#
Theoretical Investigation of the EPR G-Factor for the Axial Symmetry Ce^{3+} Center in the BaWO_{4} Single Crystal

^{*}

## Abstract

**:**

^{3+}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 BaWO

_{4}single crystal $\left(\mathrm{\Delta}\theta \approx {1.0}^{\xb0}\right)$ . The validity of the results and the changing in the local environment of the impurity–cerium ion is also discussed.

## 1. Introduction

_{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.

_{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+}and W

^{6+}sites have S

_{4}point symmetry. The Ba

^{2+}ion is coordinated by eight O

^{2-}ions in the form of dodecahedron [BaO

_{8}] made of two rotated, interpenetrated tetrahedrons. The distance Ba–O is equal to 2.7857 Å and 2.8310 Å, respectively, for two tetrahedrons. Two tetrahedrons [BaO

_{4}] make up a dodecahedron [BaO

_{8}]. Apart from dodecahedrons [BaO

_{8}], [WO

_{4}] tetrahedrons are an important part in the unit cell of the BaWO

_{4}. The [WO

_{4}] tetrahedron has an almost regular shape only slightly distorted along the

**c**(S

_{4}) axis. The distance between the W and the O ions is equal to 1.8230 Å [15]. Dodecahedrons [BaO

_{8}] are connected by their edges. Each O atom belongs to two dodecahedrons [BaO

_{8}] and one tetrahedron [WO

_{4}]. Figure 1 shows the unit cell structure of BaWO

_{4}viewed along the

**b**axis with marked dodecahedrons [BaO

_{8}] (blue). One can see that the unit cell of the BaWO

_{4}has the biggest volume among ABO

_{4}compounds (V = 401.0 Å

^{3}) [15].

_{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 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 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.

^{3+}centers, in BaWO

_{4}single crystals [25] and one other about similar, tetragonal Ce

^{3+}centers in YPO

_{4}and LuPO

_{4}crystals [26]. However, one can find that there are several papers about: (a) Ce

^{3+}and Yb

^{3+}in garnets [27,28,29], (b) Ce

^{3+}, Yb

^{3+}in YF

_{3}crystal and Ce

^{3+}in fluoride ligands [30,31], (c) Yb

^{3+}in CaWO

_{4}crystal [32], (d) Er

^{3+}in CaWO

_{4}and SrWO

_{4}crystals [33], Er

^{3+}in zircon-type compounds [34] or Er

^{3+}in PbMoO

_{4}and SrMoO

_{4}[35], Er

^{3+}at the Th

^{4+}site in ThGeO

_{4}[36], (e) Nb

^{3+}, Yb

^{3+}in double tungstate’s and molybdates [37], (f) color centers in BaWO

_{4}[38,39] and others [40,41]. All these papers are focused on analysis of spin–Hamiltonian parameters for the rare-earth ion surrounded by oxygen’s [ReO

_{8}] or fluoride’s dodecahedrons [ReF

_{8}]. It is interesting to compare our results with the results from these papers.

^{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 Re

_{Ba}substitutions gives two excess positive charges, which are compensated by one the barium vacancy (V

_{Ba}). The associated vacancies (V

_{Ba}) do not necessarily affect the surroundings of the dopant ion, dodecahedron [ReO

_{8}]. In our previous articles, we described five paramagnetic centers (Ce

^{3+}) with axial symmetry detected in the EPR spectra of four BWO monocrystals doped with cerium and/or co-doped with sodium: (1) BaWO

_{4}: 0.5% at. Ce, (2) BaWO

_{4}: 1.0% at. Ce, (3) BaWO

_{4}: 0.5% at. Ce, 1.0% at. Na and (4) BaWO

_{4}: 1.0% at. Ce, 2.0% at. Na [17,18,19,20]. One of these centers is the strongest and occurs in all four monocrystals. It is g-factors are as follows: ${g}_{\left|\right|}=1.506\left(1\right)$, ${g}_{\perp}=2.712\left(2\right)$. We will focus our analysis and calculations on this center with axial symmetry.

## 2. Calculations

^{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 scheelite-type 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 = ½, 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]:

^{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 ${\widehat{H}}_{Z}$ 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:

^{3+}ion (4f

^{1}) in tetragonal symmetry (D

_{2d}) in terms of Stevens operator equivalent can be given as follows [22,42]:

_{8}] n = 8. The parameters ${t}_{k}$ and ${\overline{A}}_{k}\left({R}_{0}\right)$ 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 ${K}_{k}^{q}\left({\theta}_{i},{\phi}_{i}\right)$ 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 [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 ${\theta}_{i}$.

