First Principle Assisted Prediction of the Birefringence Values of

Oct 14, 2014 - Qiang BianZhihua YangYanchao WangMiriding MutailipuYanming ... Bingbing Zhang , Guoqiang Shi , Zhihua Yang , Fangfang Zhang , Shilie ...
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First Principle Assisted Prediction of the Birefringence Values of Functional Inorganic Borate Materials Qiang Bian,†,‡ Zhihua Yang,*,† Lingyun Dong,†,‡ Shilie Pan,*,† Hui Zhang,†,‡ Hongping Wu,† Hongwei Yu,†,‡ Wenwu Zhao,†,§ and Qun Jing†,‡ †

Key Laboratory of Functional Materials and Devices for Special Environments of CAS; Xinjiang Key Laboratory of Electronic Information Materials and Devices, Xinjiang Technical Institute of Physics & Chemistry of CAS, 40-1 South Beijing Road, Urumqi 830011, China ‡ University of Chinese Academy of Sciences, Beijing 100049, China § Environmental and Chemical Engineering Department, Tangshan College, Hebei 063000, China S Supporting Information *

ABSTRACT: Prediction of the birefringence values of borate is very essential for developing new optical materials in UV range. In this paper, the birefringence values of five lead borates, Pb8B9O21F, PbBiBO4, Pb3BO4F, Pb6B3O10Cl, and Pb2BO3F with network B−O structure or isolated BO3 groups, are calculated by the first principle method. The calculations show that PbBiBO4, Pb3BO4F, and Pb2BO3F have the large birefringence, greater than 0.1. Pb2BO3F, especially, is the first compound with large birefringence above 0.08 among positive uniaxial borate crystals. It is found that the parallel arrangement of fundamental building units is not the only light anisotropy active character. In the further research of Pb 2 BO 3 F, polarization disproportionation via a visualized model is first put forward for identifying the origin of large birefringence, which will be helpful to search for new optical materials with suitable birefringence.

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INTRODUCTION Prediction of birefringence is vital in searching for new optical crystals because birefringence plays a principal role in phase matching of the nonlinear optical crystals1 and in modulation of the light polarization of the linear optical materials.2 Borates are the excellent materials as solid state lasers,3−5 polarizers, and beam displacers,2 since the crystals can be transparent up to the deep UV region and are highly resistive to laser damage.6−10 Now, the studies of borate birefringence have a profound impact on the new optical material design in the UV range. In general, optical materials which can be applied in industry need to have an appropriate birefringence, including nonlinear or linear optical crystals. KBe2BO3F2 (KBBF), which is the only material that can generate coherent light at wavelengths below 200 nm by direct SHG response,11 is phase-matchable in the deep-UV region because of its relatively large birefringence (Δn = 0.07−0.08).12−14 However, another famous nonlinear optical crystal, BPO4,15 which has the shortest cutoff edge of 130 nm among borates, is not phase-matchable in the deep-UV region due to the small birefringence (about 0.005). Birefringence determines partly whether a nonlinear material has the value of study. At the same time, for a birefringence crystal, no doubt that only a material which has a large birefringence can be chosen as a candidate of the birefringence crystals. The natural negative calcite crystal (Iceland spar),16 whose birefringence is as large as 0.172, is the most commonly used birefringence © 2014 American Chemical Society

crystal in the visible range. Another excellent artificial positive crystal is YVO4;17 it is widely used because of a large birefringence (about 0.202) in the transmission range of 400−5000 nm. As can be seen from the above examples, the demand for new optical materials with suitable birefringence in UV range is very urgent, and in many cases, the birefringence value can only be determined accurately through measuring the refractive indices by the minimum-deviation method in large size crystals. However, it is difficult to obtain large size crystals of optical quality, and so, the prediction of birefringence in borates is of great significance in choosing optical material with suitable birefringence in the UV range. Birefringence is related to anisotropic polarizability, which can be calculated by ab initio methods and empirical (or semiempirical) methods.1 Empirical (or semiempirical) methods are widely used to predict the refractive indices of materials. Recently, Shannon and Fischer calculated the mean refractive indices, and the relative errors of their calculation are within 5%.18 Korotkov and Atuchin used electronic polarizabilities of ions and ionic radii in crystals to make their predictions of the mean refractive indices, and the relative errors are 0−15%.19 However, these work only calculated the mean refractive Received: July 7, 2014 Revised: October 13, 2014 Published: October 14, 2014 25651

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Figure 1. Ball-and-stick models representing the structures of (a) Pb8B9O21F, (b) PbBiBO4, (c) Pb3BO4F, (d) Pb2BO3F, and (e) Pb6B3O10Cl.

