Rietveld analysis spherical harmonics

Later Von Dreele implemented the general description of texture by spherical harmonics in GSAS. Von Dreele proved that, by using this description, beside the robustness of the texture correction in the Rietveld method it is also possible to perform a reliable quantitative texture analysis. He measured by neutron time-of-flight diffraction a standard calcite sample previously used for a texture round robin. The patterns from different detector banks and sample orientations were processed by GSAS, refining the harmonic coefficients simultaneously with the structural and other parameters. Six pole distributions calculated from the refined harmonic coefficients and used as input in the... 

For a general type of texture, only a mathematical description independent of any physical model can guarantee a reliable correction in the Rietveld method. Such a description is the Fourier analysis by using symmetrized spherical harmonics. 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

This method requires only a crude structural model as a starting model. In this analysis, the starting model was a homogeneous spherical shell density for the carbon cage. As for the temperature factors of all atoms, an isotropic harmonic model was used an isotropic Gaussian distribution is presumed for a La atom in the starting model. Then, the radius of the C82 sphere was refined as structural parameter in the Rietveld refinement. 

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