Synthesis, crystal structure and charge-distribution validation of β-Na4Cu(MoO4)3 adopting the alluadite structure-type

Dridi, W.; Zid, M.F.

doi:10.1107/S2056989016010367

research communications

CRYSTALLOGRAPHIC
COMMUNICATIONS

ISSN: 2056-9890

Volume 72| Part 8| August 2016| Pages 1103-1107

https://doi.org/10.1107/S2056989016010367

Open

access

Synthesis, crystal structure and charge-distribution validation of β-Na₄Cu(MoO₄)₃ adopting the alluadite structure-type

Wassim Dridi ^a and Mohamed Faouzi Zid ^a ^*

^aLaboratoire de Matériaux et Cristallochimie, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 El Manar Tunis, Tunisia
^*Correspondence e-mail: faouzi.zid@fst.rnu.tn

Edited by T. J. Prior, University of Hull, England (Received 9 June 2016; accepted 27 June 2016; online 12 July 2016)

Single crystals of a new variety of tetrasodium copper(II) tris[molybdate(VI)], Na₄Cu(MoO₄)₃, have been synthesized by solid-state reactions and characterized by single-crystal X-ray diffraction. This alluaudite structure-type is characterized by the presence of infinite layers of composition (Cu/Na)₂Mo₃O₁₄ parallel to the (100) plane, which are linked by MoO₄ tetrahedra, forming a three-dimensional framework containing two types of hexagonal channels in which Na⁺ cations reside. The Cu²⁺ and Na²⁺ cations are located at the same general site with occupancies of 0.5. All atoms are on general positions except for one Mo, two Na (site symmetry 2) and another Na (site symmetry -1) atom. One O atom is split into two separate positions with occupancies of 0.5. The title compound is isotypic with Na₅Sc(MoO₄)₄ and Na₃In₂As₃O₁₂. The structure model is supported by bond-valence-sum (BVS) and charge-distribution CHARDI methods. β-Na₄Cu(MoO₄)₃ is compared and discussed with the K₄Cu(MoO₄)₃ and α-Na₄Cu(MoO₄)₃ structures.

Keywords: crystal structure; BVS; CHARDI; alluaudite-type; sodium; copper(II); molybdate(VI).

CCDC reference: 1487800

1. Chemical context

In recent years, a number of molybdates have received considerable attention due to their amazing properties and high application potential in various fields, such as photoluminescence (Shi et al., 2014 ) and Li-ion batteries (Reddy et al., 2013 ). For example, the copper molybdate Cu₃Mo₂O₉ doped with lithium displays high Coulombic efficiency in lithium-ion batteries and excellent charge-discharge stability (Xia et al., 2015 ). Many new molybdate phases have been synthesized and structurally characterized by X-ray diffraction, among which a large number belong to the alluaudite type, such as Na₂₅Cs₈Fe₅(MoO₄)₂₄, which presents a high electrical conductivity (Savina et al., 2014 ). The alluaudite-type structure was first determined on natural minerals by Fisher, who showed that alluaudite compounds crystallize in the monoclinic C2/c space group (Fisher, 1955 ). Moore proposed the general formula X(2)X(1)M(1)M(2)₂(PO₄)₃, in which the X and M mono-, bi- or trivalent cations are written according to their size (X are large cations and M are distorted octahedrally coordinated atoms). It represents the parental structure-type of the group referred to as alluaudite-type (Moore, 1971 ). The size of the channel and the stability of the alluaudite network led to many phases belonging to this structural type. We can totally or partially replace not only the monovalent cations, but also the central atoms of the MO₆ octahedra and TO₄ tetrahedra. It is also possible to make substitutions with cations in different oxidation states adopting the same type of coordination number (Mo⁶⁺, V⁵⁺, P⁵⁺ and As⁵⁺). Alluaudite molybdates usually have the general formula of X(2)X(1)M(1)M(2)₂(MoO₄)₃ and adopt the C2/c space group, with a ≃ 12, b ≃ 13 and c ≃ 7 Å, examples being the K_0.13Na_3.87MgMo₃O₁₂ (Ennajeh et al., 2015 ), Na₃Fe₂(MoO₄)₃ (Muessig et al., 2003 ) and Na₄Co(MoO₄)₃ (Nasri et al., 2014 ) compounds. A review of the literature also reveals the presence of other formulae, such as Na₅Sc(MoO₄)₄, Na₂Ni(MoO₄)₂ (Klevtsova et al., 1991 ) and Na_2.2Zn_0.9(MoO₄)₂ (Efremov et al., 1975 ), which crystallize in the space group C2/c with cell parameters of about a ≃ 12, b ≃ 13 and c ≃ 7 Å. All alluaudite-type compounds can be described by the general formula given by Moore (1971), but their structures can differ by the number of formula units per unit cell. They are characterized by a three-dimensional heteropolyhedral framework formed by TO₄ tetrahedra and MO₆ octahedra, delimiting two types of channels running along the c axis. A new variety of β-Na₄Cu(MoO₄)₃ formulation was obtained by a reaction in the solid state at 873 K.

