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

Exploring porous structures without crystals: advancements with pair distribution function in metal- and covalent organic frameworks

The pair distribution function (PDF) is a versatile characterisation tool in materials science, capable of retrieving atom-atom distances on a continuous scale (from a few angstroms to nanometres), without being restricted to crystalline samples. Typically, total scattering experiments are performed using high-energy synchrotron X-rays, neutrons or electrons to achieve a high atomic resolution in a short time. Recently, PDF analysis provides a powerful approach to target current characterisation challenges in the field of metal- and covalent organic frameworks. By identifying molecular interactions on the pore surfaces, tracking complex structural transformations involving disorder states, and elucidating nucleation and growth mechanisms, structural analysis using PDF has provided invaluable insights into these materials. This review article highlights the significance of PDF analysis in advancing our understanding of MOFs and COFs, paving the way for innovative applications and discoveries in porous materials research.

Evidence of Lone Pair Crafted Emphanisis in the Ruddlesden-Popper Halide Perovskite Cs2PbI2Cl2

A hallmark of typical structural transformations is an increase in symmetry upon heating due to entropic favourability. However, local symmetry breaking upon warming is recently evidenced in rare crystalline phases. Termed as emphanisis, the phenomenon implores exploration of fascinating thermodynamic nuances that drive unusual structural evolutions. Here, synchrotron X-ray total scattering measurements are presented on a Ruddlesden-Popper mixed halide perovskite, Cs2PbI2Cl2, which reveal signatures of emphanisis. The genesis of symmetry lowering upon heating is traced to a lone pair-driven cooperative local structural distortion composed of thermally actuated Pb off-centring and static Cl displacement. Mapping the thermal evolution of low-lying phonon modes with inelastic neutron scattering uncovers instances of mode hardening with picosecond lifetime and an intriguing soft mode at the X-point of the Brillouin zone-features conducive to ultralow thermal transport. Together, these observations highlight the fundamental and functional implications of chemical design in engendering unconventional phenomena in crystalline materials and associated properties.

Instrumental broadening and the radial pair distribution function with 2D detectors

The atomic pair distribution function (PDF) is a real-space representation of the structure of a material. Experimental PDFs are obtained using a Fourier transform from total scattering data which may or may not have Bragg diffraction peaks. The determination of Bragg peak resolution in scattering data from the fundamental physical parameters of the diffractometer used is well established, but after the Fourier transform from reciprocal to direct space, these contributions are harder to identify. Starting from an existing definition of the resolution function of large-area detectors for X-ray diffraction, this approach is expanded into direct space. The effect of instrumental parameters on PDF peak resolution is developed mathematically, then studied with modelling and comparison with experimental PDFs of LaB6 from measurements made in different-sized capillaries.

Toward a quantitative description of solvation structure: a framework for differential solution scattering measurements

Appreciating that the role of the solute-solvent and other outer-sphere interactions is essential for understanding chemistry and chemical dynamics in solution, experimental approaches are needed to address the structural consequences of these interactions, complementing condensed-matter simulations and coarse-grained theories. High-energy X-ray scattering (HEXS) combined with pair distribution function analysis presents the opportunity to probe these structures directly and to develop quantitative, atomistic models of molecular systems in situ in the solution phase. However, at concentrations relevant to solution-phase chemistry, the total scattering signal is dominated by the bulk solvent, prompting researchers to adopt a differential approach to eliminate this unwanted background. Though similar approaches are well established in quantitative structural studies of macromolecules in solution by small- and wide-angle X-ray scattering (SAXS/WAXS), analogous studies in the HEXS regime-where sub-ångström spatial resolution is achieved-remain underdeveloped, in part due to the lack of a rigorous theoretical description of the experiment. To address this, herein we develop a framework for differential solution scattering experiments conducted at high energies, which includes concepts of the solvent-excluded volume introduced to describe SAXS/WAXS data, as well as concepts from the time-resolved X-ray scattering community. Our theory is supported by numerical simulations and experiment and paves the way for establishing quantitative methods to determine the atomic structures of small molecules in solution with resolution approaching that of crystallography.

