Redetermination of AgPO3

Terebilenko, K.V.; Zatovsky, I.V.; Ogorodnyk, I.V.; Baumer, V.N.; Slobodyanik, N.S.

doi:10.1107/S1600536811003977

inorganic compounds

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ISSN: 2056-9890

Volume 67| Part 3| March 2011| Page i22

doi:10.1107/S1600536811003977

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Redetermination of AgPO₃

Katherina V. Terebilenko,^a ^* Igor V. Zatovsky,^a Ivan V. Ogorodnyk,^a Vyacheslav N. Baumer ^b and Nikolay S. Slobodyanik ^a

^aDepartment of Inorganic Chemistry, Taras Shevchenko National University, 64 Volodymyrska Street, 01601 Kyiv, Ukraine, and ^bSTC `Institute for Single Crystals', NAS of Ukraine, 60 Lenin Avenue, 61001 Kharkiv, Ukraine
^*Correspondence e-mail: tereb@bigmir.ru

(Received 19 January 2011; accepted 31 January 2011; online 9 February 2011)

Single crystals of silver(I) polyphosphate(V), AgPO₃, were prepared via a phosphoric acid melt method using a solution of Ag₃PO₄ in H₃PO₄. In comparison with the previous study based on single-crystal Weissenberg photographs [Jost (1961 ). Acta Cryst. 14, 779–784], the results were mainly confirmed, but with much higher precision and with all displacement parameters refined anisotropically. The structure is built up from two types of distorted edge- and corner-sharing [AgO₅] polyhedra, giving rise to multidirectional ribbons, and from two types of PO₄ tetrahedra linked into meandering chains (PO₃)_n spreading parallel to the b axis with a repeat unit of four tetrahedra. The calculated bond-valence sum value of one of the two Ag^I ions indicates a significant strain of the structure.

Related literature

For a previous crystallographic study of AgPO₃, see: Jost (1961). For the isotypic A-form of the Kurrol salt NaPO₃, see: McAdam et al. (1968 ). Properties of glassy silver phosphates have been reported by Portier et al. (1990 ) and Novita et al. (2009 ). For long-chain polyphosphates AgM^III(PO₃)₄ (M^III = La, Gd, Eu), see: El Masloumi et al. (2005 ); Naıli et al. (2006 ); Ayadi et al. (2009 ). For AgM^II (PO₃)₃ (M^II = Mg, Zn, Ba), see: Belharouak et al. (1999 ); for AgM^I(PO₃)₂ (M^I = K, Rb, Cs, Tl), see: Averbuch-Pouchot (1993 ). For background to the bond-valence method, see: Brown & Altermatt (1985 ).

Experimental

Crystal data

AgPO₃
M_r = 186.84
Monoclinic, P 2₁ /n
a = 11.9335 (3) Å
b = 6.0667 (1) Å
c = 7.3278 (2) Å
β = 93.491 (2)°
V = 529.53 (2) Å³
Z = 8
Mo Kα radiation
μ = 7.96 mm⁻¹
T = 293 K
0.10 × 0.08 × 0.04 mm

Data collection

Oxford Diffraction Xcalibur-3 CCD diffractometer
Absorption correction: multi-scan (Blessing, 1995 ) T_min = 0.465, T_max = 0.733
22720 measured reflections
2333 independent reflections
2208 reflections with I > 2σ(I)
R_int = 0.042

Refinement

R[F² > 2σ(F²)] = 0.024
wR(F²) = 0.058
S = 1.08
2333 reflections
92 parameters
Δρ_max = 1.43 e Å⁻³
Δρ_min = −1.86 e Å⁻³

Table 1
Selected geometric parameters (Å, °)

Ag1—O3ⁱ	2.441 (2)
Ag1—O1	2.460 (2)
Ag1—O6ⁱⁱ	2.491 (2)
Ag1—O1ⁱⁱⁱ	2.511 (2)
Ag1—O6	2.540 (2)
Ag2—O5^iv	2.3708 (19)
Ag2—O5^v	2.3756 (19)
Ag2—O3	2.3968 (19)
Ag2—O6ⁱⁱ	2.487 (2)
Ag2—O3^iv	2.750 (2)
P1—O1	1.490 (2)
P1—O3	1.4952 (19)
P1—O4^vi	1.5889 (17)
P1—O2	1.6033 (17)
P2—O5	1.479 (2)
P2—O6	1.4924 (19)
P2—O4	1.5909 (17)
P2—O2	1.6074 (18)

P1—O2—P2	124.88 (11)
P1^vii—O4—P2	135.91 (11)
Symmetry codes: (i) x, y-1, z; (ii) -x, -y+1, -z+1; (iii) $[-x+{\script{1\over 2}}, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ ; (iv) -x, -y+2, -z+1; (v) x, y, z+1; (vi) $[-x+{\script{1\over 2}}, y+{\script{1\over 2}}, -z+{\script{1\over 2}}]$ ; (vii) $[-x+{\script{1\over 2}}, y-{\script{1\over 2}}, -z+{\script{1\over 2}}]$ .

