Precise implications for real-space pair distribution function modeling of effects intrinsic to modern time-of-flight neutron diffractometers

Total scattering measurements at the Australian Synchrotron Powder Diffraction beamline: capabilities and limitations

This study describes the capabilities and limitations of carrying out total scattering experiments on the Powder Diffraction (PD) beamline at the Australian Synchrotron, ANSTO. A maximum instrument momentum transfer of 19 Å^-1 can be achieved if the data are collected at 21 keV. The results detail how the pair distribution function (PDF) is affected by Q_max, absorption and counting time duration at the PD beamline, and refined structural parameters exemplify how the PDF is affected by these parameters. There are considerations when performing total scattering experiments at the PD beamline, including (1) samples need to be stable during data collection, (2) highly absorbing samples with a μR > 1 always require dilution and (3) only correlation length differences >0.35 Å may be resolved. A case study comparing the PDF atom-atom correlation lengths with EXAFS-derived radial distances of Ni and Pt nanocrystals is also presented, which shows good agreement between the two techniques. The results here can be used as a guide for researchers considering total scattering experiments at the PD beamline or similarly setup beamlines.

Strategies and Considerations for Least-Squares Analysis of Total Scattering Data

The process of least-squares analysis has been applied for decades in the field of crystallography. Here, we discuss the application of this process to total scattering data, primarily in the combination of least-squares Rietveld refinements and fitting of the atomic pair distribution function (PDF). While these two approaches use the same framework, the interpretation of results from least-squares fitting of PDF data should be done with caution through carefully constructed analysis approaches. We provide strategies and considerations for applying least-squares analysis to total scattering data, combining both crystallographic Rietveld and fitting of PDF data, given in context with recent examples from the literature. This perspective is aimed to be an accessible document for those new to the total scattering approach, as well as a reflective framework for the total scattering expert.

Effects of Structural and Microstructural Features on the Total Scattering Pattern of Nanocrystalline Materials

Atomic- and nanometer-scale features of nanomaterials have a strong influence on their chemical and physical properties and a detailed description of these elements is a crucial step in their characterization. Total scattering methods, in real and reciprocal spaces, have been established as fundamental techniques to retrieve this information. Although the impact of microstructural features, such as defectiveness of different kinds, has been extensively studied in reciprocal space, disentangling these effects from size- and morphology-induced properties, upon downsizing, is not a trivial task. Additionally, once the experimental pattern is Fourier transformed to calculate the pair distribution function, the direct fingerprint of structural and microstructural features is severely lost and no modification of the histogram of interatomic distances derived therefrom is clearly discussed nor considered in the currently available protocols. Hereby, starting from atomistic models of a prototypical system (cadmium selenide), we simulate multiple effects on the atomic pair distribution function, obtained from reciprocal space patterns computed through the Debye scattering equation. Size and size dispersion effects, as well as different structures, morphologies, and their interplay with several kinds of planar defects, are explored, aiming at identifying the main (measurable and informative) fingerprints of these features on the total scattering pattern in real and reciprocal spaces, highlighting how, and how much, they become evident when comparing different cases. The results shown herein have general validity and, as such, can be further extended to other classes of nanomaterials.

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.

Effects of Voigt diffraction peak profiles on the pair distribution function

Powder diffraction and pair distribution function (PDF) analysis are well established techniques for investigation of atomic configurations in crystalline materials, and the two are related by a Fourier transformation. In diffraction experiments, structural information, such as crystallite size and microstrain, is contained within the peak profile function of the diffraction peaks. However, the effects of the PXRD (powder X-ray diffraction) peak profile function on the PDF are not fully understood. Here, all the effects from a Voigt diffraction peak profile are solved analytically, and verified experimentally through a high-quality X-ray total scattering measurement on Ni powder. The Lorentzian contribution to the microstrain broadening is found to result in Voigt-shaped PDF peaks. Furthermore, it is demonstrated that an improper description of the Voigt shape during model refinement leads to overestimation of the atomic displacement parameter.

