The trigonal polymorph of strontium tetra­borate, β-SrB4O7

Vasiliev, A.D.; Cherepakhin, A.V.; Zaitsev, A.I.

doi:10.1107/S1600536810019069

inorganic compounds

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Volume 66| Part 6| June 2010| Page i48

https://doi.org/10.1107/S1600536810019069

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The trigonal polymorph of strontium tetraborate, β-SrB₄O₇

Alexander D. Vasiliev,^a ^* Alexander V. Cherepakhin ^a and Alexander I. Zaitsev ^a

^aInstitute of Physics SB RAS, Krasnoyarsk 660036, Russian Federation and Siberian Federal University, Svobodnyi st. 79, Krasnoyarsk 660041, Russian Federation
^*Correspondence e-mail: adva@iph.krasn.ru

(Received 6 May 2010; accepted 21 May 2010; online 29 May 2010)

The asymmetric unit of the title compound, β-SrB₄O₇, contains five Sr atoms (three located on a threefold rotation axis), twelve B and 21 O atoms. The structure is made up from BO₃ triangles and BO₄ tetrahedra in a 1:1 ratio. Pairs of BO₃ triangles are linked to BO₄ tetrahedra via common corners, forming chains. These chains are further linked to adjacent chains through corner-sharing, leading to a three-dimensional framework with channels running parallel to [001]. The Sr²⁺ ions reside in the channels and exhibit strongly distorted polyhedra The density of the β-polymorph is considerably lower than that of α-SrB₄O₇, which is constructed solely from BO₄ tetrahedra.

Related literature

For the orthorhombic α-polymorph, see: Block et al. (1964 ). For the physical properties of this phase, see: Oseledchik et al. (1995 ); Petrov et al. (2004 ); Zaitsev et al. (2006 ); Verwey et al. (1992 ); Machida et al. (1979 ); Pei et al. (2000 ). For other crystalline phases in the system SrO—B₂O₃ listed in the ICSD (2009 ), see: Ross & Angel (1991 ); Lin et al. (1999 ); Wei et al. (2001 ); Tang et al. (2008 ); Lapshin et al. (2007 ); Kim et al. (1996 ). For glass-phases in this system, see: Imaoka (1959 ); Polyakova & Litovchik (2008 ). For a review of B—O bond lengths in BO₃ and BO₄ units, see: Zobetz (1982 , 1990 ).

Experimental

Crystal data

SrB₄O₇
M_r = 242.86
Trigonal, P 3
a = 17.145 (1) Å
c = 4.2527 (5) Å
V = 1082.61 (16) Å³
Z = 9
Mo Kα radiation
μ = 11.19 mm⁻¹
T = 296 K
0.40 × 0.25 × 0.18 mm

Data collection

Bruker SMART CCD area-detector diffractometer
Absorption correction: multi-scan (SADABS; Sheldrick, 2004 ) T_min = 0.095, T_max = 0.242
10350 measured reflections
3709 independent reflections
3202 reflections with I > 2σ(I)
R_int = 0.054

Refinement

R[F² > 2σ(F²)] = 0.030
wR(F²) = 0.064
S = 0.85
3709 reflections
265 parameters
Δρ_max = 1.02 e Å⁻³
Δρ_min = −0.49 e Å⁻³
Absolute structure: Flack (1983 ), 1836 Friedel pairs
Flack parameter: −0.030 (7)

Data collection: SMART (Bruker, 2001 ); cell refinement: SAINT-Plus (Bruker, 2001); data reduction: SAINT-Plus; program(s) used to solve structure: SHELXTL (Sheldrick, 2008 ); program(s) used to refine structure: SHELXTL; molecular graphics: SHELXTL; software used to prepare material for publication: SHELXTL.

Supporting information

Crystal structure: contains datablocks global, I. DOI: https://doi.org/10.1107/S1600536810019069/wm2345sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S1600536810019069/wm2345Isup2.hkl

checkCIF report

Comment top

The orthorhombic phase of strontium tetraborate, α-SrB₄O₇ (I), is known for a long time (Block et al., 1964). This compound has attracted attention owing to its interesting physical properties, namely an unprecedented fundamental optical-absorption edge among oxide compounds (~130 nm), high non-linear optical coefficients (Oseledchik et al., 1995; Petrov et al., 2004; Zaitsev et al., 2006), good luminescent characteristics and an ability to stabilize rare-earth elements in divalent state (Verwey et al., 1992; Pei et al., 2000; Machida et al., 1979).

SrB₄O₇ falls in a glass-forming range within the SrO—B₂O₃ system and can simply be obtained as a glass (Imaoka, 1959). The process of glass re-crystallization occurs through complex mechanisms with probabilistic formation of other crystalline phases, specifically of metastable phases. Such a phase was in fact observed and designated as β-SrB₄O₇ (Polyakova & Litovchik, 2008). However, X-ray powder diffraction data of this phase and of two other new compounds described by these authors were not analysed because of impure samples.

The FIZ/NIST Inorganic Crystal Structure Database (release 2009; ICSD, 2009) reveals six phases in the SrO—B₂O₃ system besides (I): strontium diborate, (IIa), SrB₂O₄ (Kim et al., 1996), its high-pressure form (Ross & Angel, 1991), (IIb), distrontium diborate, Sr₂B₂O₅, (III), (Lin et al., 1999), tristrontium tetraborate, Sr₃B₂O₆, (IV), (Wei et al., 2001), distrontium hexadecaborate, Sr₂B₁₆O₂₆, (V), (Tang et al., 2008) and tetrastrontium tetradecaborate, Sr₄B₁₄O₂₅, (VI), (Lapshin et al., 2007). Only two of these phases crystallize in non-centrosymmetric space groups, viz. (I, Pmn2₁ and VI, Cmc2₁). The main feature of all these structures are BO_x units (x = 3,4). Isolated (IV) or flat pairs (III) of BO₃ triangles, a framework of BO₄ tetrahedra with shared vertices (I, II) and a framework of triangles and tetrahedra with shared vertices (V, VI) are found in these structures.

