International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.3, p. 264

Section 3.3.2.2. Constant-wavelength powder profile asymmetry

R. B. Von Dreelea*

aAdvanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA
Correspondence e-mail: vondreele@anl.gov

3.3.2.2. Constant-wavelength powder profile asymmetry

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Rietveld (1969[link]) 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[link]; Howard, 1982[link]; Hastings et al., 1984[link]) 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[link]), 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[link], this gives peak intensity beginning at 2ϕmin < 2θ via scattering from only the ends of the sample; at 2ϕinfl 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[link]). This approach was then considered by Finger et al. (1994[link]) 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)[link]] for use by many Rietveld refinement codes. Although originally formulated for parallel-beam neutron optics, it was shown by Finger et al. (1994[link]) 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.

[Figure 3.3.1]

Figure 3.3.1 | top | pdf |

The band of intensity diffracted by a sample with height 2S, as seen by a detector with opening 2H and a detector angle 2ϕ moving in the detector cylinder. For angles below 2ϕmin no intensity is seen. For angles between 2ϕinfl and 2θ, scattering from the entire sample can be seen by the detector. Figure and caption adapted from Finger et al. (1994[link]).

[Figure 3.3.2]

Figure 3.3.2 | top | pdf |

Low-angle synchrotron powder diffraction line (2θ ≃ 4.1°) fitted by the Finger et al. (1994[link]) axial divergence powder line-shape function. The observed points (+), calculated curve, background and difference curves are shown. Note the offset of the peak top from the Bragg 2θ position (vertical line).

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[link] .

References

Cooper, M. J. & Sayer, J. P. (1975). The asymmetry of neutron powder diffraction peaks. J. Appl. Cryst. 8, 615–618.Google Scholar
Finger, L. W., Cox, D. E. & Jephcoat, A. P. (1994). A correction for powder diffraction peak asymmetry due to axial divergence. J. Appl. Cryst. 27, 892–900.Google Scholar
Hastings, J. B., Thomlinson, W. & Cox, D. E. (1984). Synchrotron X-ray powder diffraction. J. Appl. Cryst. 17, 85–95.Google Scholar
Howard, C. J. (1982). The approximation of asymmetric neutron powder diffraction peaks by sums of Gaussians. J. Appl. Cryst. 15, 615–620.Google Scholar
Laar, B. van & Yelon, W. B. (1984). The peak in neutron powder diffraction. J. Appl. Cryst. 17, 47–54.Google Scholar
Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.Google Scholar








































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