International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 5.2, p. 496

The recent large increase in the use of powder samples for crystalstructure refinement and analysis has also stimulated interest in latticeparameter determinations, which are derived during the course of the calculation. The most frequently used method is that of Rietveld (1967, 1969) described in Chapter 8.6 . In outline, a profilefitting function containing adjustable parameters to vary the width and shape with is selected. The parameters corresponding to the atomic positions, multiplicity, lattice parameters, etc. of the selected structure model are varied until the best leastsquares fit between the whole observed diffraction pattern and the whole calculated pattern of the model is obtained. There is no detailed published study of the accuracy of the lattice parameters that is attained but the estimated standard deviations quoted in a number of papers (see, for example, Young, 1988) appear to be comparable with those published for simple structures having no overlapped reflections. In this type of calculation, the accuracy of the lattice parameters is tied to the accuracy of the refined structure because it includes the model errors in the leastsquares residuals.
An alternative to the Rietveld method is the patterndecomposition method in which the integrated intensities are derived from profile fitting and the data used in a powder leastsquaresrefinement program. The reflections may be fitted individually or in small clusters (Parrish & Huang, 1980) or the whole pattern can be fitted (Pawley, 1981; Langford, Louër, Sonneveld & Visser, 1986; Toraya, 1986, 1988); unlike the Rietveld method, no crystalstructure model is required and only the first stage is used for lattice parameters. The Pawley method was developed for neutrondiffraction data and uses slack constraints to handle the problem of leastsquares illconditioning due to overlapping reflections, and the positions of the reflections are constrained by the lattice parameters. The refinement also determines the zeroangle calibration correction.
Toraya extended the Pawley method to Xray powder diffractometry. He first determined the profile shapes and peak positions of several standard samples by individual profile fittings to generate a curve relating the peak shifts to . A pair of split Pearson VII profiles was used for conventional patterns to handle the doublets and the profile asymmetries, and a pseudoVoight function for the nearly symmetrical synchrotronradiation profiles. The program is set up so that the parameters of the fitting function are varied with to account for the increasing widths and the peak shifts and the whole pattern is automatically fitted. The positions of the individual reflections are a function of the calculated lattice parameters, which are refined together with the integrated intensities as independent variables. This method also permits simultaneous refinement of several phases present in the pattern. Unit cells calculated from wholepattern profile fitting and incorporating the peakshift corrections had estimated standard deviations an order of magnitude smaller than those not using the systematic error correction. It is also possible to use an internal standard and to make the corrections by refining the cell parameters of the sample and holding constant the parameters of the standard.
Good results can also be obtained using selected peaks rather than the whole pattern (Parrish & Huang, 1980). Peak search or profile fitting is used to determine the observed peak positions. The leastsquares refinement is used to minimize (observed − calculated). It also determines the average and the standard deviation of all the d's and 's. In principle, all the aberrations causing shifts can be incorporated in the refinement. There are, however, large correlations between aberrations with similar angular dependencies. In practice, the zeroangle calibration correction is always determined, and the specimensurface displacement shift is usually included.
The latticeparameter determination requires an indexed pattern in which the peak angles have been determined by peak search or profile fitting. Reflections known to have poor precision because of very low intensity or close overlapping should be omitted. The estimated standard deviation is dependent on the number of reflections used and it is better to use all the well measured peaks. There is the question of using a weighting scheme in which the highangle reflections are given greater weight because of their higher accuracy for a given error. As noted in Subsection 5.2.13, higherorder reflections usually have low intensities and much overlapping. Some judgement and critical tests are often required.
References
Langford, J. I., Louër, D., Sonneveld, E. J. & Visser, J. W. (1986). Applications of total pattern fitting to a study of crystallite size and strain in powder zinc oxide. Powder Diffr. 1, 211–221.Parrish, W. & Huang, T. C. (1980). Accuracy of the profile fitting method for Xray polycrystalline diffractometry. Natl Bur. Stand. (US) Spec. Publ. No. 457, pp. 95–110.
Pawley, G. S. (1981). Unitcell refinement from powder diffraction scans. J. Appl. Cryst. 14, 357–361.
Rietveld, H. M. (1967). Line profiles of neutron powder diffraction peaks for structure refinement. Acta Cryst. 22, 151–152.
Rietveld, H. M. (1969). A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2, 65–71.
Toraya, H. (1986). Wholepowderpattern fitting without reference to a structural model: application to Xray powder diffractometer data. J. Appl. Cryst. 19, 440–447.
Toraya, H. (1988). The deconvolution of overlapping reflections by the procedure of direct fitting. J. Appl. Cryst. 21, 192–196.
Young, R. A. (1988). Pressing the limits of Rietveld refinement. Aust. J. Phys. 41, 294–310.