Correlated disorder and crystal structure of β-VOSO₄

Pauflerite β-VOSO₄ has recently been identified as a one-dimensional S = ½ Heisenberg system, of interest both from a fundamental point of view and a potential material for future spintronics applications. The observation of diffuse scattering in a synthetic β-VOSO₄ provides a microscopic interpretation of the underlying correlated disorder, which is linked to the inversion of the short–long V—O distance pairs along VO₆ chains, forming a local defect state. Direct Monte Carlo modeling indicates that such defects form thin layers with a positive inter-layer correlation, forming small domains with inverted vanadyl bonding patterns. Two-dimensional defects in anisotropic magnetic systems may perturb, or even destroy, long-range magnetic ordering leading to unusual interactions. In particular, the lack of inversion symmetry in the defect layers opens up the possibility for the Dzyaloshinskii–Moriya interaction (DMI) and, consequently, chiral magnetism localized in the defect planes. The defect β-VOSO₄ structure, therefore, opens up new possibilities for the study of low-dimensional magnetic systems.

1. Introduction

Disorder in bulk materials, particularly those with low-dimensional structural building blocks, is an important ingredient of the physics underpinning their unique and technologically relevant properties. Prominent examples of this include superconductivity in systems of weakly bound 1D chains which can be enhanced by disorder (Petrovic et al., 2016), the appearance of spin-Peierls instabilities in S = ½ finite spin chains (Seidel et al., 2003), the creation of edge states in S = , two spin chains, and broken hidden symmetries in S = 1 spin chains (Qin et al., 1995; Fang et al., 2021), as well as strong effects in the transport and magnetic properties of 1D and 2D systems (Chávez et al., 2021; Xiao et al., 2020). These low-dimensional magnetic systems are of interest both from the fundamental point of view (Vasiliev et al., 2018; Haldane, 1983), and as prospective materials for future quantum information technologies (Marchukov et al., 2016; Mathew et al., 2020; Laurell et al., 2021), for which understanding structure–property relationships is a key component of development.

Various V⁴⁺-containing materials are known to exhibit such low-dimensional magnetic interactions (O'Connor et al., 1994), and the compound β-VOSO₄, with natural analog pauflerite (Krivovichev et al., 2007), presents a particularly interesting case. Having been previously described as showing long-range antiferromagnetic ordering by Longo & Arnott (1970) and ferrimagnetic behavior by Villeneuve et al. (1989), β-VOSO₄ has been recently identified as a textbook S = ½ Heisenberg antiferromagnet (Quintero-Castro et al., 2022, in preparation).

The crystal structure is built from distorted VO₆ octahedra with one short (vanadyl bond, 1.607 Å), four intermediate (1.992–1.996 Å), and one long (2.267 Å) V—O bond distances. The corner-sharing octahedra form folded chains propagating along the a-direction, with a strict alternating short-long bonding pattern. SO₄ tetrahedra link neighboring chains which have opposite bonding patterns, in agreement with the inversion symmetry (Fig. 1). Magnetic interactions are present both within the VO₆ chains, and between neighboring chains, mediated by V–SO₄–V superexchange. The latter interactions are very sensitive to geometrical distortions, in particular to small variations of the V—O—S bond angles, and vary from ferromagnetic for α-VOSO₄ to antiferromagnetic for VOSO₄ ˙H₂0 (Villeneuve et al., 1989). These structural distortions and associated changes in magnetic interactions can be easily affected by structural disorder, that, when present, may become an important ingredient of the low-dimensional magnetism.

Figure 1 The crystal structure of β-VOSO4. (a) A single distorted VO6 octahedron with short (vanadyl, S) and long (L) bonds. The octahedra are connected in chains, propagating along the a-axis with alternating bonding patterns. (b) Schematics of the bonding patterns for the two neighboring chains, gray crossed dots indicate inversion centers. (c) The VO6 substructure, viewed down the a-axis, showing the alternating vanadyl bonding pattern.

Here we report on the observation of the diffuse scattering in a synthetic β-VOSO₄, and provide a microscopic interpretation of the underlying correlated disorder. The very nature of the disorder is linked to the inversion of the short-long V—O distance pairs along the chains, forming a local defect state. Diffuse scattering indicates that such defects form thin layers, destroying the alternation of the V—O bonding pattern in the neighboring chains. We present an atomistic realization of the disordered crystal structure derived from Monte Carlo (MC) modeling, and discuss the possible interplay between disorder and magnetic properties.