Data collection: CrysAlis CCD (Oxford Diffraction, 2006 $[Oxford Diffraction (2006). CrysAlis CCD and CrysAlis RED. Oxford Diffraction Ltd, Abingdon, England.]$ ); cell refinement: CrysAlis CCD; data reduction: CrysAlis RED (Oxford Diffraction, 2006 $[Oxford Diffraction (2006). CrysAlis CCD and CrysAlis RED. Oxford Diffraction Ltd, Abingdon, England.]$ ); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 1999 ); software used to prepare material for publication: WinGX (Farrugia, 1999 ) and enCIFer (Allen et al., 2004 ).

Supporting information

Crystal structure: contains datablocks global, I. DOI: 10.1107/S1600536811003977/wm2454sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536811003977/wm2454Isup2.hkl

checkCIF report

Comment top

Glassy silver polyphosphates as a part of complex oxide-chalcogenide systems have various applications in the field of solid electrolytes (Portier et al., 1990; Novita et al., 2009). Much attention has therefore been paid to phase equilibrium studies within AgPO₃–M(PO₃)_n systems, where M is a rare earth (Ayadi et al., 2009; Naili et al., 2006, El Masloumi et al., 2005), a divalent (Belharouak et al., 1999) or a monovalent metal (Averbuch-Pouchot, 1993).

The title compound is isotypic with the A-form of the Kurrol salt NaPO₃ (McAdam et al., 1968). AgPO₃ has been previously structurally studied based on single crystal Weissenberg photographs (Jost, 1961). The current study mainly confirms the previous results, but with significantly higher precision and with anisotropic displacement parameters refined for all atoms.

The structure of AgPO₃ features two types of penta-coordinated Ag^I ions: Ag1 is located in a distorted trigonal bipyramid and Ag2 in a irregular tetragonal pyramid of oxygen atoms (Fig. 1). Calculated values of bond valence sums (BVS; Brown & Altermatt, 1985) were found to be quite different: 0.96 valence units (v.u.) for Ag2 and 0.87 v.u. for Ag1. The observed deviation of the BVS of the latter atom from its chemical valence (expected 1) is due to a high polyhedral distortion caused by the presence of rigid (PO₃)_n chains. In this case the BVS values may be seen as a degree of silver "underbonding" and have a concomitant effect on physical properties of the title compound or glassy materials containing silver polyphosphate. The next-nearest coordination spheres of Ag1 include six adjacent silver atoms, resulting in edge-sharing Ag1 and Ag2 polyhedra and other four polyhedra connected through corners. As a result of this linkage, two polyhedral ribbons appear. One spreads parallel to the a-axis through interconnected [Ag1O₅] polyhedra by sharing a common edge and vertex alternatively, and another spreads parallel to the b-axis and consist of corner-sharing [Ag1O₅] polyhedra. Ag2 is remotely surrounded by one Ag1 and two Ag2, resulting in a ribbon of edge-sharing [Ag2O₅] polyhedra that run parallel to the c-axis. Intersecting these ribbons leads to a three-dimensional network penetrated with tunnels having eight-sided windows. Helical chains (PO₃)_n with a repeating unit of four phosphate tetrahedra are located within the tunnels and spiral along the 2₁ axes parallel to the b-axis. Due to the centrosymmetric nature of the structure, adjacent chains are left- and right-helices. As is characteristic for catena-polyphosphates (Averbuch-Pouchot, 1993), each of the two PO₄ tetrahedra displays two types of P—O bond lengths: P—O terminal ranging from 1.479 (2) to 1.4952 (19) Å and P—O bridging from 1.5889 (17) to 1.6074 (18) Å. The corresponding BVS are 4.92 v.u. and 4.95 v.u. for P1 and P2, respectively. The BVS per doubled formula (corresponds to the crystallographically independent atoms in a cell) of positively charged atoms equals to 11.72 v.u., while the chemical charge of the remaining O atoms is equal to -12 valences. This difference is an additional indication of the strain in the structure caused by the presence of rigid phosphate chains.

