When can we trust structural models derived from pair distribution function measurements?

The pair distribution function (PDF) is an important metric for characterising structure in complex materials, but it is well known that meaningfully different structural models can sometimes give rise to equivalent PDFs. In this paper, we discuss the use of model likelihoods as a general approach for discriminating between such homometric structure solutions. Drawing on two main case studies-one concerning the structure of a small peptide and the other amorphous calcium carbonate-we show how consideration of model likelihood can help drive robust structure solution, even in cases where the PDF is particularly information-poor. The obvious thread of these individual case studies is the potential role for machine-learning approaches to help guide structure determination from the PDF, and our paper finishes with some forward-looking discussion along these lines.

In situ/Operando Synchrotron Radiation Analytical Techniques for CO2/CO Reduction Reaction: From Atomic Scales to Mesoscales

Electrocatalytic carbon dioxide/carbon monoxide reduction reaction (CO(2)RR) has emerged as a prospective and appealing strategy to realize carbon neutrality for manufacturing sustainable chemical products. Developing highly active electrocatalysts and stable devices has been demonstrated as effective approach to enhance the conversion efficiency of CO(2)RR. In order to rationally design electrocatalysts and devices, a comprehensive understanding of the intrinsic structure evolution within catalysts and micro-environment change around electrode interface, particularly under operation conditions, is indispensable. Synchrotron radiation has been recognized as a versatile characterization platform, garnering widespread attention owing to its high brightness, elevated flux, excellent directivity, strong polarization and exceptional stability. This review systematically introduces the applications of synchrotron radiation technologies classified by radiation sources with varying wavelengths in CO(2)RR. By virtue of in situ/operando synchrotron radiationanalytical techniques, we also summarize relevant dynamic evolution processes from electronic structure, atomic configuration, molecular adsorption, crystal lattice and devices, spanning scales from the angstrom to the micrometer. The merits and limitations of diverse synchrotron characterization techniques are summarized, and their applicable scenarios in CO(2)RR are further presented. On the basis of the state-of-the-art fourth-generation synchrotron facilities, a perspective for further deeper understanding of the CO(2)RR process using synchrotron radiation analytical techniques is proposed.

Local Structure and Dynamics in MPt(CN)6 Prussian Blue Analogues

We use a combination of X-ray pair distribution function (PDF) measurements, lattice dynamical calculations, and ab initio density functional theory (DFT) calculations to study the local structure and dynamics in various MPt(CN)6 Prussian blue analogues. In order to link directly the local distortions captured by the PDF with the lattice dynamics of this family, we develop and apply a new "interaction-space" PDF refinement approach. This approach yields effective harmonic force constants, from which the (experiment-derived) low-energy phonon dispersion relations can be approximated. Calculation of the corresponding Grüneisen parameters allows us to identify the key modes responsible for negative thermal expansion (NTE) as arising from correlated tilts of coordination octahedra. We compare our results against the phonon dispersion relations determined using DFT calculations, which identify the same NTE mechanism.

Quest for Multifunctionality: Current Progress in the Characterization of Heterometallic Metal-Organic Frameworks

Metal-organic frameworks (MOFs) are a class of porous, crystalline materials that have been systematically developed for a broad range of applications. Incorporation of two or more metals into a single crystalline phase to generate heterometallic MOFs has been shown to lead to synergistic effects, in which the whole is oftentimes greater than the sum of its parts. Because geometric proximity is typically required for metals to function cooperatively, deciphering and controlling metal distributions in heterometallic MOFs is crucial to establish structure-function relationships. However, determination of short- and long-range metal distributions is nontrivial and requires the use of specialized characterization techniques. Advancements in the characterization of metal distributions and interactions at these length scales is key to rapid advancement and rational design of functional heterometallic MOFs. This perspective summarizes the state-of-the-art in the characterization of heterometallic MOFs, with a focus on techniques that allow metal distributions to be better understood. Using complementary analyses, in conjunction with computational methods, is critical as this field moves toward increasingly complex, multifunctional systems.

Evolution of Short-Range Order of Amorphous GeTe Upon Structural Relaxation Obtained by TEM Diffractometry and RMC Methods

Glasses frequently reveal structural relaxation that leads to changes in their physical properties including enthalpy, specific volume, and resistivity. Analyzing the short-range order (SRO) obtained from electron diffraction by transmission electron microscopy (TEM) in combination with Reverse-Monte-Carlo (RMC) simulations is shown to provide information on the atomic arrangement. The technique elaborated here features several benefits including reliability, accessibility, and allows for obtaining detailed structural data quickly. This is demonstrated with a detailed view of the structural changes in the as-deposited amorphous phase change material (PCM) GeTe. The data show a significant increase in the average bond angle upon thermal treatment. At the same time the fraction of tetrahedrally coordinated Ge atoms decreases due to an increase in octahedrally distorted and pyramidal motifs. This finding provides further evidence for the atomic processes that govern structural relaxation in amorphous GeTe and other PCMs. A thorough literature review finally unveils possible origins of the large discrepancies reported on the structure of amorphous GeTe.

