Terbium(III) hydrogendiphosphate(V) tetra­hydrate

Fejfarová, K.; Essehli, R.; El Bali, B.; Dušek, M.

doi:10.1107/S1600536808001992

inorganic compounds

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ISSN: 2056-9890

Volume 64| Part 2| February 2008| Page i15

doi:10.1107/S1600536808001992

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Terbium(III) hydrogendiphosphate(V) tetrahydrate

Karla Fejfarová,^a Rachid Essehli,^b Brahim El Bali ^b and Michal Dušek ^a ^*

^aInstitute of Physics, Na Slovance 2, 182 21 Praha 8, Czech Republic, and ^bDepartment of Chemistry, Faculty of Sciences, University Mohammed 1st, PO Box 524, 60 000 Oujda, Morocco
^*Correspondence e-mail: fejfarov@fzu.cz

(Received 19 December 2007; accepted 18 January 2008; online 23 January 2008)

The Tb atom of the title compound, TbHP₂O₇·4H₂O, is coordinated by the O atoms of three symmetrically independent water molecules and by five O atoms belonging to HP₂O₇⁻ groups. The TbO₈ polyhedra are interconnected by the diphospate anions, forming a three-dimensional network which is additionally stabilized by O—H⋯O hydrogen bonding between water molecules and O atoms of the HP₂O₇⁻ anions. Uncoordinated water molecules are situated in channels and are connected via hydrogen bonds with the framework.

Related literature

Isostructural compounds of the type REHP₂O₇·4H₂O were reported for RE = Sm by Chehimi-Moumen et al. (2002 ), for RE = Gd by Hraiech et al. (2005 ) and for RE = Eu by Anna-Rabah et al. (2006 ).

Experimental

Crystal data

TbHP₂O₇·4H₂O
M_r = 405.9
Monoclinic, P 2₁ /n
a = 6.6006 (6) Å
b = 11.4744 (9) Å
c = 11.7252 (13) Å
β = 92.150 (8)°
V = 887.42 (15) Å³
Z = 4
Mo Kα radiation
μ = 8.38 mm⁻¹
T = 295 K
0.14 × 0.06 × 0.03 mm

Data collection

Oxford Diffraction Xcalibur 2 diffractometer with Sapphire 2 CCD area-detector
Absorption correction: analytical [CrysAlis RED (Oxford Diffraction, 2005 $[Oxford Diffraction (2005). CrysAlis CCD and CrysAlis RED. Oxford Diffraction Ltd, Abingdon, Oxfordshire, England.]$ ), using a multifaceted crystal model based on expressions derived by Clark & Reid (1995 )] T_min = 0.329, T_max = 0.635
8671 measured reflections
1850 independent reflections
1628 reflections with I > 3σ(I)
R_int = 0.019

Refinement

R[F² > 2σ(F²)] = 0.011
wR(F²) = 0.034
S = 1.21
1850 reflections
155 parameters
10 restraints
H atoms treated by a mixture of independent and constrained refinement
Δρ_max = 0.29 e Å⁻³
Δρ_min = −0.21 e Å⁻³

Table 1
Selected bond lengths (Å)

Tb1—O1	2.3145 (17)
Tb1—O2ⁱ	2.2877 (17)
Tb1—O3ⁱⁱ	2.3514 (18)
Tb1—O5	2.3718 (18)
Tb1—O6ⁱⁱⁱ	2.3842 (17)
Tb1—O8	2.605 (2)
Tb1—O9	2.433 (2)
Tb1—O10	2.421 (2)
P1—O1	1.5129 (18)
P1—O2	1.5063 (18)
P1—O3	1.5169 (19)
P1—O7	1.6201 (18)
P2—O4	1.562 (2)
P2—O5	1.4957 (19)
P2—O6	1.4891 (18)
P2—O7	1.6116 (18)
Symmetry codes: (i) x+1, y, z; (ii) -x+1, -y+1, -z+2; (iii) $[-x+{\script{3\over 2}}, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ .

