Instrumental Resolution Function Research Articles

Upgrade of D+ software for hierarchical modeling of X-ray scattering data from complex structures in solution, fibers and single orientations.

This article presents an upgrade of the D+ software [Ginsburg et al. (2019 ▸). J. Appl. Cryst. 52, 219-242], expanding its hierarchical solution X-ray scattering modeling capabilities for fiber diffraction and single crystallographic orientations. This upgrade was carried out using the reciprocal grid algorithm [Ginsburg et al. (2016 ▸). J. Chem. Inf. Model. 56, 1518-1527], providing D+ its computational strength. Furthermore, the extensive modifications made to the Python API of D+ are described, broadening the X-ray analysis performed with D+ to account for the effects of the instrument-resolution function and polydispersity. In addition, structure-factor and radial-distribution-function modules were added, taking into account the effects of thermal fluctuations and intermolecular interactions. Finally, numerical examples demonstrate the usage and potential of the added features.

Accurate extrinsic and intrinsic peak broadening modelling for time-resolved in situ ball milling reactions via synchrotron powder X-ray diffraction.

The debate on the mechanisms which underpin mechanochemical reactions via ball mill grinding is still open. Our ability to accurately measure the microstructural (crystal size and microstrain) evolution of materials under milling conditions as well as their phase composition as a function of time is key to the in-depth understanding of the kinetics and driving forces of mechanochemical transformations. Furthermore, all ball milling reactions end with a steady state or milling equilibrium - represented by a specific phase composition and relative microstructure - that does not change as long as the milling conditions are maintained. The use of a standard sample is essential to determine the instrumental contribution to the X-ray powder diffraction (XRPD) peak broadening for time-resolved in situ (TRIS) monitoring of mechanochemical reactions under in operando conditions. Using TRIS-XRPD on a ball milling setup, coupled with low-energy synchrotron radiation, we investigated different data acquisition and analysis strategies on a silicon standard powder. The diffraction geometry and the microstructural evolution of the standard itself have been studied to model the instrumental contribution to XRPD peak broadening throughout the grinding activity. Previously proposed functions are here challenged and further developed. Importantly, we show that minor drifts of the jar position do not affect the instrumental resolution function significantly. We here report and discuss the results of such investigations and their application to TRIS-XRPD datasets of inorganic and organic ball mill grinding reactions.

Quasi-analytical resolution-correction of elastic neutron scattering from proteins.

Elastic neutron scattering from proteins reflects the motional amplitudes resulting from their internal collective and single-atom dynamics and is observable if the global diffusion of whole molecules is either blocked or cannot be resolved by the spectrometer under consideration. Due to finite instrumental resolution, the measured elastic scattering amplitude always contains contaminations from quasielastic neutron scattering and some model must be assumed to extract the resolution-corrected counterpart from corresponding experimental spectra. Here, we derive a quasi-analytical method for that purpose, assuming that the intermediate scattering function relaxes with a "stretched" Mittag-Leffler function, Eα(-(t/τ)α) (0 < α < 1), toward the elastic amplitude and that the instrumental resolution function has Gaussian form. The corresponding function can be integrated into a fitting procedure and allows for eliminating the elastic intensity as a fit parameter. We illustrate the method for the analysis of two proteins in solution, the intrinsically disordered Myelin Basic Protein, confirming recently published results [Hassani et al., J. Chem. Phys. 156, 025102 (2022)], and the well-folded globular protein myoglobin. We also briefly discuss the consequences of our findings for the extraction of mean square position fluctuations from elastic scans.

Strain and Grain Size Determination of CeO2 and TiO2 Nanoparticles: Comparing Integral Breadth Methods versus Rietveld, μ-Raman, and TEM

Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from several integral breadth methods such as Scherrer, Monshi, Williamson–Hall, and their variants: (i) uniform deformation model, (ii) uniform strain deformation model, and (iii) uniform deformation energy density model. We also employed the size–strain plot and Halder–Wagner method. For this purpose, an instrumental resolution function of an Al2O3 standard was used to subtract the instrumental broadening to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination. The Rietveld whole powder pattern decomposition method was introduced for comparison purposes, being the best candidate to fit the X-ray diffraction data of metal-oxide nanoparticles. Refined microstructural parameters were obtained using the anisotropic spherical harmonic size approach and correlated with the above estimation methods and transmission electron microscopy images. In addition, μ-Raman spectra were recorded for each material, estimating the mean crystallite size for comparison by means of a phonon confinement model.

