Structural order and disorder have attracted considerable attention because of their correlation with material properties1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. Chemical (occupational) order and disorder have been studied mainly by crystal structure analysis using diffraction data. Such order and disorder are often observed among elements demonstrating similar X-ray atomic scattering factors and similar neutron scattering lengths. Here, we consider the chemical order between two elements X and Y (X/Y order) and define the Scattering Contrast Score of elements X and Y, SCS(X, Y) as a measure of the contrasts in X-ray and neutron scattering powers between the X and Y elements.

$${{{{{\rm{SCS}}}}}}\left(X,Y\right)=\left | \frac{N\left(X\right)-N\left(Y\right)}{N\left(X\right)+N\left(Y\right)}\right |+\left | \frac{{{{{{\rm{Re}}}}}}\left[b\left(X\right)\right]-{{{{{\rm{Re}}}}}}\left[b\left(Y\right)\right]}{{{{{{\rm{Re}}}}}}\left[b\left(X\right)\right]+{{{{{\rm{Re}}}}}}\left[b\left(Y\right)\right]}\right | $$

Here N(X) and Re[b(X)] are the number of electrons and real part of the coherent neutron scattering length b of atom X, respectively. There are numerous pairs of X and Y elements with low SCS values (ex. ~300 X/Y pairs with SCS lower than 0.15; red parts in Fig. 1). However, it is difficult to investigate the X/Y chemical order hidden in conventional X-ray and neutron diffraction. Thus, the chemical order is an important unresolved issue with numerous materials (Supplementary Table 1). Herein, we propose a technique to elucidate the chemical order, which is a combination of resonant X-ray powder diffraction (RXRD)17,18,19,20,21 and solid-state nuclear magnetic resonance (NMR)22,23,24,25 assisted by density functional theory (DFT) calculations26,27,28,29,30,31,32,33. Most materials are polycrystalline or powdered. In contrast to single-crystal X-ray and neutron diffraction, this combined technique can be widely applied to both polycrystalline and powdered samples. Direct evidence of the chemical order can be obtained by NMR23; however, it is difficult to quantitatively analyse the chemical order among the constituent elements. In contrast, RXRD enables the quantitative determination of the chemical order by the refinement of occupancy factors, although the refinement results using powder diffraction data are often dependent on the initial structural model. A reliable, quantitative chemical order can be obtained by the present combined technique of NMR and RXRD. We call this combined technique as RXRD/NMR method.

Fig. 1: Numerous pairs of X and Y elements having low scattering contrast score SCS(X, Y).
figure 1

Each number stands for the SCS(X, Y) value. Neutron scattering lengths are taken from the NIST website68. All the SCS(X, Y) values are available in Supplementary Data 1 and Supplementary Table 16.

In this study, we aim to elucidate the Mo/Nb order/disorder in a high ion conductor Ba7Nb4MoO20·0.15 H2O using the RXRD/NMR method (Fig. 2a). We chose Ba7Nb4MoO20·0.15 H2O, because Ba7Nb4MoO20-based oxides and related compounds are emerging materials with high ion conduction, structural disorder and high chemical stability11,34,35,36,37,38,39,40. The crystal structures of Ba7Nb4MoO20-based oxides have been extensively investigated. However, all the structural refinements were performed assuming the complete Mo/Nb disorder11,34,35,36,38,39, because the Mo6+ and Nb5+ cations have both (i) the same number of electrons leading to almost the same X-ray atomic scattering factors and (ii) almost the same neutron scattering lengths (6.715 and 7.054 fm for Mo and Nb, respectively). This indicates a small SCS value for the Mo/Nb pair of 0.037. Because the ions migrate in the oxygen-deficient c′ layers of Ba7Nb4MoO20-based oxides34,36,38,39, the determination of the chemical order/disorder of Mo and Nb atoms at the crystallographic M2 site near the c′ layer is essential (Fig. 2b). Thus, the chemical order of Mo and Nb atoms at the M2 site is an important unsolved issue. Herein, we report the chemical order of Mo atoms at the M2 site near the c′ layer, which offers unprecedented insight into the understanding of the ion diffusion mechanism in hexagonal perovskite-related oxides.

