The structure of liquid and glass is usually described by the atomic pair-distribution function (PDF), g(r), which expresses the statistical distribution of distances between atoms. The PDF can be determined by diffraction experiments using x-rays or neutrons. However, liquid is dynamic in nature, and we have to be sensible about what the PDF means for liquid. For crystalline solids the atomic structure is determined by the elastic scattering of x-rays, neutrons or electrons, because the momentum is transferred to the whole rigid body of the sample in scattering. But the elastic scattering intensity from liquid is zero because of the lack of rigidity. The scattering from liquid is purely inelastic, described by the dynamic structure factor, S(Q, ω), where Q is the momentum transfer and E = hω/2π is the energy transfer in scattering. To measure S(Q, ω) we need an elaborate inelastic scattering instruments. In particular for inelastic x-ray scattering (IXS) we need a very high energy resolution with ~ meV and ΔE/E < 10^-7. This can be achieved only with a large backscattering crystal analyzer with a long flight path. However, in regular x-ray diffraction measurement the energy resolution is poor, ~ 1 eV, far exceeding the typical energies of vibrational excitations. As a result, the measured structure function, S(Q), is the S(Q, ω) integrated over energy, thus representing the same-time correlation among atoms. Thus, the PDF, obtained by the Fourier-transformation of S(Q), is the same-time density correlation function which shows the time averaged snapshot of correlations. Therefore, the PDF does not describe the structure in a regular sense. For a long time, the PDF has been used in representing the structure, because it was the only readily available structural descriptor, and various theories have been proposed to predict dynamic properties from the PDF. On the other hand, the dynamic two-body correlation can be directly expressed by the Van Hove function, G(r, t), obtained by the double-Fourier-transformation of S(Q, ω). But, to carry out the double-Fourier-transformation accurately S(Q, ω) has to be measured over a wide Q-E space. Until recently this was unpractical, because the inelastic scattering measurement, typically done with a triple-axis-spectrometer, was extremely time-consuming. But the advent of pulsed neutron sources with large two-dimensional detector arrays and advances in the IXS instrumentation made it possible to determine the Van Hove function in a reasonable time, 4 – 12 hrs. We applied this technique to various liquids, including water, aqueous solutions of salt, metallic alloy liquids, liquid Ga, and organic electrolytes. New physical insights obtained by these measurements will be discussed. Now that this technique is available we should expand the definition of the “structure” of liquid to include the dynamic structure represented by the Van Hove function.

Pd_42.5Ni_7.5Cu₃₀P₂₀ (PNCP) has at present the most excellent glass-forming ability (GFA) among metallic glasses. The critical cooling rate (CCR) reaches of 0.067 K/s and can form a massive bulk glass with a diameter of more than 40 mm [1]. On the other hand, almost the parent alloys of Pd₄₀Ni₄₀P₂₀ (PNP) and Pd₄₀Cu₄₀P₂₀ (PCP) have worse CCRs of about 1 K/s [2] and 100 K/s [3], respectively, indicating that the mixture of Ni and Cu elements causes a better CCR in these bulk metallic glassy alloys.

In order to find a structural origin of the GFAs of these Pd-based glasses, we have carried out anomalous x-ray scattering (AXS) and neutron diffraction (ND) experiments on PNP [4], PCP [5], and PNCP (preliminary results were given in [6]), and the experimental results were analysed by using reverse Monte Carlo (RMC) modelling. The obtained atomic configurations of these alloys were discussed by using a Voronoi tessellation for the short-range atomic arrangements and a persistent homology analysis [7] for the hyper-ordered atomic structures.

Although the general features of the atomic configurations look similar to one another, i.e., most of atomic configurations around all elements are basically icosahedral-type, the main results of these analyses are as follows:

1) A large fraction of “pure” icosahedra are observed around only Ni atoms (5.8%) in PNP [4], whereas that around Cu in PCP is a half value of 2.9% [5]. Very interestingly in PNCP, large fractions of “pure” icosahedra are detected not only around the Ni atoms of 4.8%, but around the Cu atoms of 5.6%.

2) Large sizes of partial persistent homology rings are observed for the Ni/Cu atoms in all the glasses. However, the size highly depends on the GFA of the glasses, i.e., that in PNCP is slightly larger than in PNP, and much larger than in PCP.

In conclusion, the GFA of Pd-based metallic glasses is not understood as clearly characterized structures such as the existence of clusters of crystal-like fragments. It is realized through hyper-ordered structures, i.e., profound structural features in the short- and intermediate-range atomic order in the glasses.

[1] Nishiyama, N. and Inoue, A., (2002), Appl. Phys. Lett. 80, 568.[2] Drehman, A. J., Greer, A. L., and Turnbull, D., (1982). Appl. Phys. Lett. 41, 716.[3] He, Y. and Schwarz, R. B., (1997), Mater. Res. Symp. Proc. 455, 495.[4] Hosokawa, S. et al., (2019). Phys. Rev. B 100, 054204.[5] Hosokawa, S. et al., (2021), J. Non-Cryst. Solids 555, 120536.[6] Hosokawa, S. et al., (2009), Phys. Rev. B 80, 174204.[7] Hiraoka, T. et al., (2016), Proc. Natl. Acad. Sci. USA 113, 7035.