Ligand and solvent effects on the absorption spectra of CdS magic-sized clusters

The absorption spectra of congenetic wurtzite (WZ) and zincblende (ZB) CdS magic-sized clusters are investigated. We demonstrate that the exciton peak positions can be tuned by up to 500 meV by varying the strong coupling between X-type ligands and the semiconductor cores, while the addition of L-type ligands primarily affects cluster midgap states. When Z-type ligands are displaced by L-type ligands, red shifts in the absorption spectra are observed, despite the fact there is a small decrease in cluster size. Density functional theory calculations are used to explain these findings and they reveal the importance of Cd and S dangling bonds on the midgap states during the Z- to L-type ligand exchange process. Overall, ZB CdS clusters show higher chemical stability than WZ clusters but their optical properties exhibit greater sensitivity to the solvent. Conversely, WZ CdS clusters are not stable in a Lewis base-rich environment, resulting in various changes in their spectra. Our findings enable researchers to select capping ligands that modulate the optical properties of semiconductor clusters while maintaining precise control over their solvent interactions.

DeepStruc: towards structure solution from pair distribution function data using deep generative models

Structure solution of nanostructured materials that have limited long-range order remains a bottleneck in materials development. We present a deep learning algorithm, DeepStruc, that can solve a simple monometallic nanoparticle structure directly from a Pair Distribution Function (PDF) obtained from total scattering data by using a conditional variational autoencoder. We first apply DeepStruc to PDFs from seven different structure types of monometallic nanoparticles, and show that structures can be solved from both simulated and experimental PDFs, including PDFs from nanoparticles that are not present in the training distribution. We also apply DeepStruc to a system of hcp, fcc and stacking faulted nanoparticles, where DeepStruc recognizes stacking faulted nanoparticles as an interpolation between hcp and fcc nanoparticles and is able to solve stacking faulted structures from PDFs. Our findings suggests that DeepStruc is a step towards a general approach for structure solution of nanomaterials.

Multivariate analysis of disorder in metal-organic frameworks

The rational design of disordered frameworks is an appealing route to target functional materials. However, intentional realisation of such materials relies on our ability to readily characterise and quantify structural disorder. Here, we use multivariate analysis of pair distribution functions to fingerprint and quantify the disorder within a series of compositionally identical metal–organic frameworks, possessing different crystalline, disordered, and amorphous structures. We find this approach can provide powerful insight into the kinetics and mechanism of structural collapse that links these materials. Our methodology is also extended to a very different system, namely the melting of a zeolitic imidazolate framework, to demonstrate the potential generality of this approach across many areas of disordered structural chemistry.

Structural disorder in materials is challenging to characterise. Here, the authors use multivariate analysis of atomic pair distribution functions to study structural collapse and melting of metal–organic frameworks, revealing powerful mechanistic and kinetic insight.

Structural Analysis of Molecular Materials Using the Pair Distribution Function

This is a review of atomic pair distribution function (PDF) analysis as applied to the study of molecular materials. The PDF method is a powerful approach to study short- and intermediate-range order in materials on the nanoscale. It may be obtained from total scattering measurements using X-rays, neutrons, or electrons, and it provides structural details when defects, disorder, or structural ambiguities obscure their elucidation directly in reciprocal space. While its uses in the study of inorganic crystals, glasses, and nanomaterials have been recently highlighted, significant progress has also been made in its application to molecular materials such as carbons, pharmaceuticals, polymers, liquids, coordination compounds, composites, and more. Here, an overview of applications toward a wide variety of molecular compounds (organic and inorganic) and systems with molecular components is presented. We then present pedagogical descriptions and tips for further implementation. Successful utilization of the method requires an interdisciplinary consolidation of material preparation, high quality scattering experimentation, data processing, model formulation, and attentive scrutiny of the results. It is hoped that this article will provide a useful reference to practitioners for PDF applications in a wide realm of molecular sciences, and help new practitioners to get started with this technique.

Extracting interface correlations from the pair distribution function of composite materials

Using a non-negative matrix factorisation (NMF) approach, we show how the pair distribution function (PDF) of complex mixtures can be deconvolved into the contributions from the individual phase components and also the interface between phases. Our focus is on the model system Fe∥Fe3O4. We establish proof-of-concept using idealised PDF data generated from established theory-driven models of the Fe∥Fe3O4 interface. Using X-ray total scattering measurements for corroded Fe samples, and employing our newly-developed NMF analysis, we extract the experimental interface PDF ('iPDF') for this same system. We find excellent agreement between theory and experiment. The implications of our results in the broader context of interface characterisation for complex functional materials are discussed.

Structure determination of organic compounds by a fit to the pair distribution function from scratch without prior indexing

A method for the ab initio crystal structure determination of organic compounds by a fit to the pair distribution function (PDF), without prior knowledge of lattice parameters and space group, has been developed. The method is called 'PDF-Global-Fit' and is implemented by extension of the program FIDEL (fit with deviating lattice parameters). The structure solution is based on a global optimization approach starting from random structural models in selected space groups. No prior indexing of the powder data is needed. The new method requires only the molecular geometry and a carefully determined PDF. The generated random structures are compared with the experimental PDF and ranked by a similarity measure based on cross-correlation functions. The most promising structure candidates are fitted to the experimental PDF data using a restricted simulated annealing structure solution approach within the program TOPAS, followed by a structure refinement against the PDF to identify the correct crystal structure. With the PDF-Global-Fit it is possible to determine the local structure of crystalline and disordered organic materials, as well as to determine the local structure of unindexable powder patterns, such as nanocrystalline samples, by a fit to the PDF. The success of the method is demonstrated using barbituric acid as an example. The crystal structure of barbituric acid form IV solved and refined by the PDF-Global-Fit is in excellent agreement with the published crystal structure data.