Fig. 2: Strategies for the elucidation of the Mo/Nb order and complete crystal structure of Ba7Nb4MoO20·0.15 H2O.
figure 2

a Combined technique to determine the crystal structure and Mo/Nb order of Ba7Nb4MoO20·0.15 H2O. b Refined crystal structure showing the M1, M2, M3 and M4 sites of Mo and Nb atoms in Ba7Nb4MoO20·0.15 H2O where the Mo and Nb atoms are assumed to be completely disordered.


A single hexagonal phase of Ba7Nb4MoO20·0.15 H2O was prepared by the solid-state reactions (Supplementary Fig. 1). The lattice parameters of Ba7Nb4MoO20·0.15 H2O were determined to be a = 5.8654(3) and c = 16.5390(3) Å using the X-ray diffraction data of the mixture of Ba7Nb4MoO20·0.15 H2O sample and internal standard silicon. To determine the occupancy factors of Nb0.8Mo0.2, Ba and O atoms, preliminary Rietveld analyses of Ba7Nb4MoO20·0.15 H2O were performed using neutron diffraction (ND) data and conventional synchrotron X-ray diffraction (SXRD) data recorded with 0.6994806(5) Å X-ray far from the Nb K-edge, based on the Mo/Nb disordered model (Supplementary Note 1 and Supplementary Fig. 2 for details). Ba7Nb4MoO20·0.15 H2O was confirmed to be a \(P\bar{3}m1\) hexagonal perovskite polytype 7H with four Mo/Nb cation sites (M1, M2, M3 and M4) (Fig. 2b). The occupancy factors were determined as follows:

$$ g ({{{\rm{Nb}}}}; Mi)+g({{{\rm{Mo}}}};Mi)=g ({{{\rm{Ba}}}};{{{\rm{Ba}}}} j)=g({{{\rm{O}}}};{{{\rm{O}}}} k)=1,\\ g({{{\rm{Nb}}}}; M 2)+g({{{\rm{Mo}}}};M 2)=0.92,\\ g ({{{\rm{Nb}}}}; M 4)+g ({{{\rm{Mo}}}}; M 4)=0.08,\\ (i=1 \, {{{\rm{and}}}} \,3; \, j=1,\, 2,\, 3,\, {{{\rm{and}}}} \, 4; k=2,\, 3,\, {{{\rm{and}}}} \, 4)$$

Here, g(Nb; Mi) + g(Mo; Mi) = g(Nb0.8Mo0.2; Mi), and the g(X; s) denotes the occupancy factor of X atoms at s site. The refined crystal parameters of Ba7Nb4MoO20·0.15 H2O were consistent with those reported in the literature11,38.

Direct experimental evidence for Mo order at M2 site by NMR

We performed 93Nb and 95Mo solid-state NMR experiments on Ba7Nb4MoO20·0.15 H2O at a high magnetic field (18.8 T), which enabled the selective observation of Nb and Mo cations, respectively23. Figure 3a and Supplementary Fig. 3 show two-dimensional (2D) 93Nb triple-quantum magic angle spinning (3QMAS) and one-dimensional (1D) 93Nb magic angle spinning (MAS) NMR spectra, respectively. Three peaks are observed in each 93Nb spectrum, indicating the presence of three Nb sites in Ba7Nb4MoO20·0.15 H2O. In contrast, in the 1D 95Mo MAS NMR spectrum, only one peak is observed (Fig. 3b), indicating a single Mo site and Mo order in Ba7Nb4MoO20·0.15 H2O.

