Theoretical Investigation of the EPR G-Factor for the Axial Symmetry Ce 3+ Center in the BaWO 4 Single Crystal

: The parameters g-factor ( g || and g ⊥ ) together with the local structure of the Ce3+ center in BaWO4 single crystal (scheelite structure crystals) were theoretically investigated using a complete diagonalization procedure of energy matrix (CDM method). The intrinsic parameters were calculated. It is shown that the experimental and the calculated values of the g-factors are in good agreement. The angular distortion has also been calculated. It was found that the polar angles of the impurity–ligand bonding are smaller than in BaWO4 single crystal (cid:0) ∆ θ ≈ 1.0 0 (cid:1) . The validity of the results and the changing in the local environment of the impurity–cerium ion is also discussed.


Introduction
The scheelite structure crystals, tungstate AWO 4 (A = Ba, Sr, Ca, Pb) doped with trivalent rare-earth ions (Re 3+ ), have received much interest thanks to their unusual properties such as luminescence, nonlinear optical activity, or scintillation [1][2][3][4][5][6] and for their application. The barium tungstate (BaWO 4 , BWO) is a very interesting inorganic optical material. Its potential applications include stimulated Raman scattering [7], scintillators and X-ray phosphor [8]. The blue and green PL emissions of BaWO 4 are widely discussed for their importance in future optical applications [9][10][11]. The barium tungstate crystals (BaWO 4 ) can also be used as a material for designing all solid-state lasers, especially for a wide variety of pump pulse durations in Raman laser pulses [12,13]. The barium tungstate (BaWO 4 ) single crystals and nanocrystals have recently become the subject of intense scientific research [10,14]. More generally, many tungstate crystals like BaWO 4 , PbWO 4 , CaWO 4 , or SrWO 4 are the subject of intense research because of their possible applications in optical devices such as lasers, or scintillators. Doping of tungstate crystals with rare earth elements (RE 3+ ) like cerium (Ce 3+ ), ytterbium (Yb 3+ ), erbium (Er 3+ ), etc., is a method of increasing their optical activity and their future applications.
The wolframite and scheelite structures are common structure types for ABO 4 compounds [15]. The BaWO 4 crystal crystallizes in tetragonal space group with C 6 4h (I4 1 /a) [16]. Lattice parameters of BWO are: a = b = 5.6148 Å, c = 12.721 Å [15,16]. Both Ba 2+    The present paper is a continuation of our research on the local structure of the BaWO4 single crystals doped and co-doped with ions of various elements. There are a few papers that have been investigating a local structure of BaWO4 single crystals doped and co-doped with, Ce, Na, Pr using EPR method [17][18][19]. Five paramagnetic centers with axial symmetry and about ten centers with low symmetry (C2) were found [17,18]. In the next step, we were investigated the connection between the g-shift and the environment of the Ce 3+ centers with axial symmetry for four BaWO4 doped with Ce (cerium) and codoped with Na (sodium) with different concentrations [20]. We also determined dislocations of Ce 3+ ions. We were used a simplified method proposed by D. J. Newman [21]. Structural information can be obtained from spin-Hamiltonian parameters (and/or g-parameters) using a superposition model (SPM) [22,23] or perturbation methods (PM) up to second-order [24]. Previously, we used a simplified Newman model, a g-shift model [20]. In this paper, we choose to use superposition model (SPM) to obtain structural information on trivalent cerium ion (Ce 3+ ) and its surroundings. Our investigations were focused on the most intense paramagnetic centers with axial symmetry presented in all four BaWO4 single crystals doped with Ce and co-doped with Na [17][18][19]. Obtained results will be compared with our previous results and other investigations.