Related literature top

For a previous crystallographic study of AgPO₃, see: Jost (1961). For the isotypic A-form of the Kurrol salt NaPO₃, see: McAdam et al. (1968). Properties of glassy silver phosphates have been reported by Portier et al. (1990) and Novita et al. (2009). For long-chain polyphosphates AgM^III(PO₃)₄ (M^III = La, Gd, Eu), see: El Masloumi et al. (2005); Naıli et al. (2006); Ayadi et al. (2009). For AgM^II (PO₃)₃ (M^II = Mg, Zn, Ba), see: Belharouak et al. (1999); for AgM^I(PO₃)₂ (M^I = K, Rb, Cs, Tl), see: Averbuch-Pouchot (1993). For background to the bond-valence method, see: Brown & Altermatt (1985).

Experimental top

AgPO₃ was prepared by crystallizing a solution of Ag₃PO₄ in H₃PO₄ (84 %_wt) at a molar ratio Ag/P = 0.01. The thermal treatment included heating the mixture in a graphite crucible at 473 K for 6 h and then cooling to room temperature. After leaching with water, the product consisted of colorless prismatic crystals.

Computing details top

Data collection: CrysAlis CCD (Oxford Diffraction, 2006); cell refinement: CrysAlis CCD (Oxford Diffraction, 2006); data reduction: CrysAlis RED (Oxford Diffraction, 2006); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 1999); software used to prepare material for publication: WinGX (Farrugia, 1999) and enCIFer (Allen et al., 2004).

Figures top

	Fig. 1. The connectivity of the Ag and P atoms in the structure of AgPO₃ with displacement ellipsoids disolayed at the 50% probability level. [Symmetry codes: (i) 0.5 - x, 1/2 + y, 0.5 - z; (ii) -x, -1 + y, z; (iii) 0.5 - x,1/2 + y, 1.5 - z; (iv) -x, 1 - y, 1 - z; (v) x, y, 1 + z; (vi) -x, 2 - y, 1 - z;].
	Fig. 2. A view of the crystal structure of AgPO₃ down the b-axis, emphasizing the tunnels with eight-sided windows where the helical polyphosphate chains reside.

Silver(I) polyphosphate top

Crystal data top

AgPO₃	F(000) = 688
M_r = 186.84	D_x = 4.687 Mg m⁻³
Monoclinic, P2₁/n	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2yn	Cell parameters from 22720 reflections
a = 11.9335 (3) Å	θ = 3.2–35.0°
b = 6.0667 (1) Å	µ = 7.96 mm⁻¹
c = 7.3278 (2) Å	T = 293 K
β = 93.491 (2)°	Prism, colorless
V = 529.53 (2) Å³	0.10 × 0.08 × 0.04 mm
Z = 8

Data collection top

Oxford Diffraction Xcalibur-3 CCD diffractometer	2333 independent reflections
Radiation source: fine-focus sealed tube	2208 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.042
ϕ and ω scans	θ_max = 35°, θ_min = 3.2°
Absorption correction: multi-scan (Blessing, 1995)	h = −19→19
T_min = 0.465, T_max = 0.733	k = −9→9
22720 measured reflections	l = −11→11

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.024	w = 1/[σ²(F_o²) + (0.0212P)² + 1.7313P] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.058	(Δ/σ)_max = 0.001
S = 1.08	Δρ_max = 1.43 e Å⁻³
2333 reflections	Δρ_min = −1.86 e Å⁻³
92 parameters	Extinction correction: SHELXL97 (Sheldrick, 2008)
0 restraints	Extinction coefficient: 0.0034 (3)

Crystal data top

AgPO₃	V = 529.53 (2) Å³
M_r = 186.84	Z = 8
Monoclinic, P2₁/n	Mo Kα radiation
a = 11.9335 (3) Å	µ = 7.96 mm⁻¹
b = 6.0667 (1) Å	T = 293 K
c = 7.3278 (2) Å	0.10 × 0.08 × 0.04 mm
β = 93.491 (2)°

Data collection top

Oxford Diffraction Xcalibur-3 CCD diffractometer	2333 independent reflections
Absorption correction: multi-scan (Blessing, 1995)	2208 reflections with I > 2σ(I)
T_min = 0.465, T_max = 0.733	R_int = 0.042
22720 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.024	92 parameters
wR(F²) = 0.058	0 restraints
S = 1.08	Δρ_max = 1.43 e Å⁻³
2333 reflections	Δρ_min = −1.86 e Å⁻³