2. Experiment

Single crystal samples of β-VOSO₄ were synthesized through the reaction V₂O₅ + 2H₂SO₄ → 2β-VOSO₄ + 2H₂O + 0.5O₂, following Sieverts & Mueller (1928) as described by Quintero-Castro et al. (2022, in preparation). The resultant single crystals have a dark green color and needle-like shape (extending to a couple of mm), growing along the crystallographic a-direction.

Single crystal Bragg and diffuse scattering data were collected at the BM01 end station of the Swiss–Norwegian Beamlines at the ESRF (Grenoble, France), on the Pilatus@SNBL diffractometer (Dyadkin et al., 2016). For the Bragg data, the wavelength was set to λ = 0.69027 Å (17.96 keV), the detector was in its closest position (0.139 m) and the exposure time was 2 s per 1° of rotation. For the diffuse scattering, the wavelength was 0.68949 Å the sample–detector distance was 0.639 m, and data were collected for 10 s every 0.1°. Additional datasets were measured at the ESRF ID28 side station (Girard et al., 2019) with a wavelength of 0.980 Å and 0.5 s per 0.25° sampling. Each dataset used a single axis full rotation around ω.

The Bragg data were processed with CrysAlis Pro software (Rigaku Oxford Diffraction, 2015) and the structure solved with SHELXT, then refined with SHELXL (Sheldrick, 2008). Representative parameters characterizing the Bragg data and structural refinement are given in Table 1.

Table 1
Crystal data and structure refinement at 293 K

Empirical formula	VOSO4
Formula weight	163.00
Wavelength (Å)	0.69027
Crystal system, space group	Orthorhombic, Pnma
Unit-cell dimensions (Å)	a = 7.39080 (10)
	b = 6.28640 (10)
	c = 7.09810 (10)
Volume (Å^3)	329.789 (8)
Z	4
Calculated density (Mg m^−3)	3.283
Absorption coefficient (mm^−1)	3.209
F(000)	316
Crystal size (mm)	0.02 × 0.02 × 0.05
Theta range (°) for data collection	3.866–32.873
No. of reflections collected, unique, Rint	2593, 664, 0.0306
Completeness to θ = 24.467°	98.8
Refinement method	Full-matrix least-squares on F^2
No. of data, restraints, parameters	664, 0, 44
Goodness-of-fit on F^2	1.177
Final R indices [I > 2σ(I)]	R1 = 0.0264, wR2 = 0.0702
R indices (all data)	R1 = 0.0268, wR2 = 0.0706
Largest difference peak and hole (e Å^−3)	0.846 and −0.86

3. Results and discussion

The refined average structure agrees well with previous structural reports, with one minor but significant addition: some low residual density positioned 0.6 Å away from vanadium and opposite to the vanadyl bond (Fig. 2). Inspection of reciprocal space shows no indications of non-/partial-merohedral twinning, and the refined average structure has no unusual anisotropy in the ADPs, thus this additional site likely originates from a structural feature and not twinning. This suggests the presence of disorder on the vanadium site, for example, a minor portion of the vanadyl ions having an orientation that is opposite to the majority. The addition of another vanadium atom at the position of the residual improved the refinement, and converged with an occupancy of 2%, suggesting that a small number of VO₆ octahedra have an inverted bonding pattern, and form local defects. The presence of the defects significantly deforms the V—O—S bond angles between neighboring chains, increasing from 136.6° to 137–149°.

Figure 2 The refined average structure of VOSO4, with blue and yellow polyhedra representing VO6 and SO4 units, respectively.

3.1. Disorder in β-VOSO₄

The defects can either be distributed randomly or in a correlated fashion, and this information is encoded in the diffuse scattering data. As illustrated in Fig. 3, we observe rod-like diffuse scattering propagating along b^*. The diffuse rods are centered at the Bragg nodes of the average structure with some intensity modulation along the rods. In particular, for the 3kl layer, the diffuse rods are present for all nodes except l = 0, whereas in the 4kl layer, the diffuse rods are only observed for the nodes with k + l = 2n − 1. Reconstructions of additional sections are available in the supporting information. Since the structure is centrosymmetric, we do not expect any violation of Friedel's law, therefore the apparent asymmetry in the intensity of the rods normal to c* is due to limitations in the finite precision of the orientation matrix (given the extreme sharpness of the diffuse features), beam inhomogeneity and anisotropic crystal sizes.

