Composition variations in line broadening analysis

A. Leineweber, E. J. Mittemeijer

¹Max Planck Institute for Metals Research, Heisenbergstraße 3, 70569 Stuttgart, Germany

In the course of an analysis of broadened powder diffraction-line profiles the separation of different contributions to the overall physical line broadening is required. E. g. several methods exist to separate size and microstrain broadening due to their different dependencies on the reflection order.

The most prominent source of microstrain broadening are microstresses around extended defects like dislocations. Another, less frequently considered origin of microstrain broadening is local variation in composition [1] which leads to local variations in the lattice parameters. In the simplest case composition varies only between different coherently diffracting domains, i.e. each domain has its own characteristic lattice parameters which are themselves a usually monotonous function of composition. In such cases the diffraction patterns are simple superpositions of the diffraction patterns originating from the different compositions.

In general the probability density function of composition is expressed in the diffraction line profile of each reflection hkl, according to the dependence of the d-spacing d_hkl on composition. This compositional contribution to the line broadening can be very complicated and has to be convoluted with other occurring sources of line broadening and the instrumental resolution.

For relatively narrow and symmetrical unimodal composition distributions around an average composition the widths B_hkl of the reflections on the diffraction angle 2q_hkl scale vary for a certain direction of the diffraction vector like

(1)

with A(hkl) being an anisotropy factor which varies with the direction of the diffraction vector but not with its length (i.e. with 2q_hkl). A(hkl) can be expressed as

. (2)

The parameters D_HKL can be calculated from the dependencies of the reciprocal metrical matrix components on composition and the width of the composition distribution function. The symmetry restrictions for the dependencies of the reciprocal metrical matrix components on composition reflect the crystal system and thus impose symmetry restrictions on D_HKL, since the influence of the composition distribution on the line width is the same for all reflections. This leads to one parameter D_HKL for the cubic system (i.e. the line broadening is isotropic) and six parameters D_HKL for the triclinic system. For sufficiently (pseudo‑)Voigt-like composition distributions anisotropic line broadening according to Eqs. 1-2 can conveniently be incorporated into a Thompson-Cox-Hastings pseudo-Voigt function profile function [2] in the course of a Rietveld refinement.

The anisotropic line broadening as given by Eq. 1-2 constitutes a physically founded special case of a previously proposed phenomenological model for anisotropic microstrain broadening [3,4] having an anisotropy factor of

(3)

which is to be combined with Eq. 1. In order that Eq. 3 is identical with Eq. 2, additional symmetry restrictions on S_HKL have to be introduced, only then Eq. 3 could be used to describe the anisotropic line broadening due to composition variations as well.

Different examples of diffraction line broadening from more or less inhomogeneous solid solution samples will be presented. Possibilities to distinguish compositional microstrain broadening from other types of line broadening, as well as problems encountered in such analyses will be presented.

[1] Leineweber, A. & Mittemeijer, E.J., J. Appl. Crystallogr. 37 (2004) 123.

[2] Thompson, P., Cox, D. E. & Hastings, J. B, J. Appl. Crystallogr. 20 (1987) 79-83.

[3] Stephens, P.W., J. Appl. Crystallogr. 32 (1999) 281.

[4] Popa, N.C., J. Appl. Crystallogr. 31 (1998) 176.