2. Structural commentary

The structural unit in β-Na₄Cu(MoO₄)₃ is formed by MO₆ (M = Cu1/Na1) octahedra linked by sharing vertices with Mo1O₄ tetrahedra and two slightly different Mo2O₄ tetrahedra, with a partially occupied (0.5 occupancy) Mo2 site. Atom O4 is split into two separate positions, with occupancies of 0.5 for the O4 and O41 atoms. The charge compensation is provided by Na⁺ cations (Fig. 1). The essential building units of the structure are M₂O₁₀ units obtained from two edge-sharing MO₆ octahedra. These units are connected by Mo1O₄ tetrahedra through vertex-sharing via Mo—O—M mixed bridges. This results in M₂Mo₂O₁₆ units. Each unit is connected to six other identical units by the sharing of vertices, leading to an infinite layer of the M₂Mo₃O₁₄ type parallel to the (100) plane (Fig. 2). The linkage of these layers is ensured by the two slightly different Mo2O₄ tetrahedra, linking via corners. This results in a three-dimensional framework delimited by two kinds of channels running along the c axis at ( $[1 \over 2]$ , 0, z) and (0, 0, z). These channels are occupied by Na⁺ cations (Fig. 3). In the anionic framework, each Mo2O₄ tetrahedron shares its O atoms with four different M₂O₁₀ dimers belonging to two adjacent layers. The Mo1O₄ tetrahedron shares only three O atoms with three M₂O₁₀ units belonging to the same layer, the other O atom being free and pointing towards the channels where the Na3 cations are located (Fig. 4). There is some compositional flexibility in the alluaudite structure and the studied material is isostructural with Na₅Sc(MoO₄)₄ (Klevtsova et al., 1975 ) and Na₃In₂As₃O₁₂ (Khorari et al., 1997 ). The two crystallographically independent Mo atoms have tetrahedral coordination, with an average Mo—O distance of 1.761 Å for Mo1 and 1.777 Å for Mo2, which is in a good agreement with those typically observed in Rb₂Cu₂(MoO₄)₃ (Solodovnikov & Solodovnikova, 1997 ). The Na⁺ cations occupy three crystallographically independent sites with different O-atom environments. The Na2, Na3 and Na4 atoms are surrounded by four, eight and six O atoms, respectively (Table 1). The Cu1 and Na1 cations are located at the same general site, with occupancies of 0.5, and have an octahedral environment with an average distance of 2.214 Å. This distance presents a mean between the Na—O distances of Na₂Cu(PO₃)₄ (Laügt et al., 1972 ) and the Cu—O distances encountered in Ag₂Cu₂(MoO₄)₃ (Tsyrenova et al., 2009 ). The proposed structural model is confirmed by two validation models: (i) the bond-valence approach using the empirical formula of Brown (Brown & Altermatt, 1985 ) and (ii) the charge-distribution method Chardi (Nespolo, 2015 , 2016 ). The charge distribution method is the most recent development of Pauling's concept of bond strength (Pauling, 1929 ). Instead of empirical parameters used in the bond-valence approach, it exploits the experimental bond lengths deduced from the structural study to compute a non-integer coordination number, ECoN (effective coordination number), around a PC-atom (atom placed at the center of a polyhedron, q > 0), which is coordinated by V atoms (atoms located at the vertices, q < 0); q is the formal oxidation number. ECoN takes into account not only the number of V atoms around a given PC atom, but also their weight in terms of relative distances. Calculated charges Q(i) and valences V(i) are in good agreement with the formal oxidation number (q) multiplied by occupancy rates. The dispersion factor MAPD, which measures the mean absolute percentage deviation, is 2.2% for the calculated cationic charges. The variation of the ECoN value to the traditional coordination indicates the degree of distortion. The two validation models results are summarized in Table 2. Comparing our structure with that of a similar formulation, i.e. K₄Cu(MoO₄)₃ (Menard et al., 2011 ), we found a clear difference, on the one hand, in the crystal symmetry and, on the other hand, in the arrangement of polyhedra. K₄Cu(MoO₄)₃ crystallizes in the Pnma space group. Its structure can be described as being composed of a distorted square-planar CuO₄ polyhedron bound by shared vertices to two Mo1O₄ tetrahedra to form CuMo₂O₁₀-type units. These units are interconnected, on the one hand, by insertion of two Mo2O₄ tetrahedra which share a face with a partial occupation (0.5 occupancy) of Mo2 atoms, and secondly by forming a mixed bridge of the Mo—O—Cu type. This forms ribbons arranged parallel to the [100] direction. This results in a one-dimensional structure in which K⁺ atoms reside in the inter-ribbon spaces (Fig. 5). The structure of our new variety β-Na₄Cu(MoO₄)₃ is compared with the α variety. Indeed, α-Na₄Cu(MoO₄)₃ (Klevtsova et al., 1991) crystallizes in the triclinic system, space group P $[\overline{1}]$ , and its structure is formed by the same Cu₂O₁₀ dimers present in our structure (here present as mixed-occupied M₂O₁₀ units). The latter connects via Mo—O—Cu double composite bridges with two bidentate tetrahedra MoO₄ and by Mo—O—Cu simple bridges with monodentate MoO₄ tetrahedra to form Cu₂Mo₄O₂₀ units. The Cu₂Mo₄O₂₀ units are connected by MoO₄ tetrahedra and the pooling of vertices to form ribbons arranged in the [010] direction. All the ribbons form a one-dimensional framework with inter-ribbon spaces containing monovalent Na⁺ cations (Fig. 6). This structure has the same arrangement of structural units found in the one-dimensional structure of K₃Mn(MoO₄)₃ (Solodovnikov et al., 1998 ) (Fig. 7).

