Real-space calculation of powder diffraction patterns on graphics processing units

Implications of size dispersion on X-ray scattering of crystalline nanoparticles: CeO2 as a case study

Controlling the shape and size dispersivity and crystallinity of nanoparticles (NPs) has been a challenge in identifying these parameters' role in the physical and chemical properties of NPs. The need for reliable quantitative tools for analyzing the dispersivity and crystallinity of NPs is a considerable problem in optimizing scalable synthesis routes capable of controlling NP properties. The most common tools are electron microscopy (EM) and X-ray scattering techniques. However, each technique has different susceptibility to these parameters, implying that more than one technique is necessary to characterize NP systems with maximum reliability. Wide-angle X-ray scattering (WAXS) is mandatory to access information on crystallinity. In contrast, EM or small-angle X-ray scattering (SAXS) is required to access information on whole NP sizes. EM provides average values on relatively small ensembles in contrast to the bulk values accessed by X-ray techniques. Besides the fact that the SAXS and WAXS techniques have different susceptibilities to size distributions, SAXS is easily affected by NP-NP interaction distances. Because of all the variables involved, there have yet to be proposed methodologies for cross-analyzing data from two techniques that can provide reliable quantitative results of dispersivity and crystallinity. In this work, a SAXS/WAXS-based methodology is proposed for simultaneously quantifying size distribution and degree of crystallinity of NPs. The most reliable easy-to-access size result for each technique is demonstrated by computer simulation. Strategies on how to compare these results and how to identify NP-NP interaction effects underneath the SAXS intensity curve are presented. Experimental results are shown for cubic-like CeO2 NPs. WAXS size results from two analytical procedures are compared, line-profile fitting of individual diffraction peaks in opposition to whole pattern fitting. The impact of shape dispersivity is also evaluated. Extension of the proposed methodology for cross-analyzing EM and WAXS data is possible.

GPU-accelerated multitiered iterative phasing algorithm for fluctuation X-ray scattering

The multitiered iterative phasing (MTIP) algorithm is used to determine the biological structures of macromolecules from fluctuation scattering data. It is an iterative algorithm that reconstructs the electron density of the sample by matching the computed fluctuation X-ray scattering data to the external observations, and by simultaneously enforcing constraints in real and Fourier space. This paper presents the first ever MTIP algorithm acceleration efforts on contemporary graphics processing units (GPUs). The Compute Unified Device Architecture (CUDA) programming model is used to accelerate the MTIP algorithm on NVIDIA GPUs. The computational performance of the CUDA-based MTIP algorithm implementation outperforms the CPU-based version by an order of magnitude. Furthermore, the Heterogeneous-Compute Interface for Portability (HIP) runtime APIs are used to demonstrate portability by accelerating the MTIP algorithm across NVIDIA and AMD GPUs.

GALLOP: accelerated molecular crystal structure determination from powder diffraction data

A new GPU-accelerated algorithm delivers state-of-the-art performance for molecular crystal structure determination from powder diffraction data.

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.

Whole-nanoparticle atomistic modeling of the schwertmannite structure from total scattering data

Schwertmannite is a poorly crystalline nanometric iron sulfate oxyhydroxide. This mineral shows a structural variability under different environments. Because of that, the determination of its structure and, consequently, of its physical–chemical properties is quite challenging. This article presents a detailed structural investigation of the structure of schwertmannite conducted under different approaches: X-ray absorption spectroscopy, Rietveld refinement, and a combined reverse Monte Carlo and Debye function analysis of the whole nanoparticle structure. The schwertmannite model presented here is, to the auhors' knowledge, the most complete model so far reported.

Debye-Waller coefficient of heavily deformed nanocrystalline iron

Extensive deformation of an iron alloy powder increases the static disorder contribution to the thermal factor, with an increase of ∼20% in the Debye–Waller coefficient observed by both X-ray diffraction and extended X-ray absorption fine structure. Molecular dynamics simulations shed light on the underlying mechanisms, confirming the major role played by the grain boundary.

