English

• 1. INTRODUCTION

  With the development of society, the optoelectronic functional crystals play a crucial role in the industrial, medical and scientific fields, especially the ultraviolet (UV) even deep-UV (DUV) optoelectronic functional crystals due to their applications in the short wavelength laser areas^[1-3]. In recent years, a large number of new UV/DUV optical crystals have been reported, including carbonates^[4], silicates^[5], sulfates^[6], borates^[7-10], phosphates^[11-13], and so on.

  Among them, borates are excellent candidates among these systems due to their abundant structures. The B atoms can coordinate with O atoms to form simple [BO₃]^3– and [BO₄]^5– groups, and these two kinds of groups further connect together to form many different polymerization clusters ([B₂O₄]^2–, [B₃O₅]^–, [B₄O₁₁]^10–, [B₇O₁₄]^7– etc.), which could reduce certain terminal oxygens in the structure to make the UV cut-off wavelength blue-shift, so it is available to obtain DUV optical materials in borate^[14]. Besides, the BO₃ groups have a large π-conjugated orbital electron structure, which is beneficial for obtaining new crystals with outstanding optical performance^[15]. Based on the previous researches, many new borate crystals with excellent optical properties have been reported, such as β-BaB₂O₄ (β-BBO)^[16], KBe₂BO₃F₂ (KBBF)^[17] and LiB₃O₅ (LBO)^[18] which can be used as DUV nonlinear optical (NLO) crystals, and α-BaB₂O₄ (α-BBO)^[19], Na₃Ba₂(B₃O₆)₂F^[20] and Ca(BO₂)₂^[21] can serve as DUV birefringent crystals.

  Although many efforts proved that the type and arrangement of anionic groups have main contribution to the optical properties of crystals, the metal cation is of great significance for the optical properties^[22]. And some design strategies have been brought up to obtain new borates. First, the cations with lone-pair electrons (Pb²⁺, Sn²⁺, etc.) or d⁰ electrons (Nb⁵⁺, V⁵⁺, Mo⁶⁺, etc.) can enhance the optical properties for crystals effectively, but easily narrow the light transmission range. For example, Pb₂B₅O₉I possesses the largest SHG response in borate crystals, but the light transmission range will be limited in the ultraviolet (UV) region^[23]. Second, the alkali and alkali-earth metals have no d-d or f-f electron transition, which is also beneficial for UV cut-off wavelength blue-shift, as well as the introduction of cations with large atomic radii (Rb or Cs) may lead to special crystal structures. So, the alkali metals benefit for the DUV optical functional crystal^[24].

  Guided by the above ideas, we focused on the Li-Rb-B-O system to explore new UV optoelectronic functional crystals, and our efforts produced a new crystal, LiRbB₈O₁₃, which owns large bandgap and birefringence. Herein, the single crystal structure, synthesis, IR spectrum and UV-Vis-NIR diffuse reflectance spectrum of LiRbB₈O₁₃ are presented. Moreover, theoretical calculations are performed to understand its electronic structure and optical properties.

  2. EXPERIMENTAL

  2.1 Synthesis

  The LiRbB₈O₁₃ crystal was grown in the sealed system. First, the mixture of LiF (0.052 g), RbF (0.052 g) and B₂O₃ (0.174 g) was loaded into a tidy quartz tube which was then flame-sealed under 10⁻³ Pa. After that, this sealed tube was heated from room temperature to 550 ℃ in 600 min, and kept at this temperature for 600 min. Subsequently, it was cooled to 400 ℃ at a rate of 1 ℃/h, then further cooled to 300 ℃ at a rate of 4 ℃/h, finally lowered to room temperature fast. During the cooling process, the target crystals were obtained. Colorless crystal was separated from the quartz tube for single-crystal X-ray diffraction (XRD) measurements.

  The polycrystalline sample of LiRbB₈O₁₃ was synthesized by solid-state reaction. The mixture of LiNO₃, Rb₂CO₃ and H₃BO₃ with the molar ratio of 1:0.5:8 was ground evenly. Subsequently, the mixture was heated to 580 ℃ slowly and held for 14 days at this temperature, in which the mixture was ground for several times. Finally, it was cooled to room temperature naturally. In this process, the target sample was obtained.