Table 1
Selected bond lengths (Å)

Mo1—O1	1.746 (3)	Na2—O5^iv	2.462 (3)
Mo1—O5	1.762 (3)	Na2—O5	2.549 (3)
Mo1—O2ⁱ	1.764 (3)	Na2—O5ⁱⁱ	2.549 (3)
Mo1—O6	1.774 (3)	Na3—O41^vii	2.443 (7)
Mo2—Mo2ⁱⁱ	0.447 (2)	Na3—O41^viii	2.443 (7)
Mo2—O41	1.456 (7)	Na3—O1^ix	2.493 (3)
Mo2—O3ⁱⁱ	1.738 (3)	Na3—O1ⁱⁱ	2.493 (3)
Mo2—O41ⁱⁱ	1.740 (7)	Na3—O1^vi	2.675 (3)
Mo2—O4	1.787 (7)	Na3—O1^x	2.675 (3)
Mo2—O3	1.822 (3)	Na3—O4^vii	2.749 (7)
Mo2—O4ⁱⁱ	2.150 (6)	Na3—O4^viii	2.749 (7)
Cu1—O4ⁱⁱⁱ	1.884 (6)	Na3—O41^xi	3.008 (7)
Cu1—O3	2.098 (3)	Na3—O41^xii	3.008 (7)
Cu1—O2	2.116 (3)	Na4—O3	2.337 (2)
Cu1—O6	2.152 (3)	Na4—O3^xiii	2.337 (2)
Cu1—O5^iv	2.317 (3)	Na4—O2	2.424 (3)
Cu1—O6^v	2.464 (4)	Na4—O2^xiii	2.424 (3)
Cu1—O41ⁱⁱⁱ	2.467 (4)	Na4—O1^v	2.490 (4)
Na2—O5^vi	2.462 (3)	Na4—O1^xiv	2.490 (4)
Symmetry codes: (i) x, y, z-1; (ii) $[-x+1, y, -z+{\script{1\over 2}}]$ ; (iii) $[x+{\script{1\over 2}}, -y+{\script{1\over 2}}, z+{\script{1\over 2}}]$ ; (iv) $[x, -y, z+{\script{1\over 2}}]$ ; (v) $[-x+{\script{3\over 2}}, -y+{\script{1\over 2}}, -z+1]$ ; (vi) -x+1, -y, -z; (vii) $[x-{\script{1\over 2}}, -y+{\script{1\over 2}}, z-{\script{1\over 2}}]$ ; (viii) $[-x+{\script{1\over 2}}, y-{\script{1\over 2}}, -z+{\script{1\over 2}}]$ ; (ix) $[x-1, -y, z-{\script{1\over 2}}]$ ; (x) x-1, y, z; (xi) $[-x+{\script{1\over 2}}, -y+{\script{1\over 2}}, -z]$ ; (xii) $[x-{\script{1\over 2}}, y-{\script{1\over 2}}, z]$ ; (xiii) $[-x+1, y, -z+{\script{3\over 2}}]$ ; (xiv) $[x-{\script{1\over 2}}, -y+{\script{1\over 2}}, z+{\script{1\over 2}}]$ .