It is assumed that dopant ions such as rare earth ions (Re 3+ like Ce 3+ , Er 3+ , or Yb 3+ ) substitute in place of barium ions (Ba 2+ ) [24][25][26][27][28][29][30]. Hence, two ReBa substitutions gives two The present paper is a continuation of our research on the local structure of the BaWO 4 single crystals doped and co-doped with ions of various elements. There are a few papers that have been investigating a local structure of BaWO 4 single crystals doped and co-doped with, Ce, Na, Pr using EPR method [17][18][19]. Five paramagnetic centers with axial symmetry and about ten centers with low symmetry (C 2 ) were found [17,18]. In the next step, we were investigated the connection between the g-shift and the environment of the Ce 3+ centers with axial symmetry for four BaWO 4 doped with Ce (cerium) and co-doped with Na (sodium) with different concentrations [20]. We also determined dislocations of Ce 3+ ions. We were used a simplified method proposed by D. J. Newman [21]. Structural information can be obtained from spin-Hamiltonian parameters (and/or g-parameters) using a superposition model (SPM) [22,23] or perturbation methods (PM) up to secondorder [24]. Previously, we used a simplified Newman model, a g-shift model [20]. In this paper, we choose to use superposition model (SPM) to obtain structural information on trivalent cerium ion (Ce 3+ ) and its surroundings. Our investigations were focused on the most intense paramagnetic centers with axial symmetry presented in all four BaWO 4 single crystals doped with Ce and co-doped with Na [17][18][19]. Obtained results will be compared with our previous results and other investigations.

Calculations
Rare-earth ions (like Ce 3+ ) substitute Ba 2+ site in the elementary cell of the barium tungstate (BaWO 4 ) [17][18][19][20]25,26,38]. The local symmetry of the Ce Ba site (Ba 2+ place occupied by Ce 3+ ) is S 4 (tetragonal symmetry). However, the D 2d symmetry approximation is often taken for many other rare-earth impurity ions (e.g., Yb 3+ , Er 3+ ) in many scheelitetype oxygen and fluoride compounds because of the small distortion [25,26,38,39]. We will apply the D 2d symmetry approximation here. Cerium ion (4f 1 electronic configuration) in tetragonal symmetry has 2 F 5/2 ground state and 2 F 7/2 exited state. The crystal field with D 2d symmetry splits the ground and exited states into three and four doublets, respectively. The Ce 3+ ion has an effective spin S = 1 2 , because only the lowest doublet is populated. The effective spin-Hamiltonian for Ce 3+ ion in an external magnetic fields can be written as [42]: whereĤ f is free ion term,Ĥ SO = ξ L ·Ŝ denotes spin-orbit interaction term, with ξ-the spin-orbit coupling parameter,Ĥ CF -crystal field term, andĤ Z is Zeeman term. The Zeeman term is usually expressed as:Ĥ whereĴ is total angular momentum in the 2S+1 L J manifold. Therefore, we obtain 14 × 14 energy matrix and the energy levels (e.g., eigenvalues) can be calculated after diagonalization of spin-Hamiltonian matrices (1). Usually, the spin-Hamiltonian parameters for 4f 1 ion are calculated using so-called complete diagonalization method (CDM), where the Zeeman term is not included to spin-Hamiltonian [26]. In this paper, the Zeeman term H Z is added to spin-Hamiltonian and next the full spin-Hamiltonian is subject of complete diagonalization method (CDM) according H.G. Liu et al. [26]. Thus, no perturbation formula is needed. We get only g-factor parameters obtained from EPR measurements for Ce 3+ in BaWO 4 [17][18][19][20]. The g-factor can be obtained using CDM method using the following equations: E Z and E X are the Zeeman splitting between two the lowest energy levels (doublet) obtained by complete diagonalization the energy matrix from Equation (1), where the magnetic field is directed along the Z and the X axes, respectively. The crystal field interaction termĤ CF for Ce 3+ ion (4f 1 ) in tetragonal symmetry (D 2d ) in terms of Stevens operator equivalent can be given as follows [22,42]: B q k there are the crystal field parameters, k = 2, 4, 6, |q| ≤ k. According the superposition model (SPM), the crystal field parameters B q k can be calculated as follows [21][22][23]: the summation is only over the nearest neighbor ligands. In the case of the dodecahedron [CeO 8 ] n = 8. The parameters t k and A k (R 0 ) are the power law exponents and the intrinsic parameters, respectively. R 0 , called the reference distance, is often taken as a usual distance between the metal ion and the ligands. The parameters K q k (θ i , ϕ i ) are the geometric coordination factors [23,24,26,43]. The Ce 3+ ion is surrounded by eight O 2ions, and only these oxygens O 2ligands were included in the SPM model [20,28]. The dodecahedron [BaO 8 ] is made in the form of two interpenetrated and rotated tetrahedrons. Figure 2 shows fragments of the BWO cell structure: a schematic view of the dodecahedron [BaO 8 ] consisting of two rotated tetrahedrons with marked crystallographic axis Z (Z || c) and two polar angles θ i .