In the process of glass re-crystallization of a strontium tetraborate composition at 973–983 K during one day, we obtained β-SrB₄O₇ crystals with dimensions of ~200–400 µm. The crystals were located on the glass surface and were optically homogeneous (i.e. crystals showed homogeneous extinction when observed under a polarizing microscope). The crystals possess strong anisotropy when abrased, and crystals with an elongated ellipsoidal shape were obtained in such a way. As it turned out, the crystals are most firm along the c-direction.

The crystal structure of (I) is built up from a three-dimensional framework of connected boron-oxygen tetrahedra. The asymmetric unit of the title structure contains five Sr (three on special positions), twelve B and 21 O atoms. Alternatively, the structural formula of the title compound can thus be written as Sr₃B₁₂O₂₁. It consists of BO₃ triangles and BO₄ tetrahedra in an 1:1 ratio (3:1 for structure V and 3:4 for structure VI). They form a three-dimensional framework constructed via common vertices. The BO₃ triangles are linked to one another so that two of their vertices and the bridging O atom are located on a straight line (see O1, O3, O5; O6, O7, O10 and O11, O13, O15 in Fig.1). The plane of one triangle in such a pair is tilted relatively to the other one about the line with an angle of ~20°. The remaining two vertices are common with the same tetrahedron (e.g. see O2 and O4 in Fig.1). The BO₄ tetrahedra are connected to one another via common vertices and form chains along the c-direction (Fig. 2). These chains are connected with pairs of BO₃ triangles, leading to the formation of channels in the structure. The channels are filled with strontium ions (Fig. 3). The coordination polyhedra around the strontium ions are non-regular and defined by six O atoms in the range 2.479 (3)–2.786 (3) Å when a distance < 2.8 Å is considered as relevant.

All vertices in the anionic framework are shared so that every oxygen atom is connected to two boron atoms. The B—O distances fall into the interval 1.323 (6)–1.420 (6)Å (average is 1.367 (6) Å) for BO₃ triangles and into the interval 1.425 (6)–1.538 (6) Å (average is 1.474 (6) Å) for BO₄ tetrahedra. These values compare well with the mean bond lengths calculated for various borate structures (Zobetz, 1982, 1990).

In comparison with α-SrB₄O₇ which is constructued solely from BO₄ tetrahedra, the density of the β-polymorph is considerably lower.

Related literature top

For the orthorhombic α-polymorph, see: Block et al. (1964). For the physical properties of this phase, see: Oseledchik et al. (1995); Petrov et al. (2004); Zaitsev et al. (2006); Verwey et al. (1992); Machida et al. (1979); Pei et al. (2000). For other crystalline phases in the system SrO—B₂O₃ listed in the ICSD (2009), see: Ross & Angel (1991); Lin et al. (1999); Wei et al. (2001); Tang et al. (2008); Lapshin et al. (2007); Kim et al. (1996). For glass-phases in this system, see: Imaoka (1959); Polyakova & Litovchik (2008). For a review of B—O bond lengths in BO₃ and BO₄ units, see: Zobetz (1982, 1990).

Experimental top

Crystals were extracted out of glass by careful dissolving of the latter in a 2% HNO₃ solution. The initial glass has been made from a mixture of SrCO₃ (99.8%) and H₃BO₃ (99.98%) in a 1:4 ratio. The mixture was heated up to 353—363 K with addition of a small amount of water and careful mixing until CO₂ gas evolution had stopped. Then the temperature was increased slowly up to 573 K to yield a anhydrous phase. The derived mixture was then placed into a glass-carbon crucible and kept in a molten state at 1323 K during 6 h in a nitrogen atmosphere. The flux was cooled in air down to 773 K and the glass was finally annealed at 723 K during a day to remove strain.

Structure description top

In comparison with α-SrB₄O₇ which is constructued solely from BO₄ tetrahedra, the density of the β-polymorph is considerably lower.

For the orthorhombic α-polymorph, see: Block et al. (1964). For the physical properties of this phase, see: Oseledchik et al. (1995); Petrov et al. (2004); Zaitsev et al. (2006); Verwey et al. (1992); Machida et al. (1979); Pei et al. (2000). For other crystalline phases in the system SrO—B₂O₃ listed in the ICSD (2009), see: Ross & Angel (1991); Lin et al. (1999); Wei et al. (2001); Tang et al. (2008); Lapshin et al. (2007); Kim et al. (1996). For glass-phases in this system, see: Imaoka (1959); Polyakova & Litovchik (2008). For a review of B—O bond lengths in BO₃ and BO₄ units, see: Zobetz (1982, 1990).

Computing details top

Data collection: SMART (Bruker, 2001); cell refinement: SAINT-Plus (Bruker, 2001); data reduction: SAINT-Plus (Bruker, 2001); program(s) used to solve structure: SHELXTL (Sheldrick, 2008); program(s) used to refine structure: SHELXTL (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXTL (Sheldrick, 2008).

Figures top

	Fig. 1. View of the basic structural motif in the title compound with the atom numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. Symmetry codes: (i) x, y, z-1; (ii) 1-y, 1+x-y, 1+z; (iii) -y, x-y, 1+z; (iv) 1-y, x-y, 1+z.
	Fig. 2. Columns of BO₄ tetrahedra with shared vertices and attached BO₃ triangles extending along the c-direction.
	Fig. 3. View down [001] of the framework structure of the title compound showing the formation of channels that are filled with Sr²⁺ ions.

strontium tetraborate top

Crystal data top

SrB₄O₇	D_x = 3.353 (1) Mg m⁻³
M_r = 242.86	Mo Kα radiation, λ = 0.71073 Å
Trigonal, P3	Cell parameters from 2820 reflections
Hall symbol: P 3	θ = 2.4–29.3°
a = 17.145 (1) Å	µ = 11.19 mm⁻¹
c = 4.2527 (5) Å	T = 296 K
V = 1082.61 (16) Å³	Ellipsoidal, colorless
Z = 9	0.40 × 0.25 × 0.18 mm
F(000) = 1026

Data collection top

Bruker SMART CCD area-detector diffractometer	3709 independent reflections
Radiation source: fine-focus sealed tube	3202 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.054
φ and ω scans	θ_max = 28.7°, θ_min = 2.4°
Absorption correction: multi-scan (SADABS; Sheldrick, 2004)	h = −22→23
T_min = 0.095, T_max = 0.242	k = −23→23
10350 measured reflections	l = −5→5

