Common volume functions and diffraction line profiles of polyhedral domains

Abstract

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 positions via the Debye scattering equation.

On the Modeling of the Diffraction Pattern from Metal Nanocrystals

“…[10,22,23] The key to use a convolution of effects is to exploit the convolution theorem for Fourier Transforms, which turns a computationally complex problem of folding into a simple multiplication of different terms. As of the most recent developments, [24] WPPM can use virtually any crystalline domain shape and strain models for dislocations in any crystal system. [25] Nanocrystals can then be studied down to small sizes, with the highest precision for the spherical domain shape.…”

“…[29] In the following, we use a spherical nanocrystal model, including Debye-Waller factor and Thermal Diffuse Scattering (TDS) for the spherical shape. [24] The complex and anisotropic atomic displacement caused by surface relaxation is treated in a simplified way, providing for a shift of the average unit cell parameter and a flexible, although not entirely rigorous, r.m.s. atomic displacement (microstrain).…”

Atomistic Model of Metal Nanocrystals with Line Defects: Contribution to Diffraction Line Profile

“…Figure 10 shows the modeling results for two different powders of cylindrical domains, respectively D = 40/H = 28.7 nm ( Figures 10A-C) and D = 20/H = 28.7 nm (Figures 10D-F). The result for ideal cylindrical domains (Figures 10A,D), i.e., with Pd atoms positioned according to ideal fcc structure, no dislocations, and no surface relaxation, is very good, as expected in case of domain size broadening effects only (Leonardi et al, 2012). Small deviations between DSE pattern and WPPM are expected, owing to the different hypotheses underlying DSE and WPMM, as the former is based on an intrinsically discrete, atomistic model, whereas, the last considers crystalline domains as ideal solid models, i.e., cylinders with a smooth surface [details can be found in Beyerlein et al (2011)].…”

“…Besides using Eq. 1 for refining the dislocation parameters (like ρ, R e , and C hkl ), WPPM can model nanocrystalline domains of virtually any size and shape (Leonardi et al, 2012), also considering the presence of stacking faults (Scardi and Leoni, 2002;Scardi, 2008) and other microstructural features responsible for line broadening effects (Scardi, 2008). For example, the complex effect of relaxation of the nanocrystal surface can be described by an additional "strain" profile component, with a corresponding FT given by…”

1-D Paracrystalline Model to Simulate a Bragg Reflection: Computation of Crystallite Size and Lattice Strain

“…The accurate powder diffraction peak profile by numerical procedure has been reported [21]. An elegant method of computation of common volume fraction of any type of polyhedra has been reported [22]. In this article it is very clear that the crystallite size is different along different Bragg angles and assumption of symmetric crystallite size as done in various other methods is not appropriate.…”