0.08 among positive uniaxial borate crystals, and a visualized model is established to explain this phenomenon in Pb2BO3F. The optically positive or negative borates with a large birefringence can be also explained by the visualized model. In addition, the influence of heavy metal cations such as Pb and Bi on the refractive indices and birefringence are also analyzed in this paper.

indices without consideration of birefringence. More recently, Qin and Li reported their prediction of the isotropic refractive indices based on electronic polarizabilities of anionic groups in crystals, and the relative errors in each direction are within 3.5%.1 Shortly afterward, Li published his calculation results for the refractive indices of borate crystals based on anisotropic polarizabilities of anionic groups with errors not more than 3% in comparison with experimental values, especially in successful explanation of the enhanced anisotropy of the KBe2BO3F family crystals.20 To our best knowledge, investigation into the contribution of heavy metal cations with lone pairs and B−O anionic groups for birefringence was still lacking until now and the relationship of optical anisotropy and the structure anisotropy still remains reclusive. Another way to predict the birefringence is ab initio method which is widely applied for the calculation of crystals. In the present paper, birefringence of borates with heavy atoms is predicted by ab initio method based on density functional theorem to account for the exchange-correlation interactions among the electrons. The correlation of optical anisotropy and structure was discussed. The influences of the arrangement of heavy atoms and the boron group were investigated using ab initio method and a visualized model. Five lead borates are selected as representations to explore the prediction of birefringence. Among them, Pb8B9O21F21 has a network B−O structure, PbBiBO4,22 and Pb3BO4F23 Pb6B3O10Cl,24 and Pb2BO3F25 share the same isolated BO3 anionic partial structures but with different BO3 coplanar configurations. Different structures are found to exhibit different refractive indices and different birefringence. It is worth mentioning that Pb2BO3F is the first compound with large birefringence above

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EXPERIMENTAL AND CALCULATION METHODOLOGY A. Experimental Details. The polycrystalline samples of Pb8B9O21F, Pb3BO4F, Pb2BO3F, Pb6B3O10Cl, and PbBiBO4 were prepared using the earlier described methods.21−25 Experimental details are presented in the Supporting Information, and the results are consistent with the previous reports. The UV−vis−IR optical diffuse reflectance spectrum was measured at room temperature using a Shimadzu Solid Spec-3700DUV spectrophotometer. Data in the energy range of 190−2600 nm were collected, and reflectance spectrum was converted to absorbance with the Kubelka− Munk function.26 The optical band gaps, which are shown in Figure S1 of the Supporting Information, are obtained via the extrapolation method with the values of 3.64, 3.21, 4.04, 3.71, and 3.36 eV for Pb8B9O21F, Pb3BO4F, Pb2BO3F, Pb6B3O10Cl, and PbBiBO4, respectively. B. Theoretical Details. Our first-principles calculations of electronic structure, band structure, and optical properties were based on DFT within the generalized gradient approximation (GGA) in the scheme of Perdew−Burke−Ernzerhof,27 as implemented in the CASTEP code.28 The crystallographic data used were obtained from the single crystal X-ray diffraction 25652

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analysis. All atoms were refined with anisotropic displacement parameters. Norm-conserving pseudopotentials29 were employed for each atomic species with the following valence configurations: Pb 5d106s26p2, F 2s25p5, B 2s22p1, O 2s22p4, and Bi 5d106s26p3, respectively. A plane-wave cutoff energy more than 900 eV was employed throughout the calculations. To evaluate integrals in reciprocal space, the Monk-horst-Pack kpoint mesh Pb8B9O21F (4 × 4 × 1), PbBiBO4(4 × 4 × 4), Pb3BO4F (4 × 6 × 4), Pb6B3O10Cl (6 × 2 × 4), and Pb2BO3F (4 × 4 × 4) in the primitive cell were chosen to achieve a wellconverged electronic structure and optical properties. Our tests reveal that the above-mentioned computational parameters are sufficiently accurate for the present calculations. On the basis of the electronic structures, the imaginary part of dielectric function ε2 can be calculated, and its real part is determined by the Kramers−Kronig transform, from which the refractive indices (and the birefringence Δn) were obtained.30 The calculation of optical properties was scissor-corrected by the difference between the calculated and measured energy gaps. The scissor operators of these five compounds are 0.06 eV (Pb8B9O21F), 0.61 eV (Pb3BO4F), 0.78 eV (Pb2BO3F), 0.87 eV (Pb6B3O10Cl), and 0.53 eV (PbBiBO4). A real-space atomcutting technique was applied to analyze the contribution of an ion (or ionic group) to the first-order susceptibility χ(1).31 The contribution of ion A to the first-order polarizabilities is denoted as χ(1)(A), it can be obtained by cutting all ions except A from the original wave functions χ(1) (A) = χ(1) (all ions except A are cut). Before exploring the title compounds, several materials were calculated in the same way, and the calculated values were in good agreement with the experimental ones. The results are listed in Table S1 of the Supporting Information. The agreement proves the validity of our theoretical model within the pseudopotential method.