Special details top

Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'s 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
Ag1	0.127313 (18)	0.33468 (3)	0.61230 (3)	0.02461 (6)
Ag2	0.03023 (2)	0.89757 (4)	0.77035 (3)	0.02748 (6)
P1	0.22766 (5)	0.81768 (10)	0.48624 (7)	0.01372 (10)
P2	0.11152 (5)	0.61417 (10)	0.18002 (8)	0.01331 (10)
O1	0.25346 (18)	0.6554 (3)	0.6355 (3)	0.0241 (4)
O2	0.22388 (14)	0.6980 (3)	0.2909 (2)	0.0179 (3)
O3	0.12509 (15)	0.9578 (3)	0.4956 (3)	0.0218 (3)
O4	0.16402 (14)	0.4679 (3)	0.0265 (2)	0.0169 (3)
O5	0.05294 (17)	0.7997 (3)	0.0843 (3)	0.0234 (3)
O6	0.04724 (16)	0.4744 (3)	0.3047 (3)	0.0220 (3)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Ag1	0.02707 (10)	0.02043 (10)	0.02611 (10)	0.00056 (7)	−0.00023 (7)	0.00252 (7)
Ag2	0.03883 (12)	0.02340 (11)	0.02121 (9)	0.00423 (8)	0.01011 (8)	0.00068 (7)
P1	0.0138 (2)	0.0153 (2)	0.0121 (2)	−0.00206 (17)	0.00169 (16)	0.00000 (17)
P2	0.0124 (2)	0.0141 (2)	0.0134 (2)	0.00133 (17)	0.00096 (16)	−0.00069 (17)
O1	0.0312 (9)	0.0223 (9)	0.0183 (7)	−0.0081 (7)	−0.0021 (7)	0.0064 (6)
O2	0.0155 (6)	0.0225 (8)	0.0157 (6)	−0.0010 (6)	0.0008 (5)	−0.0059 (6)
O3	0.0169 (7)	0.0268 (9)	0.0220 (8)	0.0031 (6)	0.0041 (6)	−0.0047 (7)
O4	0.0161 (6)	0.0200 (8)	0.0145 (6)	0.0058 (6)	0.0000 (5)	−0.0042 (5)
O5	0.0278 (9)	0.0208 (8)	0.0211 (8)	0.0108 (7)	−0.0020 (6)	0.0000 (6)
O6	0.0218 (8)	0.0243 (9)	0.0204 (7)	−0.0050 (7)	0.0061 (6)	0.0000 (7)

Geometric parameters (Å, º) top

Ag1—O3ⁱ	2.441 (2)	P1—O4^vii	1.5889 (17)
Ag1—O1	2.460 (2)	P1—O2	1.6033 (17)
Ag1—O6ⁱⁱ	2.491 (2)	P2—O5	1.479 (2)
Ag1—O1ⁱⁱⁱ	2.511 (2)	P2—O6	1.4924 (19)
Ag1—O6	2.540 (2)	P2—O4	1.5909 (17)
Ag1—Ag2ⁱ	3.1431 (3)	P2—O2	1.6074 (18)
Ag2—O5^iv	2.3708 (19)	O1—Ag1^viii	2.511 (2)
Ag2—O5^v	2.3756 (19)	O3—Ag1^vi	2.441 (2)
Ag2—O3	2.3968 (19)	O4—P1^ix	1.5889 (17)
Ag2—O6ⁱⁱ	2.487 (2)	O5—Ag2^iv	2.3708 (19)
Ag2—O3^iv	2.750 (2)	O5—Ag2^x	2.3756 (19)
Ag2—Ag1^vi	3.1431 (3)	O6—Ag2ⁱⁱ	2.487 (2)
P1—O1	1.490 (2)	O6—Ag1ⁱⁱ	2.491 (2)
P1—O3	1.4952 (19)