  2.2 Structural determination

  The single-crystal XRD data of LiRbB₈O₁₃ were collected by using a Bruker SMART APEX II 4K CCD diffractometer equipped with MoKα radiation (λ = 0.71073 Å) at room temperature, and the data were integrated with a SAINT program^[25]. The direct methods and full-matrix least-squares program SHELXL were used to solve and refine the crystal structure, respectively^[26]. The structure was checked for missing symmetry elements with PLATON^[27]. The final R = 0.0547, wR = 0.1224. The selected bond lengths and bond angles are listed in Table 1.

  Table 1

  

  Table 1.  Selected Bond Lengths and Bond Angles for LiRbB₈O₁₃

  DownLoad: CSV

  Bond	Dist.		Bond	Dist.		Bond	Dist.
  Li(1)–O(8)	1.882(12)		B(1)–O(3)	1.371(8)		B(5)–O(4)	1.378(8)
  Li(1)–O(10)#5	1.890(11)		B(1)–O(6)	1.405(7)		B(5)–O(6)	1.379(8)
  Li(1)–O(7)#6	1.913(11)		B(2)–O(2)	1.356(8)		B(6)–O(8)	1.361(7)
  Li(1)–O(11)#6	1.999(12)		B(2)–O(10)	1.359(8)		B(6)–O(13)	1.362(8)
  Rb(1)–O(4)#1	2.836(4)		B(2)–O(3)	1.391(7)		B(6)–O(12)	1.381(8)
  Rb(1)–O(1)#2	2.978(4)		B(3)–O(5)	1.450(7)		B(7)–O(7)	1.443(7)
  Rb(1)–O(9)#3	2.986(4)		B(3)–O(9)#10	1.468(7)		B(7)–O(2)#11	1.446(7)
  Rb(1)–O(9)#2	2.993(4)		B(3)–O(1)	1.477(7)		B(7)–O(11)#12	1.448(7)
  Rb(1)–O(12)#1	3.013(4)		B(3)–O(10)#10	1.479(8)		B(7)–O(8)#12	1.479(7)
  Rb(1)–O(1)#3	3.041(4)		B(4)–O(5)	1.336(8)		B(8)–O(7)	1.346(8)
  Rb(1)–O(3)	3.130(4)		B(4)–O(4)	1.378(8)		B(8)–O(11)	1.394(8)
  Rb(1)–O(6)#1	3.326(4)		B(4)–O(13)	1.381(7)		B(8)–O(12)	1.400(8)
  B(1)–O(9)	1.326(8)		B(5)–O(1)	1.342(8)
  Angle	(°)		Angle	(°)		Angle	(°)
  O(4)#1–Rb(1)–O(5)#4	96.89(11)		O(9)–B(1)–O(6)	120.9(5)		O(1)–B(5)–O(6)	122.6(5)
  O(1)#2–Rb(1)–O(5)#4	92.63(10)		O(3)–B(1)–O(6)	117.0(5)		O(4)–B(5)–O(6)	116.3(6)
  O(9)#3–Rb(1)–O(5)#4	119.22(11)		O(2)–B(2)–O(10)	122.1(5)		O(8)–B(6)–O(13)	117.3(5)
  O(9)#2–Rb(1)–O(5)#4	43.23(10)		O(2)–B(2)–O(3)	117.5(5)		O(8)–B(6)–O(12)	120.3(5)
  O(12)#1–Rb(1)–O(5)#4	58.31(11)		O(10)–B(2)–O(3)	120.3(5)		O(13)–B(6)–O(12)	122.4(5)
  O(1)#3–Rb(1)–O(5)#4	146.57(10)		O(5)–B(3)–O(9)#10	110.1(5)		O(7)–B(7)–O(2)#11	109.9(5)
  O(3)–Rb(1)–O(5)#4	59.83(10)		O(5)–B(3)–O(1)	111.1(5)		O(7)–B(7)–O(11)#12	109.5(5)
  O(6)#1–Rb(1)–O(5)#4	134.80(9)		O(9)#10–B(3)–O(1)	108.3(5)		O(2)#11–B(7)–O(11)#12	109.6(5)
  O(8)–Li(1)–O(10)#5	105.5(5)		O(5)–B(3)–O(10)#10	108.8(5)		O(7)–B(7)–O(8)#12	106.6(5)
  O(8)–Li(1)–O(7)#6	108.2(5)		O(9)#10–B(3)–O(10)#10	109.7(5)		O(2)#11–B(7)–O(8)#12	112.3(5)
  O(10)#5–Li(1)–O(7)#6	125.5(6)		O(1)–B(3)–O(10)#10	108.9(5)		O(11)#12–B(7)–O(8)#12	108.7(5)
  O(8)–Li(1)–O(11)#6	134.9(6)		O(5)–B(4)–O(4)	122.9(5)		O(7)–B(8)–O(11)	113.8(5)
  O(10)#5–Li(1)–O(11)#6	110.2(5)		O(5)–B(4)–O(13)	117.4(5)		O(7)–B(8)–O(12)	124.2(6)
  O(7)#6–Li(1)–O(11)#6	71.8(4)		O(4)–B(4)–O(13)	119.7(5)		O(11)–B(8)–O(12)	122.0(5)
  O(9)–B(1)–O(3)	122.1(5)		O(1)–B(5)–O(4)	121.1(6)
  Symmetry transformations used to generate the equivalent atoms: #1: –x + 1, y + 1/2, –z + 1/2; #2: x, –y + 3/2, z – 1/2; #3: –x + 1, –y + 2, –z + 1; #4: –x + 1, –y + 1, –z + 1; #5: x + 1, y – 1, z; #6: x, –y + 1/2, z + 1/2; #7: x + 1, –y + 3/2, z + 1/2; #8: x, –y + 3/2, z + 1/2; #9: x – 1, y + 1, z; #10: –x + 1, y – 1/2, –z + 3/2; #11: –x + 1, y – 1/2, –z + 1/2; #12: –x + 2, y + 1/2, –z + 1/2; #13: x, –y + 1/2, z – 1/2; #14: –x + 2, y – 1/2, –z + 1/2; #15: –x + 1, y + 1/2, –z + 3/2; #16: x, –y + 5/2, z + 1/2