Refinement top

Refinement on F²	Secondary atom site location: difference Fourier map
Least-squares matrix: full	w = 1/[σ²(F_o²) + (0.P)²] where P = (F_o² + 2F_c²)/3
R[F² > 2σ(F²)] = 0.030	(Δ/σ)_max < 0.001
wR(F²) = 0.064	Δρ_max = 1.02 e Å⁻³
S = 0.85	Δρ_min = −0.49 e Å⁻³
3709 reflections	Extinction correction: SHELXTL (Sheldrick, 2008), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
265 parameters	Extinction coefficient: 0.0794 (13)
0 restraints	Absolute structure: Flack (1983), 1836 Friedel pairs
Primary atom site location: structure-invariant direct methods	Absolute structure parameter: −0.030 (7)

Crystal data top

SrB₄O₇	Z = 9
M_r = 242.86	Mo Kα radiation
Trigonal, P3	µ = 11.19 mm⁻¹
a = 17.145 (1) Å	T = 296 K
c = 4.2527 (5) Å	0.40 × 0.25 × 0.18 mm
V = 1082.61 (16) Å³

Data collection top

Bruker SMART CCD area-detector diffractometer	3709 independent reflections
Absorption correction: multi-scan (SADABS; Sheldrick, 2004)	3202 reflections with I > 2σ(I)
T_min = 0.095, T_max = 0.242	R_int = 0.054
10350 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.030	0 restraints
wR(F²) = 0.064	Δρ_max = 1.02 e Å⁻³
S = 0.85	Δρ_min = −0.49 e Å⁻³
3709 reflections	Absolute structure: Flack (1983), 1836 Friedel pairs
265 parameters	Absolute structure parameter: −0.030 (7)

Special details top

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

Refinement. Refinement of F² against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F², conventional R-factors R are based on F, with F set to zero for negative F². The threshold expression of F² > σ(F²) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F² are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Sr1	0.0000	0.0000	0.5000	0.0106 (2)
Sr2	0.6667	0.3333	0.7047 (2)	0.0097 (2)
Sr3	0.3333	0.6667	0.6084 (5)	0.01172 (15)
Sr4	0.32408 (4)	0.33158 (4)	0.7001 (4)	0.00732 (14)
Sr5	−0.00569 (3)	0.33541 (4)	0.5183 (4)	0.00743 (15)
O1	0.3441 (2)	0.5031 (2)	0.5634 (9)	0.0115 (7)
O2	0.2472 (2)	0.3798 (2)	0.2476 (8)	0.0094 (7)
O3	0.2591 (2)	0.5240 (2)	0.1833 (8)	0.0108 (7)
O4	0.1239 (2)	0.3939 (2)	0.0012 (9)	0.0109 (7)
O5	0.1760 (2)	0.5441 (2)	−0.1939 (8)	0.0090 (7)
O6	0.1571 (2)	0.1803 (2)	0.4810 (9)	0.0119 (8)
O7	0.2841 (2)	0.2150 (2)	0.1685 (8)	0.0075 (6)
O8	0.1428 (2)	0.0790 (2)	0.0952 (8)	0.0109 (7)
O9	0.2767 (2)	0.0826 (2)	−0.0917 (8)	0.0083 (7)
O10	0.1278 (2)	−0.0221 (2)	−0.2863 (8)	0.0095 (7)
O11	0.5201 (2)	0.3007 (2)	0.4426 (8)	0.0082 (7)
O12	0.4709 (2)	0.4061 (2)	0.3108 (8)	0.0077 (7)
O13	0.6000 (2)	0.4102 (2)	0.0726 (8)	0.0105 (7)
O14	0.5899 (2)	0.5436 (2)	0.0455 (8)	0.0065 (6)
O15	0.6814 (2)	0.5195 (2)	−0.3106 (8)	0.0082 (7)
O16	0.0899 (2)	0.2724 (2)	0.3487 (8)	0.0084 (7)
O17	0.1632 (2)	0.2814 (2)	0.8498 (8)	0.0085 (7)
O18	0.3864 (2)	0.1604 (2)	0.2995 (8)	0.0082 (7)
O19	0.3945 (2)	0.2362 (2)	0.7992 (8)	0.0062 (6)
O20	0.5007 (2)	0.5578 (2)	0.4336 (8)	0.0073 (7)
O21	0.4320 (2)	0.4828 (2)	0.9349 (8)	0.0067 (7)
B1	0.2862 (4)	0.4681 (4)	0.3246 (14)	0.0079 (10)*
B2	0.1811 (3)	0.4849 (3)	−0.0018 (13)	0.0059 (10)*
B3	0.1948 (3)	0.1620 (3)	0.2422 (13)	0.0075 (10)*
B4	0.1853 (4)	0.0444 (3)	−0.0934 (13)	0.0071 (10)*
B5	0.5250 (3)	0.3704 (3)	0.2794 (12)	0.0052 (9)*
B6	0.6277 (3)	0.4939 (3)	−0.0547 (13)	0.0063 (10)*
B7	0.1537 (4)	0.3283 (3)	0.1104 (13)	0.0048 (10)*
B8	0.1082 (3)	0.2266 (3)	0.5981 (12)	0.0052 (10)*
B9	0.3394 (4)	0.1748 (4)	0.0454 (12)	0.0064 (11)*
B10	0.4409 (3)	0.2160 (3)	0.5607 (12)	0.0063 (10)*
B11	0.4960 (4)	0.4991 (4)	0.1768 (13)	0.0076 (12)*
B12	0.4379 (3)	0.5431 (3)	0.6829 (12)	0.0061 (10)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Sr1	0.0098 (3)	0.0098 (3)	0.0121 (5)	0.00490 (15)	0.000	0.000
Sr2	0.0093 (3)	0.0093 (3)	0.0106 (5)	0.00466 (16)	0.000	0.000
Sr3	0.0100 (2)	0.0100 (2)	0.0151 (4)	0.00502 (10)	0.000	0.000
Sr4	0.0072 (3)	0.0066 (3)	0.0083 (3)	0.0036 (2)	0.0001 (3)	0.0014 (2)
Sr5	0.0072 (3)	0.0076 (3)	0.0082 (3)	0.0043 (2)	0.0007 (2)	0.0015 (2)
O1	0.0055 (16)	0.0191 (19)	0.0082 (17)	0.0048 (15)	−0.0004 (14)	−0.0017 (14)
O2	0.0069 (16)	0.0074 (16)	0.0109 (17)	0.0014 (13)	−0.0039 (13)	0.0001 (13)
O3	0.0099 (16)	0.0065 (16)	0.0141 (18)	0.0027 (14)	−0.0075 (14)	−0.0033 (14)
O4	0.0090 (16)	0.0103 (17)	0.0153 (19)	0.0062 (14)	−0.0051 (14)	−0.0006 (14)
O5	0.0074 (16)	0.0100 (17)	0.0091 (18)	0.0039 (14)	−0.0009 (13)	0.0018 (13)
O6	0.0139 (19)	0.0164 (18)	0.0101 (18)	0.0110 (16)	0.0039 (14)	0.0042 (15)
O7	0.0080 (16)	0.0090 (16)	0.0058 (16)	0.0046 (14)	−0.0001 (13)	−0.0004 (13)
O8	0.0065 (16)	0.0099 (16)	0.0148 (19)	0.0029 (14)	0.0034 (14)	−0.0030 (14)
O9	0.0076 (16)	0.0109 (16)	0.0081 (16)	0.0059 (14)	0.0012 (13)	−0.0006 (13)
O10	0.0087 (17)	0.0060 (16)	0.0148 (19)	0.0044 (14)	−0.0046 (14)	−0.0022 (14)
O11	0.0061 (16)	0.0095 (16)	0.0093 (17)	0.0042 (14)	0.0016 (13)	0.0018 (13)
O12	0.0090 (16)	0.0090 (16)	0.0069 (16)	0.0057 (14)	0.0031 (13)	0.0025 (13)
O13	0.0104 (17)	0.0078 (16)	0.0125 (18)	0.0039 (14)	0.0087 (13)	0.0050 (13)
O14	0.0077 (15)	0.0062 (15)	0.0060 (16)	0.0039 (13)	0.0001 (13)	−0.0009 (12)
O15	0.0086 (18)	0.0114 (18)	0.0048 (17)	0.0053 (15)	0.0040 (14)	0.0046 (14)
O16	0.0069 (16)	0.0130 (17)	0.0075 (16)	0.0066 (14)	0.0036 (13)	0.0026 (13)
O17	0.0082 (16)	0.0092 (16)	0.0077 (17)	0.0041 (14)	−0.0004 (13)	−0.0023 (13)
O18	0.0081 (16)	0.0112 (16)	0.0079 (17)	0.0068 (14)	−0.0016 (13)	0.0003 (13)
O19	0.0089 (15)	0.0083 (16)	0.0043 (16)	0.0065 (13)	0.0021 (12)	0.0017 (12)
O20	0.0099 (16)	0.0099 (16)	0.0038 (15)	0.0061 (14)	0.0024 (13)	0.0000 (12)
O21	0.0057 (15)	0.0066 (15)	0.0076 (17)	0.0029 (13)	−0.0009 (12)	0.0029 (13)

