The non-accessible range for selected wavelength and angular ranges is shaded in gray.Īnomalous scattering effects are often disregarded for simplicity, but they become extremely important if the wavelength used is in the vicinity of an absorption edge of an atomic species in the sample. Lower charts: real and imaginary parts of the anomalous dispersion coefficient and the complex phase shift introduced by the imaginary part of the anomalous dispersion coefficient. Upper chart: absolute form factor without correction for anomalous scattering (red line), corrected for anomalous scattering (blue line), corrected for Debye–Waller factor (introduced in Section 3 ) (red dashed line), the real part of the anomalous dispersion coefficient (green line) and the imaginary part of the anomalous dispersion coefficient (orange line) depending on the scattering length s. Screenshot of a Mathematica script for determining the individual atomic form factor for fixed-wavelength data for atoms ranging from order number 1 (H) to 92 (U). In general, the scattering vector is given as d*, q or s, which are used interchangeably given the following relations: Finally, the intensity distribution of a powder pattern is demonstrated for a nanocrystalline material, following two alternative approaches based on (i) the structure factor and common volume function (CVF), including the effect of small-angle scattering for spherical particles, and (ii) total scattering from a single crystallite, with atomic distances used in the Debye scattering equation. Then, a series of correction factors for step-scan and integrated intensities are discussed in detail, including the Lorentz and polarization factors, multiplicity, various absorption effects, the overspill effect, and preferred orientation. This is followed by a discussion of the complex structure factor and the effect of thermal diffuse scattering on a powder pattern. We begin by introducing the scripts that visualize the complex atomic form factor for angular- and energy-dispersive X-ray diffraction and the displacement factor due to thermal motion. Every model is an oversimplification of the underlying physics, but different models can be useful for studying various phenomena or increasing the precision of the investigation. The idea is to `learn by doing' one may gain intuition for how a given mathematical model performs for describing diffraction peaks in an experimental powder pattern and what the limitations of the said model are. When possible, parameter values from real-life examples are given as the default inputs. Bugs and problems should be reported to In particular, the `Manipulate' command is extensively used to visualize the impact of parameters in an interactive manner. Non-subscribers of Mathematica can run the scripts using the freely available Wolfram Player at. They are freely available at the website. All scripts have been written in Wolfram Mathematica, version 13.0.0.0, and are constantly updated. Accompanying each part is a collection of user-friendly, interactive and freely distributable Mathematica (Wolfram Research, ) teaching scripts. This series of papers deals with the description and visualization of mathematical functions used to describe a powder pattern.
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