^{2+}ion in the first and the second tetrahedron, respectively (see 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 (Ba

^{2+}). However, the local lattice relaxation arising from the size mismatch can be sufficiently good approximated from the equation [25,26]:

^{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 ${\theta}_{i}$ and the azimuthal angles ${\phi}_{i}$ 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.

^{3+}ion embedded in the Cs

_{2}NaYCl

_{6}crystal host ${\xi}_{4f}\approx 624.1$ [cm

^{−1}] [45]. Jun Wen et al. in the study of the Ce

^{3+}ion doped in the various fluoride compounds estimated the value of ${\xi}_{4f}\approx 614.9$ [cm

^{−1}] [46]. H.G. Liu et al. in the investigation of the tetragonal Ce

^{3+}centers in the YPO

_{4}and LuPO

_{4}crystals calculated the spin–orbit coupling parameter as $\xi \approx 608$ [cm

^{−1}] and $\xi \approx 604$ [cm

^{−1}] for YPO

_{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 $\xi \approx 606$ [cm

^{−1}].

^{3+}center in BaWO

_{4}single crystal get values:

^{3+}center in BaWO

_{4}single crystal are in quite a good agreement with each other, especially in the values of g parallel ${g}_{\left|\right|,calc}\approx {g}_{\left|\right|,exp}$. In the next step, we tried to enhancing the fit, by varying the polar angles ${\theta}_{i}$. The host metal–ligand polar angles ${\theta}_{i}$ (host–ligands) were replaced by polar angles ${\theta}_{i}^{\prime}$ (impurity–ligands) with small distortion: ${\theta}_{i}^{\prime}\approx {\theta}_{i}+\mathrm{\Delta}\theta $, $\mathrm{\Delta}\theta $ there is the local angular distortion. We obtained the angular distortion:

## 3. Discussion

^{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.

^{−1}], and ${\overline{A}}_{6}\left({R}_{0}\right)$ has a value of a few [cm

^{−1}]. Wu Shao-Yi et al. set the values of intrinsic parameters ${\overline{A}}_{2}\left({R}_{0}\right)\approx 400,{\overline{A}}_{4}\left({R}_{0}\right)\approx 50,{\overline{A}}_{6}\left({R}_{0}\right)\approx 17\left[{\mathrm{cm}}^{-1}\right]$ for Er

^{3+}centers in BaWO

_{4}, CaWO

_{4}, and SrWO

_{4}crystals [25,33]. H.G. Liu, W.C. Zheng and W.L. Feng calculated the intrinsic parameters ${\overline{A}}_{2}\left({R}_{0}\right)\approx 290,{\overline{A}}_{4}\left({R}_{0}\right)\approx 48,{\overline{A}}_{6}\left({R}_{0}\right)\approx 45\left[{\mathrm{cm}}^{-1}\right]$ from optical spectra for LuPO

_{4}: Ce, and YPO

_{4}: Ce single crystals with zircon structure [26]. We did not found other data regarding cerium ion, but there are many papers on rare-earth impurities, like Er

^{3+}, Yb

^{3+}, and Nd

^{3+}in different oxide and fluoride crystals. The values of the ${\overline{A}}_{2}\left({R}_{0}\right)$ (second rank) reach one thousand and more inverse centimeters (even ${\overline{A}}_{2}\left({R}_{0}\right)\approx 1200\left[{\mathrm{cm}}^{-1}\right]$ [37]), while sixth rank intrinsic parameter reach a few inverse centimeters (${\overline{A}}_{6}\left({R}_{0}\right)\approx 3\left[{\mathrm{cm}}^{-1}\right])$ [37,40,41]. Therefore, the calculated values of intrinsic parameters (rank second, fourth and sixth) should be considered as reasonable and contained within the range of data.