Erratum: "Setup for meV-resolution inelastic X-ray scattering measurements and X-ray diffraction at the Matter in Extreme Conditions endstation at the Linac Coherent Light Source" [Rev. Sci. Instrum. 89, 10F104 (2018)

We describe a setup for performing inelastic X-ray scattering measurements at the Matter in Extreme Conditions (MEC) endstation of the Linac Coherent Light Source (LCLS). This technique is capable of performing high-, meV-resolution measurements of dynamic ion features in both crystalline and non-crystalline materials. A four-bounce silicon (533) monochromator was used in conjunction with three silicon (533) diced crystal analyzers to provide an energy resolution of ~50 meV over a range of ~500 meV in single shot measurements. In addition to the instrument resolution function, we demonstrate the measurement of longitudinal acoustic phonon modes in polycrystalline diamond. Furthermore, this setup may be combined with the high intensity laser drivers available at MEC to create warm dense matter, and subsequently measure ion acoustic modes.

Setup for meV-resolution inelastic X-ray scattering measurements and X-ray diffraction at the Matter in Extreme Conditions endstation at the Linac Coherent Light Source.

We describe a setup for performing inelastic X-ray scattering and X-ray diffraction measurements at the Matter in Extreme Conditions (MEC) endstation of the Linac Coherent Light Source. This technique is capable of performing high-, meV-resolution measurements of dynamic ion features in both crystalline and non-crystalline materials. A four-bounce silicon (533) monochromator was used in conjunction with three silicon (533) diced crystal analyzers to provide an energy resolution of ∼50 meV over a range of ∼500 meV in single shot measurements. In addition to the instrument resolution function, we demonstrate the measurement of longitudinal acoustic phonon modes in polycrystalline diamond. Furthermore, this setup may be combined with the high intensity laser drivers available at MEC to create warm dense matter and subsequently measure ion acoustic modes.

Laboratory X-ray characterization of a surface acoustic wave on GaAs: the critical role of instrumental convolution

Surface acoustic waves of micrometre wavelength travelling on a monocrystal give diffraction satellites around each Bragg peak in an X-ray diffraction diagram. By using a four-crystal monochromator, a secondary two-crystal analyser and masks reducing the footprint to the part of the crystal containing the acoustic modulation, it is possible to observe these satellites on a GaAs (001) surface using a laboratory diffractometer. The finite extension of the satellite diffraction rods and of the crystal truncation rod perpendicular to the surface leads to geometrical correction factors when convoluted with the instrumental resolution function, which had previously been ignored. The calculation of these geometrical correction factors in the framework of the kinematic approximation allows the determination of the surface acoustic wave amplitude, and the study of its attenuation and its dependence on radiofrequency power and duty cycle. The ability to perform such determinations with a laboratory diffractometer should prove useful in optimizing surface acoustic waves, which are presently used in a broad range of condensed matter physics studies.

Momentum and energy dependent resolution function of the ARCS neutron chopper spectrometer at high momentum transfer: Comparing simulation and experiment