When sample conditions for conventional crystallography are not met (i.e. a large, well-ordered crystal) then x-ray diffraction techniques often do not yield an unambiguous 3D atomic structure. This can occur in powder diffraction and small-angle x-ray scattering (SAXS), where ensembles of crystals or particles in random orientations produce isotropic diffraction around the beam axis. It also occurs for disordered materials, such amorphous solids and liquids, where randomness at the molecular scale has a similar suppression of accessible structural information via x-ray scattering. The accessible structural information is the distribution of atom-pair distances (known as the pair distribution function or PDF). The PDF has no information about local angular structure, such as bond angles, and in many cases does not uniquely determine the 3D structure.

Fluctuation x-ray scattering (FXS) [1,2] aims to measure the local angular structure in disordered materials using a small x-ray beam to enhance angular scattering fluctuations. We have developed a method of inverting FXS data to recover a sum of three- and four-atom distributions in real-space[3]. We call this 3D function the Pair-Angle Distribution Function (PADF). It is a natural generalisation of the widely used PDF to higher dimensions. The PADF contains, for example, a bond angle distribution and massively increases the amount of structural information beyond that of the PDF.

There are exciting opportunities to combine PADF analysis with crystallography, powder diffraction and SAXS. It could yield new routes to crystal structures, nanoscale disorder, amorphous structure and liquid structure. Here we give an introduction to the PADF and report on our early experimental results with synchrotron, x-ray free-electron lasers and electron microscopes. These include applications to self-assembled lipids[4], disordered carbon materials[5], protein crystals[6] and liquids.

[1] Kurta, R.P., Altarelli, M. and Vartanyants, I.A. (2016). “Structural analysis by x-ray intensity angular cross-correlations” in Advances in Chemical Physics (eds S.A. Rice and A.R. Dinner).

[2] Kam, Z. (1977). Macromolecules, 10(5), 927–934.

[3] Martin, A. V. (2017). IUCrJ, 4, 24–36.

[4] Martin, A. V., et al., (2020). Small, 2000828, 1–6

[5] Martin, A. V., et al., (2020). Communications Materials, 1(40), 1–8.

[6] Adams, P.,, et al., (2020). Crystals, 10, 724.

Commonly classical nucleation theory has been used to explain nucleation, but this is now being challenged as atomic scale techniques has been developed to study solutions showing larger clusters before nucleation [1, 2]. Thus, a new theory including these clusters with predictive value is needed. To achieve this, it is essential to investigate the atomic structure of precursors across different elements as well as chemical environments.

In this study the precursors of group 13 metal oxides have been examined. Al, In and Ga form similar oxides and hydroxides such as M(OH)₃, MOOH and M₂O₃ in solvothermal synthesis. The individual systems exhibit complex polymorphism, which can be controlled with different synthesis parameters such as solvent and temperature, however, the actual mechanisms are unknown.

The precursor structures of the group 13 metal oxides have been determined by combining PDF and EXAFS analysis of the three metal nitrates in various solvents. Across element and solvents the structures were determined to be octahedrally coordinated metal-oxygen with further structure.[3] For the gallium system variation of pH, anions and concentration were further investigated using PDF analysis revealing the diverse solution chemistry of gallium [4].

Based on the results, the formation mechanisms of the group 13 metal oxides are discussed, for example reason for the production of AlOOH at most synthesis conditions instead of the desirable γ-Al₂O₃ phase [3,5].

Figure 1. Modelling of both EXAFS and PDF data for the same models.

[1] Bøjesen, E. D. & Iversen, B. B. (2016). CrystEngComm. 43, 8332-8353 [2] Gebauer, D., Kellermeier, M., Gale, J. D., Bergström, L. & Cölfen, H. (2014). Chem. Soc. Rev. 43, 2348-2371. [3] Sommer, S., Nielsen, I. G. & Iversen, B. B. (2020). Chem. – Eur. J. 26, 1022-1026. [4] Nielsen, I. G., Sommer, S., Dippel, A.-C. Skibsted, J. & Iversen, B. B. (2021). Submitted to JACS. [5] Nielsen, I. G., Sommer, S. & Iversen, B. B. (2021). Nanoscale 13, 4038-4050.

Due to its sensitivity to local structure, X-ray absorption spectroscopy is a powerful tool to study disordered systems. One of the most interesting property of XAFS is the sensitivity not only to pair distribution function, but also to three-body distribution, which contains information on bond angles between nearest neighbours. Reverse Monte Carlo (RMC) is a structural modelling method from which this information can be obtained [1], but it requires to know the density of the system being investigated, which may not be available especially in extreme thermodynamic conditions. Being a simulation method, it is also costly in terms of time. In recent years, neural networks (NN) have become a widely used tool to tackle different problems and have also been applied to the analysis of EXAFS data [2]. We wanted to investigate whether the same methodology could be applied to disordered systems and whether it would be possible to obtain information beyond the pair distribution function.