  2.3 Optical characterization

  The powder X-ray diffraction (PXRD) analysis of the polycrystalline sample was carried out by a Bruker D2 PHASER diffractometer with CuKα radiation at room temperature. The scan step width and fixed counting time are 0.02° and 1 s/step, respectively. The diffraction patterns were performed in the angular range of 2θ = 5~75°. The measurement result for the title compound is shown in Fig. 1, which indicates that the experimental result agrees well with the theoretical one except for several impurity peaks.

  Figure 1

  

  Figure 1.  Calculated and experimental powder X-ray diffraction pattern for LiRbB₈O₁₃

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  We mixed the powder sample with dried KBr and ground evenly. And the IR spectrum was collected by Shimadzu IRAffinity-1 Fourier transform IR spectrometer with a resolution of 2 cm⁻¹ in the range of 400~4000 cm⁻¹ at room temperature.

  The UV-Vis-NIR diffuse reflectance spectrum was obtained at room temperature by using polytetrafluoroethylene as a standard to collect the data of the title compound on the Shimadzu SolidSpec-3700DUV spectrophotometer, and the wavelength range was from 190 to 2500 nm.

  2.4 Theoretical calculations

  The electronic structure and optical properties were calculated for LiRbB₈O₁₃ by the CASTEP software^[28], a plane-wave pseudopotential DFT package, with the normconserving pseudopotentials^[29]. The exchange-correlation functional was Perdew-Burke-Emzerhof (PBE) functional within the generalized gradient approximation (GGA)^[30]. The plane-wave energy cutoff was set at 850.0 eV. Selfconsistent field (SCF) calculations were performed with a convergence criterion of 1 × 10⁻⁶ eV/atom on the total energy. The k-point separation for each material was set as 0.07 Å^-1 in the Brillouin zone. The valence electrons of the elements in LiRbB₈O₁₃ were calculated as follows: Li 2s¹, Rb 4s²4p⁶5s¹, B 2s²2p¹ and O 2s²2p⁴, respectively. The empty bands were set as 3 times the valence bands in the calculation to ensure the convergence of optical properties. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization technique was employed in geometry optimization during the calculation and the residual forces on the atoms, the displacements and the energy change of atoms are less than 0.01 eV/Å, 5 × 10^–4 Å and 5.0 × 10^–6 eV, respectively. The default values of the CASTEP code were retained for other parameters and convergent criteria.