Table 2
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O4—H41⋯O11	0.813 (13)	1.736 (14)	2.546 (3)	174 (3)
O8—H81⋯O1ⁱⁱ	0.82 (2)	1.98 (3)	2.762 (3)	159 (3)
O8—H82⋯O3ⁱ	0.822 (18)	2.231 (14)	2.972 (3)	150 (3)
O9—H91⋯O4^iv	0.83 (3)	2.06 (3)	2.851 (3)	161 (3)
O9—H92⋯O8ⁱⁱⁱ	0.828 (18)	1.95 (2)	2.750 (3)	164 (3)
O10—H101⋯O5ⁱⁱⁱ	0.82 (2)	2.16 (3)	2.880 (3)	147 (3)
O10—H102⋯O7^v	0.812 (16)	2.28 (2)	2.991 (3)	147 (3)
O11—H111⋯O3^v	0.80 (2)	2.18 (2)	2.941 (3)	157 (3)
Symmetry codes: (i) x+1, y, z; (ii) -x+1, -y+1, -z+2; (iii) $[-x+{\script{3\over 2}}, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ ; (iv) -x+1, -y+1, -z+1; (v) $[-x+{\script{1\over 2}}, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ .

Data collection: CrysAlis CCD (Oxford Diffraction, 2005 $[Oxford Diffraction (2005). CrysAlis CCD and CrysAlis RED. Oxford Diffraction Ltd, Abingdon, Oxfordshire, England.]$ ); cell refinement: CrysAlis RED (Oxford Diffraction, 2005 $[Oxford Diffraction (2005). CrysAlis CCD and CrysAlis RED. Oxford Diffraction Ltd, Abingdon, Oxfordshire, England.]$ ); data reduction: CrysAlis RED; program(s) used to solve structure: SIR2002 (Burla et al., 2003 ); program(s) used to refine structure: JANA2000 (Petříček et al., 2007 ); molecular graphics: DIAMOND (Brandenburg & Putz, 2005 ); software used to prepare material for publication: JANA2000.

Supporting information

Crystal structure: contains datablocks global, I. DOI: 10.1107/S1600536808001992/wm2172sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536808001992/wm2172Isup2.hkl

checkCIF report

Comment top

Acidic rare earth diphosphates of general formula REHP₂O₇^.nH₂O exhibit interesting luminescent and optical properties (Hraiech et al., 2005 and references herein). The title compound is isostructural with other compounds of formula type REHP₂O₇^.4H₂O, RE = Sm (Chehimi-Moumen et al., 2002), Gd (Hraiech et al., 2005), and Eu (Anna-Rabah et al., 2006).

The structure of TbHP₂O₇^.4H₂O is made up of TbO₈ polyhedra and HP₂O₇ groups that form a three-dimensional framework. In channels running along a (Fig. 2) free water molecules are located which are connected via hydrogen bonds with the framework (see hydrogen-bonding Table).

The P₂O₇ group is protonated, with the H atom located at O4 (Fig. 1), as also indicated by elongation of the corresponding P—O distance. The bridging angle P1—O7—P2 between the two PO₄ tetrahedra is 130.73 (11)°.

Related literature top

Isostructural compounds of the type REHP₂O₇.4H₂O were reported for RE = Sm by Chehimi-Moumen et al. (2002), for RE = Gd by Hraiech et al. (2005) and for RE = Eu by Anna-Rabah et al. (2006).

Experimental top

An aqueous solution of TbCl₃^.6H₂O (0.1M) was added dropwise to anhydrous Na₄P₂O₇ dissolved in destilled water (0.1M). The pH of the mixture was controlled with diluted hydrochloric acid to be slightly acidic, and the solution was stirred for two h at room temperature. Prismatic-shaped colourless crystals with a maximal size of 0.3 mm formed after a few days on slow evaporation.

Computing details top

Data collection: CrysAlis CCD (Oxford Diffraction, 2005); cell refinement: CrysAlis RED (Oxford Diffraction, 2005); data reduction: CrysAlis RED (Oxford Diffraction, 2005); program(s) used to solve structure: SIR2002 (Burla et al., 2003); program(s) used to refine structure: JANA2000 (Petříček et al., 2007); molecular graphics: DIAMOND (Brandenburg & Putz, 2005); software used to prepare material for publication: JANA2000 (Petříček et al., 2007).

