# Reciprocal space imaging of ionic correlations in intercalation compounds

## Abstract

The intercalation of alkali ions into layered materials has played an essential role in battery technology since the development of the first lithium-ion electrodes. Coulomb repulsion between the intercalants leads to ordering of the intercalant sublattice, which hinders ionic diffusion and impacts battery performance. While conventional diffraction can identify the long-range order that can occur at discrete intercalant concentrations during the charging cycle, it cannot determine short-range order at other concentrations that also disrupt ionic mobility. In this Article, we show that the use of real-space transforms of single-crystal diffuse scattering, measured with high-energy synchrotron X-rays, allows a model-independent measurement of the temperature dependence of the length scale of ionic correlations along each of the crystallographic axes in sodium-intercalated V2O5. The techniques described here provide a new way of probing the evolution of structural ordering in crystalline materials.

## Main

The cathodes of the first lithium-ion batteries developed over 40 years ago were materials with van der Waals-bonded layers between which the lithium ions could be rapidly intercalated and deintercalated1,2. Since the first tests on LixTiS2, many other intercalation compounds have been investigated as potential electrodes because of their rapid charging rates and inherent stability over multiple charge–discharge cycles3. Insertion of alkali ions between such weakly bonded layers is highly reversible because the weak coupling of the intercalants to the host produces only minor structural modifications during the electrochemical cycle. The potential use of multivalent ions in batteries has renewed interest in such materials4,5,6, although weak coupling can compromise a battery’s gravimetric and volumetric energy density. The transition metal oxides most frequently used as cathodes in consumer electronics, such as LiCoO2 (ref. 7), are also layered materials, albeit with stronger host–intercalant interactions8.

Even if the interactions of the alkali ions with the host lattice are weak, the Coulomb interactions between the intercalants cannot be ignored, especially at high concentrations9. Voltage anomalies as a function of lithiation in TiS2 were quickly ascribed to correlations between the intercalants induced by Coulomb repulsion10, causing changes in the configurational entropy and consequently the chemical potential for ion insertion11. Berlinsky et al. showed that order–disorder transitions can be inferred from extrema in the voltage derivatives versus composition9. Such anomalies have also been observed in transition metal oxide cathodes12,13, where they have been modelled using both lattice gas13,14,15 and first-principles methods16,17,18. Ordering of intercalants impacts battery performance by disrupting ionic mobility, and there is evidence that suppressing ordering can improve charging rates19.

Order–disorder transitions are usually associated with discrete stoichiometries, corresponding to commensurate ordering of the intercalant sublattice. In LixCoO2 and other transition metal oxides, ordering occurs at $$x = \frac{1}{2}$$ and a number of other rational fractions11,20,21,22. Although it is possible to infer order–disorder transitions at these values from bulk measurements, direct evidence can only be obtained by the observation of (often weak) superlattice peaks from unit-cell expansion in diffraction measurements. The low cross section of lithium makes this difficult with X-ray scattering, unless associated changes in the host lattice space group are observable. For this reason, electron diffraction provided the first conclusive evidence of lithium-vacancy ordering in Li0.5CoO2 (ref. 23), showing unit-cell doubling both within and perpendicular to the planes. However, conventional Bragg diffraction is only sensitive to long-range order and cannot be used to probe short-range correlations, which will also play a role in decreasing ionic mobility across the whole composition range. Understanding such correlations is important in modelling diffusion kinetics, for example as a function of concentration fluctuations close to the electrolyte interface.

The most powerful technique to probe short-range correlations is single-crystal diffuse scattering, using either X-rays or neutrons, which comprises all the scattering that results from nanoscale deviations from the average crystalline structure24,25,26. The average occupancies of all the atomic sites within the translationally invariant unit cell of a crystal can be determined from Bragg peak intensities using standard crystallographic techniques, but, apart from thermal diffuse scattering from lattice vibrations, all the scattering between (and under) the Bragg peaks results from the presence of local disorder, often in the form of point defects, such as interstitials and vacancies, or of short-range order from defect–defect correlations. When measured over a sufficiently large volume of reciprocal space, diffuse scattering contains a true thermodynamic average of disorder scattering from the entire sample and so is complementary to techniques that utilize X-ray coherence to reconstruct real-space images of atomic correlations over a limited coherence volume.