Synchrotron radiation X-ray diffraction (XRD) patterns from an extensively ball-milled iron alloy powder were collected at 100, 200 and 300 K. The results were analysed together with those using extended X-ray absorption fine structure, measured on the same sample at liquid nitrogen temperature (77 K) and at room temperature (300 K), to assess the contribution of static disorder to the Debye–Waller coefficient (B_iso). Both techniques give an increase of ∼20% with respect to bulk reference iron, a noticeably smaller difference than reported by most of the literature for similar systems. Besides good quality XRD patterns, proper consideration of the temperature diffuse scattering seems to be the key to accurate values of the Debye–Waller coefficient. Molecular dynamics simulations of nanocrystalline iron aggregates, mapped on the evidence provided by XRD in terms of domain size distribution, shed light on the origin of the observed B_iso increase. The main contribution to the static disorder is given by the grain boundary, while line and point defects have a much smaller effect.

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.

High-performance powder diffraction pattern simulation for large-scale atomistic models via full-precision pair distribution function computation

A new full-precision algorithm to solve the Debye scattering equation has been developed for high-performance computing of powder diffraction line profiles from large-scale atomistic models of nanomaterials. The Debye function was evaluated using a pair distribution function computed with high accuracy, exploiting the series expansion of the error between calculated and equispace-sampled pair distances of atoms. The intensity uncertainty (standard deviation) of the computed diffraction profile was estimated as a function of the algorithm-intrinsic approximations and coordinate precision of the atomic positions, confirming the high accuracy of the simulated pattern. Based on the propagation of uncertainty, the new algorithm provides a more accurate powder diffraction profile than a brute-force calculation. Indeed, the precision of floating-point numbers employed in brute-force computations is worse than the estimated accuracy provided by the new algorithm. A software application, ROSE-X, has been implemented for parallel computing on CPU/GPU multi-core processors and distributed clusters. The computing performance is directly proportional to the total processor speed of the devices. An average speed of ∼30 × 109 computed pair distances per second was measured, allowing simulation of the powder diffraction pattern of an ∼23 million atom microstructure in a couple of hours. Moreover, the pair distribution function was recorded and reused to evaluate powder diffraction profiles of the same system with different properties (i.e. Q rather than 2θ range, step and wavelength), avoiding additional pair distance computations. This approach was used to investigate a large collection of monoatomic and polyatomic microstructures, isolating the contribution from atoms belonging to different moieties (e.g. different species or crystalline domains).

Vibrational Properties of Nanocrystals from the Debye Scattering Equation

One hundred years after the original formulation by Petrus J.W. Debije (aka Peter Debye), the Debye Scattering Equation (DSE) is still the most accurate expression to model the diffraction pattern from nanoparticle systems. A major limitation in the original form of the DSE is that it refers to a static domain, so that including thermal disorder usually requires rescaling the equation by a Debye-Waller thermal factor. The last is taken from the traditional diffraction theory developed in Reciprocal Space (RS), which is opposed to the atomistic paradigm of the DSE, usually referred to as Direct Space (DS) approach. Besides being a hybrid of DS and RS expressions, rescaling the DSE by the Debye-Waller factor is an approximation which completely misses the contribution of Temperature Diffuse Scattering (TDS). The present work proposes a solution to include thermal effects coherently with the atomistic approach of the DSE. A deeper insight into the vibrational dynamics of nanostructured materials can be obtained with few changes with respect to the standard formulation of the DSE, providing information on the correlated displacement of vibrating atoms.

Size and shape of graphene layers in commercial carbon blacks established by Debye refinement

The size and the shape of graphene layers in commercial conductive carbon blacks, Super P and Super S, have been determined from powder X-ray diffraction data. Using a refinement procedure based on the fundamental diffraction equation of Debye, it is shown that the ordered regions within the layers of both materials are of elliptical shape, curved in a cylindrical fashion along the longer axis of the ellipse. The regions are greater in Super P, ellipse axes 5.4 and 2.2 nm, than in Super S, 4.6 and 2.1 nm, and less curved (curvature radii 12.7 and 11.7 nm, respectively). There is no crystallographic registry between layers, which are equidistantly stacked into concentric groups of six or seven, on average.

A new parallel and GPU version of aTREOR-based algorithm for indexing powder diffraction data

One of the key parts of the crystal structure solution process from powder diffraction data is indexing – the determination of the lattice parameters from experimental data. This paper presents a modification of theTREORindexing method that makes the algorithm suitable and efficient for execution on graphics processing units. TheTREORalgorithm was implemented in its pure form, which can be simply described as a `brute-force' approach. The effectiveness and time consumption of such an algorithm was tested on several data sets including monoclinic and triclinic examples. The results show the potential of using GPUs for indexing powder diffraction data.