Sub-second pair distribution function using a broad bandwidth monochromator

Here the use of a broad energy bandwidth monochromator, i.e. a pair of B4C/W multilayer mirrors (MLMs), is demonstrated for X-ray total scattering (TS) measurements and pair distribution function (PDF) analysis. Data are collected both on powder samples and from metal oxo clusters in aqueous solution at various concentrations. A comparison between the MLM PDFs and those obtained using a standard Si(111) double-crystal monochromator shows that the measurements yield MLM PDFs of high quality which are suitable for structure refinement. Moreover, the effects of time resolution and concentration on the quality of the resulting PDFs of the metal oxo clusters are investigated. PDFs of heptamolybdate clusters and tungsten α-Keggin clusters from X-ray TS data were obtained with a time resolution down to 3 ms and still showed a similar level of Fourier ripples to PDFs obtained from 1 s measurements. This type of measurement could thus open up faster time-resolved TS and PDF studies.

Structural water in amorphous carbonate minerals: ab initio molecular dynamics simulations of X-ray pair distribution experiments

Water is known to play a controlling role in directing mineralization pathways and stabilizing metastable amorphous intermediates in hydrous carbonate mineral MCO₃·nH₂O systems, where M²⁺ is a divalent metal cation. Despite this recognition, the nature of the controls on crystallization is poorly understood, largely owing to the difficulty in characterizing the dynamically disordered structures of amorphous intermediates at the atomic scale. Here, we present a series of atomistic models, derived from ab initio molecular dynamics simulation, across a range of experimentally relevant cations (M = Ca, Mg, Sr) and hydration levels (0 ≤ n ≤ 2). Theoretical simulations of the dependence of the X-ray pair distribution function on the hydration level n show good agreement with available experimental data and thus provide further evidence for a lack of significant nanoscale structure in amorphous carbonates. Upon dehydration, the metal coordination number does not change significantly, but the relative extent of water dissociation increases, indicating that a thermodynamic driving force exists for water dissociation to accompany dehydration. Mg strongly favors monodentate conformation of carbonate ligands and shows a marked preference to exchange monodentate carbonate O for water O upon hydration, whereas Ca and Sr exchange mono- and bidentate carbonate ligands with comparable frequency. Water forms an extensive hydrogen bond network among both water and carbonate groups that exhibits frequent proton transfers for all three cations considered suggesting that proton mobility is likely predominantly due to water dissociation and proton transfer reactions rather than molecular water diffusion.

Structural characterisation of nanoalloys for (photo)catalytic applications with the Sapphire library

A non-trivial interplay rules the relationship between the structure and the chemophysical properties of a nanoparticle. In this context, characterization experiments, molecular dynamics simulations and electronic structure calculations may allow the variables that determine a given property to be pinpointed. Conversely, a rigorous computational characterization of the geometry and chemical ordering of metallic nanoparticles and nanoalloys enables discrimination of which descriptors could be linked with their stability and performance. To this end, we introduce a modular and open-source library, Sapphire, which may classify the structural characteristics of a given nanoparticle through several structural analysis techniques and order parameters. A special focus is geared towards using geometrical descriptors to make predictions on a given nanoparticle's catalytic activity.