Inelastic neutron scattering at high momentum transfers (i.e. Q≥20A˚), commonly known as deep inelastic neutron scattering (DINS), provides direct observation of the momentum distribution of light atoms, making it a powerful probe for studying single-particle motions in liquids and solids. The quantitative analysis of DINS data requires an accurate knowledge of the instrument resolution function Ri(Q,E) at each momentum Q and energy transfer E, where the label i indicates whether the resolution was experimentally observed i=obs or simulated i=sim. Here, we describe two independent methods for determining the total resolution function Ri(Q,E) of the ARCS neutron instrument at the Spallation Neutron Source, Oak Ridge National Laboratory. The first method uses experimental data from an archetypical system (liquid 4He) studied with DINS, which are then numerically deconvoluted using its previously determined intrinsic scattering function to yield Robs(Q,E). The second approach uses accurate Monte Carlo simulations of the ARCS spectrometer, which account for all instrument contributions, coupled to a representative scattering kernel to reproduce the experimentally observed response S(Q,E). Using a delta function as scattering kernel, the simulation yields a resolution function Rsim(Q,E) with comparable lineshape and features as Robs(Q,E), but somewhat narrower due to the ideal nature of the model. Using each of these two Ri(Q,E) separately, we extract characteristic parameters of liquid 4He such as the intrinsic linewidth α2 (which sets the atomic kinetic energy 〈K〉∼α2) in the normal liquid and the Bose–Einstein condensate parameter n0 in the superfluid phase. The extracted α2 values agree well with previous measurements at saturated vapor pressure (SVP) as well as at elevated pressure (24bars) within experimental precision, independent of which Ri(Q,y) is used to analyze the data. The actual observed n0 values at each Q vary little with the model Ri(Q,E), and the effective Q-averaged n0 values are consistent with each other, and with previously reported values.

Contrasting dynamics of fragile and non-fragile polyalcohols through the glass, and dynamical, transitions: A comparison of neutron scattering and dielectric relaxation data for sorbitol and glycerol

BackgroundGlycerol and sorbitol are glass-forming hydrogen-bonded systems characterized by intriguing properties which make these systems very interesting also from the applications point of view. The goal of this work is to relate the hydrogen-bonded features, relaxation dynamics, glass transition properties and fragility of these systems, in particular to seek insight into their very different liquid fragilities. MethodsThe comparison between glycerol and sorbitol is carried out by collecting the elastic incoherent neutron scattering (EINS) intensity as a function of temperature and of the instrumental energy resolution. ResultsIntensity data vs temperature and resolution are analyzed in terms of thermal restraint and Resolution Elastic Neutron Scattering (RENS) approaches. ConclusionsThe number of OH groups, which are related to the connecting sites, is a significant parameter both in the glass transition and in the dynamical transition. On the other hand, the disordered nature of sorbitol is confirmed by the existence of different relaxation processes. General significanceFrom the applications point of view, glycerol and sorbitol have remarkable bioprotectant properties which make these systems useful in different technological and industrial fields. Furthermore, polyols are rich in glassforming liquid phenomenology and highly deserving of study in their own right. The comparison of EINS and calorimetric data on glycerol and sorbitol helps provide a connection between structural relaxation, dynamical transition, glass transition, and fragility. The evaluation of the inflection point in the elastic intensity behavior as a function of temperature and instrumental energy resolution provides a confirmation of the validity of the RENS approach.This article is part of a Special Issue entitled “Science for Life” Guest Editor: Dr. Austen Angell, Dr. Salvatore Magazù and Dr. Federica Migliardo

Size and spatial correlation of defective domains in yttrium-doped CeO2

The size of dopant-rich nanodomains was assessed in four samples of Ce1−μYμO2−μ/2 through systematic pair distribution function (PDF) refinements. Experimental G(r) curves were fitted by different structural models with the aim of finding a description which balanced precise structure parameterization and reasonable number of parameters. The most reliable model was a single Y2O3-like phase, which best accommodated to the close relationship between the fluorite (CeO2-like) and C-type (Y2O3-like) structures. In this model, a refined cation coordinate, x(M2), measured the relative occurrence in the G(r) of the chemical environment of Y and Ce at any value of r. The r-value at which x(M2) vanished, i.e. at which the refined C-type cell becomes a redundant, low-symmetry description of a fluorite cell, was assumed as the size of a C-type domain. Subtle features in G(r) could be attributed to the fluorite or C-type phase up to ~500 Å thanks to the narrow instrumental resolution function of the ID31 beamline (now ID22) at the ESRF, which allows us to get high resolution PDF data.

