1. Introduction

Yttrium aluminium borate YAl₃(BO₃)₄ crystals(YAB) have excellent physical and chemical properties such as high laser damage threshold, good chemical stability, broad transparency range, large nonlinear optical coefficients, proper refractive index dispersion for phase matching, high thermal and mechanical resistance, and so on [¹-³]. It is interesting that YAB can be expediently doped with transition or rare earth impurity ions. In general, these above properties are closely related to the local structure and electronic states of the impurity ion in the host, which can be effectively investigated by means of electron paramagnetic resonance (EPR) technique. So many related experimental and theoretical works have been done in the past years [⁴-⁸]. For instance, EPR studies were carried out for Yb³⁺ doped YAl₃(BO₃)₄, the g factors g _∥ and g _⊥ and hyperfine structure constants were also measured for the trigonal Yb³⁺ center recently [⁸]. Dammk et al. calculated the EPR parameters by the method of crystal field theory. In the calculation, they directly used crystal field (CF) parameter which is not connected with the local crystal structure [⁹]. There are some mistakes found in their paper, for example, the wrong representation of the irreducible tensor operators lead directly to various errors in their calculation values of EPR parameters [¹⁰]. Li et al. not only recalculated the EPR values using the same CF parameters, but also theoretically studied the EPR parameters by the means of the first-principles [¹⁰]. Above theoretical results are also poor agreement with the experimental data, see Table I. As well known, the superposition model is mostly used in analysis of experimentally determined crystal field parameters. When the crystal structure of a magnetic ion is available, the SH principle is very expediently employed to study the local physical properties of the magnetic ion with its surrounding ligands. So in this paper, the EPR parameters are explained by the aid of the SH model and the perturbation formulas of 4f¹³ ion in trigonal symmetry. From these formulas, the EPR parameters g _∥ g _⊥ A _∥ and A _⊥ for Yb³⁺ center in YAl₃(BO₃)₄ crystal are reasonably explained and the angle distortion of impurity Yb³⁺ center is suggested. The results are discussed.

2. Calculation

The crystal structure of YAl₃(BO₃)₄ belongs to the space group R32 with three molecules per unit cell [¹¹]. In this structure, the coordination polyhedron of Y³⁺, Al³⁺, and B³⁺ is trigonal prism, octahedra and triangles surrounded by the oxygen ions. The Yb³⁺ ionic radius (0.858 A) is close˚ to that of Y³⁺ (0.893 Å), whereas it is much larger than that of Al³⁺ (0.51 Å) or B ³⁺ (0.23 Å) [¹²-¹³]. When Yb³⁺ is doped into the lattice of YAl₃(BO₃)₄ crystal, it can substitute for the octahedral Y³⁺ site and conserve the local trigonal symmetry, because of their similar ionic size, and no charge compensation is required [¹²]. In the following, the local structures of the Yb³⁺ centers are to be theoretically studied from the perturbation formulas of the EPR parameters.

The free Yb³⁺ ion has a 4f¹³ electronic configuration with a 2F_7/2 ground state and a 2F_5/2 excited state [¹⁴]. When Yb³⁺ ion is located on the Y³⁺ site of YAl₃(BO₃)₄ crystal, the free ion ground ²F _7/2 and excited ²F _5/2 states of free-ion splits into three and four Kramers doublets under trigonal symmetry crystal field, respectively. Because of the J-mixing between J = 7/2 and J = 5/2 states via crystalfield interaction, the basis wave function of ground doublet Γγ(or Γγ') may be gained by diagonalizing the 14 × 14 energy matrix for 4f¹³ ion in trigonal symmetry field. Thus, one obtains

where the subscript γ or γ’ denote the two components of Γ irreducible representation. M _J1 and M _J2 are halfintegers in the ranges −7/2 to 7/2 and −5/2 to 5/2, respectively [¹⁵]. The coefficients C(² F _7/2 ; Γγ(γ’)M _J1 ) or C(² F _5/2 ;Γγ(γ’)M _J2 ) can be determined by diagonalizing the 14 × 14 energy matrix containing the ² F ₇ /2 and ² F ₅ /2 states.

The perturbation Hamiltonian for the rare earth ion in the crystal under an external magnetic field can be expressed as [¹⁴]

where Ĥ _SO is the spin-orbit coupling interaction and Ĥ _CF is the crystal field Hamiltonian. Ĥ _SO can be written as:

where ζ is the spin-orbit coupling coefficient, here ζ ≈ 2907 cm⁻¹ [¹⁵], L^ and Ŝ are the orbital and spin momentum operators, respectively.

The crystal-field interaction Hamiltonian Ĥ _CF for a 4f¹³ ion may be written in terms of the irreducible tensor operators under trigonal symmetry [¹⁴]:

Where Bkqk=2,4 and 6;q≤k are the crystal-field parameters.

The Zeeman interaction Ĥ _Z can be expressed in terms of the Lande factor g _J and the angular momentum operator Ĵ as [¹⁴]

and the hyperfine interaction term can be denoted as Ĥ _hf = PN _J N^, where P is the dipolar hyperfine structure constant, i.e., P(¹⁷¹Yb) = 388.4(7) × 10⁻⁴ cm⁻¹ and P(¹⁷³Yb) = −106.5(2) × 10⁻⁴ cm⁻¹, the free ion values [¹⁵], N _J is the diagonal matrix element for ^2S+1 LJ state [¹⁴, ¹⁵].