Figure 3 Reciprocal layers reconstructed from the ID28 data, showing the rod-like diffuse scattering measured with β-VOSO4 crystals. The white gaps originate from the spacing between modules in the detectors.

The observed rod-like diffuse scattering corresponds to planar objects in real space, oriented normal to the crystallographic b-axis. This suggests that the defects are correlated such that they form thin layers in the ac-plane. Within the plane, the bonding pattern of the ordered structure is maintained, but layers of defects destroy the ideal alternating bonding pattern along the b-axis (Fig. 4), introducing disorder and leading to the diffuse maxima along b*. Correlations between these layers along the b-axis are also possible, and information about these can be extracted from the intensity modulations of the rods.

Figure 4 The creation of the models for diffuse scattering calculation. The average structure is expanded to a supercell (left, the SO4 is left out for clearer visualization) where the light and dark polyhedra represent the + and − vanadyl bonding patterns, respectively. Planar defects are introduced into the structure by inverting the bonding pattern of all the VO6 octahedra in the ac-plane, as shown by the black box. Each layer of octahedra (stacked along the b-axis) is assigned a pseudo-spin as indicated by the 1/−1 at the bottom, depending on their orientation relative to the ideal, therefore a complete ac-plane can be represented by one spin variable. The whole defect configuration can then be visualized as shown on the right, with gray and red circles representing regular and defect layers, respectively.

Atomistic MC modeling is one method through which these correlations can be explored. Here, large disordered models with a defined correlation between layers are created and the calculated diffuse scattering patterns are qualitatively compared with the measured scattering patterns. The construction of models is illustrated in Fig. 4. The average unit cell from Bragg diffraction was expanded to a 40 × 100 × 40 supercell ( 296 × 630 × 285 ų), and 10% of the layers were randomly converted to defect layers by changing the orientation of the vanadyl bond (Fig. 4, left-hand view). The change was propagated throughout the ac-plane, making them perfectly periodic in accordance with the observed diffuse rods. To simplify the model further, each ac-plane can be represented as either 1 (for a normal layer) or −1 (for a defect layer), corresponding to the numbers underneath each layer in the supercell in Fig. 4. The configuration of the defects can then be visualized, as in Fig. 4 (right-hand view) as a chain along the b-axis with gray circles representing layers with the correct bonding pattern (1), and those in red representing defect layers (−1). Note that a larger percentage of defects (10% rather than 2%) was used in the modeling as we are limited to fairly small boxes (100 unit cells = 200 layers) so investigation of different ordering patterns would not be possible with such a low level of defects.

Correlations between defect layers were created using an MC algorithm. Layers were chosen at random and their vanadyl bonding orientation swapped. MC moves were then accepted based on the Ising-like energy function:

where σ = +/ − 1 are the pseudo-spin variables assigned to each layer to represent their orientation and k_ij is the interaction energy between spins i and j. The correlation along the b-axis is defined by the Warren–Cowley short-range order parameter

where is the probability of finding spins i and j next to each other along the inter-layer vector, v, and m_i and m_j are the average concentrations of each spin. To drive the system towards the desired correlation, the interaction energies were updated at the end of each MC cycle, to make moves more/less likely to be accepted depending on the difference between the target and current correlations.

The values of α_v vary between −1 and 1. Positive correlations correspond to the clustering of defect layers, which would correspond to the formation of domains of inverted bonding in the crystal. A correlation of zero indicates a random distribution of defects throughout the crystal, and negative correlations mean defects prefer not to be next to each other. Because there is not a 1:1 ratio between normal and defect layers in this model, multiple different configurations satisfy a negative correlation. Both configurations in Figs. 5(c) and (d) have a nearest neighbor correlation of −0.1, which is the most negative this concentration of defects can achieve. If every defect is isolated, will be equal to the average defect concentration and thus, for the models used here, α = 1 − 0.1/(0.1*0.9) = −0.1111. Additionally, the configuration in Fig. 5(c) looks essentially the same as the random one above it, again reflecting the relatively low level of defects in the model. To distinguish between different configurations with negative correlations and move away from randomness towards defects being as far apart as possible, additional nearest neighbors need to be included in the model. For example, specifying a negative nearest neighbor correlation and a positive next-nearest neighbor correlation results in the alternating configuration in Fig. 5(d), and specifying ten layers of neighbors all with a negative correlation would lead to defects being evenly distributed as far apart as possible.