Geometric parameters (Å, º) top

Sr1—O10ⁱ	2.569 (3)	Sr5—B11^xii	3.171 (6)
Sr1—O10ⁱⁱ	2.569 (3)	O1—B1	1.335 (6)
Sr1—O10ⁱⁱⁱ	2.569 (3)	O1—B12	1.486 (6)
Sr1—O8^iv	2.734 (3)	O2—B1	1.354 (6)
Sr1—O8^v	2.734 (3)	O2—B7	1.509 (6)
Sr1—O8	2.734 (3)	O2—Sr4^xiv	2.988 (3)
Sr1—O6	2.913 (4)	O3—B1	1.392 (6)
Sr1—O6^v	2.913 (4)	O3—B2	1.401 (6)
Sr1—O6^iv	2.913 (4)	O3—Sr3^xiv	3.236 (3)
Sr1—B3	3.285 (5)	O4—B2	1.366 (6)
Sr1—B3^iv	3.285 (5)	O4—B7	1.521 (6)
Sr1—B3^v	3.285 (5)	O4—Sr5^xiv	2.816 (3)
Sr2—O11^vi	2.542 (3)	O5—B2	1.340 (6)
Sr2—O11^vii	2.542 (3)	O5—B12^xv	1.489 (6)
Sr2—O11	2.542 (3)	O5—Sr3^xiv	2.594 (3)
Sr2—O13^viii	2.644 (3)	O6—B3	1.323 (6)
Sr2—O13^ix	2.644 (3)	O6—B8	1.499 (6)
Sr2—O13ⁱⁱ	2.644 (3)	O7—B3	1.370 (6)
Sr2—O15^viii	3.074 (3)	O7—B9	1.517 (6)
Sr2—O15^ix	3.074 (3)	O7—Sr4^xiv	2.657 (3)
Sr2—O15ⁱⁱ	3.074 (3)	O8—B3	1.394 (6)
Sr2—B6^viii	3.303 (5)	O8—B4	1.399 (6)
Sr2—B6^ix	3.303 (5)	O8—Sr1^xiv	3.304 (3)
Sr2—B6ⁱⁱ	3.303 (5)	O9—B4	1.364 (6)
Sr3—O5^x	2.594 (3)	O9—B9	1.516 (6)
Sr3—O5ⁱⁱ	2.594 (3)	O9—Sr5^xvi	2.676 (3)
Sr3—O5^xi	2.594 (3)	O10—B4	1.349 (6)
Sr3—O3^xii	2.786 (3)	O10—B8^xvi	1.487 (5)
Sr3—O3	2.786 (3)	O10—Sr1^xiv	2.569 (3)
Sr3—O3^xiii	2.786 (3)	O11—B5	1.348 (6)
Sr3—O1^xii	2.908 (3)	O11—B10	1.494 (6)
Sr3—O1^xiii	2.908 (3)	O12—B5	1.350 (6)
Sr3—O1	2.908 (3)	O12—B11	1.537 (6)
Sr3—O3^x	3.236 (3)	O13—B6	1.377 (6)
Sr3—O3^xi	3.236 (4)	O13—B5	1.419 (6)
Sr3—O3ⁱⁱ	3.236 (4)	O13—Sr2^xiv	2.644 (3)
Sr4—O19	2.507 (3)	O14—B6	1.371 (6)
Sr4—O21	2.519 (3)	O14—B11	1.503 (6)
Sr4—O17	2.526 (3)	O14—Sr5^xiii	2.657 (3)
Sr4—O7ⁱⁱ	2.657 (3)	O14—Sr5^xvii	2.836 (3)
Sr4—O2	2.685 (3)	O15—B6	1.349 (6)
Sr4—O12	2.737 (3)	O15—B10^xviii	1.501 (6)
Sr4—O1	2.845 (4)	O15—Sr5^xvii	2.649 (3)
Sr4—O7	2.864 (3)	O15—Sr2^xiv	3.074 (3)
Sr4—O6	2.893 (4)	O16—B8	1.445 (6)
Sr4—O2ⁱⁱ	2.988 (3)	O16—B7	1.447 (6)
Sr4—B12	3.145 (5)	O17—B8	1.425 (6)
Sr4—B1	3.158 (5)	O17—B7ⁱⁱ	1.426 (6)
Sr5—O16	2.479 (3)	O18—B9	1.441 (6)
Sr5—O20^xii	2.482 (3)	O18—B10	1.458 (6)
Sr5—O18^v	2.539 (3)	O18—Sr5^iv	2.539 (3)
Sr5—O15^x	2.649 (3)	O19—B10	1.434 (6)
Sr5—O14^xii	2.657 (3)	O19—B9ⁱⁱ	1.450 (5)
Sr5—O9ⁱ	2.676 (3)	O20—B12	1.440 (6)
Sr5—O4ⁱⁱ	2.816 (3)	O20—B11	1.460 (6)
Sr5—O14^x	2.836 (3)	O20—Sr5^xiii	2.482 (3)
Sr5—O4	2.924 (4)	O21—B11ⁱⁱ	1.425 (6)
Sr5—B10^v	3.129 (5)	O21—B12	1.458 (6)
Sr5—B6^x	3.149 (5)