  3. RESULTS AND DISCUSSION

  3.1 Structural description

  LiRbB₈O₁₃ is isomorphic with α-LiKB₈O₁₃^[31], and crystallizes in space group P2₁/c (No. 14) of monoclinic crystal system. The asymmetric unit of LiRbB₈O₁₃ includes one unique Li atom, one unique Rb atom, eight unique B atoms and thirteen unique O atoms. The B(3) and B(7) atoms coordinate with four oxygens to form the [BO₄]^5- groups, and the other six kinds of B atoms are bound to three oxygens to form the [BO₃]^3– groups. The [BO₃]^3– and [BO₄]^5– groups are further connected to form the two groups by sharing oxygens, [B₅O₁₀]^5– and [B₃O₇]^5–, which are linked together by sharing one oxygen to build the fundamental building block (FBB) of the title compound, [B₈O₁₆]^8- unit (Fig. 2a). And the FBBs are connected to construct the 3D [B₈O₁₃]_∞ framework by common oxygen atoms (Fig. 2b), which has 18-, 10- and 6-membered rings, respectively. From Fig. 2c it can be seen that the Rb atoms are inside the 18-membered rings and on the edge of the 6 rings, while Li atoms lie inside the 10-membered rings and on the edge of the 18-membered rings.

  Figure 2

  

  Figure 2.  (a) [B₈O₁₆]^8- unit; (b) Framework of _∞³[B₈O₁₃] viewed in the ac plane; (c) Crystal structure of LiRbB₈O₁₃; (d) Crystal structure within the red border rotate certain angle

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  In the structure, the B–O bond lengths of BO₃ triangles are from 1.326(8) to 1.405(7) Å and from 1.443(7) to 1.479(8) Å in the BO₄ tetrahedra (Table 1). All the bond lengths are consistent with those observed in other borates^[32-34]. The coordination environments of the Li and Rb atoms are shown in Fig. 2c and 2d, respectively. It can be seen that the Li and Rb atoms are bound to four and eight O atoms to form the LiO₄ and RbO₈ polyhedra, respectively. The Li–O bond lengths fall in the range of 1.882(12)~1.999(12) Å and the Rb–O bond lengths vary from 2.836(4) to 3.326(4) Å (Table 1)^[35].

  In order to understand the 3D framework of LiRbB₈O₁₃, we provide the topological picture, in which the [B₈O₁₆]^8– units (FBBs) act as 6-connected nodes (Fig. 3). The Schläfli symbol, as analyzed by the TOPOS 4.0 program, is {4⁹·6⁶}^[36]. From the topological structure, it is easy to find that the structure has two independent interpenetrating 3D frameworks.

  Figure 3

  

  Figure 3.  Topological illustration of LiRbB₈O₁₃

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  3.2 Optical analysis

  The infrared (IR) spectrum for LiRbB₈O₁₃ is given in Fig. S1 in the Supporting Information, which shows the obvious stretching and bending vibration peaks for B–O bonds. According to the previous literatures^{[37, 38]}, the observed absorption peaks in the ranges of 1255~1213 and 1093~1041 cm^–1 belong to the asymmetric vibrations of [BO₃]^3– and [BO₄]^5– groups, respectively. The symmetric stretching peaks of [BO₃]^3– and [BO₄]^5– groups can be observed at 941 and 783 cm^–1, respectively. The peaks below 683 cm^–1 are attributed to bending vibrations of B–O bonds. The IR spectrum further confirms the existence of B–O groups in the structure of the title compound, which agrees with the result from single-crystal X-ray diffraction.

  The UV-vis-IR diffuse reflectance spectrum of LiRbB₈O₁₃ is shown in Fig. 4, in which the wavelength ranges from 200 to 2500 nm. It is clear that the transmittance is about 70% at 200 nm, so the cut-off edge of LiRbB₈O₁₃ is below 200 nm, but a weak absorption peak can be observed at 242 nm, which may be due to the minor impurity.