Fig. 3: Solid-state NMR spectra of Ba7Nb4MoO20·0.15 H2O, showing the site assignment and Mo order.
figure 3

a 2D 93Nb 3QMAS NMR spectrum and b 1D 95Mo MAS NMR spectrum of Ba7Nb4MoO20·0.15 H2O. An asterisk * denotes spinning sidebands.

To assign the NMR peaks to different crystallographic sites, we performed gauge-including projector augmented wave (GIPAW) DFT calculations of NMR parameters26,27,28,29 with the VASP programme41. To validate this method, the calculated 93Nb and 95Mo NMR parameters were computed for 13 niobates and 11 molybdates (Supplementary Tables 2, 3). The experimental and calculated 93Nb and 95Mo NMR parameters show good correlations (Supplementary Fig. 5). Thus, we can assign the Nb and Mo peaks by comparing the experimental and calculated NMR parameters of Ba7Nb4MoO20·0.15 H2O. For this purpose, the atomic positions in ten possible structural models with different Nb and Mo configurations were optimised by DFT calculations with the P1 space group (Supplementary Figs. 6, 7). The NMR parameters of the optimised structures were estimated by the GIPAW DFT calculations. The calculated peak positions for (Mo2)O4 tetrahedron of Ba7Nb4MoO20 ranged from –29 to –36 ppm depending on the structural model, which is close to the experimental peak position of –47 ppm for Ba7Nb4MoO20·0.15 H2O. (Table 1), where Mo2 is the Mo atom at the M2 site. The calculated quadrupolar coupling constant |CQ | values ranged from 0.36 to 0.90 depending on the structural model, which is consistent with the observed value (≤ 2 MHz). In contrast, the peaks calculated for different sites were not observed in the experimental 95Mo NMR spectrum. Thus, the single 95Mo NMR peak was assigned to the M2 site. Similarly, observed 93Nb peaks at isotropic chemical shifts δiso = –748, –952 and –928 ppm can be assigned to the M1, M2 and M3 sites, respectively (Supplementary Table 4). These results lead us to conclude that the Mo cations are located at the M2 site near the ion-conducting c′ layer, indicating Mo order in Ba7Nb4MoO20·0.15 H2O.

Table 1 Calculated and experimental 95Mo NMR parameters of Ba7Nb4MoO20, showing the peak assignment to the M2 site

Quantitative determination of the occupancy factors of Mo and Nb atoms by resonant X-ray diffraction

We used resonant X-ray diffraction (RXRD) to quantify the occupancy factors of the Mo and Nb atoms in Ba7Nb4MoO20·0.15 H2O. We measured the X-ray absorption near edge structure (XANES) spectra of Ba7Nb4MoO20·0.15 H2O and the resonant (anomalous) scattering factors of Nb atoms (Supplementary Table 5) were determined by Kramers–Kronig transformation from the XANES spectra42 (Supplementary Fig. 8). In the Rietveld analyses of the RXRD data of Ba7Nb4MoO20·0.15 H2O, we used the linear constraints Eq. (2), which were obtained in the preliminary analyses of the ND and conventional SXRD data. The occupancy factors of the Nb and Mo atoms at the M1, M2, M3 and M4 sites were not simultaneously refined because of strong correlations. Therefore, we carefully examined the residual sum of squares (RSS) in the Rietveld analysis for fixed occupancy values of Mo atoms at the Mi site g(Mo; Mi) step-by-step (0.005 step interval for the finest case). Here the RSS is defined as

$${{{{{\rm{RSS}}}}}}=\mathop{\sum }\limits_{i=1}^{N}\frac{1}{{y}_{i}^{{{{{{\rm{obs}}}}}}}}{\left({y}_{i}^{{{{{{\rm{obs}}}}}}}-{y}_{i}^{{{{{{\rm{cal}}}}}}}\right)}^{2}$$