Table 2
CHARDI and BVS analysis of cation polyhedra in β-Na₄Cu(MoO₄)₃

Cation	q(i)·sof(i)	Q(i)	V(i).sof(i)	CN(i)	ECoN(i)	d_ar	d_med
Mo1	6.00	6.24	5.93	4	4.00	1.761	1.761
Mo2	3.00	2.48	3.06	4	3.52	1.777	1.776
M	1.50	1.75	1.64	6	4.97	2.214	2.214
Na2	1.00	0.98	0.79	4	4.49	2.505	2.703
Na3	1.00	0.83	0.85	8	8.19	2.587	2.670
Na4	1.00	1.24	1.11	6	5.86	2.417	2.417
Notes: M = Cu1/Na1, q(i) = formal oxidation number, sof(i) = site-occupation factor, Q(i) = calculated charges, CN = coordination number, ECoN = number of effective coordination, MAPD = 100/NΣ_i^N.\|qi - Qi/qi\|, d_ar = arithmetic average distance and d_med = weighted average distance.

Figure 1
Representation of the coordination polyhedra in the structural unit of β-Na₄Cu(MoO₄)₃, showing (a) full atomic, (b) polyhedral. All atoms are represented as displacement ellipsoids at the 50% probability level. [Symmetry codes: (i) x, y, z − 1; (ii) −x + 1, y, −z + $[{1\over 2}]$ ; (iii) x + $[{1\over 2}]$ , −y + $[{1\over 2}]$ , z + $[{1\over 2}]$ ; (iv) x, −y, z + $[{1\over 2}]$ ; (v) −x + $[{3\over 2}]$ , −y + $[{1\over 2}]$ , −z + 1.]

Figure 2
A projection of the polyhedral layers in the bc plane.

Figure 3
A projection of the β-Na₄Cu(MoO₄)₃ structure, viewed normal to (001), showing the channels where monovalent cations are located.

Figure 4
The association modes of M₂O₁₀-based units by the (a) Mo(1)O₄ and (b) Mo(2)O₄ tetrahedra. For clarity, we present only one atom of molybdenum, Mo2.

Figure 5
A projection of the K₄Cu(MoO₄)₃ structure, viewed along the [001] direction.

Figure 6
A projection of the α-Na₄Cu(MoO₄)₃ structure, viewed in the (010) plane.

Figure 7
A projection of the K₄Mn(MoO₄)₃ structure, viewed normal to (010).

3. Synthesis and crystallization

β-Na₄Cu(MoO₄)₃ crystals were obtained by a solid-state reaction from the following reagents: Na₂CO₃ (Prolabo, 70128), Cu(CO₂CH₃)·H₂O (Sigma–Aldrich, C5893) and (NH₄)₆Mo₇O₂₄·4H₂O (Sigma–Aldrich, 13301) with an Na:Cu:Mo molar ratio of 4:1:3. The resulting mixture was milled in an agate mortar, placed in a porcelain crucible and then preheated slowly in air at 623 K for 24 h, in order to eliminate volatile products. Thereafter, it was heated to a temperature close to that of the fusion at 873 K. It was left at this temperature for 20 d to induce nucleation and crystal growth. The final residue was first cooled slowly (5 K per half day) to 823 K and then rapidly (50 K h⁻¹) to room temperature. Green crystals of sufficient size for the measurement of intensities were obtained.