Erratum: Illustrated formalisms for total scattering data: a guide for new practitioners. Corrigendum and addendum

Errors and ambiguities in the article by Peterson, Olds, McDonnell & Page [J. Appl. Cryst. (2021), 54, 317–332] are corrected and clarified, respectively.

Errors and ambiguities in the article by Peterson, Olds, McDonnell & Page [J. Appl. Cryst. (2021), 54, 317–332] are corrected and clarified, respectively.

Simulating and benchmarking neutron total scattering instrumentation from inception of events to reduced and fitted data

In the design and realization of modern neutron scattering instrumentation, particularly when designing beamline concepts from the ground up, it is desirable to fully benchmark against realistically simulated data. This is especially true for total scattering beamlines, where the future deliverable data is to be analysed in both reciprocal- and real-space representations, and needs must be carefully balanced to ensure sufficient range, resolution and flux of the instrument. An approach to optimize the design of neutron scattering instrumentation via a workflow including ray-tracing simulations, event-based data reduction, heuristic analysis and fitting against realistically simulated spectra is demonstrated here. The case of the DISCOVER beamline concept at the Spallation Neutron Source is used as an example. The results of the calculations are benchmarked through simulation of existing instrumentation and subsequent direct comparison with measured data. On the basis of the validated models, the ability to explore design characteristics for future beamline concepts or future instrument improvements is demonstrated through the examples of detector tube size and detector layout.

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.

Local and long-range atomic/magnetic structure of non-stoichiometric spinel iron oxide nanocrystallites

Spinel iron oxide nanoparticles of different mean sizes in the range 10-25 nm have been prepared by surfactant-free up-scalable near- and super-critical hydro-thermal synthesis pathways and characterized using a wide range of advanced structural characterization methods to provide a highly detailed structural description. The atomic structure is examined by combined Rietveld analysis of synchrotron powder X-ray diffraction (PXRD) data and time-of-flight neutron powder-diffraction (NPD) data. The local atomic ordering is further analysed by pair distribution function (PDF) analysis of both X-ray and neutron total-scattering data. It is observed that a non-stoichiometric structural model based on a tetragonal γ-Fe2O3 phase with vacancy ordering in the structure (space group P43212) yields the best fit to the PXRD and total-scattering data. Detailed peak-profile analysis reveals a shorter coherence length for the superstructure, which may be attributed to the vacancy-ordered domains being smaller than the size of the crystallites and/or the presence of anti-phase boundaries, faulting or other disorder effects. The intermediate stoichiometry between that of γ-Fe2O3 and Fe3O4 is confirmed by refinement of the Fe/O stoichiometry in the scattering data and quantitative analysis of Mössbauer spectra. The structural characterization is complemented by nano/micro-structural analysis using transmission electron microscopy (TEM), elemental mapping using scanning TEM, energy-dispersive X-ray spectroscopy and the measurement of macroscopic magnetic properties using vibrating sample magnetometry. Notably, no evidence is found of a Fe3O4/γ-Fe2O3 core-shell nanostructure being present, which had previously been suggested for non-stoichiometric spinel iron oxide nanoparticles. Finally, the study is concluded using the magnetic PDF (mPDF) method to model the neutron total-scattering data and determine the local magnetic ordering and magnetic domain sizes in the iron oxide nanoparticles. The mPDF data analysis reveals ferrimagnetic collinear ordering of the spins in the structure and the magnetic domain sizes to be ∼60-70% of the total nanoparticle sizes. The present study is the first in which mPDF analysis has been applied to magnetic nanoparticles, establishing a successful precedent for future studies of magnetic nanoparticles using this technique.