Figure 5 Views (a), (b), (c) and (d) show four different correlation models, where gray and red circles indicate normal and defect layers stacked along the b-axis, and the corresponding calculated diffuse scattering patterns for selected sections. For (d), the two correlation values correspond to that of the nearest neighbors and next-nearest neighbors.

Several different correlation models were tested in order to produce various local defect structures. For each, 20 atomic configurations were generated, and reciprocal space sections of the diffuse scattering were calculated using the program Scatty (Paddison, 2019). The scattering from each of the configurations was averaged to improve the statistics and Lanczos resampling was employed to aid noise reduction. The Bragg scattering was subtracted from all reciprocal sections and the resulting diffuse scattering sections are shown in Fig. 5 along with examples of the defect configurations used to simulate them.

All the models reproduce the narrow diffuse rods in the correct places, indicating that the planar defects inferred from the diffuse scattering are reasonable models of the disorder in this material. It is also evident that the intensity modulations along b* are very sensitive to the inter-layer correlations, even the subtle difference between the models in Fig. 5(c) and 5(d). The differences between each of these models can be more easily visualized in the plots in Fig. 6. Here, the left panel is a histogram of the local defect concentration in the model. A small section of the model (e.g. 20 layers) is examined and the defect concentration is calculated as a percentage. This is repeated, taking random sections each time, to build up the histogram which represents how the defects are distributed throughout the model. In the random model, we see a roughly Gaussian distribution of defects around the global average (10%). The corresponding calculated and experimental diffuse scattering for the 4k1 rod is then shown on the right for the different models. For the random model, the calculated intensity is featureless because there are no inter-layer correlations.

Figure 6 (a) Histograms of the local defect concentration for various correlation models, with the correlations quantified by α in equation (2). The labels −0.1 1NN and −0.1 4NN correspond to models with correlations of −0.1 spanning one and four nearest neighbors, respectively. (b) From top to bottom, the experimental scattering from ID28 and the calculated diffuse scattering patterns for positive and negative correlations along the 4k1 direction.

For positive inter-layer correlations, large portions of the model contain no defects, as they cluster together, therefore the peak in the histogram shifts towards lower concentrations. These models produce diffuse maxima at the correct Bragg nodes (k + l = 2n − 1 for 4kl) with respect to the experimental scattering, and they get narrower and more intense as the correlation becomes more positive, i.e. the antiphase domains in the crystal become larger. For negative correlations, an alternating configuration [as in Fig. 5(d)] produces two peaks in the histogram: one at 50% and one at 0%, showing that the model is made of sections where there is a strictly alternating pattern of defects and normal layers and sections which have no defects. The simulated diffuse scattering shows strong maxima at every Bragg node. As defects move from a random distribution to explicitly avoiding each other (achieved in the model by including more and more nearest neighbors with a negative correlation), the histogram peak remains centered on the global average but becomes narrower as the defects tend toward being evenly distributed in the model. The diffuse scattering is particularly sensitive to this and evolves non-trivially as we approach this limit.

A comparison of the experimental and simulated scattering suggests that a positive correlation gives the best description of the local defect structure in this material. Since the defects are in a higher concentration in the model and the correlations are not being refined, further analysis would need to be done before being able to extract quantitative parameters. In future work, improved model statistics could be achieved using a 3D approach, similar to Chan et al. (2010), and the correlation could be refined against the measured data, assuming the Bragg and diffuse scattering can be separated. Regardless, the positive correlation is enough to conclude that small domains of inverted bonding form in the crystal.

4. Discussion

Disorder in vanadyl bonds has previously been noted for several compounds, such as Ba₂V₃O₉ (Dhaussy et al., 1996), BiCu₂(P_1–xV_x)O₆ (Mentré et al., 2006), Pb₂V₃O₉, Pb₂As₂VO₉ (Mentré et al., 1999), (VO)₂P₂O₇ (Hiroi et al., 1999) and Na_xV₂O₅ (Krogstad et al., 2020). However, none of these studies have been conclusive in this regard, due to the influence of very low-symmetry space groups, chemical impurities, crystal quality, and ion diffusion. Here we present clear indications of a new type of defect in high-quality β-VOSO₄ single crystals, namely 2D planar objects of the same vanadyl bonding pattern.