4. Refinement

Crystal data, data collection and structure refinement details are summarized in Table 3. We used of EADP and EXYZ constraints within SHELXL2014 (Sheldrick, 2015 ) for Cu1/Na1 located at the same crystallographic site. Atom O4 was split over two sites (O4 and O41) as this displayed a very elongated displacement ellipsoid. The occupancies of O4 and O41 were set to 0.5 in line with the occupany of Mo2; the two separate O-atom sites (O4 and O41) correspond to two different orientations of the Mo2O₄ tetrahedron related by symmetry. Refining atomic occupancies leads to a value of 0.497 (4) for the Cu atom. For conditions of electrical neutrality, we set the occupancy of the Cu atom as 0.5. This leads to well-defined ellipsoids. The maximum and minimum electron densities in the final Fourier difference map are acceptable and located at 0.77 and 0.82 Å, respectively, from the Na2 and Mo1 atoms.

Table 3
Experimental details

Crystal data
Chemical formula	Na₄Cu(MoO₄)₃
M_r	635.32
Crystal system, space group	Monoclinic, C2/c
Temperature (K)	298
a, b, c (Å)	12.5318 (9), 13.8181 (9), 7.1159 (7)
β (°)	111.95 (2)
V (Å³)	1142.9 (2)
Z	4
Radiation type	Mo Kα
μ (mm⁻¹)	5.26
Crystal size (mm)	0.28 × 0.22 × 0.18

Data collection
Diffractometer	Enraf–Nonius CAD-4
Absorption correction	ψ scan (North et al., 1968 )
T_min, T_max	0.224, 0.387
No. of measured, independent and observed [I > 2σ(I)] reflections	2678, 1238, 1208
R_int	0.030
(sin θ/λ)_max (Å⁻¹)	0.638

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.023, 0.058, 1.17
No. of reflections	1238
No. of parameters	104
Δρ_max, Δρ_min (e Å⁻³)	0.80, −0.72
Computer programs: CAD-4 EXPRESS (Duisenberg, 1992 ), XCAD4 (Harms & Wocadlo, 1995 ), SHELXS97 (Sheldrick, 2008 ), SHELXL2014 (Sheldrick, 2015), DIAMOND (Brandenburg & Putz, 2001 ), WinGX (Farrugia, 2012 ) and publCIF (Westrip, 2010 ).

Supporting information

CCDC reference: 1487800

Crystal structure: contains datablock I. DOI: https://doi.org/10.1107/S2056989016010367/pj2032sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2056989016010367/pj2032Isup2.hkl

checkCIF report

Computing details top

Data collection: CAD-4 EXPRESS (Duisenberg, 1992); cell refinement: CAD-4 EXPRESS (Duisenberg, 1992); data reduction: XCAD4 (Harms & Wocadlo, 1995); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL2014 (Sheldrick, 2015); molecular graphics: DIAMOND (Brandenburg & Putz, 2001); software used to prepare material for publication: WinGX (Farrugia, 2012) and publCIF (Westrip, 2010).

Tetrasodium copper(II) tris[molybdate(VI)] top

Crystal data top

Na₄Cu(MoO₄)₃	F(000) = 1180
M_r = 635.32	D_x = 3.692 Mg m⁻³
Monoclinic, C2/c	Mo Kα radiation, λ = 0.71073 Å
a = 12.5318 (9) Å	Cell parameters from 25 reflections
b = 13.8181 (9) Å	θ = 10–15°
c = 7.1159 (7) Å	µ = 5.26 mm⁻¹
β = 111.95 (2)°	T = 298 K
V = 1142.9 (2) Å³	Prism, green
Z = 4	0.28 × 0.22 × 0.18 mm

Data collection top

Enraf–Nonius CAD-4 diffractometer	1208 reflections with I > 2σ(I)
ω/2θ scans	R_int = 0.030
Absorption correction: ψ scan (North et al., 1968)	θ_max = 27.0°, θ_min = 2.3°
T_min = 0.224, T_max = 0.387	h = −15→15
2678 measured reflections	k = −1→17
1238 independent reflections	l = −9→9

Refinement top

Refinement on F²	0 restraints
Least-squares matrix: full	w = 1/[σ²(F_o²) + (0.0173P)² + 5.2182P] where P = (F_o² + 2F_c²)/3
R[F² > 2σ(F²)] = 0.023	(Δ/σ)_max = 0.001
wR(F²) = 0.058	Δρ_max = 0.80 e Å⁻³
S = 1.17	Δρ_min = −0.72 e Å⁻³
1238 reflections	Extinction correction: SHELXL2014 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
104 parameters	Extinction coefficient: 0.00081 (11)

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.