Figure 2. Calculated charge density of PbBiBO4. (a) Bi2O2 group, (b) BO3 group, and (c) Pb cations and O anions.

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RESULTS AND DISCUSSION A. Structure and Electronic Structure. 1. Fundamental Building Blocks. Pb8B9O21F, Pb3BO4F, Pb2BO3F, Pb6B3O10Cl, and PbBiBO4 exhibit either network or one-dimensional anionic partial structure (shown in Figure 1). As the fundamental building blocks (FBBs), B−O groups in these five borates are very distinctive in structure. The B−O FBBs in Pb8B9O21F are B9O21, which is a network B−O structure. However, the B−O groups in the other four compounds are isolated BO3 groups. The BO3 groups in PbBiBO4 and Pb3BO4F are all locating in a plane. The arrangement of BO3 groups in PbBiBO4 and Pb3BO4F is similar, and the groups are almost parallel to each other. While in Pb6B3O10Cl, the BO3 groups are partial-coplanar, some BO3 groups are coplanar and some are at an acute angle to the YZ plane. In Pb2BO3F, the planar BO3 groups form an equilateral triangle, and they are parallel to the z axis. In addition, the calculated charge densities also show the overlap behavior of the B−O groups in these five borates, as shown in Figure 2 and Figures S2 and S3 of the Supporting Information. The Pb−O electronic states in Pb3BO4F and Pb6B3O10Cl and Bi−O electronic states in PbBiBO4 exhibit hybridization and indicate the covalent chemical bonding. 2. Band Structure and DOS. As shown in Figure 3 and Figure S4 of the Supporting Information, Pb8B9O21F and PbBiBO4 are indirect-gap materials with the band gaps 3.58 and 2.83 eV, whereas Pb3BO4F, Pb6B3O10Cl, and Pb2BO3F are direct-gap materials with the calculated band gaps 2.60, 2.84,

Figure 3. Band structure and density of states of (a and b) PbBiBO4 and (c and d) Pb2BO3F.

and 3.26 eV. The calculated band gaps are all in good agreement with the values obtained from the diffuse-reflectance spectra.21−25 Analysis of the partial density of states (PDOS) is helpful to understand the transitions from occupied states to unoccupied states which mainly determine the optical properties.32 The electronic dispersions of the five lead borates are quite similar, especially for the electronic states close to the energy band gap. The upper region of the valence band (VB) is mainly composed of the 2p orbitals of boron, oxygen, and halogen atoms and exhibits a wide hybridization from −5 to 0 eV between B 2p states and O 2p states, which reveal the B−O bonds to be the main contributors to the top of VB in the five lead borates. The bottom of the conduction band (CB) is mainly composed of the orbitals of oxygen, boron, and heavy metal atoms (Pb and Bi). Evidently, the BO3 anionic groups are the main contributors to the energy band gaps and optical anisotropy in the five lead borates. Owing to the repulsion interactions of the lone pairs of Pb2+ cations, the Pb2+ cations may also lead to a little contribution to the optical anisotropy. Otherwise, because the fluorine anions occupy little to the top of VB and the bottom of the CB, its contribution to the optical anisotropy is negligible. 25653