These defect layers could significantly affect the magnetic properties of the compound through multiple mechanisms. Firstly, the local magnetization of vanadium will be slightly deformed by the polarization effects of π-bonding in the vanadyl ions (Ballhausen & Gray, 1962). Secondly, the change of the bonding patterns in the neighboring chains connected via SO₄ might affect the intra-chain magnetic interaction, cutting the spin chain and creating free S = ½ contributions. In theory, at high concentrations, these defects could induce a spin-Peierls instability (Seidel et al., 2003) driving the compound to dimerization, but it is unclear what the defects convey at low concentrations. This is especially relevant at high magnetic fields near the field-induced long-range magnetic order, where spin-nematic and spin-glass phases have been proposed (Büttgen et al., 2014). It is also likely that these defects have an influence on the lack of long-range magnetic ordering reported down to 0.1 K (Quintero-Castro et al., 2022, in preparation). Furthermore, the lack of inversion symmetry at the boundaries between defect layers opens the possibility for Dzyaloshinskii–Moriya interactions (DMI) to arise, which would open a magnetic field-dependent spin gap modifying the system's magnetic phase diagram (Oshikawa & Affleck, 1997).

5. Conclusions

The crystal structure and disorder in β-VOSO₄ have been elucidated through single-crystal diffraction and atomistic modeling. The structure was found to contain defects, whereby 2% of the VO₆ octahedra have inverted vanadyl bonding patterns. Diffuse scattering analysis suggested that these defects are correlated, forming planes of defects normal to the crystallographic b-axis and disrupting the alternating bonding pattern. Atomistic modeling shows that further correlations between defect planes are present in β-VOSO₄ such that defect layers are clustered together forming small antiphase domains of inverted vanadyl bonding patterns. These defects are not mobile at ambient conditions and we expect them to be created during crystallization. The 2D defect layers represent a new type of defect in these systems, one which is likely to impact the magnetic properties and potentially facilitate unusual magnetic interactions. In this respect, the defect β-VOSO₄ structure offers a new and as yet unexplored playground for the study of the relationships between disorder and magnetic properties.

Supporting information

Computing details

Program(s) used to refine structure: SHELXL2014/7 (Sheldrick, 2014).

(I)

Crystal data

O5SV	Dx = 3.283 Mg m^−3
Mr = 163.00	Synchrotron radiation, λ = 0.69027 Å
Orthorhombic, Pnma	Cell parameters from 2031 reflections
a = 7.3908 (1) Å	θ = 3.9–32.9°
b = 6.2864 (1) Å	µ = 3.21 mm^−1
c = 7.0981 (1) Å	T = 293 K
V = 329.79 (1) Å^3	Cube-like, green
Z = 4	0.05 × 0.03 × 0.02 mm
F(000) = 316

Data collection

Dectris-CrysAlisPro-abstract goniometer imported dectris images diffractometer	663 independent reflections
Radiation source: synchrotron	644 reflections with I > 2σ(I)
Synchrotron monochromator	Rint = 0.031
Detector resolution: 5.8140 pixels mm^-1	θmax = 32.9°, θmin = 3.9°
Absorption correction: multi-scan CrysAlisPro 1.171.40.67a (Rigaku Oxford Diffraction, 2019) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm.	h = −10→10
Tmin = 0.298, Tmax = 1.000	k = −9→9
2591 measured reflections	l = −10→10

Refinement

Refinement on F^2	0 restraints
Least-squares matrix: full	w = 1/[σ^2(Fo^2) + (0.0372P)^2 + 0.3329P] where P = (Fo^2 + 2Fc^2)/3
R[F^2 > 2σ(F^2)] = 0.025	(Δ/σ)max < 0.001
wR(F^2) = 0.069	Δρmax = 0.87 e Å^−3
S = 1.15	Δρmin = −0.75 e Å^−3
663 reflections	Extinction correction: SHELXL-2014/7 (Sheldrick 2014, Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
44 parameters	Extinction coefficient: 0.024 (4)

Special details

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 (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
V1	0.33361 (18)	0.2500	0.26749 (7)	0.00686 (19)	0.979 (5)
V2	0.391 (8)	0.2500	0.253 (3)	0.00686 (19)	0.021 (5)
S1	0.37713 (7)	0.7500	0.36813 (7)	0.00697 (14)
O1A	0.3747 (2)	0.56303 (18)	0.24372 (15)	0.0116 (2)
O1B	0.5447 (2)	0.7500	0.4805 (2)	0.0135 (3)
O1C	0.2157 (2)	0.7500	0.4904 (2)	0.0127 (3)
O2	0.1265 (2)	0.2500	0.3323 (2)	0.0125 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
V1	0.0044 (5)	0.00812 (17)	0.0080 (2)	0.000	−0.00067 (15)	0.000
V2	0.0044 (5)	0.00812 (17)	0.0080 (2)	0.000	−0.00067 (15)	0.000
S1	0.0062 (2)	0.0079 (2)	0.0068 (2)	0.000	−0.00018 (14)	0.000
O1A	0.0158 (5)	0.0079 (4)	0.0110 (5)	−0.0004 (4)	0.0004 (3)	−0.0017 (3)
O1B	0.0080 (6)	0.0227 (7)	0.0099 (6)	0.000	−0.0031 (5)	0.000
O1C	0.0086 (6)	0.0201 (7)	0.0093 (6)	0.000	0.0020 (5)	0.000
O2	0.0087 (7)	0.0164 (7)	0.0125 (7)	0.000	0.0003 (5)	0.000