where N, \({y}_{i}^{{{{{{\rm{obs}}}}}}}\) and \({y}_{i}^{{{{{{\rm{cal}}}}}}}\) are the total number of intensity data, the observed and calculated intensities for the ith step, respectively, of the RXRD data. Figure 4 shows RXRD results obtained with 0.6527887(5) Å X-ray at the BL02B2 beamline of SPring-8, which indicates that the occupancies of Mo atoms are 0.00 at the M1, M3 and M4 sites and 0.50 at the M2 site:

$$g ({{{\rm{Mo}}}} ; M 1)=g ({{{\rm{Mo}}}}; M 3)=g ({{{\rm{Mo}}}}; M 4)=0.00,\,\, g ({{{\rm{Mo}}}}; M 2)=0.50$$
$${{{\mathrm{Thus}}}} ,\ g({{{\mathrm{Nb}}}}; M1)=g({{{\mathrm{Nb}}}}; M3)=1.00,\\ g({{{\mathrm{Nb}}}}; M2)=0.42,\,\, g({{{\mathrm{Nb}}}}; M4)=0.08$$
Fig. 4: Determination of the Mo occupancy factors in Ba7Nb4MoO20·0.15 H2O.
figure 4

Variation of the residual sum of squares (RSS; see the definition of Eq. (3)) with the occupancy factor of Mo atom at the a M1, b M2, c M3 and d M4 sites in the Rietveld analyses for the RXRD data measured with 0.6527887(5) Å X-ray at the BL02B2 beamline.

The same values were also obtained also in the Rietveld analyses for the RXRD data taken with a 0.6523630(5) Å X-ray at the different beamline BL19B2 of SPring-8 (Supplementary Fig. 9), validating the Mo occupancy values of Eq. (4). In preliminary analyses, the refined occupancy factors g(Mo; Mi) (i = 1, 3 and 4) were negative (Supplementary Tables 68), supporting the Mo occupancy factors of Eq. (4). These results clearly indicate the Mo chemical order at the M2 site near the ion-conducting c′ layer, which is consistent with the NMR results previously discussed.

Complete crystal structure of Ba7Nb4MoO20·0.15 H2O

To accurately refine the structural parameters of hydrogen and oxygen atoms, we have analysed the crystal structure of Ba7Nb4MoO20·0.15 H2O using neutron diffraction (ND) data collected at 30 and 300 K. During this process, the occupancy factors of Mo and Nb atoms were fixed to the values of Eqs. (4) and (4′), respectively, which were obtained from the analysis of the RXRD data. Excellent fittings were obtained for both ND and RXRD data (Fig. 5 and Supplementary Fig. 10). The crystallographic parameters refined using ND and RXRD data were consistent with each other (Table 2 and Supplementary Table 9). The water content x in bulk crystalline Ba7Nb4MoO20–x(OH)2x (= Ba7Nb4MoO20+xH2x = Ba7Nb4MoO20·x H2O) was calculated to be x = 0.151(5) using the refined occupancy factors at 30 K (Supplementary Table 10), which is consistent with the water content estimated from the thermogravimetric-mass spectroscopic (TG-MS) analyses (Supplementary Fig. 11). The O1–H distance was estimated to be 1.07(4) Å using the refined crystal structure of Ba7Nb4MoO20·x H2O at 300 K, which agrees with the O–H distance of 0.99738(8) Å obtained from its Raman spectrum (Supplementary Fig. 12), indicating the presence of hydroxide ions formed by the hydration. The bond-valence sums (BVSs) of cations and anions for the refined structure of Ba7Nb4MoO20·0.15 H2O agree with their formal charges (Table 2). These results confirm the validity of the refined crystal structure of Ba7Nb4MoO20·0.15 H2O.

Fig. 5: Rietveld fitting patterns of Ba7Nb4MoO20·0.15 H2O.
figure 5

a Resonant X-ray diffraction (RXRD) data measured at 297 K with 0.6527887(5) Å X-ray at the BL02B2 beamline. b Neutron diffraction data at 300 K. The observed and calculated intensities and difference plots are shown by red cross marks, blue solid lines, and blue dots, respectively. Green tick marks denote the calculated Bragg peak positions.