Tunable Broad Light Emission from 3D "Hollow" Bromide Perovskites through Defect Engineering

Hybrid halide perovskites consisting of corner-sharing metal halide octahedra and small cuboctahedral cages filled with counter cations have proven to be prominent candidates for many high-performance optoelectronic devices. The stability limits of their three-dimensional perovskite framework are defined by the size range of the cations present in the cages of the structure. In some cases, the stability of the perovskite-type structure can be extended even when the counterions violate the size and shape requirements, as is the case in the so-called "hollow" perovskites. In this work, we engineered a new family of 3D highly defective yet crystalline "hollow" bromide perovskites with general formula (FA)_1-x(en)_x(Pb)_1-0.7x(Br)_3-0.4x (FA = formamidinium (FA⁺), en = ethylenediammonium (en²⁺), x = 0-0.44). Pair distribution function analysis shed light on the local structural coherence, revealing a wide distribution of Pb-Pb distances in the crystal structure as a consequence of the Pb/Br-deficient nature and en inclusion in the lattice. By manipulating the number of Pb/Br vacancies, we finely tune the optical properties of the pristine FAPbBr₃ by blue shifting the band gap from 2.20 to 2.60 eV for the x = 0.42 en sample. A most unexpected outcome was that at x> 0.33 en incorporation, the material exhibits strong broad light emission (1% photoluminescence quantum yield (PLQY)) that is maintained after exposure to air for more than a year. This is the first example of strong broad light emission from a 3D hybrid halide perovskite, demonstrating that meticulous defect engineering is an excellent tool for customizing the optical properties of these semiconductors.

Mixed hierarchical local structure in a disordered metal-organic framework

Amorphous metal-organic frameworks (MOFs) are an emerging class of materials. However, their structural characterisation represents a significant challenge. Fe-BTC, and the commercial equivalent Basolite® F300, are MOFs with incredibly diverse catalytic ability, yet their disordered structures remain poorly understood. Here, we use advanced electron microscopy to identify a nanocomposite structure of Fe-BTC where nanocrystalline domains are embedded within an amorphous matrix, whilst synchrotron total scattering measurements reveal the extent of local atomic order within Fe-BTC. We use a polymerisation-based algorithm to generate an atomistic structure for Fe-BTC, the first example of this methodology applied to the amorphous MOF field outside the well-studied zeolitic imidazolate framework family. This demonstrates the applicability of this computational approach towards the modelling of other amorphous MOF systems with potential generality towards all MOF chemistries and connectivities. We find that the structures of Fe-BTC and Basolite® F300 can be represented by models containing a mixture of short- and medium-range order with a greater proportion of medium-range order in Basolite® F300 than in Fe-BTC. We conclude by discussing how our approach may allow for high-throughput computational discovery of functional, amorphous MOFs.

Comparison and evaluation of pair distribution functions, using a similarity measure based on cross-correlation functions

An approach for the comparison of pair distribution functions (PDFs) has been developed using a similarity measure based on cross-correlation functions. The PDF is very sensitive to changes in the local structure, i.e. small deviations in the structure can cause large signal shifts and significant discrepancies between the PDFs. Therefore, a comparison based on pointwise differences (e.g. R values and difference curves) may lead to the assumption that the investigated PDFs as well as the corresponding structural models are not in agreement at all, whereas a careful visual inspection of the investigated structural models and corresponding PDFs may reveal a relatively good match. To quantify the agreement of different PDFs for those cases an alternative approach is introduced: the similarity measure based on cross-correlation functions. In this paper, the power of this application of the similarity measure to the analysis of PDFs is highlighted. The similarity measure is compared with the classical R wp values as representative of the comparison based on pointwise differences as well as with the Pearson product-moment correlation coefficient, using polymorph IV of barbituric acid as an example.

Bridging Structural Inhomogeneity to Functionality: Pair Distribution Function Methods for Functional Materials Development

The correlation between structure and function lies at the heart of materials science and engineering. Especially, modern functional materials usually contain inhomogeneities at an atomic level, endowing them with interesting properties regarding electrons, phonons, and magnetic moments. Over the past few decades, many of the key developments in functional materials have been driven by the rapid advances in short-range crystallographic techniques. Among them, pair distribution function (PDF) technique, capable of utilizing the entire Bragg and diffuse scattering signals, stands out as a powerful tool for detecting local structure away from average. With the advent of synchrotron X-rays, spallation neutrons, and advanced computing power, the PDF can quantitatively encode a local structure and in turn guide atomic-scale engineering in the functional materials. Here, the PDF investigations in a range of functional materials are reviewed, including ferroelectrics/thermoelectrics, colossal magnetoresistance (CMR) magnets, high-temperature superconductors (HTSC), quantum dots (QDs), nano-catalysts, and energy storage materials, where the links between functions and structural inhomogeneities are prominent. For each application, a brief description of the structure-function coupling will be given, followed by selected cases of PDF investigations. Before that, an overview of the theory, methodology, and unique power of the PDF method will be also presented.

Illustrated formalisms for total scattering data: a guide for new practitioners

This article provides a detailed and visual presentation of the derivations of and relationships between many of the commonly employed functional forms of real- and reciprocal-space data employed by the worldwide total scattering community.

