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The concept of a variable-counting-time (VCT) strategy for use in Rietveld analysis of X-ray powder diffraction data was introduced by Madsen & Hill [Adv. X-ray Anal. (1992), 35, 39–47]. This strategy is based on a function that increases the counting time used at each step in the scan in a manner that is inversely proportional to the decline in reflection intensity that inevitably results from the combined effects of Lp factor, scattering factor and thermal vibration. The present work extends the VCT function to include the effects of reflection multiplicity, cylindrical-sample (capillary) absorption and monochromator polarization. The new algorithm has been incorporated into a PC computer program and applied to the collection of data from samples of LaB6, tourmaline, forsterite and boehmite. Subsequent analysis of the data using the Rietveld method has shown that VCT data can produce more accurate atomic coordinates and site occupancies, lower residual 'noise' in difference Fourier maps and more stable refinement of `light' atoms.

Recent developments in the Rietveld method for the analysis of powder diffraction data have seen the method evolve from its original purpose of crystal structure refinement to nclude the determination of phase abundance in polycrysalline mixtures and the estimation of crystal size and strain parameters. However, the Rietveld method is not easy to use and may deter many powder diffractionists, who are not inerested in structure refinement per se, from using the method in its non-structural applications.In order to overcome the difficulties in using the Rietveld method, a program, QPDA (for Quantitative Powder Diffraction Analysis), has been written that sets the conditions necessary for a single or multi-phase refinement, runs the Rietveld program and extracts phase abundance and size/strain information from the refined parameters. The program comprises a user-friendly, default-driven system of subroutines, written initially in VAX Fortran, and operates from a database of inorganic materials frequently encountered in a wide range of minerals and materials science industries.Equations (3) and (5) should be corrected to read as follows

Crystal structure parameters have been obtained for α-Al2O3, β-PbO2 and (Mg, Fe)2SiO4 by Rietveld analysis of Cu Kα X-ray powder diffraction data collected on a conventional diffractometer using counting times ranging from 0.01 to 5 s per step. For all but the 0.01 s data collected on (Mg, Fe)2SiO4 the structural parameters obtained at different counting times are statistically identical at the 3 σ level, and the spread in the values is essentially the same as that obtained by sample repacking at a fixed counting time of 1 s per step. The parameter e.s.d.'s and conventional agreement indices Rwp and RB decrease to values limited by residual model errors as the counting time increases, but the goodness-of-fit parameter becomes unacceptably larger than its ideal value of unity. When more than a few thousand counts are accumulated for the maximum step intensity in the diffraction profile a weighting scheme based solely on counting variance is inappropriate and the parameter e.s.d.'s are no longer a reflection of their accuracy.

Crystal-structure parameters and their associated estimated standard deviations (e.s.d.'s) have been determined for α-Al2O3 by Rietveld analysis of Cu Kα X-ray powder diffraction data collected on a standard diffractometer with profile step widths ranging from 0.01 to 0.32° 2θ. Estimates of the true precision of the parameters have been obtained by replication of the experiment five times. The degree of serial correlation between adjacent points in the profiles has been quantitatively determined by the use of the Durbin–Watson d statistic. Values of the conventional profile agreement indices, Rp, Rwp and goodness of fit, are not influenced by the scan-point density, but the Bragg R-factor decreases to a limiting value as the step width is decreased. The structural parameters obtained at all step widths are statistically indistinguishable at the 1σ level, although there is some loss of accuracy relative to single-crystal values at widths larger than 0.16°. The least-squares e.s.d.'s decrease linearly with the reciprocal square root of the number of steps in the pattern, as expected, but the e.s.d.'s of the unit-cell dimensions are the only ones that are significantly smaller than estimates of their true precision obtained from repetitions of the experiment. In the case of α-Al2O3, and possibly in general, the optimum step width for Rietveld analysis corresponds approximately to the minimum value of the full-width at half-maximum for well resolved peaks.