Scattering functions of Platonic solids

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.

Precise size control of hydrophobic gold nanoparticles in the 2-5 nm range

We report a seed-mediated synthesis strategy to control the size of gold nanoparticles at the atomic scale in the 2-5 nm size range. Starting from 2 nm seeds, a regrowth in organic solvent with a designed amount of precursor can achieve in a predictive fashion a precise mean size with a 0.3 nm resolution. We show that these monodisperse nanoparticles assemble into a 2D hexagonal lattice over a distance that can span tens of micrometers.

Numerically stable form factor of any polygon and polyhedron

Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes's theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near the singularities causes a loss of precision that can be avoided by using series expansions. An important application domain is small-angle scattering by nanocrystals.

Unwarping GISAXS data

Grazing-incidence small-angle X-ray scattering (GISAXS) is a powerful technique for measuring the nanostructure of coatings and thin films. However, GISAXS data are plagued by distortions that complicate data analysis. The detector image is a warped representation of reciprocal space because of refraction, and overlapping scattering patterns appear because of reflection. A method is presented to unwarp GISAXS data, recovering an estimate of the true undistorted scattering pattern. The method consists of first generating a guess for the structure of the reciprocal-space scattering by solving for a mutually consistent prediction from the transmission and reflection sub-components. This initial guess is then iteratively refined by fitting experimental GISAXS images at multiple incident angles, using the distorted-wave Born approximation (DWBA) to convert between reciprocal space and detector space. This method converges to a high-quality reconstruction for the undistorted scattering, as validated by comparing with grazing-transmission scattering data. This new method for unwarping GISAXS images will broaden the applicability of grazing-incidence techniques, allowing experimenters to inspect undistorted visualizations of their data and allowing a broader range of analysis methods to be applied to GI data.

Monitoring the Discontinuous Dodecamer-Icosamer Transition of a Calix[4]arene-Derived Surfactant by Time-Resolved Small-Angle X-ray Scattering

Calix[4]arene-derived surfactants form monodisperse micelles with a well-defined aggregation number (N_agg ) of 4, 6, 8, 12, or 20, corresponding to the Platonic solids. This feature is in strong contrast to conventional micelles. In this study, a transition from a dodecamer (N_agg =12) to an icosamer (N_agg =20) was induced by a rapid increase in the NaCl concentration (C_NaCl ) using a stopped-flow device and directly observed by time-resolved small-angle X-ray scattering. The N_agg remained unchanged during the first 60 s after the increase in C_NaCl  , and then abruptly increased to 20. This feature resembles phase transitions in supersaturated or supercooled states, or highly cooperative phenomena. We surmise that this finding may be due to the fact that only a few N_agg values are thermodynamically allowed when N_agg is sufficiently small. This is the first observation of such an induction time in micellar aggregation.

SAXS Structural Studies of Dps from Deinococcus radiodurans Highlights the Conformation of the Mobile N-Terminal Extensions

The radiation-resistant bacterium Deinococcus radiodurans contains two DNA-binding proteins from starved cells (Dps): Dps1 (DR2263) and Dps2 (DRB0092). These are suggested to play a role in DNA interaction and manganese and iron storage. The proteins assemble as a conserved dodecameric structure with structurally uncharacterised N-terminal extensions. In the case of DrDps1, these extensions have been proposed to be involved in DNA interactions, while in DrDps2, their function has yet to be established. The reported data reveal the relative position of the N-terminal extensions to the dodecameric sphere in solution for both Dps. The low-resolution small angle X-ray scattering (SAXS) results show that the N-terminal extensions protrude from the spherical shell of both proteins. The SAXS envelope of a truncated form of DrDps1 without the N-terminal extensions appears as a dodecameric sphere, contrasting strongly with the protrusions observed in the full-length models. The effect of iron incorporation into DrDps2 was investigated by static and stopped-flow SAXS measurements, revealing dynamic structural changes upon iron binding and core formation, as reflected by a quick alteration of its radius of gyration. The truncated and full-length versions of DrDps were also compared on the basis of their interaction with DNA to analyse functional roles of the N-terminal extensions. DrDps1 N-terminal protrusions appear to be directly involved with DNA, whilst those from DrDps2 are indirectly associated with DNA binding. Furthermore, detection of DrDps2 in the D. radiodurans membrane fraction suggests that the N-terminus of the protein interacts with the membrane.

Small Angle X-ray Scattering for Nanoparticle Research

X-ray scattering is a structural characterization tool that has impacted diverse fields of study. It is unique in its ability to examine materials in real time and under realistic sample environments, enabling researchers to understand morphology at nanometer and angstrom length scales using complementary small and wide angle X-ray scattering (SAXS, WAXS), respectively. Herein, we focus on the use of SAXS to examine nanoscale particulate systems. We provide a theoretical foundation for X-ray scattering, considering both form factor and structure factor, as well as the use of correlation functions, which may be used to determine a particle's size, size distribution, shape, and organization into hierarchical structures. The theory is expanded upon with contemporary use cases. Both transmission and reflection (grazing incidence) geometries are addressed, as well as the combination of SAXS with other X-ray and non-X-ray characterization tools. We conclude with an examination of several key areas of research where X-ray scattering has played a pivotal role, including in situ nanoparticle synthesis, nanoparticle assembly, and operando studies of catalysts and energy storage materials. Throughout this review we highlight the unique capabilities of X-ray scattering for structural characterization of materials in their native environment.