The total scattering method is the simultaneous study of both the real- and reciprocal-space representations of diffraction data. While conventional Bragg-scattering analysis (employing methods such as Rietveld refinement) provides insight into the average structure of the material, pair distribution function (PDF) analysis allows for a more focused study of the local atomic arrangement of a material. Generically speaking, a PDF is generated by Fourier transforming the total measured reciprocal-space diffraction data (Bragg and diffuse) into a real-space representation. However, the details of the transformation employed and, by consequence, the resultant appearance and weighting of the real-space representation of the system can vary between different research communities. As the worldwide total scattering community continues to grow, these subtle differences in nomenclature and data representation have led to conflicting and confusing descriptions of how the PDF is defined and calculated. This paper provides a consistent derivation of many of these different forms of the PDF and the transformations required to bridge between them. Some general considerations and advice for total scattering practitioners in selecting and defining the appropriate choice of PDF in their own research are presented. This contribution aims to benefit people starting in the field or trying to compare their results with those of other researchers.

A cloud platform for atomic pair distribution function analysis: PDFitc

A new web platform is presented for the pair distribution function (PDF) community to use and share advanced PDF analysis software in the cloud.

A cloud web platform for analysis and interpretation of atomic pair distribution function (PDF) data (PDFitc) is described. The platform is able to host applications for PDF analysis to help researchers study the local and nanoscale structure of nanostructured materials. The applications are designed to be powerful and easy to use and can, and will, be extended over time through community adoption and development. The currently available PDF analysis applications, structureMining, spacegroupMining and similarity­Mapping, are described. In the first and second the user uploads a single PDF and the application returns a list of best-fit candidate structures, and the most likely space group of the underlying structure, respectively. In the third, the user can upload a set of measured or calculated PDFs and the application returns a matrix of Pearson correlations, allowing assessment of the similarity between different data sets. structureMining is presented here as an example to show the easy-to-use workflow on PDFitc. In the future, as well as using the PDFitc applications for data analysis, it is hoped that the community will contribute their own codes and software to the platform.

Understanding the Instability of the Halide Perovskite CsPbI3 through Temperature-Dependent Structural Analysis

Despite the tremendous interest in halide perovskite solar cells, the structural reasons that cause the all-inorganic perovskite CsPbI₃ to be unstable at room temperature remain mysterious, especially since many tolerance-factor-based approaches predict CsPbI₃ should be stable as a perovskite. Here single-crystal X-ray diffraction and X-ray pair distribution function (PDF) measurements characterize bulk perovskite CsPbI₃ from 100 to 295 K to elucidate its thermodynamic instability. While Cs occupies a single site from 100 to 150 K, it splits between two sites from 175 to 295 K with the second site having a lower effective coordination number, which, along with other structural parameters, suggests that Cs rattles in its coordination polyhedron. PDF measurements reveal that on the length scale of the unit cell, the PbI octahedra concurrently become greatly distorted, with one of the IPbI angles approaching 82° compared to the ideal 90°. The rattling of Cs, low number of CsI contacts, and high degree of octahedral distortion cause the instability of perovskite-phase CsPbI_3. These results reveal the limitations of tolerance factors in predicting perovskite stability and provide detailed structural information that suggests methods to engineer stable CsPbI₃ -based solar cells.

There's no place like real-space: elucidating size-dependent atomic structure of nanomaterials using pair distribution function analysis

The development of new functional materials builds on an understanding of the intricate relationship between material structure and properties, and structural characterization is a crucial part of materials chemistry. However, elucidating the atomic structure of nanomaterials remains a challenge using conventional diffraction techniques due to the lack of long-range atomic order. Over the past decade, Pair Distribution Function (PDF) analysis of X-ray or neutron total scattering data has become a mature and well-established method capable of giving insight into the atomic structure in nanomaterials. Here, we review the use of PDF analysis and modelling in characterization of a range of different nanomaterials that exhibit unique atomic structure compared to the corresponding bulk materials. A brief introduction to PDF analysis and modelling is given, followed by examples of how essential structural information can be extracted from PDFs using both model-free and advanced modelling methods. We put an emphasis on how the intuitive nature of the PDF can be used for understanding important structural motifs, and on the diversity of applications of PDF analysis to nanostructure problems.

Exact and fast calculation of the X-ray pair distribution function

A fast and exact algorithm to calculate the powder pair distribution function (PDF) for the case of periodic structures is presented. The new algorithm calculates the PDF by a detour via reciprocal space. The calculated normalized total powder diffraction pattern is transferred into the PDF via the sine Fourier transform. The calculation of the PDF via the powder pattern avoids the conventional simplification of X-ray and electron atomic form factors. It is thus exact for these types of radiation, as is the conventional calculation for the case of neutron diffraction. The new algorithm further improves the calculation speed. Additional advantages are the improved detection of errors in the primary data, the handling of preferred orientation, the ease of treatment of magnetic scattering and a large improvement to accommodate more complex instrumental resolution functions.

Unveiling Nanoscale Compositional and Structural Heterogeneities of Highly Textured Mg0.7Ti0.3Hy Thin Films

Thin films often exhibit fascinating properties, but the understanding of the underlying mechanism behind such properties is not simple. This is partially because of the limited structural information available. The hurdle in obtaining such information is especially high for textured thin films such as Mg-rich Mg_xTi_1-x, a promising switchable smart coating material. Although these metastable thin films are seen as solid solution alloys by conventional crystallographic methods, their hydrogen-induced optical transition is hardly understood by a solid solution model. In this study, we collect atomic pair distribution function (PDF) data for a Mg_0.7Ti_0.3H_y thin film in situ on hydrogenation and successfully resolve TiH₂ clusters of an average size of 30 Å embedded in the Mg matrix. This supports the chemically segregated model previously proposed for this system. We also observe the emergence of a previously unknown intermediate face-centered tetragonal phase during hydrogenation of the Mg matrix. This phase appears between Mg and MgH₂ to reduce lattice mismatch, thereby preventing pulverization and facilitating rapid hydrogen uptake. This work may shed new light on the hydrogen-induced properties of Mg-rich Mg_xTi_1-x thin films.