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B. Linear Optical Properties. Pb8B9O21F, Pb3BO4F, and Pb2BO3F belong to medium crystal class and are uniaxial due to corresponding 6/m, mmm, 3̅m point groups, while PbBiBO4 and Pb6B3O10Cl are biaxial due to their mmm point group. The dispersion refractive indices calculated display different anisotropy, which can be described by the birefringence Δn. Larger birefringence values are exhibited in Figure 4 and Figure

structure: Pb 6 B 3 O 10 Cl) > (network B−O structure: Pb8B9O21F). It is well-known that light-induced polarization components (mostly electric field) perpendicular and parallel to the BO3 plane are expected to be very different and can induce a large birefringence.33,34 Then, according to the anionic group theory, the parallel configuration of anionic groups will generate a large optical anisotropy. Following this idea, Pb2BO3F should have a smaller birefringence than those of Pb3BO4F and PbBiBO4, but in fact it reveals the largest birefringence value (about 0.12 @ 589.3 nm). The structure of Pb2BO3F is very specific, and to the best of our knowledge, it is the first positive uniaxial borate crystal with so large a birefringence value, >0.08. So, why does Pb2BO3F have so large a birefringence value? The intrinsic relationship between structures and properties of Pb2BO3F should be further explored. C. Novel Birefringence Behavior in Pb2BO3F. The charge density distribution in Pb2BO3F is calculated and shown in Figure S3 of the Supporting Information. The charge densities of the Pb and F ions are almost spherical, and the Pb and F ions may contribute a little to the birefringence value in the crystal. The electron cloud of the BO3 anionic groups exhibit planar shape with conjugated electron orbitals, and thus, the BO3 anionic groups are the main source of the large birefringence in Pb2BO3F. To investigate the influence of the respective ions and atomic groups on optical response of Pb2BO3F quantitatively, the realspace atom-cutting technique is employed.31 The cutting ion radius scheme is based on the investigation of charge-density distribution between the nearest ions, following the cutting radii for the lead, boron, oxygen, and fluoride atoms as 1.20 Å, (B1:0.345 Å, B2:0.286 Å), (O1:1.092 Å, O2:1.041 Å), and (F1:1.294 Å, F2:1.588 Å), respectively. Figure 5a gives atom-cutting analysis results of Pb2BO3F, revealing that the lead and fluoride ions have a little contribution to the birefringence, while the BO3 groups become the main source of the birefringence. The contribution of the BO3 anionic groups to birefringence is about 80%, while the Pb and F ions contribute only 20%. To investigate the impact of Pb and F on the linear optical properties, the

Figure 4. Refractive indices and birefringence of PbBiBO4. (a) The calculated refractive index of PbBiBO4, (b) the birefringence and the atom-cutting results of PbBiBO4, Δn represents that the contribution to birefringence after removing PbBiBO4, cut-Pb represents that remove Pb cations, and cut-PbBiO refers to the contribution to Δn when removing Pb cations and Bi−O groups.

S5 of the Supporting Information for PbBiBO4, Pb3BO4F, Pb6B3O10Cl, and Pb2BO3F with the values of 0.11, 0.11, 0.09, and 0.12 (@ 589.3 nm), respectively, while the definitely small birefringence of Δn = 0.005 (@ 589.3 nm) is found in Pb8B9O21F. The compounds exhibit different optical anisotropy. Initially, four lead borates, except Pb2BO3F, will be investigated. Pb8B9O21F has a low optical anisotropy due to disorder orientation of B−O groups, and there are not disorder orientations existing in PbBiBO4, Pb3BO4F, and Pb6B3O10Cl crystals. The DFT-computed charge densities show high electron density configuration and strong anisotropy for B−O groups, which indicates the main contribution of the BO3 group to the optical anisotropy in PbBiBO4 , Pb 3BO 4F, and Pb6B3O10Cl. Furthermore, owing to the repulsion interactions of the lone pairs of Pb2+ cations, the spatial orientation of Pb2+ cations may also lead to a little contribution from Pb−O groups to the optical anisotropy. The Bi−O groups may also contribute to optical anisotropy due to covalent chemical bonding between the Bi and O. The real-space atom-cutting technique31 is used to confirm the analysis results mentioned above, and the results are showed in Figure 4. The BO3 groups contribute 77% and Bi2O2 groups contribute 18% of birefringence, which means that there is almost no contribution from the Pb cations in PbBiBO4. It is apparent that the B−O groups are the main source of optical anisotropy in Pb8B9O21F, PbBiBO4, Pb3BO4F, and Pb6B3O10Cl. The results are consistent with the anionic group theory. The optical anisotropy can be divided by the different arrangement of B−O groups that follow the trend of (coplanar B−O structure: PbBiBO4 and Pb3BO4F) > (partial-coplanar B−O