Diffuse X-ray scattering from ion-irradiated materials: a parallel-computing approach

A computational method for the evaluation of the two-dimensional diffuse X-ray scattering distribution from irradiated single crystals is presented. A Monte Carlo approach is used to generate the displacement field in the damaged crystal. This step makes use of vector programming and multiprocessing to accelerate the computation. Reciprocal space maps are then computed using GPU-accelerated fast Fourier transforms. It is shown that this procedure speeds up the calculation by a factor of ∼190 for a crystal containing 109unit cells. The potential of the method is illustrated with two examples: the diffuse scattering from a single crystal containing (i) a non-uniform defect depth distribution (with a potentially bimodal defect size distribution) and (ii) spatially correlated defects exhibiting either long-range or short-range ordering with varying positional disorder.

Rapid and accurate calculation of small-angle scattering profiles using the golden ratio

Calculating the scattering intensity of anN-atom system is a numerically exhaustingO(N2) task. A simple approximation technique that scales linearly with the number of atoms is presented. Using an exact expression for the scattering intensityI(q) at a given wavevectorq, the rotationally averaged intensityI(q) is computed by evaluatingI(q) in several scattering directions. The orientations of theqvectors are taken from a quasi-uniform spherical grid generated by the golden ratio. Using various biomolecules as examples, this technique is compared with an established multipole expansion method. For a given level of speed, the technique is more accurate than the multipole expansion for anisotropically shaped molecules, while comparable in accuracy for globular shapes. The processing time scales sub-linearly inNwhen the atoms are identical and lie on a lattice. The procedure is easily implemented and should accelerate the analysis of small-angle scattering data.

Combined X-ray diffraction and solid-state19F magic angle spinning NMR analysis of lattice defects in nanocrystalline CaF2

Nanocrystalline CaF2powder specimens were produced both by co-precipitation of CaCl2and NH4F and by ball milling of a coarse powder. The specimen homogeneity and a detailed picture of the lattice defects can be assessed by the simultaneous analysis of the powder diffraction pattern and of the solid-state19F magic angle spinning NMRT1relaxometry data. While diffraction line profiles provide information on domain size distribution and the content of dislocations,T1relaxometry is more sensitive to inhomogeneity of the powder (large defect-free grainsversusdefective small ones). After extensive milling it is possible to obtain fluorite domains of comparable size to the chemically synthesized CaF2(circa10–12 nm), but with a marked difference in the lattice defect types and content. It is then proved that surface defects (related to domain size), line defects (dislocations) and point (Frenkel) defects have a quite different effect on the powder pattern as well as on theT1spin-lattice relaxation time.

Using GPUs to compute fast Fourier transforms for crystal structure solution and refinement

In the past few years, new hardware tools have become available for computing using the graphical processing units (GPUs) present in modern graphics cards. These GPUs allow efficient parallel calculations with a much higher throughput than microprocessors. In this work, fast Fourier transformation calculations used inSIR2011software algorithms have been carried out using the power of the GPU, and the speed of the calculations has been compared with that achieved using normal CPUs.

Directional pair distribution function for diffraction line profile analysis of atomistic models

The concept of the directional pair distribution function is proposed for an atomistic level interpretation of the line profile broadening in powder diffraction patterns of nanocrystalline materials.

The concept of the directional pair distribution function is proposed to describe line broadening effects in powder patterns calculated from atomistic models of nano-polycrystalline microstructures. The approach provides at the same time a description of the size effect for domains of any shape and a detailed explanation of the strain effect caused by the local atomic displacement. The latter is discussed in terms of different strain types, also accounting for strain field anisotropy and grain boundary effects. The results can in addition be directly read in terms of traditional line profile analysis, such as that based on the Warren–Averbach method.

