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. 3.5, p. 282

Section 3.5.2.1. Unrestrained cell

A. Le Baila*

aUniversité du Maine, Institut des Molécules et Matériaux du Mans, UMR CNRS 6283, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
Correspondence e-mail: lebail@univ-lemans.fr

3.5.2.1. Unrestrained cell

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Without a cell hypothesis, no |Fhkl| values can be extracted; the intensity values collected will be noted by I(i) until Miller indices are attributed, enabling the multiplicity correction. Obtaining all the peak positions, areas, breadths and shape parameters as independent values for a whole powder pattern is limited to simple cases where there is not too much peak overlap. With such an approach (both cell and space group unknown or unused) one has to estimate the number of peaks to be fitted, so that the fit of a complex group of peaks will lead to large uncertainties. However, knowing the cell and space group provides at least the correct number of peaks and an estimate of their starting positions. Such calculations were made as an alternative to the Rietveld method, during the first stage of the so-called two-stage method for refinement of crystal structures (Cooper et al., 1981[link]). In the case of X-ray data, the profile shapes applied in the Rietveld method (Gaussian at the beginning for neutron data) evolved a great deal (Wiles & Young, 1981[link]), and on the WPPD side happened to be described in these two-stage approaches by a sum of Lorentzian curves, or double Gaussians (Will et al., 1983[link], 1987[link]). The computer program PROFIT (Scott, 1987[link]), derived from software for individual profile fitting (Sonneveld & Visser, 1975[link]) and extended to the whole pattern, was applied to the study of crystallite size and strain in zinc oxide (Langford et al., 1986[link]) and for the characterization of line broadening in copper oxide (Langford & Louër, 1991[link]). Studying a whole pattern can also be done in simple cases by using software designed for the characterization of single or small groups of peaks; an example is a ZnO study (Langford et al., 1993[link]) using the computer program FIT (Socabim/Bruker). WPPD on complex cases is mostly realized today by using peak positions controlled by the cell parameters, with the benefit of stronger accuracy of the |Fhkl| values, even if the lost degrees of freedom may lead to slightly worse fits, increasing the profile R factors. Before 1987, close to thirty structure determinations by powder diffractometry (SDPDs) were achieved using intensities extracted by using these old WPPD methods without cell constraints (see the SDPD database; Le Bail, 2007[link]). It can be argued that freeing the peak positions allows one to take into account subtle effects in position displacement (in stressed samples, for example). But systematic discrepancy of observed peak positions with regard to the theor­etical position, as expected from the cell parameters, can be modelled as well in modern WPPD methods or in the Rietveld method.

References

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Langford, J. I., Boultif, A., Auffrédic, J. P. & Louër, D. (1993). The use of pattern decomposition to study the combined X-ray diffraction effects of crystallite size and stacking faults in ex-oxalate zinc oxide. J. Appl. Cryst. 26, 22–33.Google Scholar
Langford, J. I. & Louër, D. (1991). High-resolution powder diffraction studies of copper(II) oxide. J. Appl. Cryst. 24, 149–155.Google Scholar
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Le Bail, A. (2007). Structure Determination from Powder Diffraction Database – SDPD. http://www.cristal.org/iniref.html .Google Scholar
Scott, H. G. (1987). PROFIT – a peak-fitting program for powder diffraction profile. IUCr Satellite Meeting on X-ray Powder Diffractometry, 20–23 August 1987, Fremantle, Western Australia. Abstract P28.Google Scholar
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