  Figure 4

  

  Figure 4.  UV-Vis-NIR diffuse reflectance spectrum of LiRbB₈O₁₃

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  3.3 Calculated optical analysis

  To deeply study the relation of structure-property, the first-principle calculations were carried out by using the CASTEP package. The calculated electronic band structure, birefringence and total and partial densities of states are shown in Fig. 5, which indicates that LiRbB₈O₁₃ is an indirect band gap compound with a value of 5.81 eV. Owing to the discontinuity of exchange-correlation energy in GGA, the calculated band gap is usually smaller than that in experiment. The birefringence of LiRbB₈O₁₃ can be known from Fig. 5b, about 0.08 at 1064 nm, which indicates the potential application of LiRbB₈O₁₃ as a DUV birefringent material. To further study the contribution of LiO₄, RbO₈, BO₃ and BO₄ groups for birefringence in the title compound, we calculated the bonding electron density difference (Δρ) of these units (Fig. 6) along the optical principal axes by the REDA (response electron distribution anisotropy) method^[39]. Fig. 6 shows that the large birefringence is mainly attributed to the BO₃ group, LiO₄ and RbO₈ groups give a little contribution, and the BO₄ group provides a negative contribution. The total and partial densities of states (Fig. 5c) show that the valence band (VB) and the conduction band (CB) are composed of O-2p, B-2p, Rb-4p and Li-1s states. Among them, the top of VB is majorly occupied by B-2p and O-2p orbitals, and the bottom by Li-1s and B-2p orbitals, respectively. Therefore, the B–O and Li–O bonds determine the VB maximum and CB minimum, which determines the band gap of the title compound.

  Figure 5

  

  Figure 5.  (a) Band structure of LiRbB₈O₁₃; (b) Birefringence calculation of LiRbB₈O₁₃; (c) Density of states (DOS) of LiRbB₈O₁₃

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  Figure 6

  

  Figure 6.  Bonding electron density difference (Δρ) of different units calculated by the REDA method

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  4. CONCLUSION

  In summary, a new complex alkali metal borate, LiRbB₈O₁₃, has been obtained successfully. The structure of LiRbB₈O₁₃ is isostructural with α-LiKB₈O₁₃, exhibiting two independent interpenetrating 3D frameworks. The experiment and the first principle calculations indicate that it has a short cutoff edge (below 200 nm) and large birefringence (0.08 at 1064 nm), indicating that it could be regarded as the DUV birefringent crystal. According to the REDA method, the birefringence is mainly contributed by the BO₃ group. Also, the research of other complex alkali metal borates is in progress.

  
  
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  Figure 1  Calculated and experimental powder X-ray diffraction pattern for LiRbB₈O₁₃

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  Figure 2  (a) [B₈O₁₆]^8- unit; (b) Framework of _∞³[B₈O₁₃] viewed in the ac plane; (c) Crystal structure of LiRbB₈O₁₃; (d) Crystal structure within the red border rotate certain angle

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  Figure 3  Topological illustration of LiRbB₈O₁₃

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  Figure 4  UV-Vis-NIR diffuse reflectance spectrum of LiRbB₈O₁₃

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  Figure 5  (a) Band structure of LiRbB₈O₁₃; (b) Birefringence calculation of LiRbB₈O₁₃; (c) Density of states (DOS) of LiRbB₈O₁₃

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  Figure 6  Bonding electron density difference (Δρ) of different units calculated by the REDA method

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  Table 1.  Selected Bond Lengths and Bond Angles for LiRbB₈O₁₃