the summation is only over the nearest neighbor ligands. In the case of the dodecahedron [CeO8] n = 8. The parameters and ̅ ( ) are the power law exponents and the intrinsic parameters, respectively. , called the reference distance, is often taken as a usual distance between the metal ion and the ligands. The parameters ( , ) are the geometric coordination factors [23,24,26,43]. The Ce 3+ ion is surrounded by eight O 2-ions, and only these oxygens O 2-ligands were included in the SPM model [20,28]. The dodecahedron [BaO8] is made in the form of two interpenetrated and rotated tetrahedrons. Figure  2 shows fragments of the BWO cell structure: a schematic view of the dodecahedron [BaO8] consisting of two rotated tetrahedrons with marked crystallographic axis Z (Z || c) and two polar angles . Therefore, we have two sets of the structural parameters: the distance and the polar angles , and the azimuthal angle , where = 1, 2 means the first and the second tetrahedron, respectively. These polar and azimuthal angles are angles between the metal-ligand distance and Z (fourfold) axis and X axis of the crystal, respectively [25,44] Figure 2) [44]. The ionic radii and the charge of the impurity rare-earth ion (like Ce 3+ ) usually are different form host, e.g., barium ion (Ba2+). However, the local lattice relaxation arising from the size mismatch can be sufficiently good approximated from the equation [25,26]: where and are the ionic radii of the impurity and the host ion, respectively. We assumed that ≈ 0.101 [nm] and ≈ 0.135 [nm] for Ce 3+ and Ba 2+ ions (both 6 coordination), respectively [43]. Now, one can estimate the distances , = 1, 2 for Ce 3+ ion in both two tetrahedrons in BaWO4 single crystals according to Equation (6). The impurityoxygens distances are changed. However, it is usually assumed that the polar angels and the azimuthal angles are the same as in the host crystal [25,44]. The structural data that were used in the calculations have been gathered in Table 1. Therefore, we have two sets of the structural parameters: the distance R H i and the polar angles θ i , and the azimuthal angle s ϕ i , where i = 1, 2 means the first and the second tetrahedron, respectively. These polar and azimuthal angles are angles between the metalligand distance R H i and Z (fourfold) axis and X axis of the crystal, respectively [25,44] Figure 2) [44]. The ionic radii and the charge of the impurity rare-earth ion (like Ce 3+ ) usually are different form host, e.g., barium ion (Ba2+). However, the local lattice relaxation arising from the size mismatch can be sufficiently good approximated from the equation [25,26]: where r I and r H are the ionic radii of the impurity and the host ion, respectively. We assumed that r I ≈ 0.101 [nm] and r H ≈ 0.135 [nm] for Ce 3+ and Ba 2+ ions (both 6 coordination), respectively [43]. Now, one can estimate the distances R i , i = 1, 2 for Ce 3+ ion in both two tetrahedrons in BaWO 4 single crystals according to Equation (6). The impurity-oxygens distances are changed. However, it is usually assumed that the polar angels θ i and the azimuthal angles ϕ i are the same as in the host crystal [25,44]. The structural data that were used in the calculations have been gathered in Table 1. R 0 (the reference distance) is taken as the average distance R 0 ≈ R = 0.2588 [nm]. The power law exponents were established as t 2 ≈ 5, t 4 ≈ 6, t 6 ≈ 10 [22,23,26,28,40,41]. We also have to estimate the value of the spin-orbit coupling parameters. H. Ramanantoanina 4 and LuPO 4 crystals, respectively [26]. We decided to take the average of the two last spin-orbit coupling parameters, because YPO 4 and LuPO 4 crystals have similar zircon-type structure and the Ce 3+ ion is surrounding by oxygens ligands, oxygens dodecahedron [CeO 8 ] [26]. Therefore, we assumed that the spin-orbit coupling parameter (spin-orbit term in spin-Hamiltonian) is equal to ξ ≈ 606 [cm −1 ].