Although it is an extremely powerful technique, single-crystal diffuse scattering has not been as widely utilized as, for example, pair-distribution-function (PDF) measurements of polycrystalline materials27, because of the challenge of both measuring and modelling large volumes of reciprocal space (Q space). By contrast, the PDF technique, which gives a one-dimensional spherical average of interatomic vector probabilities, has become a standard tool in materials science because the information can be interpreted intuitively and modelled effectively with existing software. In recent years, Thomas Weber and colleagues have shown that it is possible to extend the PDF technique to three dimensions by transforming single-crystal diffuse scattering into real space28. The result is a Patterson function, containing the summed probabilities of interatomic vectors in the crystal with contributions from both the average structure and local disorder. By excluding Bragg scattering from the real-space transform, the resulting PDF function only includes those interatomic vectors whose probabilities differ from the average (3D-ΔPDF). This is not possible in polycrystalline data because of the substantial overlap of Bragg and diffuse scattering intensity, but in single crystals it allows a determination of the length scales of short-range order extending over 100 Å or more. By exploiting a new generation of fast area detectors, it is possible to measure sufficiently complete three-dimensional volumes of reciprocal space in under 20 min using high-energy X-rays at a synchrotron source, enabling the generation of robust ΔPDFs fast enough to allow detailed studies as a function of temperature.

We have applied this technique to measure ionic correlations in single crystals of V2O5 intercalated with sodium ions. In β-NaxV2O5, the sodium ions are confined to two-leg ladders29, but the configuration of ions within the partially occupied ladders cannot be determined by powder diffraction alone. The ΔPDFs reveal the tendency for sodium ions to form zigzag configurations within each ladder, which are in phase with those in neighbouring ladders, as well as the temperature dependence of the correlation length of this ordering in all three crystallographic directions, obtained from the exponential decay of the ΔPDF intensities. It has previously been reported that this material undergoes a structural phase transition just above 200 K (ref. 30), but our results show that this temperature marks a crossover from two- to three-dimensional correlations without true long-range order being established down to 30 K. In the following Article, we describe the measurement process in more detail, the steps used to generate the 3D-ΔPDF results and the subsequent analysis to extract ionic correlation lengths, before discussing the implications of this technique for studying battery electrodes. Although the focus of this Article is on intercalation compounds, the techniques described here are applicable to any investigations of crystalline disorder.

The orthorhombic structure of pure V2O5 consists of both edge- and corner-shared square pyramids that form van der Waals-bonded planes, between which a variety of cations may be intercalated31. However, for some intercalants above a critical concentration, the structure transforms to a monoclinic β-phase (space group C2/m) with a more complex network of vanadium oxide octahedra and square pyramids that contain one-dimensional channels, into which the intercalated cations are inserted (Fig. 1)29,32,33. It has recently been proposed that the tunnel framework improves structural stability during intercalation/deintercalation34,35. In β-NaxV2O5 (0.2 x 0.5), the intercalants partially occupy a sublattice of two-leg ladders, whose legs are parallel to the crystal’s b axis and whose rungs are parallel to the c axis. The ladder sites would be filled with a value of $$x = \frac{2}{3}$$, but stereochemical constraints prevent occupations much greater than 50% since the ladder rungs are too short (<2 Å). It has been postulated that the excess sodium ions would occupy additional octahedral and tetrahedral sites for x > 0.33 in a β′-phase36, but these have never been conclusively observed (see Supplementary Fig. 15). In Supplementary notes, we discuss features in our data that provide evidence in favour of the octahedral site occupation preferred in ref. 36.