New capabilities for enhancement of RMCProfile: instrumental profiles with arbitrary peak shapes for structural refinements using the reverse Monte Carlo method

Reported here are the development and application of new capabilities in the RMCProfile software for structural refinements using the reverse Monte Carlo (RMC) method. An algorithm has been implemented to enable the use of arbitrary peak-shape functions in the modeling of Bragg diffraction patterns and instrumental resolution effects on total-scattering data. This capability eliminates the dependence of RMCProfile on preset functions, which are inadequate for data produced by some total-scattering instruments, e.g. NOMAD at the Spallation Neutron Source (SNS) at Oak Ridge, Tennessee, USA. The recently developed procedure for the instrument-resolution correction has been modified to improve its accuracy, which is critical for recovering nanoscale structure. The ability to measure fine details of local and nanoscale structures with high fidelity is required because such features are increasingly exploited in the design of materials with enhanced functional properties. The new methodology has been tested via RMC refinements of large-scale atomic configurations (distances up to 8 nm) for SrTiO3 using neutron total-scattering data collected on the Polaris and NOMAD time-of-flight powder diffractometers at the ISIS facility (Didcot, Oxfordshire, UK) and SNS, respectively. While the Polaris instrument is known to provide the high-quality data needed for RMC analysis, the similar and sound atomic configurations obtained from both instruments confirmed that the NOMAD data are also suitable for RMC refinements over a broad distance range.

Cross-examining Polyurethane Nanodomain Formation and Internal Structure

Structural and morphological interplay between hard and soft phases determine the bulk properties of thermoplastic polyurethanes. Commonly employed techniques rely on different physical or chemical phenomena for characterizing the organization of domains, but detailed structural information can be difficult to derive. Here, total scattering pair distribution function (PDF) analysis is used to determine atomic-scale insights into the connectivity and molecular ordering and compared to the domain size and morphological characteristics measured by AFM, TEM, SAXS, WAXS, and solid-state NMR 1H-1H spin-diffusion. In particular, density distribution functions are highlighted as a means to bridging the gap from the domain morphology to intradomain structural ordering. High real-space resolution PDFs are shown to provide a sensitive fingerprint for indexing aromatic, aliphatic, and polymerization-induced bonding characteristics, as well as the hard phase structure, and indicate that hard phases coexist in both ordered and disordered states.

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.

Evidence of Paracrystalline Cation Order in the Ruddlesden-Popper Phase LaSr3NiRuO8 through Neutron Total Scattering Techniques

Cation ordering in perovskite-derived phases can lead to a wealth of tunable physical properties. Ordering is typically driven by a large difference between the cation size and charge, but many Ruddlesden-Popper phases A_n+1B_nO_3n+1 appear to lack such B-site ordering, even when these differences are present. One such example is the "double" Ruddlesden-Popper n = 1 composition LaSr₃NiRuO₈. In this material, a lack of B-site ordering is observed through traditional crystallographic techniques, but antiferromagnetic ordering in the magnetism data suggests that B-site cation ordering is indeed present. Neutron total scattering, particularly analysis of the neutron pair distribution function, reveals that the structure is locally B-site-ordered below 6 Å but becomes slightly disordered in the midrange structure around 12 Å. This provides evidence for paracrystalline order in this material: cation ordering within a single perovskite sheet that lacks perfect registry within the three-dimensional stack of sheets. This work highlights the importance of employing a structural technique that can probe both the local and midrange order in addition to the crystallographic structure and provides a structural origin to the observed magnetic properties of LaSr₃NiRuO₈. Further, it is proposed that paracrystalline order is likely to be common among these layered-type oxides.