Figures top

Fig. 1. Part of the structure of HTbP₂O₇^.4H₂O drawn with displacement ellipsoids at the 50% probability level. [Symmetry codes: (i) 1 + x, y, z; (ii) 1.5 - x, -1/2 + y, 1.5 - z; (iii) 1 - x, 1 - y, 2 - z]

Fig. 2. The packing of TbHP₂O₇^.4H₂O viewed along a. Hydrogen bonds are represented by dashed lines. Colour code: Pink (P₂O₇ polyhedra), red spheres (O), grey spheres (Tb), dark grey spheres (H). All atoms are displayed with arbitrary radii. For clarity, O atoms belonging to PO₄ tetrahedra have a smaller size than O atoms of water molecules. O atoms that would obscure H atoms important for understanding the hydrogen bonding scheme are plotted semitransparently.

Terbium(III) hydrogendiphosphate(V) tetrahydrate top

Crystal data top

TbHP₂O₇·4H₂O	F(000) = 768
M_r = 405.9	D_x = 3.037 Mg m⁻³
Monoclinic, P2₁/n	Mo Kα radiation, λ = 0.71069 Å
Hall symbol: -P 2yn	Cell parameters from 972 reflections
a = 6.6006 (6) Å	θ = 2.5–26.5°
b = 11.4744 (9) Å	µ = 8.38 mm⁻¹
c = 11.7252 (13) Å	T = 295 K
β = 92.150 (8)°	Prism, colorless
V = 887.42 (15) Å³	0.14 × 0.06 × 0.03 mm
Z = 4

Data collection top

Oxford Diffraction CCD diffractometer	1850 independent reflections
Radiation source: X-ray tube	1628 reflections with I > 3σ(I)
Graphite monochromator	R_int = 0.019
Detector resolution: 8.3438 pixels mm^-1	θ_max = 26.6°, θ_min = 2.5°
ω scans	h = −8→7
Absorption correction: analytical [CrysAlis RED (Oxford Diffraction, 2005), using a multifaceted crystal model based on expressions derived by Clark & Reid (1995)]	k = −14→14
T_min = 0.329, T_max = 0.635	l = −14→14
8671 measured reflections

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F² > 2σ(F²)] = 0.011	Weighting scheme based on measured s.u.'s w = 1/[σ²(I) + 0.0004I²]
wR(F²) = 0.034	(Δ/σ)_max = 0.009
S = 1.21	Δρ_max = 0.29 e Å⁻³
1850 reflections	Δρ_min = −0.21 e Å⁻³
155 parameters	Extinction correction: B-C type 1 Lorentzian isotropic [Becker, P. J. & Coppens, P. (1974). Acta Cryst. A30, 129–147]
10 restraints	Extinction coefficient: 2.9 (8)
9 constraints

Crystal data top

TbHP₂O₇·4H₂O	V = 887.42 (15) Å³
M_r = 405.9	Z = 4
Monoclinic, P2₁/n	Mo Kα radiation
a = 6.6006 (6) Å	µ = 8.38 mm⁻¹
b = 11.4744 (9) Å	T = 295 K
c = 11.7252 (13) Å	0.14 × 0.06 × 0.03 mm
β = 92.150 (8)°

Data collection top

Oxford Diffraction CCD diffractometer	1850 independent reflections
Absorption correction: analytical [CrysAlis RED (Oxford Diffraction, 2005), using a multifaceted crystal model based on expressions derived by Clark & Reid (1995)]	1628 reflections with I > 3σ(I)
T_min = 0.329, T_max = 0.635	R_int = 0.019
8671 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.011	10 restraints
wR(F²) = 0.034	H atoms treated by a mixture of independent and constrained refinement
S = 1.21	Δρ_max = 0.29 e Å⁻³
1850 reflections	Δρ_min = −0.21 e Å⁻³
155 parameters

Special details top

Refinement. The refinement was carried out against all reflections. The conventional R-factor is always based on F. The goodness of fit as well as the weighted R-factor are based on F and F² for refinement carried out on F and F², respectively. The threshold expression is used only for calculating R-factors etc. and it is not relevant to the choice of reflections for refinement.