To study the EPR spectra and the local structure for YAl₃(BO₃)₄:Yb³⁺, the perturbation formulas of the SH parameters for a 4f¹³ ion under trigonal symmetry are adopted [¹⁶]:

Here g _J are the Lande factors for various ^2S+1 L _J configurations, which are gained from Refs. 14 and 15. The operator Ĵ ₊(= Ĵ _X +iĴ _γ ) (or N^+=N^X+iN^γ) stands for the linear combination of the X− and Y − components for the total angular momentum operator Ĵ(or N^) [¹⁴, ¹⁵].

Based on the semi-empirical superposition model [¹⁷,¹⁸], the crystal field parameters Bkq in Eq. (4) can be written as

Where the coordination factor Kkq(θJ,φj) can be obtained from the local structural parameters of the studied system, t _k is the power law exponent, and Ā _k (R ₀) is the intrinsic parameter with the reference distance R ₀, which is usually taken as the average metal-ligand bond length. For [YbO₆]⁹⁻ cluster, the superposition model parameters including t _k and Ā _k (R ₀) are t ₂ = 3.5, t ₄ = 6, t ₆ = 6, Ā₂(R ₀) ≈ −522 cm⁻¹, Ā₄(R ₀) ≈ 66.3 cm⁻¹ and Ā₆(R ₀) ≈ 4.1 cm⁻¹ [¹⁶].

In the YAl₃(BO₃)₄ crystal, the host Y³⁺ ion is coordinated by six nearest-neighbour O²⁻ ions with the cationanion distance R _H ≈ 2.302 Å [¹⁴], the local structure data are given as follows: θ ₁ = θ ₂ = θ ₃ ≈ 55.188^o, φ ₁ ≈ −157.911^o, φ ₂ ≈ −37.911^o, φ ₃ ≈ 82.089^o, θ ₄ = θ ₅ = θ ₆ ≈ 124.812^o, φ ₄ ≈ −142.089^o, φ ₅ ≈ −22.089^o, φ ₆ ≈ −97.911^o, [¹²], see Fig. 1. When a impurity ion substitutes for a host ion, R _j 6= R _H because of the different ionic radii of Yb³⁺ and the replaced Y³⁺ ion. The new cation-anion distance Rj can be reasonably estimated from the approximate formula [¹⁹, ²⁰]

FIGURE 1 Local structure for the trigonal Yb³⁺ center in YAl₃(BO₃)₄. The impurity Yb³⁺ on the octahedral Y³⁺ site experiences the angle distortion ∆θ(≈ 3.98^o). 

where r _i and r _h are the ionic radii of impurity and the host ion, respectively. For YAl₃(BO₃)₄:Yb³⁺, r _i ≈ 0.858 A, r _h ≈ 0.893 Å [¹³]. Because of the covalency of Yb³⁺-O²⁻ bonds, the orbital reduction factor k may be brought in, here k ≈ 0.866.

When the host cation is replaced by the impurity in YAl₃(BO₃)₄ crystal, it can be expected that the local structure distortion ∆θ would occur. Here it is taken as an adjustable parameter. When the above superposition model intrinsic parameters are substituted into Eqs.(6-7) and matching the calculated EPR parameters to experimental results, we have ∆θ ≈ 3.98^o. The comparisons between the calculated and experimental EPR parameters are shown in Table I.

3. Discussion

From Table I, one can find that by using the theoretical formulas of EPR parameters and the superposition model parameters given in this paper, the calculated results of EPR parameters g _∥, g _⊥, A _∥ and A _⊥ for Yb³⁺ ion at the trigonal Y³⁺ site in YAl₃(BO₃)₄ crystal based on the enhancive angle distortion ∆θ show reasonable agreement with the observed data. Thus, these experimental data are reasonably explained, suggesting that the above formulas and these parameters adopted in this paper are reasonable.

Substitute the above superposition model intrinsic parameters into Eq. (8), the crystal field parameters are obtained and shown in Table II. These crystal field parameters are comparable to previous works. As mentioned before, the superposition model intrinsic parameters t _k and Ā _k (R ₀) adopted in this paper are taken from the same impurity Yb³⁺ in Bi₄Ge₃O₁₂ crystal. Using these parameters and diagonalizing the complete 4f¹³ energy matrix in the trigonal field, the energy spectra are computed, which are the foundation of further calculation, see the Table III. One can find that the calculated energy levels reasonably coincide with the experimental results. So the data of the intrinsic parameters used in this paper can be regarded as acceptable.

We find the local structure data of impurity Yb³⁺ including the cation-anion distance and the azimuthal angle of the oxygen atoms are different from those of the host Y³⁺ ion. The cation-anion distance of Yb³⁺-O²⁻ can be reckoned by the aid of empirical equation Eq. (9). The azimuthal angle undergoes an angle distortion ∆θ ≈ 3.98o. This defect model of Yb³⁺ ion in YAl₃(BO₃)₄ is similar to other rare or transition impurity ion in the same host crystal. Obviously, the theoretical result of the ∆θ as well as the hyperfine structure constants of ¹⁷³Yb³⁺ isotopes obtained in this work still remains to be further verified with experimental studies.