Performance calculations of the X-ray powder diffraction beamline at NSLS-II

The X-ray Powder Diffraction (XPD) beamline at the National Synchrotron Light Source II is a multi-purpose high-energy X-ray diffraction beamline with high throughput and high resolution. The beamline uses a sagittally bent double-Laue crystal monochromator to provide X-rays over a large energy range (30-70 keV). In this paper the optical design and the calculated performance of the XPD beamline are presented. The damping wiggler source is simulated by the SRW code and a filter system is designed to optimize the photon flux as well as to reduce the heat load on the first optics. The final beamline performance under two operation modes is simulated using the SHADOW program. For the first time a multi-lamellar model is introduced and implemented in the ray tracing of the bent Laue crystal monochromator. The optimization and the optical properties of the vertical focusing mirror are also discussed. Finally, the instrumental resolution function of the XPD beamline is described in an analytical method.

Experimental determination of resolution function parameters from small-angle neutron scattering data of a colloidal SiO2dispersion

A dilute dispersion of charge-stabilized, monodisperse silica nanoparticles has been used to determine the instrumental resolution function parameters of a conventional small-angle neutron scattering instrument in pinhole geometry over the entire accessible range of momentum transferq. Independent determination of the structure parameters of the colloidal silica dispersion by small-angle X-ray scattering enables refinement of both theq-independent geometric and theq-dependent (wavelength spread or polychromaticity of the neutron beam) contributions to the resolution function. The procedure described is appropriate if no further instrumental characterization is available. It is demonstrated that the sample used for refining the resolution function parameters has to exhibit sharp structural scattering features such as Bragg reflections, form factor maxima and minima, or an inter-particle correlation peak for each instrumental configuration at which the measurements have been performed to cover aqrange of two orders of magnitude (0.03 &lt;q&lt; 3.2 nm−1).

Determination of the Mössbauer absorption cross section and reconstruction of hyperfinefield distribution by the maximum entropy method

Determination of the Mössbauer absorption cross section,σ(E), and accurate reconstruction of the hyperfine field distributions of the invar alloy,Fe64Ni36, by the maximum entropy method (MEM) is presented. The procedure consists of threesteps: deconvolution of the Mössbauer spectra with the instrumental resolutionfunction using MEM, nonlinear transformation of the deconvoluted spectrum intoσ(E), and reconstruction of the hyperfine field distribution. In order to test the procedure of thedeconvolution and correction for thickness effect, several simulated spectra with thickness parameter1 < t < 50 and different values of Lorentzian FWHM of the source and absorber wereanalyzed. It is shown that the procedure of the deconvolution and extraction ofσ(E) works well for spectra whose lines contain at least five experimental points per FWHM.Reconstructed distributions of hyperfine field parameters, based on the extracted Mössbauercross section of the Fe–Ni invar alloy, measured with and without application of an externalmagnetic field, are discussed. The reconstruction has been made to test the earlierpostulated non-collinear ferromagnetic state of invar without referring to any specific modelin the analysis of the Mössbauer results. It is shown that marginal probability distribution ofhyperfine magnetic field consists of the main maximum at about 28 T and a broad tailextending down to 5 T. Observed isomer shift of the main maximum is small andpositive. The isomer shift decreases with magnetic field and attains negative valuesat the lowest fields. It is shown that the magnetic texture parameter does notdepend on the hyperfine magnetic field. One thus concludes that in the invar Ni–Fealloys, in contrast to some theoretical predictions, there is no evidence for differentarrangements of the iron magnetic moments as a function of the magnetic hyperfine field.