The program used for refinement, Jana2000, uses the weighting scheme based on the experimental expectations, see _refine_ls_weighting_details, that does not force S to be one. Therefore the values of S may be larger then the ones from the SHELX program.

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

	x	y	z	U_iso*/U_eq
Tb1	0.755557 (17)	0.417027 (10)	0.836213 (10)	0.00831 (5)
P1	0.25551 (9)	0.54805 (6)	0.85787 (5)	0.00834 (18)
P2	0.47954 (9)	0.63300 (5)	0.66266 (5)	0.00860 (18)
O1	0.4213 (3)	0.45787 (15)	0.87403 (15)	0.0122 (5)
O2	0.0629 (3)	0.50007 (16)	0.80317 (16)	0.0157 (5)
O3	0.2158 (3)	0.61576 (15)	0.96588 (15)	0.0149 (5)
O4	0.3423 (3)	0.57670 (15)	0.56618 (17)	0.0169 (6)
O5	0.6553 (3)	0.55618 (15)	0.69557 (16)	0.0137 (5)
O6	0.5259 (2)	0.75465 (15)	0.62791 (15)	0.0135 (5)
O7	0.3378 (3)	0.64547 (14)	0.77094 (14)	0.0112 (5)
O8	0.7671 (3)	0.62083 (18)	0.93419 (17)	0.0177 (6)
O9	0.8201 (4)	0.34050 (17)	0.64753 (17)	0.0252 (7)
O10	0.5535 (3)	0.24098 (19)	0.8262 (2)	0.0333 (7)
O11	0.2577 (4)	0.36300 (19)	0.5980 (2)	0.0394 (9)
H81	0.697 (4)	0.614 (3)	0.9904 (18)	0.0212*
H111	0.294 (4)	0.3021 (18)	0.572 (3)	0.0472*
H82	0.881 (2)	0.643 (3)	0.955 (2)	0.0212*
H112	0.141 (2)	0.375 (3)	0.577 (3)	0.0472*
H101	0.593 (5)	0.1757 (15)	0.810 (3)	0.04*
H91	0.776 (5)	0.380 (3)	0.593 (2)	0.0303*
H92	0.777 (4)	0.2742 (13)	0.633 (3)	0.0303*
H41	0.323 (4)	0.5072 (10)	0.574 (3)	0.0203*
H102	0.434 (2)	0.243 (3)	0.807 (3)	0.04*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Tb1	0.00694 (8)	0.00827 (8)	0.00975 (8)	0.00064 (4)	0.00087 (5)	0.00041 (4)
P1	0.0067 (3)	0.0093 (3)	0.0092 (3)	0.0008 (2)	0.0021 (2)	0.0011 (2)
P2	0.0082 (3)	0.0085 (3)	0.0092 (3)	−0.0007 (2)	0.0009 (2)	0.0011 (2)
O1	0.0097 (9)	0.0127 (8)	0.0146 (9)	0.0029 (7)	0.0037 (7)	0.0037 (7)
O2	0.0095 (9)	0.0182 (9)	0.0192 (10)	−0.0040 (7)	−0.0004 (8)	0.0001 (7)
O3	0.0201 (10)	0.0137 (8)	0.0110 (10)	0.0048 (8)	0.0034 (7)	−0.0010 (7)
O4	0.0225 (11)	0.0120 (9)	0.0157 (10)	−0.0038 (8)	−0.0041 (8)	−0.0012 (7)
O5	0.0113 (9)	0.0156 (9)	0.0143 (10)	0.0031 (7)	0.0031 (8)	0.0039 (7)
O6	0.0141 (9)	0.0107 (8)	0.0159 (9)	−0.0034 (7)	0.0016 (7)	0.0022 (7)
O7	0.0115 (9)	0.0091 (8)	0.0133 (9)	0.0003 (7)	0.0055 (7)	0.0018 (7)
O8	0.0163 (11)	0.0208 (10)	0.0160 (11)	−0.0033 (9)	0.0013 (8)	0.0035 (8)
O9	0.0432 (13)	0.0160 (10)	0.0159 (11)	0.0057 (10)	−0.0061 (9)	−0.0017 (8)
O10	0.0138 (10)	0.0163 (10)	0.0697 (17)	−0.0019 (9)	−0.0005 (11)	−0.0122 (11)
O11	0.0541 (17)	0.0120 (11)	0.0542 (16)	−0.0102 (10)	0.0317 (14)	−0.0094 (10)