Structure-mining: screening structure models by automated fitting to the atomic pair distribution function over large numbers of models

A new approach is presented to obtain candidate structures from atomic pair distribution function (PDF) data in a highly automated way. It fetches, from web-based structural databases, all the structures meeting the experimenter's search criteria and performs structure refinements on them without human intervention. It supports both X-ray and neutron PDFs. Tests on various material systems show the effectiveness and robustness of the algorithm in finding the correct atomic crystal structure. It works on crystalline and nanocrystalline materials including complex oxide nanoparticles and nanowires, low-symmetry and locally distorted structures, and complicated doped and magnetic materials. This approach could greatly reduce the traditional structure searching work and enable the possibility of high-throughput real-time auto-analysis PDF experiments in the future.

Entropic elasticity and negative thermal expansion in a simple cubic crystal

While most solids expand when heated, some materials show the opposite behavior: negative thermal expansion (NTE). In polymers and biomolecules, NTE originates from the entropic elasticity of an ideal, freely jointed chain. The origin of NTE in solids has been widely believed to be different. Our neutron scattering study of a simple cubic NTE material, ScF₃, overturns this consensus. We observe that the correlation in the positions of the neighboring fluorine atoms rapidly fades on warming, indicating an uncorrelated thermal motion constrained by the rigid Sc-F bonds. This leads us to a quantitative theory of NTE in terms of entropic elasticity of a floppy network crystal, which is in remarkable agreement with experimental results. We thus reveal the formidable universality of the NTE phenomenon in soft and hard matter.

The upgraded Polaris powder diffractometer at the ISIS neutron source

This paper describes the design and operation of the Polaris time-of-flight powder neutron diffractometer at the ISIS pulsed spallation neutron source, Rutherford Appleton Laboratory, UK. Following a major upgrade to the diffractometer in 2010-2011, its detector provision now comprises five large ZnS scintillator-based banks, covering an angular range of 6° ≤ 2θ ≤ 168°, with only minimal gaps between each bank. These detectors have a substantially increased solid angle coverage (Ω ∼ 5.67 sr) compared to the previous instrument (Ω ∼ 0.82 sr), resulting in increases in count rate of between 2× and 10×, depending on 2θ angle. The benefits arising from the high count rates achieved are illustrated using selected examples of experiments studying small sample volumes and performing rapid, time-resolved investigations. In addition, the enhanced capabilities of the diffractometer in the areas of in situ studies (which are facilitated by the installation of a novel design of radial collimator around the sample position and by a complementary programme of advanced sample environment developments) and in total scattering studies (to probe the nature of short-range atomic correlations within disordered crystalline solids) are demonstrated.

Structural characterisation of amorphous solid dispersions via metropolis matrix factorisation of pair distribution function data

We measure the X-ray pair distribution functions (PDFs) of a series of felodipine:copovidone amorphous solid dispersions. Using a newly-developed Metropolis Matrix Factorisation (MMF) algorithm we extract from these data the PDF of the amorphous felodipine component in isolation. Our MMF analysis allows quantification of the degree of drug crystallinity in each sample, and structural characterisation of the amorphous drug via its PDF. Comparison with atomistic simulations reveals that the (in)accessibility of conformational rotamers distinguishes amorphous and crystalline felodipine, in turn suggesting design routes for stabilising the amorphous form. We discuss the conceptual importance of our results in the context of characterising not only amorphous pharmaceuticals, but complex mixtures in general.

A simple correction for the parallax effect in X-ray pair distribution function measurements

Total scattering and pair distribution function (PDF) analysis has created new insights that traditional powder diffraction methods have been unable to achieve in understanding the local structures of materials exhibiting disorder or complex nanostructures. Care must be taken in such analyses as subtle and discrete features in the PDF can easily be artefacts generated in the measurement process, which can result in unphysical models and interpretation. The focus of this study is an artefact called the parallax effect, which can occur in area detectors with thick detection layers during the collection of X-ray PDF data. This effect results in high-Q peak offsets, which subsequently cause an r-dependent shift in the PDF peak positions in real space. Such effects should be accounted for if a truly accurate model is to be achieved, and a simple correction that can be conducted via a Rietveld refinement against the reference data is proposed.

Algorithm for distance list extraction from pair distribution functions

An algorithm is presented to extract the distance list from atomic pair distribution functions in a highly automated way. The algorithm is constructed via curve fitting based on a Debye scattering equation model. Because of the non-convex nature of the resulting optimization problem, a number of techniques are developed to overcome various computational difficulties. A key ingredient is a new approach to obtain a reasonable initial guess based on the theoretical properties of the mathematical model. Tests on various nanostructured samples show the effectiveness of the initial guess and the accuracy and overall good performance of the extraction algorithm. This approach could be extended to any spectrum that is approximated as a sum of Gaussian functions.

