SIMPLIFIED MICROSTRUCTURAL MODELS TO ANALYZE ANISOTROPIC SIZE AND STRAIN

Juan Rodríguez-Carvajal

Laboratoire Léon Brillouin (CEA-CNRS), CEA/Saclay, 91191 Gif sur Yvette Cedex, France.

A summary of the different approaches to extract and interpret microstructural parameters from powder diffraction techniques will be presented. Special emphasis will be devoted to the so-called Voigt model for both the instrumental and the intrinsic diffraction peak shape. Under this last assumption many kinds of microstructural effects can be studied in a simplified manner. This quite general model is fully implemented and ready to be used in the computer program FullProf together with the Rietveld method. Complex anisotropic peak broadening may be due to size and strain effects, a complementary electron microscopy study is often needed to disentangle and evaluate the main (size or strain) contribution to broadening.

To treat anisotropic size effects it is extremely useful, in many cases, to use linear combinations of spherical harmonics to model the Lorentzian part of the peak broadening. The apparent sizes along different directions can be reconstructed from the refined coefficients and an average “apparent shape” of the coherent domains of the sample can be obtained. Some examples taken from battery positive electrode materials and catalysis will be presented, one of them is shown in Figure 1.

In case of dominant anisotropic broadening due to microstrains (high number of dislocations, vacancies, twin faults, solid solution effects, etc.) a phenomenological approach introduced 13 years ago [1] and based in the assumption that all the defects responsible of the broadening can be reduced to fluctuations and correlations of cell parameters, or any combination of them, has proven to be extremely useful. A convenient formulation derived from [1] when the metric parameters are the coefficients of the quadratic form in (hkl) constituting the square of a reciprocal lattice vector was proposed by Stephens [2] and a similar one, based in elasticity theory, was previously proposed by Popa [3]. We will show that there are many equivalent ways to treat anisotropic strain broadening, using the assumptions first published in [1], that can help to construct physical models for the origin of the anisotropic microstrain broadening. Some examples taken from different kind of materials (intermetallics, oxides) in different contexts (phase transitions, reducing synthesis conditions, etc) will be presented.

[1] J. Rodríguez-Carvajal, M. T. Fernández-Díaz, J. L. Martínez, J. Phys. Cond. Matter 3, 3215 (1991).

[2] P. W. Stephens, J. Appl. Cryst. 32, 281 (1999).

[3] N. C. Popa, J. Appl. Cryst. 31, 176 (1998).