We measured diffuse X-ray scattering on single crystals of β′-NaxV2O5 with x = 0.45, corresponding to 68% occupation, using high-energy monochromatic beams at two synchrotron X-ray sources, the Advanced Photon Source (APS) and the Cornell High Energy Synchrotron Source (CHESS). Advances in the dynamic range and speed of X-ray area detectors now allow both the Bragg peaks and diffuse scattering to be measured efficiently, with three-dimensional volumes of scattering in reciprocal space, S(Q), collected in under 20 min at each temperature. More details are given in Methods and Supplementary methods.

The diffuse scattering is mostly confined to rods that are parallel to H, that is, orthogonal to the ladders, occurring at half-integer values of K (Fig. 2), from which we can infer that there is predominantly two-dimensional short-range order within the ladder planes generated by ionic correlations within the sodium sublattice that tend to double the unit cell along b. A sinusoidal modulation of the rod intensities as a function of L, parallel to the direction of the ladder rungs, whose periodicity is the inverse of the rung length, indicates that site occupations across each ladder rung are strongly correlated. At room temperature, there are no sharp peaks along the rods (Fig. 2c), which are broad laterally (~0.1 Å−1 along K and ~0.13 Å−1 along L), indicating that correlation lengths are of the order of 10–20 Å. However, below ~230 K, peaks start to appear at all integer values of H (Fig. 2b), growing steadily in intensity down to the lowest measured temperature of 30 K (Fig. 2a). These peaks are evidence of the development of longer-range three-dimensional correlations within the sodium sublattice below ~230 K, in agreement with previous X-ray measurements30,37.

Most analyses of diffuse scattering require a calculation of the Q variation of the data using atomistic models that parametrize the disorder, but an alternative approach is to Fourier transform the data to generate real-space PDFs. This has the advantage of converting complicated intensity distributions in reciprocal space into discrete peaks in real space, whose positions and intensities are given by the interatomic vectors present in the disordered structure and their weighted probabilities, respectively. A further simplification is to include in the generated PDFs only those peaks whose probabilities deviate from the average structure. This is possible using the ‘punch and fill’ method pioneered by Weber and colleagues28, which utilizes the fact that the total scattering can be separated into two components: one representing the average crystal, that is, the Bragg peaks, and the other representing the diffuse scattering from crystalline defects.

$$I_{\mathrm{tot}}({\mathbf{Q}}) = F({\mathbf{Q}})F^ \ast ({\mathbf{Q}}) = |F_{HKL}({\mathbf{Q}})|^2 + |\Delta F({\mathbf{Q}})|^2$$
(1)

where Itot(Q) is the measured total scattering and F(Q) is proportional to the Fourier transform of the electron density. The Fourier transform of this scattering function is also separable.

$$\begin{array}{*{20}{l}} {P_{\mathrm{tot}}({\mathbf{r}})} \hfill & = \hfill & {{\mathrm{FT}}[|F_{HKL}({\mathbf{Q}})|^2] + {\mathrm{FT}}[|\Delta F({\mathbf{Q}})|^2]} \hfill \\ {} \hfill & = \hfill & {P_{HKL}({\mathbf{r}}) + \Delta P({\mathbf{r}})} \hfill \end{array}$$
(2)

PHKL is the Patterson function of the average structure while ΔP(r), or the 3D-ΔPDF, is the difference Patterson function due to the disorder. The punch and fill method isolates ΔP(r) by eliminating the Bragg peaks, that is, by removing the scattering in a small sphere around each Bragg peak and interpolating over the missing data, before performing the Fourier transform (Supplementary Fig. 7).

We have used this technique to produce ΔPDFs that eliminate any contribution from the framework V2O5 lattice to first order and leave only the sodium–sodium pair correlations. In the case of purely substitutional disorder, when local relaxations can be neglected, Weber and Simonov show that the 3D-ΔPDF consists of peaks at real-space vectors that connect the disordered sites (see, for example, equation (14) in ref. 28). The intensity of the peaks is given by the difference between the conditional probability that if one site is occupied the connected site is also occupied, and the average occupation. This means that ΔPDF peaks can have either positive or negative intensities. In the average structure determined by powder diffraction32 the sodium ladder sites are randomly occupied with a probability of ~0.5, so this difference can range from 0.5 to −0.5, but in a disordered structure the absolute peak intensities will in general decay with increasing distance as the correlated site occupations become increasingly random. The decay constant gives the correlation length of the substitutional ordering.