A Comparative Study of Experimental Configurations in Synchrotron Pair Distribution Function

The identification and quantification of amorphous components and nanocrystalline phases with very small crystal sizes, smaller than ~3 nm, within samples containing crystalline phases is very challenging. However, this is important as there are several types of systems that contain these matrices: building materials, glass-ceramics, some alloys, etc. The total scattering synchrotron pair distribution function (PDF) can be used to characterize the local atomic order of the nanocrystalline components and to carry out quantitative analyses in complex mixtures. Although the resolution in momentum transfer space has been widely discussed, the resolution in the interatomic distance space has not been discussed to the best of our knowledge. Here, we report synchrotron PDF data collected at three beamlines in different experimental configurations and X-ray detectors. We not only discuss the effect of the resolution in Q-space, Q_{max ins} of the recorded data and Q_max of the processed data, but we also discuss the resolution in the interatomic distance (real) space. A thorough study of single-phase crystalline nickel used as standard was carried out. Then, selected cement-related samples including anhydrous tricalcium and dicalcium silicates, and pastes derived from the hydration of tricalcium silicate and ye’elimite with bassanite were analyzed.

Time-resolved grazing-incidence pair distribution functions during deposition by radio-frequency magnetron sputtering

Characterization of local order in thin films is challenging with pair distribution function (PDF) analysis because of the minute mass of the scattering material. Here, it is demonstrated that reliable high-energy grazing-incidence total X-ray scattering data can be obtained in situ during thin-film deposition by radio-frequency magnetron sputtering. A benchmark system of Pt was investigated in a novel sputtering chamber mounted on beamline P07-EH2 at the PETRA III synchrotron. Robust and high-quality PDFs can be obtained from films as thin as 3 nm and atomistic modelling of the PDFs with a time resolution of 0.5 s is possible. In this way, it was found that a polycrystalline Pt thin film deposits with random orientation at 8 W and 2 × 10-2 mbar at room temperature. From the PDF it was found that the coherent-scattering domains grow with time. While the first layers are formed with a small tensile strain this relaxes towards the bulk value with increasing film thickness.

Local atomic structure of thin and ultrathin films via rapid high-energy X-ray total scattering at grazing incidence

Atomic pair distribution function (PDF) analysis is the most powerful technique to study the structure of condensed matter on the length scale from short- to long-range order. Today, the PDF approach is an integral part of research on amorphous, nanocrystalline and disordered materials from bulk to nanoparticle size. Thin films, however, demand specific experimental strategies for enhanced surface sensitivity and sophisticated data treatment to obtain high-quality PDF data. The approach described here is based on the surface high-energy X-ray diffraction technique applying photon energies above 60 keV at grazing incidence. In this way, reliable PDFs were extracted from films of thicknesses down to a few nanometres. Compared with recently published reports on thin-film PDF analysis from both transmission and grazing-incidence geometries, this work brought the minimum detectable film thickness down by about a factor of ten. Depending on the scattering power of the sample, the data acquisition on such ultrathin films can be completed within fractions of a second. Hence, the rapid-acquisition grazing-incidence PDF method is a major advancement in thin-film technology that opens unprecedented possibilities for in situ and operando PDF studies in complex sample environments. By uncovering how the structure of a layered material on a substrate evolves and transforms in terms of local and average ordering, this technique offers new opportunities for understanding processes such as nucleation, growth, morphology evolution, crystallization and the related kinetics on the atomic level and in real time.

Time-resolved pair distribution function analysis of disordered materials on beamlines BL04B2 and BL08W at SPring-8

A dedicated apparatus has been developed for studying structural changes in amorphous and disordered crystalline materials substantially in real time. The apparatus, which can be set up on beamlines BL04B2 and BL08W at SPring-8, mainly consists of a large two-dimensional flat-panel detector and high-energy X-rays, enabling total scattering measurements to be carried out for time-resolved pair distribution function (PDF) analysis in the temperature range from room temperature to 873 K at pressures of up to 20 bar. For successful time-resolved analysis, a newly developed program was used that can monitor and process two-dimensional image data simultaneously with the data collection. The use of time-resolved hardware and software is of great importance for obtaining a detailed understanding of the structural changes in disordered materials, as exemplified by the results of commissioned measurements carried out on both beamlines. Benchmark results obtained using amorphous silica and demonstration results for the observation of sulfide glass crystallization upon annealing are introduced.

Unraveling the Chemical Nature of the 3D "Hollow" Hybrid Halide Perovskites

The newly introduced class of 3D halide perovskites, termed "hollow" perovskites, has been recently demonstrated as light absorbing semiconductor materials for fabricating lead-free perovskite solar cells with enhanced efficiency and superior stability. Hollow perovskites derive from three-dimensional (3D) AMX₃ perovskites ( A = methylammonium (MA), formamidinium (FA); M = Sn, Pb; X = Cl, Br, I), where small molecules such as ethylenediammonium cations ( en) can be incorporated as the dication without altering the structure dimensionality. We present in this work the inherent structural properties of the hollow perovskites and expand this class of materials to the Pb-based analogues. Through a combination of physical and spectroscopic methods (XRD, gas pycnometry, ¹H NMR, TGA, SEM/EDX), we have assigned the general formula (A)_{1- x}( en) _x(M)_{1-0.7 x}(X)_{3-0.4 x} to the hollow perovskites. The incorporation of en in the 3D perovskite structure leads to massive M and X vacancies in the 3D [ MX₃] framework, thus the term hollow. The resulting materials are semiconductors with significantly blue-shifted direct band gaps from 1.25 to 1.51 eV for Sn-based perovskites and from 1.53 to 2.1 eV for the Pb-based analogues. The increased structural disorder and hollow nature were validated by single crystal X-ray diffraction analysis as well as pair distribution function (PDF) analysis. Density functional theory (DFT) calculations support the experimental trends and suggest that the observed widening of the band gap is attributed to the massive M and X vacancies, which create a less connected 3D hollow structure. The resulting materials have superior air stability, where in the case of Sn-based hollow perovskites it exceeds two orders of temporal magnitude compared to the conventional full perovskites of MASnI₃ and FASnI₃. The hollow perovskite compounds pose as a new platform of promising light absorbers that can be utilized in single junction or tandem solar cells.