Table 2 Refined crystal parameters and reliability factors in Rietveld analysis of the neutron diffraction data of Ba7Nb4MoO19.849(OH)0.302 (= Ba7Nb4MoH0.302O20.151 = Ba7Nb4MoO20.151H0.302 = Ba7Nb4MoO20·0.151 H2O) at 300 K

Figure 6 shows the refined crystal structure of Ba7Nb4MoO20·0.15 H2O, with the sequence c′hhcchh. Oxygen-deficient lattice O1 and interstitial O5 sites exist in the c′ layer. At high temperatures, oxide ions can migrate via O1–O5 diffusion pathways and the interstitialcy diffusion mechanism as shown by the maximum-entropy method (MEM) neutron scattering length density (NSLD) distribution of Ba7Nb3.9Mo1.1O20.05 at 1073 K36. Similar O1–O5 paths were visualised in MEM NSLD distribution of wet Ba7Nb4MoO20·0.87 H2O at 368 K34.

Fig. 6: Crystal structure of Ba7Nb4MoO19.849(OH)0.302.
figure 6

Refined crystal structure of Ba7Nb4MoO19.849(OH)0.302 at 300 K, which shows the Mo chemical order and site occupancies of Nb and Mo atoms.

Structural disorders have been reported in Ba7Nb4MoO20-based materials11,34,36,38,39. In contrast, a striking feature is the presence of Mo atoms only at the M2 site near the ion-conducting c′ layer, indicating Mo chemical order. DFT-optimised structures with Mo order at the M2 site have slightly lower energies than those with Mo disorder and Mo atoms at M1 and M3 sites, which supports the Mo chemical order at the M2 site (Supplementary Table 11). This is the first report on the chemical order of Mo atoms in Ba7Nb4MoO20-based materials. In the literature11,34,35,36,38,39, all structural analyses were performed based on complete Mo/Nb disorder. Meanwhile, in this study, the Mo order was indicated not only by structural refinements using RXRD data but also by NMR measurements and DFT calculations. An important question is why the Mo order occurs. The probable explanation is as follows: the M2 site has a smaller space compared with other Mi sites (i = 1, 3 and 4) (Supplementary Table 12), and the size of the Mo cation is smaller than that of the Nb cation; thus, Mo order occurs. Indeed, the BVS of Mo at M2 site 5.54 agrees with the formal charge 6 of Mo6+, which is higher than the BVS values of Mo atoms at M1 (4.63), at M3 (4.76) and at M4 (3.51) sites indicating the underbonding and instability of Mo atoms at Mi sites (i = 1, 3 and 4).


The present work has demonstrated the chemical order of Mo atoms at the M2 site near the ion-conducting c′ layer in Ba7Nb4MoO20·0.15 H2O by the combined technique of solid-state NMR, resonant XRD and DFT calculations, in addition to the neutron diffraction and conventional SXRD. The NMR spectra provided direct experimental evidence for the Mo order, while the structural analyses using the RXRD data enabled the quantitative determination of the occupancy factors of Mo and Nb atoms. This combined technique can be used to investigate the hidden chemical order in various ion-conducting hexagonal perovskite derivatives such as Ba7Nb4−xMo1+xO20+x/236, Ba7Nb4−xWxMoO20+x/238, Ba7Nb4−xCrxMoO20+x/239 and Ba3MoNbO8.56,43,44 where the Mo occupancies at the Mi sites (i = 1, 2, 3 and 4) are unknown. Here x is the dopant or excess Mo content. Beyond the limits of the combined technique of conventional X-ray diffraction and NMR (’SMARTER’ crystallography45,46), this RXRD/NMR method can be applied to numerous compounds such as thermoelectric Ag1–xCdxSbTe216 and superconducting Zr5Ir2Os47 exhibiting chemical order/disorder of atoms with both similar X-ray atomic scattering factors and similar neutron scattering lengths (Fig. 1 and Supplementary Table 1). In contrast to the single-crystal X-ray diffraction19,48 and X-ray fluorescence holography49, the RXRD/NMR method uses powders or polycrystalline samples, making it versatile and easily applicable to in situ measurements (e.g., at high temperatures). The combined technique would be useful for investigating not only the periodic average structure but also the short- and intermediate-range order/disorder hidden in conventional diffraction and total scattering.

