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Nanoparticles, exhibiting functionally relevant structural heterogeneity, are at the forefront of cutting-edge research. Now, high-throughput single-particle imaging (SPI) with x-ray free-electron lasers (XFELs) creates unprecedented opportunities for recovering the shape distributions of millions of particles that exhibit functionally relevant structural heterogeneity. To realize this potential, three challenges have to be overcome: (1) simultaneous parametrization of structural variability in real and reciprocal spaces; (2) efficiently inferring the latent parameters of each SPI measurement; (3) scaling up comparisons between $10^5$ structural models and $10^6$ XFEL-SPI measurements. Here, we describe how we overcame these three challenges to resolve the non-equilibrium shape distributions within millions of gold nanoparticles imaged at the European XFEL. These shape distributions allowed us to quantify the degree of asymmetry in these particles, discover a relatively stable `shape envelope' amongst nanoparticles, discern finite-size effects related to shape-controlling surfactants, and extrapolate nanoparticles' shapes to their idealized thermodynamic limit. Ultimately, these demonstrations show that XFEL SPI can help transform nanoparticle shape characterization from anecdotally interesting to statistically meaningful.

RevelsMD is a new open source Python library, which uses reduced variance force sampling based estimators to calculate 3D particle densities and radial distribution functions from molecular dynamics simulations. This short note describes the scientific background of the code, its utility and how it fits within the current zeitgeist in computational chemistry and materials science., Comment: 3 page note describing an open source code

Self-diffusion coefficients, $D^*$, are routinely estimated from molecular dynamics simulations by fitting a linear model to the observed mean-squared displacements (MSDs) of mobile species. MSDs derived from simulation exhibit statistical noise that causes uncertainty in the resulting estimate of $D^*$. An optimal scheme for estimating $D^*$ minimises this uncertainty, i.e., it will have high statistical efficiency, and also gives an accurate estimate of the uncertainty itself. We present a scheme for estimating $\D$ from a single simulation trajectory with high statistical efficiency and accurately estimating the uncertainty in the predicted value. The statistical distribution of MSDs observable from a given simulation is modelled as a multivariate normal distribution using an analytical covariance matrix for an equivalent system of freely diffusing particles, which we parameterise from the available simulation data. We use Bayesian regression to sample the distribution of linear models that are compatible with this multivariate normal distribution, to obtain a statistically efficient estimate of $D^*$ and an accurate estimate of the associated statistical uncertainty.

Short-range ordering in cation-disordered cathodes can have a significant effect on their electrochemical properties. Here, we characterise the cation short-range order in the antiperovskite cathode material Li$_2$FeSO, using density functional theory, Monte Carlo simulations, and synchrotron X-ray pair-distribution-function data. We predict partial short-range cation-ordering, characterised by favourable OLi$_4$Fe$_2$ oxygen coordination with a preference for polar cis-OLi$_4$Fe$_2$ over non-polar trans-OLi$_4$Fe$_2$ configurations. This preference for polar cation configurations produces long-range disorder, in agreement with experimental data. The predicted short-range-order preference contrasts with that for a simple point-charge model, which instead predicts preferential trans-OLi$_4$Fe$_2$ oxygen coordination and corresponding long-range crystallographic order. The absence of long-range order in Li$_2$FeSO can therefore be attributed to the relative stability of cis-OLi$_4$Fe$_2$ and other non-OLi$_4$Fe$_2$ oxygen-coordination motifs. We show that this effect is associated with the polarisation of oxide and sulfide anions in polar coordination environments, which stabilises these polar short-range cation orderings. We propose similar anion-polarisation-directed short-range-ordering may be present in other heterocationic materials that contain cations with different formal charges. Our analysis also illustrates the limitations of using simple point-charge models to predict the structure of cation-disordered materials, where other factors, such as anion polarisation, may play a critical role in directing both short- and long-range structural correlations.

A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal connected subgraph of $G$ that has no cut vertex. A block graph is a graph in which every block is a clique. We show that every bridgeless graph of order $n$ containing $p$ blocks has an oriented diameter of at most $n-\lfloor \frac{p}{2} \rfloor$. This bound is sharp for all $n$ and $p$ with $p \geq 2$. As a corollary, we obtain a sharp upper bound on the oriented diameter in terms of order and number of cut vertices. We also show that the oriented diameter of a bridgeless block graph of order $n$ is bounded above by $\lfloor \frac{3n}{4} \rfloor$ if $n$ is even and $\lfloor \frac{3(n+1)}{4} \rfloor$ if $n$ is odd., Comment: 15 pages, 2 figures

