local structures of glasses, due to their sensitive response to surrounding actions. For example, Cu ²⁺ ion (3d ⁹ ) with one 3d hole is normally treated as a model system of specific significance, ⁴ containing a single ground state and a single excited state under ideal octahedral crystal field and is frequently applied as probes in the EPR experiments ^5-9 Giridhar et al., performed the EPR experiments and optical absorption studies for Cu ²⁺ ion in mixed alkali cadmium phosphate glasses, ⁸ and the EPR parameters (the anisotropic g factors g_ǁ and g_┴ and the hyperfine structure constants A_║ and A_┴ ) were also measured for the center ⁸. However, until now, the local structure of the impurity center in the glasses has not been obtained, except that the observed EPR results were tentatively assigned to the tetragonally elongated Cu²⁺ center. In addition, the optical spectra were not interpreted. Since the microscopic information about the local structure and the EPR behaviors (particularly the g factors) for Cu²⁺ in mixed alkali cadmium phosphate glasses would be helpful to understand the nature and symmetry of the glasses, theoretical studies on the above EPR results are of fundamental and practical significance. In this work, the EPR parameters of Cu²⁺ center in mixed alkali cadmium phosphate (10Li₂O-10Na₂O-20CdO-59.5P₂O₅, LiNaCdP hereafter) glass are theoretically investigated by using the high order perturbation formulas of these parameters for a 3d⁹ (Cu²⁺) ion under tetragonally elongated octahedra. In these formulas, the ligand orbital contributions are considered and the energy separations are correlated with the local structure. Thus, the optical spectra can be interpreted and hence information of the local structure for the impurity ion Cu²⁺ obtained on the basis of the EPR analysis. The results are discussed.

2. Calculation

When the impurity Cu²⁺ ions are doped into mixed alkali cadmium phosphate glasses, they prefer to occupy the Cd²⁺ site due to the same charge and form the octahedral [CuO₆]^10- clusters ⁸. For a Cu²⁺ ion under octahedral symmetry, there are only two energy levels, i.e., one lower orbital doublet ²E_g $^2 and another higher orbital triplet ²T_2g. Then, the two-fold orbital degeneracy of ²E_g ground state can be canceled by the Jahn-Teller effect via vibrational interactions, which always correlates to the removal of the degeneracy of energy levels and results in lower symmetry and energy ^10-12 The doped Cu²⁺ ions are found to occur in tetragonally elongated along certain direction (C⁴ axis of the systems) and lead to the local tetragonal point symmetry ⁸. For Cu ²⁺ (3d⁹) ion in tetragonally elongated octahedra, the lower ²E_g level would be split into ²B_2g and ²A_1g, with the former being lowest ⁴. Meanwhile, the upper ²T_2g energy level would be separated into ²B_2g and ²E_g. According to the high order perturbation theory and considering the contributions from the ligand orbital and spin-orbit coupling interactions due to covalency effect between the central ion and ligand ions, the high-order perturbation formulas of the EPR parameters for a 3d⁹ ion in tetragonally elongated octahedra can be established from the cluster approach ^13,14,15

(1)

Here, g_e ≈2.0023 is the spin-only value, P and P’ are the dipolar hyperfine constant related to the interaction within t_2g states and the interaction between t_2g and e_g states.K is the core polarization constant. E₁ and E₂ are the energy separations between the excited ²B_2g , ²E_g and the ground ²B_1g states, they can be expressed in terms of the cubic field parameter D_q and the tetragonal field parameters D_s and D_t . Thus, we have:

Based on the cluster approach, the spin-orbit coupling coefficients ζ and ζ′ and the orbital reduction factors k’ and k and the dipolar hyperfine structure parameters P and P’ in the formulas of EPR parameters can be expressed as: ^13,16

(3)

In above formulas, ζ d and ζ p are, respectively the spin-orbit coupling coefficients of the free 3d⁹ and ligand ions. P⁰ (≈388X10^-4 cm^-117) is the dipolar hyperfine structure parameter of free Cu²⁺ ion. A denotes the integral R<ns|∂/∂y|np_y >, where R is the impurity-ligand distance of the present system. N_y and λ_y (or λ_s ) are the normalization factors and the orbital mixing coefficients for the cubic irreducible representations γ(=e_g or t_2g ) They can be determined from the approximate relationships: ¹³,¹⁶

(4)

And the normalization: ^13,16

Here N is the average covalency factor, characteristic of the covalency effect of the central ion in crystals. S_dpy (and S_ds ) are the group overlap integrals. In general, the orbital mixing coefficients increase with increasing the group overlap integrals, and one can approximately adopt proportionality relationship λ_e/S_dpe≈λ_s/S_s between the orbital mixing coefficients and the related group overlap integrals within the same irreducible representation e_g.