Micromechanics of bone tissue-engineering scaffolds, based on resolution error-cleared computer tomography

Synchrotron radiation micro-computed tomography (SRμCT) revealed the microstructure of a CEL2 glass–ceramic scaffold with macropores of several hundred microns characteristic length, in terms of the voxel-by-voxel 3D distribution of the attenuation coefficients throughout the scanned space. The probability density function of all attenuation coefficients related to the macroporous space inside the scaffold gives access to the tomograph-specific machine error included in the SRμCT measurements (also referred to as instrumental resolution function). After Lorentz function-based clearing of the measured CT data from the systematic resolution error, the voxel-specific attenuation information of the voxels representing the solid skeleton is translated into the composition of the material inside one voxel, in terms of the nanoporosity embedded in a dense CEL2 glass–ceramic matrix. Based on voxel-invariant elastic properties of dense CEL2 glass–ceramic, continuum micromechanics allows for translation of the voxel-specific nanoporosity into voxel-specific elastic properties. They serve as input for Finite Element analyses of the scaffold structure. Young's modulus of a specific CT-scanned macroporous scaffold sample, predicted from a Finite Element simulation of a uniaxial compression test, agrees well with the experimental value obtained from an ultrasonic test on the same sample. This highlights the satisfactory predictive capabilities of the presented approach.

Characterization of molecular motions in biomolecular systems by elastic incoherent neutron scattering

In the present work the role played by the instrumental resolution function in elastic incoherent neutron scattering (EINS) experiment is discussed. An important result consists in the definition of an equivalent time t(*), which depends both on the characteristic system time and on the resolution time, for which the spatial Fourier transform of EINS intensity profile and the self-distribution function (SDF) evaluated at t=t(*) are proportional. Then the equivalent time t(*) is introduced in the SDF procedure, an operational recipe for the mean square displacement determination. The new revised procedure is applied on data of myoglobin in trehalose dry environment and of hydrated homologous disaccharides (sucrose and trehalose).

The dynamical transition of proteins, concepts and misconceptions

The dynamics of hydrated proteins and of protein crystals can be studied within a wide temperature range, since the water of hydration does not crystallize at low temperature. Instead it turns into an amorphous glassy state below 200 K. Extending the temperature range facilitates the spectral separation of different molecular processes. The conformational motions of proteins show an abrupt enhancement near 180 K, which has been called a "dynamical transition". In this contribution various aspects of the transition are critically reviewed: the role of the instrumental resolution function in extracting displacements from neutron elastic scattering data and the question of the appropriate dynamic model, discrete transitions between states of different energy versus continuous diffusion inside a harmonic well, are discussed. A decomposition of the transition involving two motional components is performed: rotational transitions of methyl groups and small scale librations of side-chains, induced by water at the protein surface. Both processes create an enhancement of the observed amplitude. The onset occurs, when their time scale becomes compatible with the resolution of the spectrometer. The reorientational rate of hydration water follows a super-Arrhenius temperature dependence, a characteristic feature of a dynamical transition. It occurs only with hydrated proteins, while the torsional motion of methyl groups takes place also in the dehydrated or solvent-vitrified system. Finally, the role of fast hydrogen bond fluctuations contributing to the amplitude enhancement is discussed.

A comparison of modern data analysis methods for X-ray and neutron specular reflectivity data

Data analysis methods for specular X-ray or neutron reflectivity are compared. The methods that have been developed over the years can be classified into different types. The so-called classical methods are based on Parrat's or Abelès' formalism and rely on minimization using more or less evolved Levenberg–Marquardt or simplex routines. A second class uses the same formalism, but optimization is carried out using simulated annealing or genetic algorithms. A third class uses alternative expressions for the reflectivity, such as the Born approximation or distorted Born approximation. This makes it easier to invert the specular data directly, coupled or not with classical least-squares or iterative methods using over-relaxation or charge-flipping techniques. A fourth class uses mathematical methods founded in scattering theory to determine the phase of the scattered waves, but has to be coupled in certain cases with (magnetic) reference layers. The strengths and weaknesses of a number of these methods are evaluated using simulated and experimental data. It is shown that genetic algorithms are by far superior to traditional and advanced least-squares methods, but that they fail when the layers are less well defined. In the latter case, the methods from the third or fourth class are the better choice, because they permit at least a first estimate of the density profile to be obtained that can be refined using the classical methods of the first class. It is also shown that different analysis programs may calculate different reflectivities for a similar chemical system. One reason for this is that the representation of the layers is either described by chemical composition or by scattering length or electronic densities, between which the conversion of the absorptive part is not straightforward. A second important reason is that routines that describe the convolution with the instrumental resolution function are not identical.