O10ⁱ—Sr1—O10ⁱⁱ	108.20 (8)	O7—Sr4—O2ⁱⁱ	145.41 (8)
O10ⁱ—Sr1—O10ⁱⁱⁱ	108.20 (8)	O6—Sr4—O2ⁱⁱ	97.10 (9)
O10ⁱⁱ—Sr1—O10ⁱⁱⁱ	108.20 (8)	O16—Sr5—O20^xii	116.34 (10)
O10ⁱ—Sr1—O8^iv	155.68 (11)	O16—Sr5—O18^v	104.21 (10)
O10ⁱⁱ—Sr1—O8^iv	94.57 (10)	O20^xii—Sr5—O18^v	119.04 (10)
O10ⁱⁱⁱ—Sr1—O8^iv	71.17 (10)	O16—Sr5—O15^x	152.34 (10)
O10ⁱ—Sr1—O8^v	71.17 (10)	O20^xii—Sr5—O15^x	90.34 (10)
O10ⁱⁱ—Sr1—O8^v	155.68 (11)	O18^v—Sr5—O15^x	52.72 (10)
O10ⁱⁱⁱ—Sr1—O8^v	94.57 (10)	O16—Sr5—O14^xii	112.32 (11)
O8^iv—Sr1—O8^v	84.57 (11)	O20^xii—Sr5—O14^xii	54.17 (10)
O10ⁱ—Sr1—O8	94.57 (10)	O18^v—Sr5—O14^xii	69.52 (10)
O10ⁱⁱ—Sr1—O8	71.17 (10)	O15^x—Sr5—O14^xii	76.45 (10)
O10ⁱⁱⁱ—Sr1—O8	155.68 (11)	O16—Sr5—O9ⁱ	85.40 (10)
O8^iv—Sr1—O8	84.57 (11)	O20^xii—Sr5—O9ⁱ	148.56 (12)
O8^v—Sr1—O8	84.57 (11)	O18^v—Sr5—O9ⁱ	72.45 (10)
O10ⁱ—Sr1—O6	49.69 (9)	O15^x—Sr5—O9ⁱ	73.60 (10)
O10ⁱⁱ—Sr1—O6	76.23 (9)	O14^xii—Sr5—O9ⁱ	140.89 (9)
O10ⁱⁱⁱ—Sr1—O6	156.14 (11)	O16—Sr5—O4ⁱⁱ	77.96 (10)
O8^iv—Sr1—O6	132.54 (10)	O20^xii—Sr5—O4ⁱⁱ	83.34 (10)
O8^v—Sr1—O6	86.46 (9)	O18^v—Sr5—O4ⁱⁱ	151.29 (11)
O8—Sr1—O6	48.18 (9)	O15^x—Sr5—O4ⁱⁱ	114.40 (11)
O10ⁱ—Sr1—O6^v	76.23 (9)	O14^xii—Sr5—O4ⁱⁱ	136.93 (9)
O10ⁱⁱ—Sr1—O6^v	156.14 (11)	O9ⁱ—Sr5—O4ⁱⁱ	79.29 (10)
O10ⁱⁱⁱ—Sr1—O6^v	49.69 (9)	O16—Sr5—O14^x	142.36 (10)
O8^iv—Sr1—O6^v	86.46 (9)	O20^xii—Sr5—O14^x	70.68 (10)
O8^v—Sr1—O6^v	48.18 (9)	O18^v—Sr5—O14^x	102.82 (9)
O8—Sr1—O6^v	132.54 (10)	O15^x—Sr5—O14^x	50.85 (9)
O6—Sr1—O6^v	119.923 (7)	O14^xii—Sr5—O14^x	101.42 (10)
O10ⁱ—Sr1—O6^iv	156.14 (11)	O9ⁱ—Sr5—O14^x	78.32 (9)
O10ⁱⁱ—Sr1—O6^iv	49.69 (9)	O4ⁱⁱ—Sr5—O14^x	65.84 (10)
O10ⁱⁱⁱ—Sr1—O6^iv	76.23 (9)	O16—Sr5—O4	51.39 (10)
O8^iv—Sr1—O6^iv	48.18 (9)	O20^xii—Sr5—O4	71.17 (10)
O8^v—Sr1—O6^iv	132.54 (10)	O18^v—Sr5—O4	108.16 (10)
O8—Sr1—O6^iv	86.46 (9)	O15^x—Sr5—O4	142.90 (10)
O6—Sr1—O6^iv	119.923 (8)	O14^xii—Sr5—O4	66.60 (9)
O6^v—Sr1—O6^iv	119.923 (8)	O9ⁱ—Sr5—O4	136.26 (9)
O11^vi—Sr2—O11^vii	102.21 (9)	O4ⁱⁱ—Sr5—O4	95.60 (10)
O11^vi—Sr2—O11	102.21 (9)	O14^x—Sr5—O4	139.09 (9)
O11^vii—Sr2—O11	102.21 (9)	B1—O1—B12	149.0 (4)
O11^vi—Sr2—O13^viii	94.41 (10)	B1—O1—Sr4	90.7 (3)
O11^vii—Sr2—O13^viii	75.13 (10)	B12—O1—Sr4	87.2 (2)
O11—Sr2—O13^viii	163.33 (10)	B1—O1—Sr3	95.2 (3)
O11^vi—Sr2—O13^ix	75.13 (10)	B12—O1—Sr3	96.2 (3)
O11^vii—Sr2—O13^ix	163.33 (10)	Sr4—O1—Sr3	161.90 (14)
O11—Sr2—O13^ix	94.41 (10)	B1—O2—B7	122.2 (4)
O13^viii—Sr2—O13^ix	88.58 (11)	B1—O2—Sr4	97.3 (3)
O11^vi—Sr2—O13ⁱⁱ	163.33 (10)	B7—O2—Sr4	129.7 (3)
O11^vii—Sr2—O13ⁱⁱ	94.41 (10)	B1—O2—Sr4^xiv	117.3 (3)
O11—Sr2—O13ⁱⁱ	75.13 (10)	B7—O2—Sr4^xiv	91.4 (3)
O13^viii—Sr2—O13ⁱⁱ	88.58 (11)	Sr4—O2—Sr4^xiv	96.96 (10)
O13^ix—Sr2—O13ⁱⁱ	88.58 (11)	B1—O3—B2	118.8 (4)
O11^vi—Sr2—O15^viii	48.77 (9)	B1—O3—Sr3	99.3 (3)
O11^vii—Sr2—O15^viii	77.18 (10)	B2—O3—Sr3	133.8 (3)
O11—Sr2—O15^viii	148.54 (9)	B1—O3—Sr3^xiv	134.9 (3)
O13^viii—Sr2—O15^viii	47.70 (9)	B2—O3—Sr3^xiv	81.8 (3)
O13^ix—Sr2—O15^viii	89.33 (9)	Sr3—O3—Sr3^xiv	89.54 (8)
O13ⁱⁱ—Sr2—O15^viii	136.27 (10)	B2—O4—B7	122.2 (4)
