Form factor of any polyhedron: a general compact formula and its singularities

Kinematic scattering by nanocrystals

Various formulations are compared which describe diffraction from ultra-thin single-crystal films in the symmetric scattering configuration, showing that, for this thickness range, several implicit assumptions in these formulations are no longer satisfied. Consequently, the position, integrated intensity and integral breadth of a diffraction peak cannot be related to the lattice spacing of the material or the number of unit cells along the diffraction vector using traditional analysis methods. Some simple equations are proposed to obtain the correct values of these parameters for this specific sample/diffraction geometry combination. More generally, the development of rigorous formalisms for analyzing diffraction from nanocrystals is proposed.

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.

Influence of the elastic deformations on the form factor of polyhedral nanocrystals: the illustrative example of the pseudomorphic inclusion

Using an analytically tractable example, the pseudomorphic inclusion, this article examines the influence of elastic deformations on the form factor of polyhedral nanocrystals. A control parameter, the total amplitude of the variation of the complex density phase, is identified and it is shown that for low enough deformations the characteristic asymptotic behaviours as a function of the scattering vector associated with the polyhedral crystal shape are preserved, leading to a strong contrast in the dependence of the form factor on orientation. Using the sections method, it is explained why these results can be generalized to more realistic elastic situations.

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.

Form factor of rounded objects: the sections method

An analytical method, the sections method, is developed to build a close link between the singularities of the surface of a body and the asymptotic behaviour of its amplitude form factor at large scattering vector, q. In contrast with a sphere, for which the asymptotic behaviour is in q −2, surface singularities lead to both narrow regions, for which the amplitude form factor exhibits trailing behaviour, and extended regions, for which it exhibits a rapid decrease. A numerical study of a simple example, the fourfold truncated sphere, illustrates the usefulness of these analytical predictions.