The critical point of any NN is the dataset used for the training process, that should be sufficiently large and heterogeneous. For this purpose, we ran several MD simulations of mono-atomic nickel at various temperatures for different crystal configurations, varying also the first-neighbour distance. The temperature was increased past the melting point to also include liquid configurations. From each configuration, we calculated the number distribution function, bond-angle distribution of the nearest neighbours and the EXAFS signal, using GNXAS suite of programs [3]. The created dataset was then used to optimize and train a set of deep NN to estimate number and bond-angle distribution from a given EXAFS signal.

We used the NN to analyse data of nickel at different temperatures and phases. Results from each NN are averaged and standard deviation calculated to estimate errors. Obtained results show that the NN is able to distinguish between ordered and disordered configurations and is also able to detect small changes in the local ordering of liquid structure, comparable with previously published results [4].

[1] Di Cicco A., Trapananti A., Faggioni S. & Filipponi A. (2003). Phys. Rev. Lett. 91, 135505.

[2] Timoshenko J., Anspoks A., Cintins A., Kuzmin A., Purans J. & Frenkel A. I. (2018). Phys. Rev. Lett. 120, 225502.

[3] Filipponi A. & Di Cicco A. (1995). Phys. Rev. B 52, 15135.

[4] Di Cicco A., Iesari F., De Panfilis S., Celino M., Giusepponi S. & Filipponi A. (2014). Phys. Rev. B 89, 060102.

Molecules and molecular ions, unlike individual atoms, have rotational degrees of freedom. This simple observation means that they can be particularly good building blocks for disordered crystal structures. Such behaviour is not merely a crystallographic curiosity: it is responsible for many important materials properties. For instance, if, in some material, a molecule with a permanent dipole moment is free to rotate at high temperatures but freezes into place at low temperatures, the two phases will have different entropies and dielectric constants. The result will be a dielectric switching material; it is likely to be an electro- and/or barocaloric, where the phase transition responds to an external electric field or pressure; and it may also be pyro- and even ferroelectric if the low-symmetry phase is polar.

Studying such materials requires a combination of experimental and computational techniques. Traditional crystallographic methods remain vital, but entail a time and space average that can obscure the behaviour of the disordered phase. Thus it is also important to use methods such as total scattering, which are sensitive to local deviations from that average; and to study also the dynamics, or how structures change over time. Neutron scattering methods are especially appropriate because they reveal the behaviour of hydrogen atoms, which is essential to understanding these rotating molecules, and provide both structural and dynamic information.

A particular focus of recent interest has been the family of molecular perovskites, in which molecular ions on the “A” site almost always display this sort of order-disorder transition [1]. This site is a cubic interstice of the perovskite framework, which provides both structural stability and the freedom for the ions to rotate. But of course order-disorder behaviour does not require this specific coordination framework, or indeed any framework at all: similar molecular “scaffolding” can be provided by other weak interactions, including van der Waals and hydrogen bonding.

Here we compare “free” to “caged” molecules and molecular ions that undergo entropic transitions. We consider the cyanide-bridged elpasolite (double perovskite) analogues (C₃H₅N₂)₂K[M(CN)₆], C₃H₅N₂ = imidazolium, M = Fe, Co [2]; the molecular material adamantane, C₁₀H₁₆ [3], and the molecular-ionic compound ammonium sulfate, (NH₄)₂SO₄ [4]. We have studied these materials’ structure by Bragg and total neutron scattering, and their dynamics by inelastic and quasielastic neutron scattering, complemented by density-functional theory simulations. Combining these methods provides a detailed picture of the actual rotational and vibrational freedom that molecules have in these materials, and hence of the structural origins of their useful properties. In particular, we show that the limiting cases of free rotation and harmonic oscillation can both be inaccurate and even seriously misleading, with the true situation lying somewhere between these extremes. Our results will direct future attempts at “entropic engineering”: designing molecular materials to have specific order-disorder behaviour.

[1] Kieslich, G. & Goodwin, A. L. (2017). Mater. Horiz. 4, 362–366.

[2] Duncan, H. D., Beake, E. O. R., Playford, H. Y., Dove, M. T. & Phillips, A. E. (2017). CrystEngComm 19, 7316–7321; Phillips, A. E. & Fortes, A. D. (2017). Angew. Chem. Int. Ed. 56, 15950–15953; Phillips, A. E., Cai, G. & Demmel, F. (2020). Chem. Commun. 56, 11791–11794.

[3] Beake, E. O. R., Tucker, M. G., Dove, M. T. & Phillips, A. E. (2017). ChemPhysChem 18, 459–464.

[4] Cai, G. (2020). Studying orientational disorder with neutron total scattering, Ph.D. thesis, Queen Mary University of London, U.K.