Nickelate materials offer diverse functionalities for energy and computing applications. Lithium nickel oxide (LiNiO$_2$) is an archetypal layered nickelate, but the electronic structure of this correlated material is not yet fully understood. Here we investigate the temperature-dependent speciation and spin dynamics of Ni ions in LiNiO$_2$. Our ab initio simulations predict that Ni ions disproportionate into three states, which dynamically interconvert and whose populations vary with temperature. These predictions are verified using x-ray absorption spectroscopy, x-ray magnetic circular dichroism, and resonant inelastic x-ray scattering at the Ni L$_{3,2}$-edge. Charge-transfer multiplet calculations consistent with disproportionation reproduce all experimental features. Together, our experimental and computational results support a model of dynamic disproportionation that explains diverse physical observations of LiNiO$_2$, including magnetometry, thermally activated electronic conduction, diffractometry, core-level spectroscopies, and the stability of ubiquitous antisite defects. This unified understanding of the fundamental material properties of LiNiO$_2$ is important for applications of nickelate materials as battery cathodes, catalysts, and superconductors., Comment: 30 pages, 16 figures

Improvements in X-ray optics critically depend on the measurement of their optical performance. The knowledge of wavefront aberrations, for example, can be used to improve the fabrication of optical elements or to design phase correctors to compensate for these errors. Nowadays, the characterization of such optics is made using intense X-ray sources such as synchrotrons. However, the limited access to these facilities can substantially slow down the development process. Improvements in the brightness of lab-based X-ray micro-sources in combination with the development of new metrology methods, and in particular ptychographic X-ray speckle tracking, enable characterization of X-ray optics in the lab with a precision and sensitivity not possible before. Here, we present a laboratory set-up that utilizes a commercially available X-ray source and can be used to characterize different types of X-ray optics. The set-up is used in our laboratory on a routine basis to characterize multilayer Laue lenses of high numerical aperture and other optical elements. This typically includes measurements of the wavefront distortions, optimum operating photon energy and focal length of the lens. To check the sensitivity and accuracy of this laboratory set-up we compared the results to those obtained at the synchrotron and saw no significant difference. To illustrate the feedback of measurements on performance, we demonstrated the correction of the phase errors of a particular multilayer Laue lens using a 3D printed compound refractive phase plate.

Multilayer Laue lenses were used for the first time to focus x-rays from an X-ray Free Electron Laser (XFEL). In an experiment, which was performed at the European XFEL, we demonstrated focusing to a spot size of a few tens of nanometers. A series of runs in which the number of pulses per train was increased from 1 to 2, 3, 4, 5, 6, 7, 10, 20 and 30 pulses per train, all with a pulse separation of 3.55 us, was done using the same set of lenses. The increase in the number of pulses per train was accompanied with an increase of x-ray intensity (transmission) from 9% to 92% at 5 pulses per train, and then the transmission was reduced to 23.5 % when the pulses were increased further. The final working condition was 30 pulses per train and 23.5% transmission. Only at this condition we saw that the diffraction efficiency of the MLLs changed over the course of a pulse train, and this variation was reproducible from train to train. We present the procedure to align and characterize these lenses and discuss challenges working with the pulse trains from this unique x-ray source.

The electride Sr$_3$CrN$_3$ has a one-dimensional channel of electron density, which is a rare feature that offers great potential for fast ion conduction. Using density functional theory, we find that Sr$_3$CrN$_3$ is an excellent hydride conductor within this channel, with a diffusion barrier as low as 0.30 eV and an estimated diffusion coefficient of $5.37\times 10^{-8}$ cm$^2$/s. This diffusion barrier is lower than those reported for the best hydride conductors to date. We also show the most-stable amount of hydride in the host material under standard conditions and the corresponding change in electronic structure from metal to wide-gap insulator. Our results highlight the potential offered by 1D electride materials for ion-transport applications such as energy storage or gas separation.

One of the outstanding analytical problems in X-ray single particle imaging (SPI) is the classification of structural heterogeneity, which is especially difficult given the low signal-to-noise ratios of individual patterns and that even identical objects can yield patterns that vary greatly when orientation is taken into consideration. We propose two methods which explicitly account for this orientation-induced variation and can robustly determine the structural landscape of a sample ensemble. The first, termed common-line principal component analysis (PCA) provides a rough classification which is essentially parameter-free and can be run automatically on any SPI dataset. The second method, utilizing variation auto-encoders (VAEs) can generate 3D structures of the objects at any point in the structural landscape. We implement both these methods in combination with the noise-tolerant expand-maximize-compress (EMC) algorithm and demonstrate its utility by applying it to an experimental dataset from gold nanoparticles with only a few thousand photons per pattern and recover both discrete structural classes as well as continuous deformations. These developments diverge from previous approaches of extracting reproducible subsets of patterns from a dataset and open up the possibility to move beyond studying homogeneous sample sets and study open questions on topics such as nanocrystal growth and dynamics as well as phase transitions which have not been externally triggered., Comment: 29 pages, 9 figures