Figure 5. (a) Refractive indices and birefringence of Pb2BO3F, cut-Δn represents that the contribution to birefringence after removing Pb and F atoms; (b) the coordination environment of Pb in Pb2BO3F; and (c) the Pb2+ lone pair map of electronic local function for Pb2BO3F. 25654

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the conclusion that ne > n0 due to maximal polarizability along the optical axes. On the other hand, the BO3 number density of the Pb2BO3F is about 9.07 ( × 10−3), which is comparable to the KBBF [9.42 ( × 10−3)], and the high density explains further why Pb2BO3F is an optically positive crystal with a large birefringence. D. Further Empirical Investigation of the Large Birefringence in Borates. It is one of the important properties of birefringence borate crystals: positive (ne > no) or negative (no > ne) in refractive indices difference. How to determine the optically positive or negative behavior from the borate crystal structure? In this part, we investigate two kinds of structures in borates with a large birefringence. First, in uniaxial borate crystals with large birefringence, generally, the BO3 groups are always parallel to each other with a higher number density. This kind of crystal is always optically negative, for example, KBBF,35 YAl3(BO3)4,36,37Ca3(BO3)2,38 and so on. The reason why such crystals exhibit negative uniaxial phenomena can be qualitatively explained in Figure 7. The BO3 groups in the lattices are parallel to each other, so the smallest value of refractive indices arises in the direction of the z axis because of the minimum polarization perpendicular to the BO3 groups, and the maximum appears in the X−Y plane ⎯→ because F1s all locate on the planar. For a negative uniaxial borate crystal, ne is the minimum refractive index and n0 is the maximum refractive index. Therefore, the refractive index along the z axis is ne, and the refractive indices located in the X−Y plane are n0 and n0 > ne, and such crystals are certainly uniaxial negative crystals. Second, another kind of structure, which contributes to a large birefringence, is the planar anionic groups with high electronic density that form equilateral triangles (or squares), and they are parallel to the z axis. Therefore, the smallest value of refractive indices arises in the X−Y plane because the polarizabilities of the planar anionic groups are distributed very well to be a homogeneous distribution, and the maximum ⎯→ appears in the direction of the z axis because all F1s are parallel to the z direction. This kind of uniaxial crystal is always positive crystal. For instance, Pb2BO3F in borates, bastnasite in carbonates, CO(NH2)2 in organic crystals, and so on. The structures of bastnasite and CO(NH2)2 are shown in Figure S6 of the Supporting Information. Thus, Pb2BO3F, as it is found by the calculation, is related to a second kind of structures which can induce a large birefringence in borates.

electronic local function was adopted (Figure 5c). Because of lone pairs in Pb ion (Figure 5b), the localization of electron in Pb has a lower symmetry than that in F, which means that the Pb contribution to optical anisotropy is higher than F. From the above analysis, it is found that the contributions to optical anisotropy follow the trend of BO3 ≫ Pb > F. To clarify the issue why Pb2BO3F with different arrangement of BO3 groups have larger birefringence (>0.1), the structure of the crystal is re-explored by a structure model. The linear polarizability in the direction orthogonal to the BO3 triangle is smaller than that in the plane of the BO3 group. Figure 6 shows

Figure 6. Arrangements of BO3 groups in Pb2BO3F. The blue arrows ⎯→ represent the linear polarization in the planar of the BO3 groups (F1), while the green arrows represent the linear polarizability which is ⎯→ perpendicular to the BO3 groups (F2).

the arrangement of the BO3 groups in Pb2BO3F, and these are parallel to the z axis. Two vectors are set to represent the linear polarizabilities in the direction orthogonal to the BO3 triangle ⎯→ ⎯→ ⎯→ ⎯→ (F1) and in the plane of the BO3 group (F2 ), and F1 > F2. It is illustrated that the largest refractive index arises in the direction of z axis due to the maximization of superposition effect in polarizabilities along the z direction, and the smallest appears in ⎯→ the XY plane because polaribilities of the BO3 groups along F1s ⎯→ ⎯→ and F2s distribute evenly on the plane. All F1s are parallel to the z direction, so the largest polarizbility appears along this ⎯→ ⎯→ direction. However, all F1s and F2 s located on the XY plane are very well-distributed to be a homogeneous distribution so that the smallest polaribility appear on the XY plane. This makes a maximum value of birefringence situating in the z direction and the XY plane, respectively. Therefore, one can get

Figure 7. Arrangements of the BO3 groups in negative-crystals with a large birefringence. (a) KBBF and (b) Ca3(BO3)2. 25655

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Pb6B3O10Cl, and PbBiBO4. This material is available free of charge via the Internet at http://pubs.acs.org.