Dissociative Double Ionization of CO2: Dynamics, Energy Levels, and Lifetime

In a kinematically complete experiment on the dissociative double ionization of CO2 by electron impact, spontaneous and metastable decay have been observed via the channel CO2(2+) --> CO+ + O+. The metastable decay shows a lifetime of 5.8 +/- 1.5 micros. The measured kinetic energy release spectrum of the dissociation shows one broad peak. To understand the observed features, ab initio potential energy surface (PES) for the ground electronic state of CO2(2+) was computed using a multireference configuration interaction method and a correlation-consistent polarized-valence quadruple-zeta basis set, for a range of internuclear distances and O-C-O bond angles, and an analytic fit of the PES was obtained. The computed PES clearly indicates the metastability of the dication and yields a barrier height and an asymptotic limit in fair agreement with the reported data. A time-dependent quantum mechanical approach was used to compute the ground vibrational state wave function of CO2 in its ground electronic state. Assuming a Franck-Condon transition, the same function was taken to be the initial wave function at time t = 0 for the time evolution on the fitted PES for the ground electronic state of CO2(2+). The autocorrelation function was computed and Fourier transformed to obtain the excitation spectrum. Upon convolution with the instrument resolution function, the kinetic energy release spectrum was obtained, in good agreement with the experimental results, particularly at lower energies. The discrepancies at higher energies are attributed to the noninclusion of the excited states of CO2(2+) in the dynamical study.

An Improved Algorithm for the Fourier Integral of the KWW Function and Its Application to Neutron Scattering and Dielectric Data

The Kohlrausch–Williams–Watts (KWW) function is widely used to describe relaxation behavior in glass‐forming polymers and other condensed systems. The application of this time domain function to frequency domain spectroscopy is, in principle, possible through its Fourier transform. Unfortunately analytical forms for the transform exist only in limited cases, and a number of methods, which circumvent this limitation, are described in the literature. Here we have revisited the problem of evaluating the Fourier integral in the general case, but with the specific aim of producing a fast and accurate algorithm suitable for use within common spreadsheet applications available on personal computers. Two methods were examined, a corrected discrete Fourier transform and, more successfully, a two‐series expansion representation. A previous problem associated with the use of complementary series, that they may not adequately cover the whole range of evaluation of the transform, was overcome by the substitution of a Padé approximant for one of the series. This was found to provide the basis for a robust and effective algorithm whose accuracy was tested against both symbolic mathematical expressions and standard tabulated values. The use of this algorithm in iterative nonlinear least squares curve fitting routines is illustrated by reference to quasi‐elastic neutron scattering data and dielectric relaxation spectra. The former involves a convolution of the real part of the integral with the instrumental resolution function; very satisfactory fits to scattering spectra for poly(vinyl acetate) and poly(isobutylene), at temperatures above their glass transitions, was achieved and the KWW characteristic relaxation parameters established. The dielectric spectrum due to the α‐relaxation of poly(vinyl acetate) is similarly well‐described, here using both the real and the imaginary parts of the integral. The KWW description of the PVAc relaxation is compared with that using the Havriliak–Negami equation and was found to provide an equally acceptable description of the data, and by using one less adjustable parameter.

Site-specific electronic structure analysis by channeling EELS and first-principles calculations

Site-specific electronic structures were investigated by electron energy loss spectroscopy (EELS) under electron channeling conditions. The Al-K and Mn-L 2,3 electron energy loss near-edge structure (ELNES) of, respectively, NiAl 2O 4 and Mn 3O 4 were measured. Deconvolution of the raw spectra with the instrumental resolution function restored the blunt and hidden fine features, which allowed us to interpret the experimental spectral features by comparing with theoretical spectra obtained by first-principles calculations. The present method successfully revealed the electronic structures specific to the differently coordinated cationic sites.