O11^vi—Sr2—O15^ix	77.18 (10)	B2—O4—Sr5^xiv	112.8 (3)
O11^vii—Sr2—O15^ix	148.54 (9)	B7—O4—Sr5^xiv	117.1 (3)
O11—Sr2—O15^ix	48.77 (9)	B2—O4—Sr5	112.8 (3)
O13^viii—Sr2—O15^ix	136.27 (10)	B7—O4—Sr5	89.7 (2)
O13^ix—Sr2—O15^ix	47.70 (9)	Sr5^xiv—O4—Sr5	95.60 (10)
O13ⁱⁱ—Sr2—O15^ix	89.33 (9)	B2—O5—B12^xv	137.7 (4)
O15^viii—Sr2—O15^ix	119.955 (5)	B2—O5—Sr3^xiv	112.1 (3)
O11^vi—Sr2—O15ⁱⁱ	148.54 (9)	B12^xv—O5—Sr3^xiv	110.1 (2)
O11^vii—Sr2—O15ⁱⁱ	48.77 (9)	B3—O6—B8	148.2 (4)
O11—Sr2—O15ⁱⁱ	77.18 (10)	B3—O6—Sr4	94.5 (3)
O13^viii—Sr2—O15ⁱⁱ	89.33 (9)	B8—O6—Sr4	89.0 (2)
O13^ix—Sr2—O15ⁱⁱ	136.27 (10)	B3—O6—Sr1	94.1 (3)
O13ⁱⁱ—Sr2—O15ⁱⁱ	47.70 (9)	B8—O6—Sr1	95.1 (2)
O15^viii—Sr2—O15ⁱⁱ	119.955 (6)	Sr4—O6—Sr1	156.60 (14)
O15^ix—Sr2—O15ⁱⁱ	119.955 (5)	B3—O7—B9	121.5 (4)
O5^x—Sr3—O5ⁱⁱ	110.02 (8)	B3—O7—Sr4^xiv	116.9 (3)
O5^x—Sr3—O5^xi	110.02 (8)	B9—O7—Sr4^xiv	95.6 (2)
O5ⁱⁱ—Sr3—O5^xi	110.02 (8)	B3—O7—Sr4	94.6 (3)
O5^x—Sr3—O3^xii	70.47 (10)	B9—O7—Sr4	127.1 (3)
O5ⁱⁱ—Sr3—O3^xii	94.31 (9)	Sr4^xiv—O7—Sr4	100.68 (10)
O5^xi—Sr3—O3^xii	152.90 (11)	B3—O8—B4	119.2 (4)
O5^x—Sr3—O3	152.90 (11)	B3—O8—Sr1	100.4 (3)
O5ⁱⁱ—Sr3—O3	70.47 (10)	B4—O8—Sr1	132.7 (3)
O5^xi—Sr3—O3	94.31 (9)	B3—O8—Sr1^xiv	136.6 (3)
O3^xii—Sr3—O3	82.43 (11)	B4—O8—Sr1^xiv	79.8 (3)
O5^x—Sr3—O3^xiii	94.31 (9)	Sr1—O8—Sr1^xiv	89.02 (8)
O5ⁱⁱ—Sr3—O3^xiii	152.90 (11)	B4—O9—B9	123.2 (4)
O5^xi—Sr3—O3^xiii	70.47 (10)	B4—O9—Sr5^xvi	114.8 (3)
O3^xii—Sr3—O3^xiii	82.43 (11)	B9—O9—Sr5^xvi	120.0 (2)
O3—Sr3—O3^xiii	82.43 (11)	B4—O10—B8^xvi	135.4 (4)
O5^x—Sr3—O1^xii	77.41 (9)	B4—O10—Sr1^xiv	114.0 (3)
O5ⁱⁱ—Sr3—O1^xii	49.47 (9)	B8^xvi—O10—Sr1^xiv	110.6 (3)
O5^xi—Sr3—O1^xii	158.70 (11)	B5—O11—B10	131.2 (4)
O3^xii—Sr3—O1^xii	48.06 (9)	B5—O11—Sr2	114.5 (3)
O3—Sr3—O1^xii	84.14 (10)	B10—O11—Sr2	112.6 (3)
O3^xiii—Sr3—O1^xii	129.98 (11)	B5—O12—B11	122.3 (4)
O5^x—Sr3—O1^xiii	49.47 (9)	B5—O12—Sr4	121.9 (3)
O5ⁱⁱ—Sr3—O1^xiii	158.70 (10)	B11—O12—Sr4	115.1 (3)
O5^xi—Sr3—O1^xiii	77.41 (9)	B6—O13—B5	120.1 (4)
O3^xii—Sr3—O1^xiii	84.14 (10)	B6—O13—Sr2^xiv	106.1 (3)
O3—Sr3—O1^xiii	129.98 (11)	B5—O13—Sr2^xiv	129.8 (3)
O3^xiii—Sr3—O1^xiii	48.06 (9)	B6—O14—B11	121.3 (4)
O1^xii—Sr3—O1^xiii	119.572 (18)	B6—O14—Sr5^xiii	119.7 (3)
O5^x—Sr3—O1	158.70 (11)	B11—O14—Sr5^xiii	95.3 (3)
O5ⁱⁱ—Sr3—O1	77.41 (9)	B6—O14—Sr5^xvii	89.9 (3)
O5^xi—Sr3—O1	49.47 (9)	B11—O14—Sr5^xvii	130.0 (3)
O3^xii—Sr3—O1	129.98 (11)	Sr5^xiii—O14—Sr5^xvii	101.42 (10)
O3—Sr3—O1	48.06 (9)	B6—O15—B10^xviii	147.5 (4)
O3^xiii—Sr3—O1	84.14 (9)	B6—O15—Sr5^xvii	98.7 (3)
O1^xii—Sr3—O1	119.572 (19)	B10^xviii—O15—Sr5^xvii	93.8 (2)
O1^xiii—Sr3—O1	119.572 (19)	B6—O15—Sr2^xiv	87.5 (3)
O5^x—Sr3—O3^x	44.45 (9)	B10^xviii—O15—Sr2^xiv	89.5 (2)
O5ⁱⁱ—Sr3—O3^x	68.82 (9)	Sr5^xvii—O15—Sr2^xiv	162.45 (13)
O5^xi—Sr3—O3^x	109.89 (10)	B8—O16—B7	125.3 (4)
O3^xii—Sr3—O3^x	89.54 (8)	B8—O16—Sr5	113.2 (3)
O3—Sr3—O3^x	137.69 (4)	B7—O16—Sr5	111.1 (3)
O3^xiii—Sr3—O3^x	137.69 (4)	B8—O17—B7ⁱⁱ	137.0 (4)
O1^xii—Sr3—O3^x	60.73 (9)	B8—O17—Sr4	106.5 (3)
O1^xiii—Sr3—O3^x	89.91 (9)	B7ⁱⁱ—O17—Sr4	114.7 (3)