Geometric parameters (Å, º)

V1—O2	1.599 (2)	V2—O2	2.04 (6)
V1—O1A	1.9983 (12)	S1—O1A	1.4702 (11)
V1—O1A^i	1.9983 (12)	S1—O1A^v	1.4702 (11)
V1—O1C^ii	1.9999 (17)	S1—O1B	1.4730 (16)
V1—O1B^iii	2.0022 (16)	S1—O1C	1.4755 (16)
V1—O2^iv	2.277 (2)	O1B—V2^iii	1.95 (2)
V2—O2^iv	1.84 (6)	O1B—V1^iii	2.0022 (16)
V2—O1B^iii	1.95 (2)	O1C—V1^vi	1.9999 (17)
V2—O1A	1.973 (3)	O1C—V2^vi	2.03 (2)
V2—O1A^i	1.973 (3)	O2—V2^vii	1.84 (6)
V2—O1C^ii	2.03 (2)	O2—V1^vii	2.277 (2)
O2—V1—O1A	99.79 (6)	O1B^iii—V2—O2	88.1 (17)
O2—V1—O1A^i	99.79 (6)	O1A—V2—O2	87.1 (17)
O1A—V1—O1A^i	159.96 (11)	O1A^i—V2—O2	87.1 (17)
O2—V1—O1C^ii	96.23 (9)	O1C^ii—V2—O2	83.0 (18)
O1A—V1—O1C^ii	86.83 (3)	O1A—S1—O1A^v	106.16 (9)
O1A^i—V1—O1C^ii	86.83 (3)	O1A—S1—O1B	109.59 (7)
O2—V1—O1B^iii	99.96 (9)	O1A^v—S1—O1B	109.59 (7)
O1A—V1—O1B^iii	90.41 (4)	O1A—S1—O1C	110.10 (7)
O1A^i—V1—O1B^iii	90.41 (4)	O1A^v—S1—O1C	110.10 (7)
O1C^ii—V1—O1B^iii	163.81 (10)	O1B—S1—O1C	111.17 (10)
O2—V1—O2^iv	178.61 (4)	S1—O1A—V2	140.9 (6)
O1A—V1—O2^iv	80.16 (6)	S1—O1A—V1	137.59 (7)
O1A^i—V1—O2^iv	80.16 (6)	V2—O1A—V1	12.6 (17)
O1C^ii—V1—O2^iv	82.37 (7)	S1—O1B—V2^iii	136.9 (17)
O1B^iii—V1—O2^iv	81.43 (8)	S1—O1B—V1^iii	149.48 (12)
O2^iv—V2—O1B^iii	95 (2)	V2^iii—O1B—V1^iii	12.6 (17)
O2^iv—V2—O1A	92.7 (17)	S1—O1C—V1^vi	136.55 (11)
O1B^iii—V2—O1A	92.8 (6)	S1—O1C—V2^vi	149.0 (16)
O2^iv—V2—O1A^i	92.7 (17)	V1^vi—O1C—V2^vi	12.4 (16)
O1B^iii—V2—O1A^i	92.8 (6)	V1—O2—V2^vii	144.0 (7)
O1A—V2—O1A^i	172 (3)	V1—O2—V2	0.8 (6)
O2^iv—V2—O1C^ii	93.7 (16)	V2^vii—O2—V2	144.73 (16)
O1B^iii—V2—O1C^ii	171 (3)	V1—O2—V1^vii	145.14 (10)
O1A—V2—O1C^ii	86.8 (7)	V2^vii—O2—V1^vii	1.2 (7)
O1A^i—V2—O1C^ii	86.8 (7)	V2—O2—V1^vii	145.9 (6)
O2^iv—V2—O2	176.7 (12)

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