Next, we discuss the influences of Mo chemical order on the material properties of Ba7Nb4MoO20. The flexibility of the coordination of M2 atoms near the c′ layer was suggested to determine the high ion conduction in Ba7Nb4MoO20·0.5 H2O from the ab initio molecular dynamics simulations34. Since the present work has indicated that Mo cations are localised at the M2 site near the ion-conducting c′ layer in Ba7Nb4MoO20·0.15 H2O, the flexibility of Mo atoms is important for the high ion conduction in Ba7Nb4MoO20-based materials as well as in other Mo-containing ionic conductors such as La2Mo2O950. Therefore, the bulk conductivity of Ba7Nb4−xMo1+xO20+x/2 increases with increasing the excess amount of Mo atoms x from x = 0 to 0.136, which is ascribed to not only a larger amount of excess oxygen atoms but also the larger amount of Mo atoms.

The energy barriers for oxide-ion migration Eb/O of Mo-ordered and virtual Mo-disordered Ba7Nb4MoO20·0.15 H2O were calculated using the bond-valence method51,52. The Eb/O along the c axis in Mo-ordered Ba7Nb4MoO20·0.15 H2O (1.93 eV) is higher than that in the virtual Mo-disordered Ba7Nb4MoO20·0.15 H2O (1.60 eV) [Supplementary Table 13], which is attributable to the narrower bottleneck for oxide-ion migration along the c axis due to the higher occupancy factor of larger Nb cations at the bottleneck triangle (Supplementary Fig. 14).

The substitution of Nb with Mo improves the oxide-ion conductivity because of the larger number of interstitial oxygen atoms (higher carrier concentration). The formation energies ΔHf of the Mo-ordered and virtual Mo-disordered Ba7Nb3.5Mo1.5O20.25 oxides were calculated using the DFT method. The calculated ΔHf values of Mo-ordered models are lower than those of virtual Mo-disordered ones (Supplementary Table 15), which indicates that Mo ordering stabilises Ba7Nb3.5Mo1.5O20.25 with interstitial oxygen atoms more efficiently than Mo disordering, leading to higher oxide-ion conductivity. The hydration enthalpies ΔHhyd of Mo-ordered and Mo-disordered Ba7Nb4MoO20 were also investigated by DFT calculations, because the hydration is important for proton conduction in Ba7Nb4MoO20. Compared with the calculated ΔHhyd of the Mo-disordered Ba7Nb4MoO20 (1.70 kJ mol−1), that of Mo-ordered Ba7Nb4MoO20 (−22.7 kJ mol−1) is close to the experimental value below 300 °C (−24 kJ mol−1)34. The calculated ΔHhyd for the Mo-ordered system (−22.7 kJ mol−1) is lower than that of the Mo-disordered one (1.70 kJ mol−1), indicating that the Mo ordering also stabilises the hydrated Ba7Nb4MoO20 more efficiently compared with Mo disordering. These results demonstrate that the Mo order in Ba7Nb4MoO20 affects the material properties. The present findings represent a major advance in the fundamental understanding of the correlation between the crystal structure and material properties of ionic conductors.