O1—Sr3—O3^x	129.76 (10)	B9—O18—B10	133.6 (4)
O5^x—Sr3—O3^xi	68.82 (9)	B9—O18—Sr5^iv	123.3 (3)
O5ⁱⁱ—Sr3—O3^xi	109.89 (10)	B10—O18—Sr5^iv	99.5 (3)
O5^xi—Sr3—O3^xi	44.45 (9)	B10—O19—B9ⁱⁱ	125.1 (4)
O3^xii—Sr3—O3^xi	137.69 (4)	B10—O19—Sr4	123.0 (3)
O3—Sr3—O3^xi	137.69 (4)	B9ⁱⁱ—O19—Sr4	104.1 (3)
O3^xiii—Sr3—O3^xi	89.54 (8)	B12—O20—B11	130.6 (4)
O1^xii—Sr3—O3^xi	129.76 (10)	B12—O20—Sr5^xiii	120.1 (3)
O1^xiii—Sr3—O3^xi	60.73 (9)	B11—O20—Sr5^xiii	104.0 (3)
O1—Sr3—O3^xi	89.91 (9)	B11ⁱⁱ—O21—B12	128.2 (4)
O3^x—Sr3—O3^xi	69.12 (9)	O1—B1—O2	121.8 (5)
O5^x—Sr3—O3ⁱⁱ	109.89 (10)	O1—B1—O3	116.7 (4)
O5ⁱⁱ—Sr3—O3ⁱⁱ	44.45 (9)	O2—B1—O3	121.1 (4)
O5^xi—Sr3—O3ⁱⁱ	68.82 (9)	O5—B2—O4	126.2 (4)
O3^xii—Sr3—O3ⁱⁱ	137.69 (4)	O5—B2—O3	112.9 (4)
O3—Sr3—O3ⁱⁱ	89.54 (8)	O4—B2—O3	120.6 (4)
O3^xiii—Sr3—O3ⁱⁱ	137.69 (4)	O6—B3—O7	122.4 (4)
O1^xii—Sr3—O3ⁱⁱ	89.91 (9)	O6—B3—O8	116.6 (4)
O1^xiii—Sr3—O3ⁱⁱ	129.76 (10)	O7—B3—O8	120.5 (4)
O1—Sr3—O3ⁱⁱ	60.73 (9)	O10—B4—O9	125.9 (4)
O3^x—Sr3—O3ⁱⁱ	69.12 (9)	O10—B4—O8	113.1 (4)
O3^xi—Sr3—O3ⁱⁱ	69.12 (9)	O9—B4—O8	120.7 (4)
O19—Sr4—O21	105.01 (10)	O11—B5—O12	126.8 (4)
O19—Sr4—O17	122.38 (10)	O11—B5—O13	112.6 (4)
O21—Sr4—O17	111.46 (10)	O12—B5—O13	120.4 (4)
O19—Sr4—O7ⁱⁱ	53.90 (9)	O15—B6—O14	120.4 (4)
O21—Sr4—O7ⁱⁱ	104.90 (10)	O15—B6—O13	118.6 (4)
O17—Sr4—O7ⁱⁱ	74.20 (10)	O14—B6—O13	120.1 (4)
O19—Sr4—O2	142.35 (11)	O17^xiv—B7—O16	115.6 (4)
O21—Sr4—O2	100.85 (10)	O17^xiv—B7—O2	104.0 (4)
O17—Sr4—O2	70.44 (10)	O16—B7—O2	110.3 (4)
O7ⁱⁱ—Sr4—O2	141.90 (9)	O17^xiv—B7—O4	110.8 (4)
O19—Sr4—O12	77.05 (9)	O16—B7—O4	106.4 (4)
O21—Sr4—O12	74.24 (10)	O2—B7—O4	109.7 (4)
O17—Sr4—O12	154.69 (11)	O17—B8—O16	116.4 (4)
O7ⁱⁱ—Sr4—O12	129.48 (9)	O17—B8—O10ⁱ	111.8 (4)
O2—Sr4—O12	84.31 (10)	O16—B8—O10ⁱ	109.5 (4)
O19—Sr4—O1	149.31 (10)	O17—B8—O6	103.8 (4)
O21—Sr4—O1	50.97 (9)	O16—B8—O6	112.0 (4)
O17—Sr4—O1	87.17 (10)	O10ⁱ—B8—O6	102.2 (3)
O7ⁱⁱ—Sr4—O1	141.53 (10)	O18—B9—O19^xiv	116.7 (4)
O2—Sr4—O1	50.18 (10)	O18—B9—O9	105.9 (4)
O12—Sr4—O1	77.65 (9)	O19^xiv—B9—O9	110.2 (4)
O19—Sr4—O7	72.72 (9)	O18—B9—O7	110.4 (4)
O21—Sr4—O7	145.82 (10)	O19^xiv—B9—O7	104.4 (3)
O17—Sr4—O7	96.98 (10)	O9—B9—O7	109.2 (4)
O7ⁱⁱ—Sr4—O7	100.68 (10)	O19—B10—O18	116.4 (4)
O2—Sr4—O7	70.49 (10)	O19—B10—O11	110.3 (4)
O12—Sr4—O7	72.05 (9)	O18—B10—O11	110.7 (4)
O1—Sr4—O7	115.02 (10)	O19—B10—O15^ix	111.6 (4)
O19—Sr4—O6	93.03 (10)	O18—B10—O15^ix	102.4 (3)
O21—Sr4—O6	159.89 (10)	O11—B10—O15^ix	104.5 (3)
O17—Sr4—O6	49.69 (10)	O21^xiv—B11—O20	116.5 (4)
O7ⁱⁱ—Sr4—O6	78.44 (10)	O21^xiv—B11—O14	110.8 (4)
O2—Sr4—O6	68.12 (10)	O20—B11—O14	104.6 (4)
O12—Sr4—O6	119.38 (10)	O21^xiv—B11—O12	106.0 (4)
O1—Sr4—O6	114.52 (9)	O20—B11—O12	109.2 (4)
O7—Sr4—O6	48.38 (9)	O14—B11—O12	109.7 (4)
O19—Sr4—O2ⁱⁱ	118.18 (10)	O20—B12—O21	116.5 (4)
O21—Sr4—O2ⁱⁱ	66.73 (10)	O20—B12—O1	111.7 (4)
O17—Sr4—O2ⁱⁱ	48.74 (9)	O21—B12—O1	104.4 (4)
O7ⁱⁱ—Sr4—O2ⁱⁱ	68.92 (9)	O20—B12—O5^xi	109.0 (4)
O2—Sr4—O2ⁱⁱ	96.96 (10)	O21—B12—O5^xi	111.8 (4)
O12—Sr4—O2ⁱⁱ	140.51 (9)	O1—B12—O5^xi	102.5 (3)
O1—Sr4—O2ⁱⁱ	73.46 (9)