Highly versatile laboratory X-ray scattering instrument enabling (nano-)material structure analysis on multiple length scales by covering a scattering vector range of almost five decades

A compact laboratory X-ray scattering platform that uniquely enables for high-performance ultra-small-angle X-ray scattering (USAXS), small- and wide-angle X-ray scattering (SAXS/WAXS), and total scattering (atomic pair distribution function analysis; PDF) experiments was developed. It covers Bragg spacings from sub-Angstroms to 1.7 μm, thus allowing the analysis of dimensions and complex structures in (nano-)materials on multiple length scales. The accessible scattering vector q-range spans over almost five decades (q_min = 0.0036 nm^-1, q_max = 215 nm^-1), without any gaps. Whereas SAXS is suitable to characterize materials on a length scale of 1-100 nm, with USAXS, this range can be significantly extended to the micrometer range. On the other end, from WAXS and particularly from PDF measurements, information about the local atomic order and disorder can be obtained. The high performance, exceptional versatility, and ease-of-use of the instrument are enabled by a high-resolution 2-circle goniometer with kinematic mounts, a modular concept based on prealigned, quickly interchangeable X-ray components, and advanced detector technology. For USAXS measurements, a modified Bonse-Hart experimental setup with single crystal collimator and analyzer optics is used. SAXS/WAXS measurements are enabled by focusing optics, an evacuated beam path, and a 2D detector. For total scattering experiments, a high-energy X-ray source is used in combination with a hybrid pixel array detector that is based on a CdTe sensor for the highest counting efficiency. To ensure high resolution and sensitivity in these various applications, special care is taken to suppress any type of background scattering signal. The high resolution that can be achieved with the USAXS collimation system is demonstrated on a set of monodisperse, colloidal silica dispersions and derived colloidal crystals, with particle diameters in the range of hundreds of nanometers up to 1.6 µm. USAXS and SAXS results are shown to be consistent with those obtained by static light scattering (SLS) and dynamic light scattering. It is demonstrated that the obtainable USAXS data bridge the gap in q between SAXS and SLS. The capabilities of the instrument to acquire high-quality total scattering data for PDF analysis are demonstrated on amorphous SiO₂ nanoparticles as well as on NaYF₄ upconversion nanocrystals. To the best of our knowledge, it is for the first time that we present a single laboratory instrument that enables measurements of high-quality X-ray scattering data within such a wide q-range, by combining four complementary elastic X-ray scattering techniques. The modular design concept of the instrument allows for incremental improvements as well as to add more applications in the future.

POWGEN: rebuild of a third-generation powder diffractometer at the Spallation Neutron Source

This work describes the design principles and upgrade of the neutron powder diffractometer POWGEN at the Spallation Neutron Source.

The neutron powder diffractometer POWGEN at the Spallation Neutron Source has recently (2017–2018) undergone an upgrade which resulted in an increased detector complement along with a full overhaul of the structural design of the instrument. The current instrument has a solid angular coverage of 1.2 steradians and maintains the original third-generation concept, providing a single-histogram data set over a wide d-spacing range and high resolution to access large unit cells, detailed structural refinements and in situ/operando measurements.

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.

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.

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.

A numerical method for deriving shape functions of nanoparticles for pair distribution function refinements

In the structural refinement of nanoparticles, discrete atomistic modeling can be used for small nanocrystals (< 15 nm), but becomes computationally unfeasible at larger sizes, where instead unit-cell-based small-box modeling is usually employed. However, the effect of the nanocrystal's shape is often ignored or accounted for with a spherical model regardless of the actual shape due to the complexities of solving and implementing accurate shape effects. Recent advancements have provided a way to determine the shape function directly from a pair distribution function calculated from a discrete atomistic model of any given shape, including both regular polyhedra (e.g. cubes, spheres, octahedra) and anisotropic shapes (e.g. rods, discs, ellipsoids) [Olds et al. (2015). J. Appl. Cryst. 48, 1651-1659], although this approach is still limited to small size regimes due to computational demands. In order to accurately account for the effects of nanoparticle size and shape in small-box refinements, a numerical or analytical description is needed. This article presents a methodology to derive numerical approximations of nanoparticle shape functions by fitting to a training set of known shape functions; the numerical approximations can then be employed on larger sizes yielding a more accurate and physically meaningful refined nanoparticle size. The method is demonstrated on a series of simulated and real data sets, and a table of pre-calculated shape function expressions for a selection of common shapes is provided.