Synthesis and characterisation

The Ba7Nb4MoO20·0.15 H2O samples were prepared by the solid-state reaction method. High-purity (>99.9%) BaCO3, Nb2O5 and MoO3 were mixed as ethanol slurries and ground as dry powders using an agate mortar and pestle. The obtained powders were calcined at 900 °C for 12 h for decarbonation. The materials thus obtained were crushed and ground into fine powders in an agate mortar for 1 h as dried powders and ethanol slurries. The resultant powders were uniaxially pressed at 150 MPa and then sintered in air at 1100 °C for 24 h. The sintered pellets were crushed and ground into fine powders for X-ray powder diffraction (XRD), inductively coupled plasma atomic emission spectroscopy (ICP-AES, Shimadzu ICPS-8100 spectrometer), and TG-MS measurements. The ICP-AES results indicated that the cation molar ratio of Ba7Nb4MoO20·0.15 H2O was Ba: Nb: Mo = 6.89(12): 4.078(18): 1.034(10), which is consistent with the nominal composition. TG-MS analyses of Ba7Nb4MoO20·0.15 H2O were performed using RIGAKU Thermo Mass Photo under He flows at a heating rate of 20 K min–1 up to 900 °C. The Raman spectrum of Ba7Nb4MoO20·0.15 H2O was collected using an NRS-4100 (JASCO Co.) instrument with an excitation wavelength of 532 nm.

Synchrotron X-ray and neutron diffraction experiments and data analysis

Synchrotron X-ray diffraction (SXRD) experiments were performed at beamlines BL02B2 (297 K)53 and BL19B2 (300 K)54 of SPring-8. X-ray wavelengths for resonant X-ray diffraction experiments were selected from the spectrum of Nb K-edge X-ray absorption near edge structure (XANES) for a Ba7Nb4MoO20·0.15 H2O powder. X-ray wavelengths were determined from the X-ray diffraction data of standard silicon powder (SRM 640c) using FullProf software55. Conventional SXRD data were recorded with a 0.6994806(5) Å X-ray. RXRD measurements were performed using a 0.6527887(5) Å X-ray at the BL02B2 beamline and a 0.6523630(5) Å X-ray at BL19B2. Both the conventional SXRD and RXRD data were analysed using the Rietveld method with the computer programme RIETAN-FP56. We used atomic scattering factors in the form of f = f0 + f’′ + if”, where f0 is the Thomson scattering factor and f’ and f” are the resonant (anomalous) scattering factors. The f’ and f” factors of the Nb atom were calculated from the XANES spectrum (Supplementary Fig. 8) recorded at BL19B2 with the programme DiffKK57, and the f’ and f” factors of Ba, Mo and O atoms were obtained from the theoretical values reported by Cromer and Libermann58 (Supplementary Table 5).

Time-of-flight (TOF) neutron diffraction data of Ba7Nb4MoO20·0.15 H2O were obtained at 30 and 300 K using a high-intensity total diffractometer NOVA (BL-21) in the MLF of J-PARC. Rietveld analyses were performed using Z-Rietveld59,60 using neutron diffraction data obtained from the backscattering bank of the NOVA.

The bond-valence-based energy (BVE) landscapes for a test oxide ion and proton in Ba7Nb4MoO20·0.15 H2O were calculated using refined crystal parameters at 300 K using the SoftBV programme51,52. The refined structures and BVE landscapes were depicted using the VESTA 361.

