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. 263
Section 3.3.1. Introduction^{a}Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA |
The analysis of a powder diffraction pattern usually involves the fitting of a model to the set of peaks that are found in that pattern. The desired result may be accurate peak positions to be used as input for an indexing procedure, or extraction of the suite of reflection intensities for crystal structure determination or a Rietveld refinement. In any case, a good description of the shape of the powder peak profile and how it varies across the entire pattern is of paramount importance for obtaining the highest-quality results, and this topic was briefly reviewed in Volume C of International Tables for Crystallography (Parrish, 1992).
The fitting is a least-squares procedure in which the model used to calculate the intensity of the profile iswhere I_{j} is the integrated intensity of the jth peak and P is the shape function for that peak, which depends on the offset (Δ = x − T_{j}) of its position T_{j} from the observation point x. The sum is over all reflections that could contribute to the profile and B(x) is a background intensity function. The observed shape of the peaks arises from a convolution of the intrinsic source profile (G_{λ}), the various instrumental profile contributions (G_{I}) (e.g. from slits and monochromators, discussed in Chapter 3.1 ) and the characteristics of the sample (G_{S}) that broaden the idealized reciprocal-space points (see Chapter 3.6 ):
In practice, the peak profile function is usually developed by either selecting a peak-shape function that has the required shape characteristics to fit the experimental peak profiles (the semi-empirical function approach, or SFA) or by selecting a number of contributing functions and doing the requisite convolutions (the fundamental parameters approach, or FPA) (Cheary & Coelho, 1998b).
Both approaches have been used for the analysis of constant-wavelength neutron and X-ray powder diffraction data and for neutron time-of-flight (energy-dispersive) powder data. In addition, the peaks can be seen to be displaced from their expected positions given by Bragg's law. As we will see, this displacement is partially a consequence of some geometric features of the experiment but is also dependent upon the particular description of the peak profile.
References
Cheary, R. W. & Coelho, A. A. (1998b). Axial divergence in a conventional X-ray powder diffractometer. II. Realization and evaluation in a fundamental-parameter profile fitting procedure. J. Appl. Cryst. 31, 862–868.Google ScholarParrish, W. (1992). Powder and related techniques: X-ray techniques. In International Tables for Crystallography, Vol. C, edited by A. J. C. Wilson, ch. 2.3. Dordrecht: Kluwer.Google Scholar