Experimentally informed structure optimization of amorphous TiO2 films grown by atomic layer deposition

Amorphous titanium dioxide TiO₂ (a-TiO₂) has been widely studied, particularly as a protective coating layer on semiconductors to prevent corrosion and promote electron-hole conduction in photoelectrochemical reactions. The stability and longevity of a-TiO₂ is strongly affected by the thickness and structural heterogeneity, implying that understanding the structure properties of a-TiO₂ is crucial for improving the performance. This study characterized the structural and electronic properties of a-TiO₂ thin films (∼17 nm) grown on Si by atomic layer deposition (ALD). Fluctuation spectra V(k) and angular correlation functions were determined with 4-dimensional scanning transmission electron microscopy (4D-STEM), which revealed the distinctive medium-range ordering in the a-TiO₂ film. A realistic atomic model of a-TiO₂ was established guided by the medium-range ordering and the previously reported short-range ordering of a-TiO₂ film, as well as the interatomic potential. The structure was optimized by the StructOpt code using a genetic algorithm that simultaneously minimizes energy and maximizes the match to experimental short- and medium-range ordering. The StructOpt a-TiO₂ model presents improved agreements with the medium-range ordering and the k-space location of the dominant 2-fold angular correlations compared with a traditional melt-quenched model. The electronic structure of the StructOpt a-TiO₂ model was studied by ab initio calculations and compared to the crystalline phases and experimental results. This work uncovered the medium-range ordering in a-TiO₂ thin films and provided a realistic a-TiO₂ structure model for further investigation of structure-property relationships and materials design. In addition, the improved multi-objective optimization package StructOpt was provided for structure determination of complex materials guided by experiments and simulations.

Efficient solution of particle shape functions for the analysis of powder total scattering data

Structural characterization of powder samples via total scattering methods, in either real or reciprocal space, must take into account the effect of particle shape. Here, the shape contribution of a set of ideally isolated particles to the small-angle scattering (SAS) component of the intensity profile is modelled using the shape function [Svergun & Koch (2003). Rep. Prog. Phys. 66, 1735-1782]. The shape function is obtained by orientational averaging of common volume functions (CVFs) for a discrete set of directions. The effects of particle size and size dispersity are accounted for via scaling of the CVFs and their convolution with the underlying probability distribution. The method is applied to shapes with CVFs expressed analytically or by using discrete tables. The accurate calculation of SAS particle shape contributions up to large momentum transfer demonstrates the reliability and flexibility of modelling shape functions from sets of CVFs. The algorithm presented here is computationally efficient and can be directly incorporated into existing routines for analysis of powder total scattering data.

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.

On the resolution function for powder diffraction with area detectors

In a powder diffraction experiment the resolution function defines the instrumental contribution to the peak widths as a function of the Bragg angle. The Caglioti formula is frequently applied to model the instrumental broadening and used in structural refinement. The parameters in the Caglioti formula are linked to physically meaningful parameters for most diffraction geometries. However, this link is lost for the now very popular powder diffraction geometry using large 2D area detectors. Here we suggest a new physical model for the instrumental broadening specifically developed for powder diffraction data measured with large 2D area detectors. The model is verified using data from two synchrotron diffraction beamlines with the Pilatus2M and MAR345 detectors. Finally, a functional form is proposed to replace the Caglioti formula for this geometry in the Rietveld method and profile refinements.

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.

Synchrotron total-scattering data applicable to dual-space structural analysis

Synchrotron powder X-ray diffraction (PXRD) is a well established technique for investigating the atomic arrangement of crystalline materials. At modern beamlines, X-ray scattering data can be collected in a total-scattering setting, which additionally opens up the opportunity for direct-space structural analysis through the atomic pair distribution function (PDF). Modelling of PXRD and PDF data is typically carried out separately, but employing a concurrent structural model to both direct- and reciprocal-space data has the possibility to enhance total-scattering data analysis. However, total-scattering measurements applicable to such dual-space analyses are technically demanding. Recently, the technical demands have been fulfilled by a MYTHEN microstrip detector system (OHGI), which meets the stringent requirements for both techniques with respect to Q range, Q resolution and dynamic range. In the present study, we evaluate the quality of total-scattering data obtained with OHGI by separate direct- and reciprocal-space analysis of Si. Excellent agreement between structural parameters in both spaces is found, demonstrating that the total-scattering data from OHGI can be utilized in dual-space structural analysis e.g. for in situ and operando measurements.

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.