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. 3.5, p. 283
Section 3.5.2.2.1. Pawley method^{a}Université du Maine, Institut des Molécules et Matériaux du Mans, UMR CNRS 6283, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France |
The idea of removing the crystal structure refinement part in a Rietveld program and adding the potential to refine an individual intensity for every expected Bragg peak produced a new software package (named ALLHKL) allowing refinement of the cell parameters very precisely and extraction of a set of structure-factor amplitudes (Pawley, 1981). The process was much later called the `Pawley method'. Overcoming the least-squares ill conditioning due to peak overlap was achieved by using slack constraints (Waser, 1963). Pawley clearly insisted on the usefulness of the procedure for the confirmation of the indexing of a powder pattern of an unknown. Nevertheless, the structure of the C_{6}F_{10} (at 4.2 K) test case selected for demonstration purposes remained unsolved (but see Section 3.5.4.2 below). No SDPD of an unknown was realized using the Pawley method for several years (although successful tests were published corresponding to redeterminations of previously known structures). The first real SDPD of an unknown using the Pawley method seems to be that of I_{2}O_{4} (Lehmann et al., 1987); its powder pattern had been previously indexed, but the structure not determined because of the lack of a suitable single crystal. During these pioneering years, ALLHKL could not extract the intensities for more than 300 peaks, so that, in more complex cases, it was necessary to subdivide the pattern into several parts. Moreover, it was rather difficult to avoid completely the ill conditioning due to overlapping peaks. Successful fits yielded equipartitioned intensities (i.e., equal structure factors for those Bragg peaks with exact overlap). Unsuccessful fits could easily produce negative intensities which, combined with positive ones for other peak(s) at the same angle, reproduced the global positive value. Moreover, the first version to apply Gaussian peak shapes could not easily produce any SDPD because of the relatively poor resolution of constant-wavelength neutron data, so that it needed to be adapted to X-ray data, with the implementation of more complex peak shapes. Several programs were subsequently developed, based on the same principles as the original Pawley method. The first of them, by Toraya (1986), extended the use to X-ray data with non-Gaussian profile shapes, and introduced two narrow band matrices instead of a large triangular matrix, saving both computation time and memory space in a program named WPPF. Some programs were used to produce intensities in order to apply the so-called two-stage method (Cooper et al., 1981) for structure refinement, such as PROFIT (Scott, 1987) and PROFIN (Will, 1988) (no slack constraints, but equal division of the intensity between expected peaks when the overlap was severe). There was intense continuing activity on Pawley-like software with other programs such as FULFIT (Jansen et al., 1988), LSQPROF (Jansen et al., 1992) and POLISH (Byrom & Lucas, 1993). Estimation of intensities of overlapping reflections was improved in LSQPROF by applying relations between structure-factor amplitudes derived from direct methods, and the Patterson function was considered in the satellite program DOREES (Jansen et al., 1992). The question of how to determine the intensities of completely (or largely) overlapping reflections (either systematic overlap due to symmetry or fortuitous overlap) from a single powder pattern cannot have a definite simple answer, but continues to be discussed, since it is essential for improving our ability to solve structures. An early view with a probabilistic approach was given by David (1987), later introducing Bayesian statistics (Sivia & David, 1994) into the Pawley method. Early detection of preferred orientation on the basis of analysis of the E-value distribution was another way (Peschar et al., 1995) to improve the structure-factor-amplitude estimate. New computer programs based on the Pawley method continue to be written even today.
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