International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.3, p. 264
Section 3.3.2.2. Constant-wavelength powder profile asymmetry^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
Rietveld (1969) noted that at very low scattering angles the peaks displayed some asymmetry, which shifted the peak maximum to lower angles. He ascribed the effect to `vertical divergence' and proposed a purely empirical correction for it. Subsequent authors (Cooper & Sayer, 1975; Howard, 1982; Hastings et al., 1984) offered semi-empirical treatments of the profile shape that results from the intersection of a Debye–Scherrer cone with a finite receiving slit, which is described as `axial divergence'. A more complete analysis of the problem in neutron powder diffraction was offered by van Laar & Yelon (1984), who considered the effect of a finite vertical slit (2H) intercepting a set of Bragg diffraction cones generated from a finite sample length (2S) within the incident beam for a goniometer radius (L). As seen in Fig. 3.3.1, this gives peak intensity beginning at 2ϕ_{min} < 2θ via scattering from only the ends of the sample; at 2ϕ_{infl} the entire sample scatters into the detector. The resulting intensity profile is then convoluted with a Gaussian function to give the resulting asymmetric powder line profile (Fig. 3.3.2). This approach was then considered by Finger et al. (1994) for synchrotron powder diffraction and they created a Fortran code that was subsequently adopted via convolution with a pseudo-Voigt function [equation (3.3.12)] for use by many Rietveld refinement codes. Although originally formulated for parallel-beam neutron optics, it was shown by Finger et al. (1994) that it could be equally well applied to diverging X-ray and neutron optics by allowing the sample length to vary during the Rietveld refinement. They also showed that it could be applied to the asymmetry observed at low angles with Bragg–Brentano instrumentation. In that case the detector height is defined by the diffracted-beam Soller slits.
Clearly, this asymmetric peak-shape function properly represents the offset of the peak top from the peak position, in contrast to functions such as the split Pearson VII function. Consequently, single peak fits using this function will give peak positions that are more readily indexed using methods such as those described in Chapter 3.4 .
References
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