Calculation of pair distribution functions for multiphase systems

The total scattering method is becoming increasingly popular because of its ability to investigate the structures of disordered crystalline and amorphous materials. Also, in recent years, significant development of total scattering instruments and sample environments has allowed for the study of increasingly complex materials, including multiphase samples. The total scattering formalism has already been well described in the paper by Keen [J. Appl. Cryst. (2001), 34, 172–177] but it was limited to the single phase case. In the present paper the formulae for multiple phase samples (consisting of a physical mixture of two or more distinct phases) are derived for the calculation of pair distribution functions for analysis using reverse Monte Carlo and other methods. The equations for conversion between different representations of the pair distribution function are also provided.

Pair distribution function analysis of sulfide glassy electrolytes for all-solid-state batteries: Understanding the improvement of ionic conductivity under annealing condition

In general, the ionic conductivity of sulfide glasses decreases with their crystallization, although it increases for a few sulphide glasses owing to the crystallization of a highly conductive new phase (e.g., Li₇P₃S₁₁: 70Li₂S-30P₂S₅). We found that the ionic conductivity of 75Li₂S-25P₂S₅ sulfide glass, which consists of glassy and crystalline phases, is improved by optimizing the conditions of the heat treatment, i.e., annealing. A different mechanism of high ionic conductivity from the conventional mechanism is expected in the glassy phase. Here, we report the glassy structure of 75Li₂S-25P₂S₅ immediately before the crystallization by using the differential pair distribution function (d-PDF) analysis of high-energy X-ray diffraction. Even though the ionic conductivity increases during the optimum annealing, the d-PDF analysis indicated that the glassy structure undergoes no structural change in the sulfide glass-ceramic electrolyte at a crystallinity of 33.1%. We observed the formation of a nanocrystalline phase in the X-ray and electron diffraction patterns before the crystallization, which means that Bragg peaks were deformed. Thus, the ionic conductivity in the mixture of glassy and crystalline phases is improved by the coexistence of the nanocrystalline phase.

Neutron total scattering of crystalline materials in the gigapascal regime

Neutron total scattering of disordered crystalline materials provides direct experimental access to the local (short-range) structure. The ways in which this local structure agrees (or disagrees) with the long-range crystal structure can provide important insight into structure–property relationships. High-pressure neutron diffraction using a Paris–Edinburgh (P–E) pressure cell allows experimenters to explore the ways in which materials are affected by pressure, can reveal new synthetic routes to novel functional materials and has important applications in many areas, including geology, engineering and planetary science. However, the combination of these two experimental techniques poses unique challenges for both data collection and analysis. In this paper it is shown that, with only minor modifications to the standard P–E press setup, high-quality total scattering data can be obtained from crystalline materials in the gigapascal pressure regime on the PEARL diffractometer at ISIS. The quality of the data is assessed through the calculation of coordination numbers and the use of reverse Monte Carlo refinements. The time required to collect data of sufficient quality for detailed analysis is assessed and is found to be of the order of 8 h for a quartz sample. Finally, data from the perovskite LaCo0.35Mn0.65O3 are presented and reveal that PEARL total scattering data offer the potential of extracting local structural information from complex materials at high pressure.

100 years of Debye's scattering equation

Debye's scattering equation (DSE) has spanned a century of scientific development, from the dawn of quantum mechanics and the investigation of the structure of atoms and molecules to the era of nanotechnology, paving the way tototal scatteringmethods. The formulation offers the most accurate representation of the intensity scattered by randomly oriented atomic aggregates, constructed by superimposing the signal from each atomic distance in the molecule. The present paper reviews some of the milestone applications, from the interpretation of the intensity curves from gases and vapours, to aggregates of increasing size and more extended order. Important developments, aimed at mitigating the prohibitive computational complexity of the DSE, and state-of-the-art methods for the characterization of static and dynamic displacements are also discussed.

Polymorphism in magic-sized Au144(SR)60 clusters

Ultra-small, magic-sized metal nanoclusters represent an important new class of materials with properties between molecules and particles. However, their small size challenges the conventional methods for structure characterization. Here we present the structure of ultra-stable Au144(SR)60 magic-sized nanoclusters obtained from atomic pair distribution function analysis of X-ray powder diffraction data. The study reveals structural polymorphism in these archetypal nanoclusters. In addition to confirming the theoretically predicted icosahedral-cored cluster, we also find samples with a truncated decahedral core structure, with some samples exhibiting a coexistence of both cluster structures. Although the clusters are monodisperse in size, structural diversity is apparent. The discovery of polymorphism may open up a new dimension in nanoscale engineering.