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

	x	y	z	U_iso*/U_eq	Occ. (<1)
Mo1	0.72812 (2)	0.10879 (2)	0.11920 (4)	0.01991 (12)
Mo2	0.48085 (7)	0.28610 (4)	0.2400 (3)	0.0133 (3)	0.5
Cu1	0.70643 (5)	0.16100 (5)	0.61691 (9)	0.01741 (16)	0.5
Na1	0.70643 (5)	0.16100 (5)	0.61691 (9)	0.01741 (16)	0.5
Na2	0.5000	0.0169 (3)	0.2500	0.0482 (8)
Na3	0.0000	0.0000	0.0000	0.0299 (5)
Na4	0.5000	0.26934 (16)	0.7500	0.0219 (4)
O1	0.8745 (2)	0.0849 (2)	0.1859 (4)	0.0292 (6)
O2	0.6715 (2)	0.1702 (2)	0.8852 (4)	0.0337 (6)
O3	0.5390 (2)	0.2147 (2)	0.4714 (4)	0.0247 (6)
O4	0.3657 (5)	0.3629 (6)	0.2376 (11)	0.0239 (10)	0.5
O41	0.4153 (5)	0.3707 (6)	0.2551 (11)	0.0239 (10)	0.5
O5	0.6504 (3)	−0.0001 (2)	0.0904 (5)	0.0346 (7)
O6	0.7095 (3)	0.1754 (2)	0.3178 (5)	0.0373 (7)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Mo1	0.01593 (17)	0.02069 (18)	0.01973 (18)	0.00034 (10)	0.00275 (11)	−0.00104 (11)
Mo2	0.0147 (8)	0.0162 (2)	0.0079 (4)	0.0004 (2)	0.0032 (6)	0.0000 (3)
Cu1	0.0227 (3)	0.0219 (3)	0.0093 (3)	−0.0007 (3)	0.0079 (2)	0.0006 (2)
Na1	0.0227 (3)	0.0219 (3)	0.0093 (3)	−0.0007 (3)	0.0079 (2)	0.0006 (2)
Na2	0.0217 (11)	0.098 (3)	0.0203 (11)	0.000	0.0031 (9)	0.000
Na3	0.0397 (12)	0.0224 (10)	0.0179 (9)	−0.0013 (10)	−0.0004 (9)	−0.0006 (9)
Na4	0.0225 (9)	0.0270 (11)	0.0189 (9)	0.000	0.0110 (7)	0.000
O1	0.0187 (12)	0.0341 (15)	0.0307 (14)	0.0037 (11)	0.0044 (10)	0.0072 (12)
O2	0.0268 (14)	0.0330 (16)	0.0327 (15)	0.0083 (12)	0.0011 (12)	0.0017 (13)
O3	0.0275 (13)	0.0335 (15)	0.0160 (12)	−0.0003 (12)	0.0113 (10)	0.0011 (11)
O4	0.017 (3)	0.0250 (18)	0.0238 (18)	0.005 (3)	0.001 (3)	−0.0039 (16)
O41	0.017 (3)	0.0250 (18)	0.0238 (18)	0.005 (3)	0.001 (3)	−0.0039 (16)
O5	0.0357 (15)	0.0279 (14)	0.0410 (17)	−0.0059 (13)	0.0152 (13)	0.0005 (13)
O6	0.0305 (14)	0.0398 (17)	0.0385 (17)	0.0010 (14)	0.0095 (13)	−0.0084 (14)

Geometric parameters (Å, º) top

Mo1—O1	1.746 (3)	Na2—O5^iv	2.462 (3)
Mo1—O5	1.762 (3)	Na2—O5	2.549 (3)
Mo1—O2ⁱ	1.764 (3)	Na2—O5ⁱⁱ	2.549 (3)
Mo1—O6	1.774 (3)	Na3—O41^vii	2.443 (7)
Mo2—Mo2ⁱⁱ	0.447 (2)	Na3—O41^viii	2.443 (7)
Mo2—O41	1.456 (7)	Na3—O1^ix	2.493 (3)
Mo2—O3ⁱⁱ	1.738 (3)	Na3—O1ⁱⁱ	2.493 (3)
Mo2—O41ⁱⁱ	1.740 (7)	Na3—O1^vi	2.675 (3)
Mo2—O4	1.787 (7)	Na3—O1^x	2.675 (3)
Mo2—O3	1.822 (3)	Na3—O4^vii	2.749 (7)
Mo2—O4ⁱⁱ	2.150 (6)	Na3—O4^viii	2.749 (7)
Cu1—O4ⁱⁱⁱ	1.884 (6)	Na3—O41^xi	3.008 (7)
Cu1—O3	2.098 (3)	Na3—O41^xii	3.008 (7)
Cu1—O2	2.116 (3)	Na4—O3	2.337 (2)
Cu1—O6	2.152 (3)	Na4—O3^xiii	2.337 (2)
Cu1—O5^iv	2.317 (3)	Na4—O2	2.424 (3)
Cu1—O6^v	2.464 (4)	Na4—O2^xiii	2.424 (3)
Cu1—O41ⁱⁱⁱ	2.467 (4)	Na4—O1^v	2.490 (4)
Na2—O5^vi	2.462 (3)	Na4—O1^xiv	2.490 (4)