Figure 2d–f shows the ΔPDF in the bc plane, displayed in a symmetric log plot, which consists of a repeating triplet pattern of peaks with positive (red) intensity and negative (blue) intensity. With increasing distance from the origin, the absolute intensity of these peaks diminishes as expected. At 250 K (Fig. 2f) these correlations extend over only a few neighbouring sodium ladders, but as the temperature is lowered the correlation length increases substantially, as seen at 200 K (Fig. 2e). At 50 K, the correlations extend over more than 100 Å in all directions (Fig. 2d). This is the real-space corollary of the growing intensity and sharpness of the reciprocal space peaks shown in the insets to Fig. 2a–c.

Some immediate conclusions can be drawn by inspection of the ΔPDF without detailed modelling. We show in Fig. 3 a comparison of the two-leg ladders within the yz plane and the corresponding 3D-ΔPDF. The triplets of red–blue–red and blue–red–blue peaks are entirely consistent with this ladder structure, producing three-leg ladders because the origin of each interatomic vector can be on either leg. This is a consequence of analysing scattering amplitudes without phase information, so that ΔP(r) = ΔP(−r). By choosing one of the sodium sites to be occupied, we can determine the other neighbouring sites that are occupied (unoccupied) by identifying all the interatomic vectors that have positive (negative) ΔPDF intensity. In Fig. 3, we show the resulting configuration of occupied sites by indicating atoms on unoccupied sites with lighter colour. This shows that all the nearest-neighbour sites within the ladder are likely to be unoccupied, but next-nearest-neighbour sites are likely to be occupied, leading to a zigzag configuration. This is to be expected from considerations of both stereochemistry and Coulomb repulsion, but the ΔPDF also makes clear that these zigzag configurations are in phase with neighbouring ladders, since the vectors connecting sites on the same leg (left or right), that is, with ±ma ± nc, where m and n are integers, have positive probability, whereas those connecting the opposite legs have negative probability.

There are weaker ΔPDF peaks in other planes that could be associated either with additional sodium sites, such as those proposed by Galy et al.36, or with relaxations of the vanadium and oxygen ions. In the scattering data, additional superlattice peaks are also observed at $$K = \pm \frac{1}{6}$$, below ~130 K, which is the temperature of a metal–insulator transition ascribed to V4+/V5+ charge ordering37,38. These give rise to a weak modulation of the ΔPDF intensity along the y axis, but do not appear to affect the sodium correlations. The additional peaks are shown in Supplementary Fig. 5.

To quantify the growth in correlations, we have modelled the dependence of the absolute ΔPDF peak intensities on interatomic distance by analysing one-dimensional slices selected to eliminate all but allowed sodium–sodium vectors (Supplementary Figs. 11 and 12). Figure 4 shows that these intensities follow an exponential decay at high temperature, consistent with a one-dimensional Ornstein–Zernike function39. At lower temperature, it is necessary to include the effect of the finite resolution in reciprocal space, which produces an envelope function that suppresses the real-space transform at large distances. Its Gaussian width can be estimated by fitting the total PDF, that is, the data transformed without eliminating the Bragg peaks, which represents long-range crystalline order (Supplementary Figs. 8 and 9). The ΔPDF intensities in Fig. 4 have been fitted to the product of this Gaussian envelope function with a temperature-dependent exponential decay function, whose decay constant is a direct measure of the correlation length, ξ.

The temperature dependence of these fits is shown in Fig. 4 along all three crystallographic directions. As the correlation length approaches 200 Å, the effect of the finite resolution leads to an increase in the error bars, although ξ and the r = 0 amplitude of the ΔPDF correlations is measurable down to 30 K.