^{3+}in BaWO

_{4}single crystal, calculated and experimental are in good agreement (Table 2). For g parallel (${g}_{\left|\right|}$), the fit is almost perfect. One can observe a small difference in the case of g perpendicular (${g}_{\perp}$, about 0.66 %). This difference decreases with the variation of the host metal–ligand polar angles ${\theta}_{i},i=1,2$ with small distortion. The calculated angular distortion $\mathrm{\Delta}\theta \approx 1.0$° decrease a difference between experimental and calculated g perpendicular (${g}_{\perp}$) to about 0.48 %. The calculated angular distortion for Ce

^{3+}in BaWO

_{4}agrees with angular distortion for other rare-earth ions. For example: angular distortion calculated for Er

^{3+}in BaWO

_{4}($\mathrm{\Delta}\theta \approx {1.5}^{\xb0}$) [25], Ce

^{3+}in Y

_{3}Al

_{5}O

_{12}and Lu

_{3}Al

_{5}O

_{12}garnets ($\mathrm{\Delta}\theta \le {1.0}^{\xb0}$) [27], Yb

^{3+}in Na

_{3}Sc

_{2}V

_{3}O

_{12}garnet ($\mathrm{\Delta}\theta \le {1.0}^{\xb0}$) [29], Er

^{3+}in PbMoO

_{4}and SrMoO

_{4}($\mathrm{\Delta}\theta \approx {1.26}^{\xb0},{1.05}^{\xb0}$, respectively) [35]. It should be noticed that the angular distortion for Ce

^{3+}in BaWO

_{4}were calculated previously using a simplified SPM model ($\mathrm{\Delta}\theta \approx {4.0}^{\xb0}$) [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 ${\theta}_{i},i=1,2$ gives only a partial approximation to experimental values. The better approximation will be obtained by changing the azimuthal angles (${\phi}_{i},i=1,2$), 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.

_{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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Unit cell structure of BaWO

_{4}viewed along the

**b**axis with marked dodecahedrons [BaO

_{8}] (blue).

**Figure 2.**Schematic view of the dodecahedron [BaO

_{8}] in BaWO

_{4}single crystal consisting of two rotated tetrahedrons with marked crystallographic axis

**Z**(

**Z**||

**c**) and two polar angles ${\theta}_{i}$.

**Table 1.**The structural data for Ce

^{3+}ion in the barium site Ba

^{2+}in the barium tungstate (BaWO

_{4}) [44].

${\mathit{R}}_{\mathit{i}}\left[\mathbf{nm}\right]$ | ${\mathit{R}}_{\mathit{i}}^{\mathit{H}}\text{}\left[\mathbf{nm}\right]$ | ${\mathit{\theta}}_{\mathit{i}}\text{}\left[0\right]$ | ${\mathit{\phi}}_{\mathit{i}}\text{}\left[0\right]$ | |
---|---|---|---|---|

i = 1 | 0.2608 | 0.2778 | 69.05 | −35.16 |

i = 2 | 0.2568 | 0.2738 | 143.00 | −24.41 |

${\mathit{g}}_{\left|\right|}$ | ${\mathit{g}}_{\perp}$ | $\mathbf{\Delta}\mathit{g}={\mathit{g}}_{\left|\right|}-{\mathit{g}}_{\perp}$ | |
---|---|---|---|

Exp. [18,19,20] | 1.506 (1) | 2.712 (2) | −1.206 |

Calc. ^{a} | 1.506 (1) | 2.730 (20) | −1.224 |

Calc. ^{b} | 1.506 (1) | 2.725 (15) | −1.219 |

^{a}Calculated using structural polar angles ${\theta}_{i}$ in the host.

^{b}Calculated using the local polar angles ${\theta}_{i}^{\prime}$ due to the angular distortion.

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**MDPI and ACS Style**

Bodziony, T.; Kaczmarek, S.M.
Theoretical Investigation of the EPR G-Factor for the Axial Symmetry Ce^{3+} Center in the BaWO_{4} Single Crystal. *Crystals* **2021**, *11*, 804.
https://doi.org/10.3390/cryst11070804

**AMA Style**

Bodziony T, Kaczmarek SM.
Theoretical Investigation of the EPR G-Factor for the Axial Symmetry Ce^{3+} Center in the BaWO_{4} Single Crystal. *Crystals*. 2021; 11(7):804.
https://doi.org/10.3390/cryst11070804

**Chicago/Turabian Style**

Bodziony, Tomasz, and Sławomir Maksymilian Kaczmarek.
2021. "Theoretical Investigation of the EPR G-Factor for the Axial Symmetry Ce^{3+} Center in the BaWO_{4} Single Crystal" *Crystals* 11, no. 7: 804.
https://doi.org/10.3390/cryst11070804