O1—Mo1—O5	110.42 (15)	O3—Cu1—O2	85.28 (10)
O1—Mo1—O2ⁱ	111.01 (14)	O4ⁱⁱⁱ—Cu1—O6	93.4 (2)
O5—Mo1—O2ⁱ	106.95 (15)	O3—Cu1—O6	82.16 (11)
O1—Mo1—O6	108.63 (14)	O2—Cu1—O6	166.61 (12)
O5—Mo1—O6	107.69 (15)	O4ⁱⁱⁱ—Cu1—O5^iv	96.1 (2)
O2ⁱ—Mo1—O6	112.08 (15)	O3—Cu1—O5^iv	94.73 (11)
O3ⁱⁱ—Mo2—O4	118.4 (2)	O2—Cu1—O5^iv	88.43 (12)
O3ⁱⁱ—Mo2—O3	110.75 (17)	O6—Cu1—O5^iv	97.18 (12)
O4—Mo2—O3	112.3 (3)	O6^v—Cu1—O2	89.55 (14)
O3ⁱⁱ—Mo2—O4ⁱⁱ	100.3 (2)	O6^v—Cu1—O4ⁱⁱⁱ	76.78 (15)
O4—Mo2—O4ⁱⁱ	113.9 (4)	O6^v—Cu1—O5^iv	172.13 (15)
O3—Mo2—O4ⁱⁱ	99.0 (2)	O6^v—Cu1—O6	86.42 (14)
O4ⁱⁱⁱ—Cu1—O3	168.8 (2)	O6^v—Cu1—O3	92.66 (12)
O4ⁱⁱⁱ—Cu1—O2	98.1 (2)

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

CHARDI and BVS analysis of cation polyhedra in β-Na₄Cu(MoO₄)₃ top

Cation	q(i).sof(i)	Q(i)	V(i).sof(i)	CN(i)	ECoN(i)	d_ar	d_med
Mo1	6.00	6.24	5.93	4	4.00	1.761	1.761
Mo2	3.00	2.48	3.06	4	3.52	1.777	1.776
M	1.50	1.75	1.64	6	4.97	2.214	2.214
Na2	1.00	0.98	0.79	4	4.49	2.505	2.703
Na3	1.00	0.83	0.85	8	8.19	2.587	2.670
Na4	1.00	1.24	1.11	6	5.86	2.417	2.417

Notes: M = Cu1/Na1, q(i) = formal oxidation number, sof(i) = site-occupation factor, Q(i) = calculated charges, CN = coordination number, ECoN = number of effective coordination, MAPD = 100/NΣ_i^N.|qi - Qi/qi|, d_ar = arithmetic average distance and d_med = weighted average distance.