The elimination of the interatomic vector probabilities of the average structure from the PDF allows short-range order to be determined over length scales that exceed 100 Å, which means that the correlation of ions undergoing an order–disorder transition can be monitored as a function of temperature from well above to well below the transition (Fig. 5). In β-Na0.45V2O5, there is a gradual increase in the correlation lengths along the y and z directions, ξy and ξz, from 275 K to 230 K, which is easier to see in plots of ξ−1 versus temperature (Fig. 5c). At ~230 K, this induces a sharp increase in correlation length of the sodium pair correlations along the x axis, ξx, perpendicular to the sodium ladder planes, marking a crossover from two- to three-dimensional correlations (Fig. 5b). However, it does not appear to be a true second-order phase transition, since ξx saturates at 190 K with a value of only 170 Å, corresponding to approximately 10 unit cells or 20 ladder planes. The increase in ξx in turn generates a more rapid increase in ξy and ξz, again without leading to a true divergence. The maximum correlation length at 30 K is still under 200 Å. Instead of a discontinuity in the slope of $$\xi _y^{-1}$$ and $$\xi _z^{-1}$$, defining a unique transition temperature, there is a crossover between two temperature regimes at ~200 K close to where ξx saturates (Fig. 5c).

We propose that the saturation in ξx is a consequence of frustration. Due to the body-centring of the monoclinic structure, the ladder sites in the yz planes at x = 0 and $$x = \frac{1}{2}a$$ are shifted by $$\pm \frac{1}{2}b$$, which makes the strength of the Coulomb interactions independent of the relative phase of the zigzag site occupations between two neighbouring planes. This degeneracy implies that the phase must be stabilized by next-nearest-neighbour interactions along the x axis, which may be weak enough to allow stacking faults that inhibit true long-range order. Disorder of the excess sodium ions that cannot be accommodated on the ladder sites would contribute to this frustration by locally pinning the zigzag phase in each plane. The fall in ξx below 190 K indicates that any such frustration mechanism is increased by the growth in intraplanar correlations. This could be tested by further measurements on β-Na0.33V2O5 corresponding to precisely 50% occupation of the ladder sites.

The intensities of the ΔPDF peaks increase monotonically with decreasing temperature (Fig. 5a). Along the x axis, the temperature dependence of the amplitude resembles an incipient order parameter, with a value close to 0 above the transition (Supplementary Fig. 6). However, along the y and z directions, there is a much weaker temperature dependence, reflecting the strong nearest-neighbour correlations at all temperatures.

Transforming X-ray diffuse scattering data into 3D-ΔPDFs produces a remarkable simplification in how the data are represented. It is possible to interpret the short-range order in the crystal structure without a detailed simulation of the disorder, which has in the past required the optimization of a large number of parameters over a substantial volume of reciprocal space. In fact, the PDF intensities directly determined from the transformed data are related by simple analytic functions to the Warren–Cowley parameters that are frequently used to parametrize diffuse scattering models24. Even without a well defined model of the underlying disorder, our results show that the spatial dependence of the PDF intensities provides a method of extracting correlation lengths as a function of temperature or other parametric variable. We believe that this method will be of value in investigations of the evolution of structural ordering in many crystalline materials.

The ability to generate a real-space ‘image’ of interatomic vectors from reciprocal space data, albeit one without phase information, makes this an especially powerful tool in the investigation of intercalation compounds, as it is especially suited to the measurement of ionic correlations on a sublattice that is distinct from the host structure. Although we have not yet tested this technique on lithiated compounds, we believe that the elimination of the average structure of heavier elements will allow lithium–lithium correlations to be measured in spite of their low cross section. In this Article, the method has allowed us to show that the apparent order–disorder transition reported in NaxV2O5 is actually a crossover from two-dimensional to three-dimensional correlations, with interplanar correlations saturating at only ~150 Å, probably because of frustration. Such detailed insight into intercalant disorder is not possible by any other method.