Geometric parameters (Å, º) top

Tb1—O1	2.3145 (17)	P1—O1	1.5129 (18)
Tb1—O2ⁱ	2.2877 (17)	P1—O2	1.5063 (18)
Tb1—O3ⁱⁱ	2.3514 (18)	P1—O3	1.5169 (19)
Tb1—O5	2.3718 (18)	P1—O7	1.6201 (18)
Tb1—O6ⁱⁱⁱ	2.3842 (17)	P2—O4	1.562 (2)
Tb1—O8	2.605 (2)	P2—O5	1.4957 (19)
Tb1—O9	2.433 (2)	P2—O6	1.4891 (18)
Tb1—O10	2.421 (2)	P2—O7	1.6116 (18)

O1—Tb1—O2ⁱ	143.70 (6)	O6ⁱⁱⁱ—Tb1—O8	128.03 (6)
O1—Tb1—O3ⁱⁱ	83.41 (6)	O6ⁱⁱⁱ—Tb1—O9	75.70 (7)
O1—Tb1—O5	75.74 (6)	O6ⁱⁱⁱ—Tb1—O10	71.65 (6)
O1—Tb1—O6ⁱⁱⁱ	134.37 (6)	O8—Tb1—O9	136.20 (6)
O1—Tb1—O8	75.29 (6)	O8—Tb1—O10	141.01 (7)
O1—Tb1—O9	116.61 (7)	O9—Tb1—O10	76.69 (8)
O1—Tb1—O10	69.56 (7)	O1—P1—O2	113.46 (10)
O2ⁱ—Tb1—O3ⁱⁱ	101.23 (6)	O1—P1—O3	113.12 (10)
O2ⁱ—Tb1—O5	80.10 (6)	O1—P1—O7	107.03 (10)
O2ⁱ—Tb1—O6ⁱⁱⁱ	79.67 (6)	O2—P1—O3	111.92 (11)
O2ⁱ—Tb1—O8	71.90 (6)	O2—P1—O7	106.35 (10)
O2ⁱ—Tb1—O9	79.03 (7)	O3—P1—O7	104.14 (10)
O2ⁱ—Tb1—O10	146.08 (7)	O4—P2—O5	111.48 (10)
O3ⁱⁱ—Tb1—O5	143.50 (6)	O4—P2—O6	107.99 (10)
O3ⁱⁱ—Tb1—O6ⁱⁱⁱ	71.04 (6)	O4—P2—O7	105.60 (10)
O3ⁱⁱ—Tb1—O8	73.05 (6)	O5—P2—O6	117.25 (10)
O3ⁱⁱ—Tb1—O9	146.04 (7)	O5—P2—O7	108.46 (10)
O3ⁱⁱ—Tb1—O10	86.44 (8)	O6—P2—O7	105.28 (10)
O5—Tb1—O6ⁱⁱⁱ	143.13 (6)	H81—O8—H82	109 (3)
O5—Tb1—O8	72.85 (6)	H91—O9—H92	104 (3)
O5—Tb1—O9	70.39 (6)	H101—O10—H102	106 (3)
O5—Tb1—O10	112.90 (7)	H111—O11—H112	109 (3)

Symmetry codes: (i) x+1, y, z; (ii) −x+1, −y+1, −z+2; (iii) −x+3/2, y−1/2, −z+3/2.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O4—H41···O11	0.813 (13)	1.736 (14)	2.546 (3)	174 (3)
O8—H81···O1ⁱⁱ	0.82 (2)	1.98 (3)	2.762 (3)	159 (3)
O8—H82···O3ⁱ	0.822 (18)	2.231 (14)	2.972 (3)	150 (3)
O9—H91···O4^iv	0.83 (3)	2.06 (3)	2.851 (3)	161 (3)
O9—H92···O8ⁱⁱⁱ	0.828 (18)	1.95 (2)	2.750 (3)	164 (3)
O10—H101···O5ⁱⁱⁱ	0.82 (2)	2.16 (3)	2.880 (3)	147 (3)
O10—H102···O7^v	0.812 (16)	2.28 (2)	2.991 (3)	147 (3)
O11—H111···O3^v	0.80 (2)	2.18 (2)	2.941 (3)	157 (3)

Symmetry codes: (i) x+1, y, z; (ii) −x+1, −y+1, −z+2; (iii) −x+3/2, y−1/2, −z+3/2; (iv) −x+1, −y+1, −z+1; (v) −x+1/2, y−1/2, −z+3/2.