Symmetry codes: (i) −y, x−y, z+1; (ii) x, y, z+1; (iii) −x+y, −x, z+1; (iv) −x+y, −x, z; (v) −y, x−y, z; (vi) −y+1, x−y, z; (vii) −x+y+1, −x+1, z; (viii) −x+y+1, −x+1, z+1; (ix) −y+1, x−y, z+1; (x) −x+y, −x+1, z+1; (xi) −y+1, x−y+1, z+1; (xii) −x+y, −x+1, z; (xiii) −y+1, x−y+1, z; (xiv) x, y, z−1; (xv) −x+y, −x+1, z−1; (xvi) −x+y, −x, z−1; (xvii) −y+1, x−y+1, z−1; (xviii) −x+y+1, −x+1, z−1.

Experimental details

Crystal data
Chemical formula	SrB₄O₇
M_r	242.86
Crystal system, space group	Trigonal, P3
Temperature (K)	296
a, c (Å)	17.145 (1), 4.2527 (5)
V (Å³)	1082.61 (16)
Z	9
Radiation type	Mo Kα
µ (mm⁻¹)	11.19
Crystal size (mm)	0.40 × 0.25 × 0.18

Data collection
Diffractometer	Bruker SMART CCD area-detector
Absorption correction	Multi-scan (SADABS; Sheldrick, 2004)
T_min, T_max	0.095, 0.242
No. of measured, independent and observed [I > 2σ(I)] reflections	10350, 3709, 3202
R_int	0.054
(sin θ/λ)_max (Å⁻¹)	0.676

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.030, 0.064, 0.85
No. of reflections	3709
No. of parameters	265
Δρ_max, Δρ_min (e Å⁻³)	1.02, −0.49
Absolute structure	Flack (1983), 1836 Friedel pairs
Absolute structure parameter	−0.030 (7)

Computer programs: SMART (Bruker, 2001), SAINT-Plus (Bruker, 2001), SHELXTL (Sheldrick, 2008).

Acknowledgements

We thank the State Program for Support of Leading Scientific Schools (grant LS-4645.2010.2.)

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