Negative thermal expansion and associated anomalous physical properties: review of the lattice dynamics theoretical foundation

Negative thermal expansion (NTE) is the phenomenon in which materials shrink rather than expand on heating. Although NTE had been previously observed in a few simple materials at low temperature, it was the realisation in 1996 that some materials have NTE over very wide ranges of temperature that kick-started current interest in this phenomenon. Now, nearly two decades later, a number of families of ceramic NTE materials have been identified. Increasingly quantitative studies focus on the mechanism of NTE, through techniques such as high-pressure diffraction, local structure probes, inelastic neutron scattering and atomistic simulation. In this paper we review our understanding of vibrational mechanisms of NTE for a range of materials. We identify a number of different cases, some of which involve a small number of phonons that can be described as involving rotations of rigid polyhedral groups of atoms, others where there are large bands of phonons involved, and some where the transverse acoustic modes provide the main contribution to NTE. In a few cases the elasticity of NTE materials has been studied under pressure, identifying an elastic softening under pressure. We propose that this property, called pressure-induced softening, is closely linked to NTE, which we can demonstrate using a simple model to describe NTE materials. There has also been recent interest in the role of intrinsic anharmonic interactions on NTE, particularly guided by calculations of the potential energy wells for relevant phonons. We review these effects, and show how anhamonicity affects the response of the properties of NTE materials to pressure.

Recent advances in the characterization of amorphous pharmaceuticals by X-ray diffractometry

For poorly water soluble drugs, the amorphous state provides an avenue to enhance oral bioavailability. The preparation method, in addition to sample history, can dictate the nature and the stability of the amorphous phase. Conventionally, X-ray powder diffractometry is of limited utility for characterization, but structural insights into amorphous and nanocrystalline materials have been enabled by coupling X-ray total scattering with the pair distribution function. This has shown great promise for fingerprinting, quantification, and even modeling of amorphous pharmaceutical systems. A consequence of the physical instability of amorphous phases is their crystallization propensity, and recent instrumental advances have substantially enhanced our ability to detect and quantify crystallization in a variety of complex matrices. The International Centre for Diffraction Data has a collection of the X-ray diffraction patterns of amorphous drugs and excipients and, based on the available supporting information, provides a quality mark of the data.

Towards solution and refinement of organic crystal structures by fitting to the atomic pair distribution function

A method towards the solution and refinement of organic crystal structures by fitting to the atomic pair distribution function (PDF) is developed. Approximate lattice parameters and molecular geometry must be given as input. The molecule is generally treated as a rigid body. The positions and orientations of the molecules inside the unit cell are optimized starting from random values. The PDF is obtained from carefully measured X-ray powder diffraction data. The method resembles `real-space' methods for structure solution from powder data, but works with PDF data instead of the diffraction pattern itself. As such it may be used in situations where the organic compounds are not long-range-ordered, are poorly crystalline, or nanocrystalline. The procedure was applied to solve and refine the crystal structures of quinacridone (β phase), naphthalene and allopurinol. In the case of allopurinol it was even possible to successfully solve and refine the structure inP1 with four independent molecules. As an example of a flexible molecule, the crystal structure of paracetamol was refined using restraints for bond lengths, bond angles and selected torsion angles. In all cases, the resulting structures are in excellent agreement with structures from single-crystal data.

Representational analysis of extended disorder in atomistic ensembles derived from total scattering data

Representational analysis is used to characterize correlated short-range order in large atomistic ensembles. This method, analogous to tight-binding methods, enables the extraction of relevant structural parameters in an orthogonal and local basis that permits robust statistical analysis of crystalline disorder.

With the increased availability of high-intensity time-of-flight neutron and synchrotron X-ray scattering sources that can access wide ranges of momentum transfer, the pair distribution function method has become a standard analysis technique for studying disorder of local coordination spheres and at intermediate atomic separations. In some cases, rational modeling of the total scattering data (Bragg and diffuse) becomes intractable with least-squares approaches, necessitating reverse Monte Carlo simulations using large atomistic ensembles. However, the extraction of meaningful information from the resulting atomistic ensembles is challenging, especially at intermediate length scales. Representational analysis is used here to describe the displacements of atoms in reverse Monte Carlo ensembles from an ideal crystallographic structure in an approach analogous to tight-binding methods. Rewriting the displacements in terms of a local basis that is descriptive of the ideal crystallographic symmetry provides a robust approach to characterizing medium-range order (and disorder) and symmetry breaking in complex and disordered crystalline materials. This method enables the extraction of statistically relevant displacement modes (orientation, amplitude and distribution) of the crystalline disorder and provides directly meaningful information in a locally symmetry-adapted basis set that is most descriptive of the crystal chemistry and physics.

Total-scattering pair-distribution function of organic material from powder electron diffraction data

This paper shows that pair-distribution function (PDF) analyses can be carried out on organic and organometallic compounds from powder electron diffraction data. Different experimental setups are demonstrated, including selected area electron diffraction and nanodiffraction in transmission electron microscopy or nanodiffraction in scanning transmission electron microscopy modes. The methods were demonstrated on organometallic complexes (chlorinated and unchlorinated copper phthalocyanine) and on purely organic compounds (quinacridone). The PDF curves from powder electron diffraction data, called ePDF, are in good agreement with PDF curves determined from X-ray powder data demonstrating that the problems of obtaining kinematical scattering data and avoiding beam damage of the sample are possible to resolve.