X-ray powder diffraction in education. Part I. Bragg peak profiles

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. This paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes.

Evaluation of X-ray Bragg peak profiles with the variance method obtained by in situ measurement on Mg–Al alloys

The microstructural evolution in randomly oriented Mg–Al samples is investigated in situ during compression by X-ray diffraction as a function of Al concentration. The diffraction data are evaluated by the variance method, which provides information about the dislocation density and spatial distribution of the dislocations. The dislocation density increases with increasing alloying content. Since the increment of the dislocation density above the yield point is linear, the mutual dislocation interaction type is determined from the Taylor equation. The results indicate the dominance of basal–basal dislocation interactions, but at higher alloying content the share of the basal–non-basal interactions increases. It is shown that the dynamics of dislocation wall formation also depend on Al content. Transmission electron microscopy observations are in agreement with the results obtained by X-ray line profile analysis.

The Deformation Mechanisms of TWIP Steels (Fe-Mn-C) Viewed by X-Ray Diffraction

Fully austenitic steels of the Fe-Mn-C system can show extensive deformation twinning (TWIP effect). The deformed microstructure of such steels has been analysed using X-ray diffraction at the European Synchrotron Radiation Facility (ESRF). The experimental diffractograms, recorded using a 2D CCD camera, are analysed in terms of Bragg peak profiles (broadening and asymmetry) and position (shift from the reference [undeformed state] position) leading to an estimation of dislocation and stacking faults densities.

Structure determination of forms I and II of phenobarbital from X-ray powder diffraction

From pure powders of forms I and II of phenobarbital, X-ray diffraction patterns were recorded at room temperature. The starting crystal structural models were found by a Monte-Carlo simulated annealing method. The structures of the two forms were obtained through Rietveld refinements. Soft restraints were applied on bond lengths and bond angles, all H-atom positions were calculated. The cell of form I is monoclinic with the space group P21/n, Z = 12, Z′ = 3. Form II has a triclinic cell, with the space group P\bar 1, Z = 6, Z′ = 3. For both forms, the crystal cohesion is achieved by networks of N—H...O hydrogen bonds along [101]. The broadening of the Bragg peak profiles is interpreted in terms of isotropic strain effects and anisotropic size effects.

Effects of Bragg peak profiles and nanoparticle sizes on the real-space pair distribution function

A study of the effects of Bragg peak profiles and nanoparticle size broadening on the real-space pair distribution function (PDF) is presented, using `synthetic' powder diffraction data. Bragg peak profiles from both asymmetric time-of-flight (TOF) spallation neutron data and symmetric synchrotron X-ray data are considered. Due to their asymmetric peak profiles, the TOF data cause artificial shifts of the PDF peak positions towards higher pair distances. Coupled with this effect is a broadening of the PDF peak widths due to aQ-dependent spectrometer resolution, making reliable refinement of thermal parameters difficult. These effects become more pronounced as theQresolution becomes worse. By contrast, the symmetric X-ray powder diffraction data do not cause a systematic shift of the PDF peak positions, and the broadening of the PDF peak widths has a relatively minor effect on the extraction of the thermal parameters. Finally, nanoparticle size broadening of the asymmetric neutron TOF powder diffraction data causes a shift of the PDF peak positions towards lowerrvalues and smears the PDF intensities from one atomic shell to another.