References

Brandenburg, K. & Putz, H. (2001). DIAMOND. Crystal Impact GbR, Bonn, Germany. Google Scholar
Brown, I. D. & Altermatt, D. (1985). Acta Cryst. B41, 244–247. CrossRef CAS Web of Science IUCr Journals Google Scholar
Duisenberg, A. J. M. (1992). J. Appl. Cryst. 25, 92–96. CrossRef CAS Web of Science IUCr Journals Google Scholar
Efremov, V. A., Velikodnyi, Yu. A. & Trunov, V. K. (1975). Kristallografiya, 20, 287–292. CAS Google Scholar
Ennajeh, I., Georges, S., Smida, Y. B., Guesmi, A., Zid, M. F. & Boughazala, H. (2015). RSC Adv. 5, 38918–38925. Web of Science CSD CrossRef CAS Google Scholar
Farrugia, L. J. (2012). J. Appl. Cryst. 45, 849–854. Web of Science CrossRef CAS IUCr Journals Google Scholar
Fisher, D. (1955). Am. Mineral. 40, 1100–1109. CAS Google Scholar
Harms, K. & Wocadlo, S. (1995). XCAD4. University of Marburg, Germany. Google Scholar
Khorari, S., Rulmont, A. & Tarte, P. (1997). J. Solid State Chem. 134, 31–37. Web of Science CrossRef CAS Google Scholar
Klevtsova, R. F., Borisov, S. V., Bliznyuk, N. A., Glinskaya, L. A. & Klevtsov, P. V. (1991). Zh. Strukt. Khim. 32, 127–136. CAS Google Scholar
Klevtsova, R. F., Kozeeva, L. P. & Klevtsov, P. V. (1975). Kristallografiya, 20, 925–930. CAS Google Scholar
Laügt, M., Tordjman, I., Guitel, J. C. & Bassi, G. (1972). Acta Cryst. B28, 2721–2725. CrossRef IUCr Journals Web of Science Google Scholar
Menard, M. C., Ishii, R., Nakatsuji, S. & Chan, J. Y. (2011). Inorg. Chem. 50, 8767–8773. Web of Science CrossRef CAS PubMed Google Scholar
Moore, P. B. (1971). Am. Mineral. 56, 1955–1975. CAS Google Scholar
Muessig, E., Bramnik, K. G. & Ehrenberg, H. (2003). Acta Cryst. B59, 611–616. Web of Science CrossRef CAS IUCr Journals Google Scholar
Nasri, R., Fakhar Bourguiba, N., Zid, M. F. & Driss, A. (2014). Acta Cryst. E70, i47–i48. CSD CrossRef IUCr Journals Google Scholar
Nespolo, M. (2015). CHARDI-IT. Laboratoire CRM², Université de Nancy I, France. Google Scholar
Nespolo, M. (2016). Acta Cryst. B72, 51–66. Web of Science CrossRef IUCr Journals Google Scholar
North, A. C. T., Phillips, D. C. & Mathews, F. S. (1968). Acta Cryst. A24, 351–359. CrossRef IUCr Journals Web of Science Google Scholar
Pauling, L. J. (1929). J. Am. Chem. Soc. 51, 1010–1026. CrossRef CAS Google Scholar
Reddy, M. V., Subba Rao, G. V. & Chowdari, B. V. R. (2013). Chem. Rev. 113, 5364–5457. Web of Science CrossRef CAS PubMed Google Scholar
Savina, A. A., Solodovnikov, S. F., Belov, D. A., Basovich, O. M., Solodovnikova, Z. A., Belov, D. A., Basovich, O. M., Solodovnikova, Z. A., Pokholok, K. V., Stefanovich, S. Y., Lazoryak, B. I. & Khaikina, E. G. (2014). J. Solid State Chem. 220, 217–220. CrossRef CAS Google Scholar
Sheldrick, G. M. (2008). Acta Cryst. A64, 112–122. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sheldrick, G. M. (2015). Acta Cryst. C71, 3–8. Web of Science CrossRef IUCr Journals Google Scholar
Shi, P. L., Xia, Z. G., Molokeev, M. S. & Atuchin, V. V. (2014). Dalton Trans. 43, 9669–9676. Web of Science CrossRef CAS PubMed Google Scholar
Solodovnikov, S. F., Klevtsov, P. V., Solodovnikova, Z. A., Glinskaya, L. A. & Klevtsova, R. F. (1998). J. Struct. Chem. 39, 230–237. Web of Science CrossRef CAS Google Scholar
Solodovnikov, S. F. & Solodovnikova, Z. A. (1997). J. Struct. Chem. 38, 765–220. Web of Science CrossRef CAS Google Scholar
Tsyrenova, G. D., Solodovnikov, S. F., Pavlova, E. T., Khaikina, E. G. & Solodovnikova, Z. A. (2009). Russ. J. Inorg. Chem. 54, 743–750. Web of Science CrossRef Google Scholar
Westrip, S. P. (2010). J. Appl. Cryst. 43, 920–925. Web of Science CrossRef CAS IUCr Journals Google Scholar
Xia, J., Song, L. X., Liu, W., Teng, Y., Wang, Q. S., Zhao, L. & Ruanb, M. M. (2015). RSC Adv. 5, 12015–12024. Web of Science CrossRef CAS Google Scholar

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.