Otherwise, Shannon and Fischer found that the empirical electronic polarizabilities of Pb and Bi are bigger than many kinds of cations’ empirical electronic polarizabilities (Ca, Ba, Sr, Rb, and so on).18 Because of their high cations’ empirical electronic polarizabilities, the content of heavy elements (Pb and Bi) will also affect the refractive indices. For example, the refractive indices of these five crystals calculated here are all above 1.8. In accordance with the results from atom-cutting methods, the refractive indices will become lower than 2.0 if the Pb and Bi cations are cut off, as is shown in Figure S7 of the Supporting Information. Thus, the content of Pb or Bi, which has large electronic polarizabilities, is significant for high refractive indices.

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Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Tel: (86)-991- 3810816. Fax: (86)-991-3838957. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Basic Research Program of China (Grant 2014CB648400), the National Natural Science Foundation of China (Grants 11474353, 11104344, U1129301, and 21101168), the “Western Light Joint Scholar Foundation” Program of CAS (Grant LHXZ201101), the Recruitment Program of Global Experts (1000 Talent Plan, Xinjiang Special Program), the Funds for Creative Cross & Cooperation Teams of CAS, Main Direction Program of Knowledge Innovation of Chinese Academy of Sciences (Grant KJCX2-EW-H03-03), Major Program of Xinjiang Uygur Autonomous Region of China during the 12th Five-Year Plan Period (Grant 201130111), and the Science and Technology Project of Urumqi (Grant G121130002).

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CONCLUSION To reveal the source of optical anisotropy in lead borates, the structures and properties of five lead borates are systematically studied. The electronic structures and linear optical properties of PbBOFs, Pb6B3O10Cl, and PbBiBO4 are calculated by the first-principles in the framework of a plane-wave pseudopotential method. The electronic structure results, and the further atom-cutting analysis show that the B−O anionic groups are the main factor to influence band gap and linear optical properties. A different arrangement of the B−O groups results in different birefringence, which follows the trend of coplanar B−O structure > partial-coplanar B−O structure > network B− O structure. In particular, Pb2BO3F, which is the first borate with the birefringence value above 0.08 among optically positive uniaxial borate crystals, belongs to the second kind of structures, different from the coplanar arrangement configuration of the BO3 groups. A visualized model is first put forward to explain the origin of large birefringence in Pb2BO3F. The special structure, where the BO3 anionic groups form an equilateral triangle parallel to the z axis, and the high number density of the BO3 triangles are the factors governing in birefringence in Pb2BO3F. Simultaneously, because of the lone electron pairs presence, the electron density cloud in Pb ions exhibits low symmetry and produces a certain optical anisotropy. The appearance of optically positive or negative borates with a large birefringence is explained by the visualized model. Otherwise, it is found that refractive indices are related to the heavy element (Pb) content. We believe that a full understanding of the structure−property relationships and physical mechanism in borates would have great meaning in designing and searching for new optical materials with suitable birefringence in UV range, combining with the computational searching using first-principles methods and the experimental synthesis.

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AUTHOR INFORMATION

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The UV−vis−NIR diffuse reflectance spectra of Pb8B9O21F, Pb3BO4F, Pb2BO3F, Pb6B3O10Cl, and PbBiBO4. Calculated charge density of Pb8B9O21F, Pb3BO4F, Pb6B3O10Cl, and Pb2BO3F. The calculated band structure and density of states of Pb8B9O21F, Pb3BO4F, and Pb6B3O10Cl. The calculated refractive indices and birefringence of Pb8B9O21F, Pb3BO4F, and Pb6B3O10Cl. The ball-and-stick models representing the structures of bastnasite-(La) and bastnasite-(Ce). The influence of Pb and Bi in refractive indices. The comparison of the calculated and experimental values of birefringence. Crystal data and structures refinement for Pb8B9O21F, Pb3BO4F, Pb2BO3F, 25656

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dx.doi.org/10.1021/jp506744s | J. Phys. Chem. C 2014, 118, 25651−25657