Scattering functions of polyhedra

Herein, a general method to calculate the scattering functions of polyhedra, including both regular and semi-regular polyhedra, is presented. These calculations may be achieved by breaking a polyhedron into sets of congruent pieces, thereby reducing computation time by taking advantage of Fourier transforms and inversion symmetry. Each piece belonging to a set or subunit can be generated by either rotation or translation. Further, general strategies to compute truncated, concave and stellated polyhedra are provided. Using this method, the asymptotic behaviors of the polyhedral scattering functions are compared with that of a sphere. It is shown that, for a regular polyhedron, the form factor oscillation at highqis correlated with the face-to-face distance. In addition, polydispersity affects the Porod constant. The ideas presented herein will be important for the characterization of nanomaterials using small-angle scattering.

The correlation functions of the regular tetrahedron and octahedron

The expressions of the autocorrelation functions (CFs) of the regular tetrahedron and the regular octahedron are reported. They have an algebraic form that involves the arctangent function and rational functions of r and (a + br 2)1/2, a and b being appropriate integers and r a distance. The CF expressions make the numerical determination of the corresponding scattering intensities fast and accurate even in the presence of a size dispersion.

The chord-length probability density of the regular octahedron

The chord-length probability density of the regular octahedron is separated into three contributions, relating to the pairs of facets opposite to each other or sharing an edge or a vertex. Each of these contributions is explicitly evaluated throughout the full range of distances and the final expressions only involve inverse trigonometric functions of elementary algebraic functions. Since the chord-length probability density is proportional to the second derivative of the correlation function, knowledge of the chord-length probability density makes the numerical evaluation of the associated small-angle scattering intensity very fast and accurate.

New insight into the structure of RNA in red clover necrotic mosaic virus and the role of divalent cations revealed by small-angle neutron scattering

Red clover necrotic mosaic virus (RCNMV) is a 36-nm-diameter, T = 3 icosahedral plant virus with a genome that is split between two single-stranded RNA molecules of approximately 3.9 kb and 1.5 kb, as well as a 400-nucleotide degradation product. The structure of the virus capsid and its response to removing Ca(2+) and Mg(2+) was previously studied by cryo-electron microscopy (cryo-EM) (Sherman et al. J Virol 80:10395-10406, 2006) but the structure of the RNA was only partially resolved in that study. To better understand the organization of the RNA and conformational changes resulting from the removal of divalent cations, small-angle neutron scattering with contrast variation experiments were performed. The results expand upon the cryo-EM results by clearly showing that virtually all of the RNA is contained in a thin shell that is in contact with the interior domains of the viral capsid protein, and they provide new insight into changes in the RNA packing that result from removal of divalent cations.

Formfactors of Hollow and Massive Rectangular Parallelepipeds at Variable Degree of Anisometry

The present work is a comparative study on the scattering behaviour of hollow and filled rectangular parallelepipeds. Comparison is based on the model formfactors of rectangular parallelepipeds, which cover the entire regime of variable wall thickness. The entire regime of wall thicknesses has been made available by the present work, which completed the set of formulas by calculating the formfactor for the limit of hollow parallelepipeds with infinitely thin walls, which was still lacking. The formfactors are expressed as a function of the momentum transfer q. Discrimination between massive and hollow structures by means of the q-dependent scattering data SF(q) gets possible once the Guinier radius and the particle volume can be established at the lower limit of q or a power law of SF(q) ∼ q α can be extracted towards large q. Whereas the former requires extrapolation of the scattering data to q = 0 with high accuracy, the latter needs experiments over a very broad q-regime. If experimental data is restricted to a q-regime which includes only the first peak in a Kratky representation of q 2 SF(q) vs. q without giving way to an accurate extrapolation to q = 0 nor to a clear power law in q, discrimination between a hollow and a massive structure gets extremely difficult. Yet, the effect of anisometry is striking and enables extraction of a crude guess of the degree of anisometry already from the first Kratky peak. This could be achieved by introducing a dimensionless parameter established from the width and location of the first Kratky peak.

Contrast variation in spin-echo small angle neutron scattering

The use of contrast variation in spin-echo small angle neutron scattering (SESANS) experiments is discussed for the case of colloidal structural investigation. On the basis of calculations for several model systems, we find that the contrast variation SESANS technique, in terms of the measured SESANS correlation function G(z), is not sensitive to the structural characteristics of colloidal suspensions consisting of particles with uniform scattering length density profiles. However, its ability to resolve structural heterogeneity, at both intra-colloidal and inter-colloidal length scales, is clearly demonstrated. The prospect of using this new technique to investigate structural information that is difficult to probe in other ways is also explored.