In first step, by substituting all these parameters into Equations (1)-(6), we can estimate the intrinsic parameters A 2 (R 0 ), A 4 (R 0 ), A 6 (R 0 ) by fitting the EPR g-factors calculated using the new CDM method to the experimental values. Experimental and calculated values of the g-factor were calculated and have been gathered in Table 2, in the first and the second row. The differences between parallel and the perpendicular g-factor (∆g = g || − g ⊥ ) have also been presented. a Calculated using structural polar angles θ i in the host. b Calculated using the local polar angles θ i due to the angular distortion.
The calculated intrinsic parameters for Ce 3+ center in BaWO 4 single crystal get values: Table 2 shows that the calculated and observed g factor for Ce 3+ center in BaWO 4 single crystal are in quite a good agreement with each other, especially in the values of g parallel g ||, calc ≈ g ||, exp . In the next step, we tried to enhancing the fit, by varying the polar angles θ i . The host metal-ligand polar angles θ i (host-ligands) were replaced by polar angles θ i (impurity-ligands) with small distortion: θ i ≈ θ i + ∆θ, ∆θ there is the local angular distortion. We obtained the angular distortion: It means that after decreasing the polar angles by one degree, we obtained the better fit to the experimental g-factors ( Table 2, third row). The change is not so large, but visible. The difference in percentage between calculated and experimental values for the g perpendicular drop from 0.66 % to 0.48% (see Table 2).

Discussion
The calculated g factors for above values of intrinsic parameters A 2 (R 0 ), A 4 (R 0 ), A 6 (R 0 ) show good agreement with experimental values (see Table 2). However, the calculation based on the local distortion θ i of the polar angels gives a slightly better fit to observed values of the g-factors. Hence, it is justifiable to say that the g-factor for tetragonal Ce 3+ center in BaWO 4 single crystal is quite well explained with additional information about the local structure of the Ce 3+ ion in dodecahedron [CeO 8 ]. There are several points that need further discussion.
The values of g-factors (g || and g ⊥ ) for Ce 3+ in BaWO 4 single crystal, calculated and experimental are in good agreement (Table 2). For g parallel (g || ), the fit is almost perfect. One can observe a small difference in the case of g perpendicular ( g ⊥ , about 0.66 %). This difference decreases with the variation of the host metal-ligand polar angles θ i , i = 1, 2 with small distortion. The calculated angular distortion ∆θ ≈  [20], from which follows that even a simplified SPM model can give a quite good angular distortion result useful for structural analysis. Our calculation shows that angular distortion of polar angles θ i , i = 1, 2 gives only a partial approximation to experimental values. The better approximation will be obtained by changing the azimuthal angles ( ϕ i , i = 1, 2), too. It means that oxygen ligands are not only coming slightly closer to a fourfold axis (Z or c axis), but oxygen tetrahedrons are also twisted (see Figure 2). Further calculations and structural analysis are required for further comprehension. It should be pointed out that calculated intrinsic parameters were obtained only from the fitting procedure using SPM method to experimental EPR g-factors (g || and g ⊥ ).
These results should be considered as a first approximation and confirmed by fitting to results of the optical measurements. Unfortunately, we did not find the results of optical measurement of barium tungstate doped with cerium (BaWO 4 : Ce 3+ ). If optical bands of BaWO 4 : Ce 3+ will be recorded, our results could be a useful starting point. Our results could also be a useful help to structural analysis a local surroundings of the rare-earth ion (Ce 3+ ), which substitute a Ba 2+ site in oxygens dodecahedron [BaO 8 ]. It may be important in further application of BaWO 4 : Ce 3+ single crystals, for example in optical devices.

Informed Consent Statement: Not applicable.
Data Availability Statement: The study did not report any data.

Conflicts of Interest:
The authors declare no conflict of interest.