Although we have focused on layered materials, the technique can be utilized in more three-dimensional insertion compounds such as the oxide spinels15, as well as other disordered materials. The principal limitation of this technique is the need for single crystals, which makes in operando experiments within electrochemical cells challenging, although not impossible. Sample thicknesses can be less than 100 μm, and provided the rest of the cell is polycrystalline the real-space transform will only contain discrete peaks that correspond to the crystalline electrode. Measuring the phase diagram of order–disorder transitions as a function of both temperature and intercalant concentration will allow the strength of interionic interactions to be determined and improve our understanding of the how they limit ionic mobility. Such measurements could also be used to evaluate the effectiveness of strategies for mitigating these limitations, for example by co-intercalation with aliovalent cations40 or substitution of transition metals on the vanadium sites41, to disrupt ordering phenomena.

## Methods

### Synthesis

NaxV2O5 crystals were grown using a self-flux technique. Five grams of V2O5 (Aldrich, >98%) in a nickel crucible was placed in an oven inside an Ar-atmosphere glovebox. The sample was heated to 720 °C. While the sample was liquid, the crucible was removed from the furnace and a stoichiometric amount of NaI (Aldrich, 99.5%) was added to the liquid. The sample was then put back in the furnace and cooled to 650 °C over 1 h, then radiatively cooled to room temperature. Single crystals were isolated from the flux. EDX analysis showed that the sodium concentration, when normalized to the nominal vanadium stoichiometry, was x = 0.45 ± 0.02.

### X-ray scattering

Three-dimensional volumes of diffuse X-ray scattering were collected at APS and CHESS. The APS data were measured on sector 6-ID-D using an incident energy of 87.1 keV and a Dectris Pilatus 2M with a 1-mm-thick CdTe sensor layer. The CHESS data were measured on beamline A2 using an incident beam energy of 27.3 keV and a Dectris Pilatus 6M detector with a 1-mm-thick Si sensor layer. The data were collected from 30 K to 300 K, with samples cooled by flowing He gas below 100 K and N2 gas above 100 K. During the measurements, the samples were continuously rotated about an axis perpendicular to the beam at 1° s−1 over 370°, with images read out every 0.1 s. Three sets of rotation images were collected for each sample at each temperature to fill in gaps between the detector chips. The resulting images were stacked into a three-dimensional array, oriented using an automated peak search algorithm and transformed in reciprocal space coordinates using the software package CCTW (Crystal Coordinate Transformation Workflow)42, allowing S(Q) to be determined over a range of ~±15 Å−1 in all directions. Further details are given in Supplementary methods.

## Data availability

Files containing the datasets used in this Article are available for download from the Materials Data Facility43 (https://doi.org/10.18126/ooin-ce23) as NeXus files stored in the HDF5 format44. The files for each measured temperature contain S(Q), ΔPDF and, at three temperatures, the total PDF results. The data can be plotted using the Python package NeXpy (http://nexpy.github.io/nexpy/).

## Change history

• ### 30 October 2019

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

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## Acknowledgements

This work was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division and Scientific User Facilities Division. X-ray experiments were performed at APS, which is supported by the Office of Basic Energy Sciences under contract no. DE-AC02-06CH11357, and CHESS, which is supported by the NSF and NIH/NIGMS via NSF award DMR-1332208. Computational developments were supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the US Department of Energy, Office of Science, and the National Nuclear Security Administration. We thank D. Robinson and X. Zhang for technical support during the experiments, A. Rettie for performing the EDX analysis, T. Weber and A. Simonov for discussions about the ΔPDF technique, B. Campbell for help with the formalism of transforming the data to reciprocal space and P. Zapol and C. Haley for discussions about interpreting the results. Crystal structure images were generated using CrystalMaker, CrystalMaker Software Ltd, http://www.crystalmaker.com.

## Author information

Samples were prepared by J.T.V. and prepared for measurement by M.J.K. The experiments were devised by M.J.K., S.R. and R.O. The X-ray experiments were performed by M.J.K., S.R., J.P.C.R., J.M.W. and R.O. The data were analysed by M.J.K., R.O., J.M.W. and G.J., using software written by G.J., M.J.K., R.O. and J.M.W. The manuscript and Supplementary information were written by R.O. with input from all the authors.

Correspondence to Raymond Osborn.

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## Supplementary information

### Supplementary Information

Supplementary methods, Figs. 1–15, notes and references.

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