Analytical formulation of the variance method of line-broadening analysis for Voigtian X-ray diffraction peaks

Abstract

An alternative formulation of the variance method for the line-broadening analysis of polycrystalline materials is presented. It maintains the theoretical basis of the earlier formulations of the variance method, but differs in the manner of calculating the variance coefficients of the line profiles. In the proposed formulation, these are evaluated analytically in terms of the shape parameters of Voigt functions fitted to the X-ray diffraction data. Explicit expressions are thus derived for calculating the (surface-weighted) crystal sizes and (root-mean-square) lattice microstrains from the integral breadths of the Gauss and Lorentz components of the Voigt functions that model the experimental and instrumental line profiles.

Crystallite sizes of LiH before and after ball milling and thermal exposure

“…With increasing ball milling time, it is observed that the integral breadth of the Lorentz component (β L ) increases from 0.0004 • for the as-purchased LiH to 0.2132 • for the 180 min ball-milled LiH, whereas, on the contrary, the integral breadth of the Gauss component (β G ) remains the same at the as-purchased condition of 0.46 • . Clearly, the unchanged β G is incompatible with the introduction of lattice defects within the LiH crystals during ball milling, since the attendant internal stresses would induce additional Gauss broadening [41]. On the other hand, the increased β L is entirely consistent with the crystallite size refinement from the micrometer range to the nanometer range, since the presence of LiH nano-crystals would induce additional Lorentz broadening [41].…”

“…Clearly, the unchanged β G is incompatible with the introduction of lattice defects within the LiH crystals during ball milling, since the attendant internal stresses would induce additional Gauss broadening [41]. On the other hand, the increased β L is entirely consistent with the crystallite size refinement from the micrometer range to the nanometer range, since the presence of LiH nano-crystals would induce additional Lorentz broadening [41].…”

“…θ Si was evaluated by subtracting the position of the 1 1 1 Si peak given in the 27-1402 reference PDF card from the one measured experimentally.The surface-weighted average crystal size (D S ) and the root-mean-square (e RMS ) microstrain of LiH in the ball-milled powders were calculated by the variance method through the following relations [41]:…”

Effect of Ar or N2 sintering atmosphere on the high-temperature oxidation behaviour of pressureless liquid-phase-sintered α-SiC in air

“…36,37 Second, the nitrogen solutes have introduced structural defects into the lattice of the YAG crystals because their peaks in PLPS SiC-N 2 exhibit additional Gaussian broadening with respect to PLPS SiC-Ar, and the XRD theory establishes that lattice defects introduce Gaussian broadening whereas the nanocrystallites induce Lorentzian broadening. 38 Fig. 3 are the specific mass-change curves for PLPS SiC-Ar and PLPS SiC-N 2 isothermally oxidized in air at 1100-1450 • C. As can be seen, there exist appreciable differences between the oxidation kinetics of the two materials.…”

Behavior of explosive compacted/consolidated of nanometric copper powders

“…The crystallite size and the lattice microstrains of 3D copper were evaluated by analytical formulation of the variance method of line-broadening analysis of Voigt functions using as fitting curves of X-ray diffraction peaks [9]. The X-ray diffraction (XRD) (X-Pert, Phillips) had a Bragg-Brentano configuration and Co(K␣) anticathode.…”