Experimental details

Crystal data
Chemical formula	TbHP₂O₇·4H₂O
M_r	405.9
Crystal system, space group	Monoclinic, P2₁/n
Temperature (K)	295
a, b, c (Å)	6.6006 (6), 11.4744 (9), 11.7252 (13)
β (°)	92.150 (8)
V (Å³)	887.42 (15)
Z	4
Radiation type	Mo Kα
µ (mm⁻¹)	8.38
Crystal size (mm)	0.14 × 0.06 × 0.03

Data collection
Diffractometer	Oxford Diffraction CCD diffractometer
Absorption correction	Analytical [CrysAlis RED (Oxford Diffraction, 2005), using a multifaceted crystal model based on expressions derived by Clark & Reid (1995)]
T_min, T_max	0.329, 0.635
No. of measured, independent and observed [I > 3σ(I)] reflections	8671, 1850, 1628
R_int	0.019
(sin θ/λ)_max (Å⁻¹)	0.629

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.011, 0.034, 1.21
No. of reflections	1850
No. of parameters	155
No. of restraints	10
H-atom treatment	H atoms treated by a mixture of independent and constrained refinement
Δρ_max, Δρ_min (e Å⁻³)	0.29, −0.21

Computer programs: CrysAlis CCD (Oxford Diffraction, 2005), CrysAlis RED (Oxford Diffraction, 2005), SIR2002 (Burla et al., 2003), JANA2000 (Petříček et al., 2007), DIAMOND (Brandenburg & Putz, 2005).

Selected bond lengths (Å) top

Tb1—O1	2.3145 (17)	P1—O1	1.5129 (18)
Tb1—O2ⁱ	2.2877 (17)	P1—O2	1.5063 (18)
Tb1—O3ⁱⁱ	2.3514 (18)	P1—O3	1.5169 (19)
Tb1—O5	2.3718 (18)	P1—O7	1.6201 (18)
Tb1—O6ⁱⁱⁱ	2.3842 (17)	P2—O4	1.562 (2)
Tb1—O8	2.605 (2)	P2—O5	1.4957 (19)
Tb1—O9	2.433 (2)	P2—O6	1.4891 (18)
Tb1—O10	2.421 (2)	P2—O7	1.6116 (18)

Symmetry codes: (i) x+1, y, z; (ii) −x+1, −y+1, −z+2; (iii) −x+3/2, y−1/2, −z+3/2.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O4—H41···O11	0.813 (13)	1.736 (14)	2.546 (3)	174 (3)
O8—H81···O1ⁱⁱ	0.82 (2)	1.98 (3)	2.762 (3)	159 (3)
O8—H82···O3ⁱ	0.822 (18)	2.231 (14)	2.972 (3)	150 (3)
O9—H91···O4^iv	0.83 (3)	2.06 (3)	2.851 (3)	161 (3)
O9—H92···O8ⁱⁱⁱ	0.828 (18)	1.95 (2)	2.750 (3)	164 (3)
O10—H101···O5ⁱⁱⁱ	0.82 (2)	2.16 (3)	2.880 (3)	147 (3)
O10—H102···O7^v	0.812 (16)	2.28 (2)	2.991 (3)	147 (3)
O11—H111···O3^v	0.80 (2)	2.18 (2)	2.941 (3)	157 (3)

Symmetry codes: (i) x+1, y, z; (ii) −x+1, −y+1, −z+2; (iii) −x+3/2, y−1/2, −z+3/2; (iv) −x+1, −y+1, −z+1; (v) −x+1/2, y−1/2, −z+3/2.

Acknowledgements

This work was supported by the Grant Agency of the Czech Republic (grant No. 202/05/0421).

References

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