As mentioned before, the Jahn-Teller Cu²⁺ ion can suffer the Jahn-Teller effect via stretching the two Cu²⁺O^2- bonds along the C₄ axial direction and then lead to the local tetragonal point symmetry. So, the parallel and perpendicular Cu²⁺-O² bond lengths can be given in terms of the reference distance R and the elongation ΔZ along the C₄ axis as R_║=R+∆Z and R _┴=R-∆Z (see Fig. 1). Thus, the cubic and tetragonal field parameters (D_q, D_s and D_t ) can be determined from the superposition model18 and the geometrical relationship of the studied impurity Cu²⁺ center:^13-15

FIGURE 1 The local structure of Cu²⁺ in LiNaCdP glass 

(6)

Here t₂ (≈3) and t₄ (≈5) are the power-law exponents, Ᾱ₂ (R) and Ᾱ₄ (R) are the intrinsic parameters with the reference bonding length R, here we estimate reasonably the reference distance R(≈2.04Å) for the Cu²⁺ center to be close to the sum of ionic radii of Cu²⁺ and O ^2- ions ¹⁹, due to the electrostatic attraction between two ions ^20,21,22. For 3d ⁿ ions in octahedra, the relationship Ᾱ₂ (R)≈Ᾱ₄ (R) is proved to be valid and is reasonably applied here ^14,23.

From the distance R and the Slater-type SCF functions,^24,25 the group overlap integrals S_dpt , S_dpe , and S_ds and the integral A are calculated and shown in Table I. Then, the molecular orbital coefficients N_y and λ_y can be obtained from Eq. (4), Eq. (5) and the covalency factor N(≈0.86) which can be determined from the relationship: N²≈1-h(L)k(M) (here the parameter h(L)(≈1) is the characteristic of the ligand O ^2- and k(M) (≈0.26) is the characteristic of the central metal ion Cu ²⁺ ) ^26,27. The calculated results are also shown in Table I. The spin-orbit coupling coefficients ζ and ζ′ and the orbital reduction factors k and k’ can be determined from Eq. (3) and the corresponding free-ion values ζ d (Cu ²⁺ )≈829 cm ^-1 , ¹¹ ζ p (O ^2- )≈151°cm^-1 , ¹⁶ they are collected in Table II.

TABLE I The group overlap (and A) integrals, the molecular orbital coefficients Nγ and λγ and λs for Cu²⁺ center in LiNaCdP glass. 

TABLE II Spin-orbit coupling coefficients (in cm⁻¹), the orbital reduction factors and the dipolar hyperfine structure parameters for Cu²⁺ center in LiNaCdP glass. 

In the formulas of the hyperfine structure constants, the core polarization constant k can be determined from the relationship: k≈-2X/(3<r^-3 >), ^16,17,26 here χ is the characteristic of the density of unpaired spins at the nucleus of the central ion and<r^-3 > is the expectation value of inverse cube of the 3d radial wave function. From the data <r^-3 >≈8.252 a.u. ⁴ and X≈-3.40 a.u. ¹⁷ for Cu²⁺ in some oxides, one can estimate k≈0.275 for the studied system which is close to the expectation value (≈0.3) for the 3d ⁿ ions incrystals ^11,12,17 and can be regarded as reasonable.

Thus, there are only two unknown parameters, the tetragonal elongation ∆Z and the intrinsic parameter Ᾱ₄(R) in the formulas of the EPR parameters. Substituting the related values into Eq. (1) and fitting the theoretical results to the experimental data, we have:

Accordingly, the parallel and perpendicular Cu-O distance R_║ and R _┴ in the [CuO₆]^10- cluster are 2.051 Å and 2.029 Å respectively. The corresponding theoretical results of the optical absorption bands and the EPR parameters are shown in Table III. For comparison, the theoretical results based on the elongation ∆Z in Eq. (7) and neglecting the ligand contributions (i.e., ζ=ζ=N ζ d and =k’=N) are also collected in Table III.

TABLE III The optical spectral band positions (in cm⁻¹) and EPR parameters (A constants are in units of 10⁻⁴ cm⁻¹) for Cu²⁺ ions in LiNaCdP glass. 

^a Calculation based on the elongation ∆Z and neglecting the ligand contributions.

^b Calculation based on the elongation ∆Z and inclusion of the ligand contributions.

^c Reference 8

3. Discussion

From Table III, one can find that the calculated results (i.e., optical absorption spectra and EPR parameters) for Cu ²⁺ in LiNaCdP glass based on the elongation ∆Z and the intrinsic parameters Ᾱ₄(R) in Eq. (7) show good agreement with the experimental data. Thus, the optical and EPR spectra for the impurity Cu ²⁺ in the studied glass are satisfactorily and uniformly interpreted.

4. Conclusions

The EPR parameters and the local structure of the tetragonal Cu ²⁺ center in mixed alkali cadmium phosphate glasses are theoretically investigated from the high-order perturbation formulas for a 3d ⁹ ion in a tetragonally elongated octahedron. Based on the studies, the oxygen octahedral around Cu ²⁺ ions are found to suffer the tetragonal distortion ∆R(=R_║-R_┴ ) of about 0.022 Å due to the Jahn-Teller effect. The d-d transition optical spectra of the Cu ²⁺ center are theoretically interpreted. The signs of A _║ and A _┴ are suggested.