Bragg peak microstructure: X-ray powder diffraction in education​
​by Robert Dinnebier​
This is the first part of three-part series for this work.
Second part: https://community.wolfram.com/groups/-/m/t/2409355​
Third part: https://community.wolfram.com/groups/-/m/t/2409873
ABSTRACT: A collection of scholarly scripts dealing with the mathematics and physics of peak profile functions in X-ray powder diffraction has been written using the Wolfram language in Mathematica. Common distribution functions, the concept of convolution in real and Fourier space, instrumental aberrations and microstructural effects are visualized in an interactive manner and explained in detail. The corresponding paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes.
The following script deals with the influence of microstructural effects (microstrain and domain size) on the profile of powder diffraction peaks.

Size & Pseudo-VoigtMicrostrain profiles

ClearAll["Global`*"];
The following .m files and the notebooks that generate them are attached at the end of this post.
In[]:=
Get["Voigt_functions_general.m"]​​Get["SizeStrain_functions_general.m"]
Out[]=
{FourierParameters{0,1}}

Fourier transform of size distribution in the cubic crystal system:

Fourier transform of the line profile given by spherical (1), cubic (2), tetrahedral (3) and octahedral (4) shaped particles in a single [hkl] direction (diameter for sphere, and edge for cube, tetrahedron and octahedron) for particles of uniform size (upper graph) and for a lognormal distribution (lower graphs) of sizes characterized by lognormal mean and lognormal standard deviation. See paper for details.

Code


Interface (interactive version in the attached notebook)

Fourier transform of size distribution - different hkl’s in the cubic crystal system:

Fourier transform of the line profile given by either spherical, cubic, tetrahedral, or octahedral shaped particles for a set of different [hkl] directions (diameter for sphere, and edge for cube, tetrahedron and octahedron) for particles of uniform size (upper graph) and for a lognormal distribution (lower graphs) of sizes characterized by lognormal mean and lognormal standard deviation. See paper for details.

Code


Interface (interactive version in the attached notebook)

Fourier (back)transform of Pseudo-Voigt function convoluted with lognormal domain size distribution in the cubic crystal system:

Line profile of a Bragg reflection (and corresponding Fourier transform) given by the convolution of a Pseudo-Voigt function with either spherical, cubic, tetrahedral, or octahedral shaped particles in a single [hkl] direction (diameter for sphere, and edge for cube, tetrahedron and octahedron) for a for a given lognormal distribution of a chosen shape. The corresponding line profiles are shown in reciprocal space and in 2θ space. See paper for details.

Code


Interface (interactive version in the attached notebook)

Fourier (back)transform of Pseudo-Voigt function convoluted with microstrain in the cubic crystal system:

Line profile of a Bragg reflection (and corresponding Fourier transform) given by the convolution of a Pseudo-Voigt function with microstrain considering KW and PAH models, for the cases of ferritic iron, fluorite and free input. The corresponding line profiles are shown in reciprocal space and in 2θ space. The corresponding Warren plot is also shown. See paper for details.

Code


Interface (interactive version in the attached notebook)

Integral breath of Pseudo-Voigt function (as instrumental profle) convoluted with microstrain and size for cubic crystal system:

Integral breadth plot (or Williamson-Hall plot) for the cases of ferritic iron, fluorite and free input, considering contributions from instrument, domain size/shape and microstrain. The corresponding Warren plot for some representative crystallographic directions is also shown. See paper for details.

Code


Interface (interactive version in the attached notebook)