Solid-state NMR experiments

NMR experiments of Ba7Nb4MoO20·0.15 H2O were performed with a 3.2-mm homemade MAS probe at a spinning speed of 20 kHz under a magnetic field of 18.79 T, corresponding to 95Mo and 93Nb Larmor frequencies of 52.16 and 195.84 MHz, respectively. 1D 95Mo and 2D 93Nb NMR spectra were recorded using a JEOL JNM-ECA 800 spectrometer, whereas 1D 93Nb NMR spectra were obtained using a JEOL JNM-ECZ 800 R spectrometer. 95Mo chemical shifts were referenced to 2.0 M aqueous solution of Na2MoO4 at 0.00 ppm (refs. 62,63), and 93Nb chemical shifts were externally referenced to NaNbO3 at −1093 ppm (ref. 64). 95Mo NMR spectra of α-MoO3 and BaMoO4 were also obtained to investigate the relationships between the experimental and DFT-calculated NMR parameters (Supplementary Figs. 4, 5). The 1D 95Mo NMR spectra were acquired by accumulating 22,000 scans using a 1.2 μs single-pulse sequence with a relaxation delay of 20 s. The 1D 93Nb spectra were measured using a spin-echo sequence (2.0 and 4.0 μs), accumulating 1024 scans with a relaxation delay of 1 s. The 2D 93Nb 3QMAS NMR spectra were measured by the conventional three-pulse sequence with z-filter65 (2.0, 0.9 and 15 μs), and recorded with 264 transients averaged for each of the 1024 t1 points with a relaxation delay of 0.2 s. Shearing transformation66 was applied to the spectra. Here, the centre of the F1 axis was set to the centre of the F2 axis.

Density functional theory (DFT)-based calculations

Generalised gradient approximation (GGA) electronic calculations were performed using Vienna Ab initio Simulation Package (VASP)41. We used projector augmented wave (PAW) potentials for Ba, Nb, Mo and O atoms, plane-wave basis sets with a cutoff of 500 eV and the Perdew–Burke–Ernzerhof (PBE) GGA functionals. The crystal parameters refined using the neutron diffraction data of Ba7Nb4MoO20·0.15 H2O at 300 K were used as the initial parameters in the DFT structural optimisations. Atomic coordinates of Ba7Nb4MoO20 were optimised in the space group P1, with the convergence condition of 0.02 eV Å–1. The supercell programme67 was used to generate supercell models. The formation energies of Ba7Nb3.5Mo1.5O20.25 \(\Delta {H}_{{{{{{\rm{f}}}}}}}\) for the Mo-ordered and virtual Mo-disordered models were calculated according to the following equation:

$${{{{{{\rm{Ba}}}}}}}_{7}{{{{{{\rm{Nb}}}}}}}_{4}{{{{{{\rm{MoO}}}}}}}_{20}+1/2{{{{{{\rm{MoO}}}}}}}_{3}\to {{{{{{\rm{Ba}}}}}}}_{7}{{{{{{\rm{Nb}}}}}}}_{3.5}{{{{{{\rm{Mo}}}}}}}_{1.5}{{{{{{\rm{O}}}}}}}_{20.25}+1/4{{{{{{\rm{Nb}}}}}}}_{2}{{{{{{\rm{O}}}}}}}_{5}$$

The optimised structures are shown in Supplementary Fig. 15. The hydration enthalpies \(\Delta {H}_{{{{{{\rm{hyd}}}}}}}\) were also estimated for the Mo-ordered and virtual Mo-disordered models (Supplementary Fig. 16) according to the following reaction:

$${\left({{{{{{\rm{Ba}}}}}}}_{7}{{{{{{\rm{Nb}}}}}}}_{4}{{{{{{\rm{MoO}}}}}}}_{20}\right)}_{4}+{{{{{{\rm{H}}}}}}}_{2}{{{{{\rm{O}}}}}}\to {\left({{{{{{\rm{Ba}}}}}}}_{7}{{{{{{\rm{Nb}}}}}}}_{4}{{{{{{\rm{Mo}}}}}}}{{{{{{\rm{O}}}}}}}_{20.25}{{{{{{\rm{H}}}}}}}_{0.5}\right)}_{4}$$

DFT calculations of the 93Nb and 95Mo chemical magnetic shielding and electric field gradient tensors were performed using the VASP code with a cutoff energy of 700 eV for the plane-wave basis sets, where the total energy converged within 10–8 eV/atom. The GIPAW formalism28 was utilised for the calculations of the NMR chemical shielding tensors.