O3ⁱ—Ag1—O1	139.36 (7)	O3—P1—O4^vii	110.36 (11)
O3ⁱ—Ag1—O6ⁱⁱ	121.99 (6)	O1—P1—O2	110.46 (11)
O1—Ag1—O6ⁱⁱ	97.63 (7)	O3—P1—O2	108.66 (10)
O3ⁱ—Ag1—O1ⁱⁱⁱ	81.11 (6)	O4^vii—P1—O2	100.72 (9)
O1—Ag1—O1ⁱⁱⁱ	88.56 (4)	O5—P2—O6	118.48 (12)
O6ⁱⁱ—Ag1—O1ⁱⁱⁱ	117.80 (6)	O5—P2—O4	106.52 (10)
O3ⁱ—Ag1—O6	90.36 (7)	O6—P2—O4	110.79 (11)
O1—Ag1—O6	89.59 (6)	O5—P2—O2	110.76 (12)
O6ⁱⁱ—Ag1—O6	77.68 (6)	O6—P2—O2	108.33 (10)
O1ⁱⁱⁱ—Ag1—O6	164.51 (6)	O4—P2—O2	100.46 (9)
O3ⁱ—Ag1—Ag2ⁱ	48.87 (5)	P1—O1—Ag1	111.96 (11)
O1—Ag1—Ag2ⁱ	151.61 (4)	P1—O1—Ag1^viii	109.67 (10)
O6ⁱⁱ—Ag1—Ag2ⁱ	88.27 (5)	Ag1—O1—Ag1^viii	135.28 (8)
O1ⁱⁱⁱ—Ag1—Ag2ⁱ	64.42 (5)	P1—O2—P2	124.88 (11)
O6—Ag1—Ag2ⁱ	118.78 (4)	P1—O3—Ag2	112.45 (11)
O5^iv—Ag2—O5^v	77.57 (7)	P1—O3—Ag1^vi	123.86 (11)
O5^iv—Ag2—O3	119.43 (7)	Ag2—O3—Ag1^vi	81.04 (6)
O5^v—Ag2—O3	145.04 (7)	P1^ix—O4—P2	135.91 (11)
O5^iv—Ag2—O6ⁱⁱ	129.94 (7)	P2—O5—Ag2^iv	125.12 (11)
O5^v—Ag2—O6ⁱⁱ	90.36 (7)	P2—O5—Ag2^x	132.23 (11)
O3—Ag2—O6ⁱⁱ	98.10 (6)	Ag2^iv—O5—Ag2^x	102.43 (7)
O5^iv—Ag2—Ag1^vi	71.68 (5)	P2—O6—Ag2ⁱⁱ	125.33 (11)
O5^v—Ag2—Ag1^vi	123.01 (5)	P2—O6—Ag1ⁱⁱ	110.73 (11)
O3—Ag2—Ag1^vi	50.09 (5)	Ag2ⁱⁱ—O6—Ag1ⁱⁱ	99.83 (6)
O6ⁱⁱ—Ag2—Ag1^vi	145.52 (4)	P2—O6—Ag1	123.59 (11)
O1—P1—O3	118.36 (12)	Ag2ⁱⁱ—O6—Ag1	90.46 (7)
O1—P1—O4^vii	106.84 (10)	Ag1ⁱⁱ—O6—Ag1	102.32 (6)

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

Experimental details

Crystal data
Chemical formula	AgPO₃
M_r	186.84
Crystal system, space group	Monoclinic, P2₁/n
Temperature (K)	293
a, b, c (Å)	11.9335 (3), 6.0667 (1), 7.3278 (2)
β (°)	93.491 (2)
V (Å³)	529.53 (2)
Z	8
Radiation type	Mo Kα
µ (mm⁻¹)	7.96
Crystal size (mm)	0.10 × 0.08 × 0.04

Data collection
Diffractometer	Oxford Diffraction Xcalibur-3 CCD diffractometer
Absorption correction	Multi-scan (Blessing, 1995)
T_min, T_max	0.465, 0.733
No. of measured, independent and observed [I > 2σ(I)] reflections	22720, 2333, 2208
R_int	0.042
(sin θ/λ)_max (Å⁻¹)	0.807

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.024, 0.058, 1.08
No. of reflections	2333
No. of parameters	92
Δρ_max, Δρ_min (e Å⁻³)	1.43, −1.86

Computer programs: CrysAlis CCD (Oxford Diffraction, 2006), CrysAlis RED (Oxford Diffraction, 2006), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), DIAMOND (Brandenburg, 1999), WinGX (Farrugia, 1999) and enCIFer (Allen et al., 2004).

Selected geometric parameters (Å, º) top

Ag1—O3ⁱ	2.441 (2)	Ag2—O3^iv	2.750 (2)
Ag1—O1	2.460 (2)	P1—O1	1.490 (2)
Ag1—O6ⁱⁱ	2.491 (2)	P1—O3	1.4952 (19)
Ag1—O1ⁱⁱⁱ	2.511 (2)	P1—O4^vi	1.5889 (17)
Ag1—O6	2.540 (2)	P1—O2	1.6033 (17)
Ag2—O5^iv	2.3708 (19)	P2—O5	1.479 (2)
Ag2—O5^v	2.3756 (19)	P2—O6	1.4924 (19)
Ag2—O3	2.3968 (19)	P2—O4	1.5909 (17)
Ag2—O6ⁱⁱ	2.487 (2)	P2—O2	1.6074 (18)

P1—O2—P2	124.88 (11)	P1^vii—O4—P2	135.91 (11)

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

References

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