Expected values and variances of Bragg peak intensities measured in a nanocrystalline powder diffraction experiment

Abstract

A rigorous study of sampling and intensity statistics applicable for a powder diffraction experiment as a function of crystallite size is presented. This analysis yields approximate equations for the expected value, variance and standard deviations for both the number of diffracting grains and the corresponding diffracted intensity for a given Bragg peak. The classical formalism published in 1948 by Alexander, Klug & Kummer [J. Appl. Phys. (1948), 19, 742-753] appears as a special case, limited to large crystallite sizes, in the present analysis. It is observed that both the Lorentz probability expression and the statistics equations used in the classical formalism are inapplicable for nanocrystalline powder samples.2 The angular width of a diffraction peak can be described by different parameters. The most common terms are the FWHM, used here, and the integral breadth. The FWHM is also termed half-width and, less commonly, breadth in general usage.

Laser-processing of grinded and mechanically abraded Nb-surfaces

“…The peak broadening of the dominant (110) reflex increased erratically from around 0.281° for the mirror like area to 0.421° for the P2000 area with a roughness of 187 nm as result of a grain refinement at the interface. A further increase of the roughness subsequently led to a smaller peak broadening of down to 0.319° (P500) following the behavior of the maximum peak intensity, as expected from their mutual relation [19]. In total, taking into account the finite X-ray penetration depth of several micrometers in our case, it can be concluded that the mechanically treated surfaces consist of a certain polycrystalline fraction, which is relevant, in particular, for the optical absorption during the laser polishing process, as well as for the field emission behavior.…”