  Bond	Dist.		Bond	Dist.		Bond	Dist.
  Li(1)–O(8)	1.882(12)		B(1)–O(3)	1.371(8)		B(5)–O(4)	1.378(8)
  Li(1)–O(10)#5	1.890(11)		B(1)–O(6)	1.405(7)		B(5)–O(6)	1.379(8)
  Li(1)–O(7)#6	1.913(11)		B(2)–O(2)	1.356(8)		B(6)–O(8)	1.361(7)
  Li(1)–O(11)#6	1.999(12)		B(2)–O(10)	1.359(8)		B(6)–O(13)	1.362(8)
  Rb(1)–O(4)#1	2.836(4)		B(2)–O(3)	1.391(7)		B(6)–O(12)	1.381(8)
  Rb(1)–O(1)#2	2.978(4)		B(3)–O(5)	1.450(7)		B(7)–O(7)	1.443(7)
  Rb(1)–O(9)#3	2.986(4)		B(3)–O(9)#10	1.468(7)		B(7)–O(2)#11	1.446(7)
  Rb(1)–O(9)#2	2.993(4)		B(3)–O(1)	1.477(7)		B(7)–O(11)#12	1.448(7)
  Rb(1)–O(12)#1	3.013(4)		B(3)–O(10)#10	1.479(8)		B(7)–O(8)#12	1.479(7)
  Rb(1)–O(1)#3	3.041(4)		B(4)–O(5)	1.336(8)		B(8)–O(7)	1.346(8)
  Rb(1)–O(3)	3.130(4)		B(4)–O(4)	1.378(8)		B(8)–O(11)	1.394(8)
  Rb(1)–O(6)#1	3.326(4)		B(4)–O(13)	1.381(7)		B(8)–O(12)	1.400(8)
  B(1)–O(9)	1.326(8)		B(5)–O(1)	1.342(8)
  Angle	(°)		Angle	(°)		Angle	(°)
  O(4)#1–Rb(1)–O(5)#4	96.89(11)		O(9)–B(1)–O(6)	120.9(5)		O(1)–B(5)–O(6)	122.6(5)
  O(1)#2–Rb(1)–O(5)#4	92.63(10)		O(3)–B(1)–O(6)	117.0(5)		O(4)–B(5)–O(6)	116.3(6)
  O(9)#3–Rb(1)–O(5)#4	119.22(11)		O(2)–B(2)–O(10)	122.1(5)		O(8)–B(6)–O(13)	117.3(5)
  O(9)#2–Rb(1)–O(5)#4	43.23(10)		O(2)–B(2)–O(3)	117.5(5)		O(8)–B(6)–O(12)	120.3(5)
  O(12)#1–Rb(1)–O(5)#4	58.31(11)		O(10)–B(2)–O(3)	120.3(5)		O(13)–B(6)–O(12)	122.4(5)
  O(1)#3–Rb(1)–O(5)#4	146.57(10)		O(5)–B(3)–O(9)#10	110.1(5)		O(7)–B(7)–O(2)#11	109.9(5)
  O(3)–Rb(1)–O(5)#4	59.83(10)		O(5)–B(3)–O(1)	111.1(5)		O(7)–B(7)–O(11)#12	109.5(5)
  O(6)#1–Rb(1)–O(5)#4	134.80(9)		O(9)#10–B(3)–O(1)	108.3(5)		O(2)#11–B(7)–O(11)#12	109.6(5)
  O(8)–Li(1)–O(10)#5	105.5(5)		O(5)–B(3)–O(10)#10	108.8(5)		O(7)–B(7)–O(8)#12	106.6(5)
  O(8)–Li(1)–O(7)#6	108.2(5)		O(9)#10–B(3)–O(10)#10	109.7(5)		O(2)#11–B(7)–O(8)#12	112.3(5)
  O(10)#5–Li(1)–O(7)#6	125.5(6)		O(1)–B(3)–O(10)#10	108.9(5)		O(11)#12–B(7)–O(8)#12	108.7(5)
  O(8)–Li(1)–O(11)#6	134.9(6)		O(5)–B(4)–O(4)	122.9(5)		O(7)–B(8)–O(11)	113.8(5)
  O(10)#5–Li(1)–O(11)#6	110.2(5)		O(5)–B(4)–O(13)	117.4(5)		O(7)–B(8)–O(12)	124.2(6)
  O(7)#6–Li(1)–O(11)#6	71.8(4)		O(4)–B(4)–O(13)	119.7(5)		O(11)–B(8)–O(12)	122.0(5)
  O(9)–B(1)–O(3)	122.1(5)		O(1)–B(5)–O(4)	121.1(6)
  Symmetry transformations used to generate the equivalent atoms: #1: –x + 1, y + 1/2, –z + 1/2; #2: x, –y + 3/2, z – 1/2; #3: –x + 1, –y + 2, –z + 1; #4: –x + 1, –y + 1, –z + 1; #5: x + 1, y – 1, z; #6: x, –y + 1/2, z + 1/2; #7: x + 1, –y + 3/2, z + 1/2; #8: x, –y + 3/2, z + 1/2; #9: x – 1, y + 1, z; #10: –x + 1, y – 1/2, –z + 3/2; #11: –x + 1, y – 1/2, –z + 1/2; #12: –x + 2, y + 1/2, –z + 1/2; #13: x, –y + 1/2, z – 1/2; #14: –x + 2, y – 1/2, –z + 1/2; #15: –x + 1, y + 1/2, –z + 3/2; #16: x, –y + 5/2, z + 1/2

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