Short-range and long-range order of phyllomanganate nanoparticles determined using high-energy X-ray scattering

High-energy X-ray scattering (HEXS) is used to explore the pH-dependent structure of randomly stacked manganese oxide nanosheets of nominal formula δ-MnO2. Data are simulated in real space by pair distribution function (PDF) analysis and in reciprocal space by both the Bragg-rod method and the Debye equation in order to maximize the information gained from the total scattering measurements. The essential new features of this triple-analysis approach are (1) the use of a two-dimensional supercell in PDF modeling to describe local distortions around Mn layer vacancies, (2) the implementation in Bragg-rod calculations of a lognormal crystal size distribution in the layer plane and an empirical function for the effect of strain, and (3) the incorporation into the model used with the Debye equation of an explicit elastic deformation of the two-dimensional nanocrystals. The PDF analysis reveals steady migration at acidic pH of the Mn atoms from layer to interlayer sites, either above or below the Mn layer vacancies, and important displacement of the remaining in-layer Mn atoms toward vacancies. The increased density of the vacancy–interlayer Mn pairs at low pH causes their mutual repulsion and results in short-range ordering. The layer microstructure, responsible for the long-range lateral disorder, is modeled with spherically and cylindrically bent crystallites having volume-averaged radii of 20–40 Å. Thebunit-cell parameter from the hexagonal layer has different values in PDF, Bragg-rod and Debye equation modeling, because of the use of different weighting contributions from long-range and short-range distances in each method. The PDFbparameter is in effect a measure of the average inlayer Mn...Mn distance and consistently deviates from the average structure value determined by the Bragg-rod method by 0.02 Å at low pH, as a result of the local relaxation induced by vacancies. The layer curvature increases the Bragg-rod value by 0.01–0.02 Å with the cylindrical model and as much as 0.04–0.05 Å with the spherical model. Therefore, in principle, the diffraction alone can unambiguously determine with good accuracy only a volume-averaged apparent layer dimension of the manganese oxide nanosheets. Thebparameter is model dependent and has no single straightforward interpretation, so comparison ofbbetween different samples only makes sense if done in the context of a single specified model.

Common volume functions and diffraction line profiles of polyhedral domains

A general numerical algorithm is proposed for the fast computation of the common volume function (CVF) of any polyhedral object, from which the diffraction pattern of a corresponding powder can be obtained. The theoretical description of the algorithm is supported by examples ranging from simple equilibrium shapes in cubic materials (Wulff polyhedra) to more exotic non-convex shapes, such as tripods or hollow cubes. Excellent agreement is shown between patterns simulated using the CVF and the corresponding ones calculated from the atomic positionsviathe Debye scattering equation.

Fast computation of scattering maps of nanostructures using graphical processing units

Scattering maps from strained or disordered nanostructures around a Bragg reflection can be either computed quickly using approximations and a (fast) Fourier transform or obtained using individual atomic positions. In this article, it is shown that it is possible to compute up to 4 × 1010 reflections atoms s−1using a single graphics card, and the manner in which this speed depends on the number of atoms and points in reciprocal space is evaluated. An open-source software library (PyNX) allowing easy scattering computations (including grazing-incidence conditions) in the Python language is described, with examples of scattering from non-ideal nanostructures.

DEBUSSY: a Debye user system for nanocrystalline materials

DEBUSSYis a new free open-source package, written in Fortran95 and devoted to the application of the Debye function analysis (DFA) of powder diffraction data from nanocrystalline, defective and/or non-periodic materials through the use of sampled interatomic distance databases. The suite includes a main program, taking the name of the package,DEBUSSY, and dealing with the DFA of X-ray, neutron and electron experimental data, and a suite of 11 programs, namedCLAUDE, enabling users to create their own databases for nanosized crystalline materials, starting from the list of space-group generators and the asymmetric unit content. A new implementation of the Debye formula is adopted inDEBUSSY, which makes the approach fast enough to deal with the pattern calculation of hundreds of nanocrystals, to sum up their contributions to the total pattern and to perform iterative algorithms for optimizing the parameters of the pattern model. The package strategy uses the sampled-distance database(s) created previously byCLAUDEand combines, for any phase, a log-normal or a bivariate log-normal function to deal with the sample-size distribution; four different functions are implemented to manage possible lattice expansions/contractions as a function of crystal size. A number of output ASCII files are produced to supply some statistics and data suitable for graphical use. The databases of sampled interatomic non-dimensional distances for cuboctahedral, decahedral and icosahedral structure types, suitable for dealing with noble metal nanoparticles, are also available.