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. 3.9, p. 368
Section 3.9.10.3.4. Whole-pattern-refinement effects^{a}CSIRO Mineral Resources, Private Bag 10, Clayton South 3169, Victoria, Australia,^{b}TU Bergakademie Freiberg, Institut für Mineralogie, Brennhausgasse 14, Freiberg, D-09596, Germany, and ^{c}Bruker AXS GmbH, Oestliche Rheinbrückenstr. 49, 76187 Karlsruhe, Germany |
One of the distinct advantages of structure-based whole-pattern fitting for QPA is that no standards need to be prepared because the structure for each phase provides the phase constant ZMV; the unit-cell dimensions allow the calculation of the cell volume V and the unit-cell contents provide the mass ZM (Bish & Howard, 1988; Hill & Howard, 1987). These values are used, along with the Rietveld scale factor S, in equation (3.9.26) to derive the phase abundance. This is especially useful for complex systems where the preparation of multiple standards would add considerably to the analytical complexity.
An additional advantage is the ability to refine the crystal structure (unit-cell dimensions and site-occupation factors, for example), when the data are of sufficiently high quality, in order to obtain the best fit between observed and calculated patterns. In addition to updating the ZMV value, the site occupancies are contained in the structure-factor calculation and, therefore, will change the relative reflection intensities and have an impact on the scale factor and QPA. Other structural parameters that have a strong effect on the scale factor and QPA are the atomic displacement parameters (ADPs). Strong correlation between the ADPs and amorphous material concentration has been shown by Gualtieri (2000) and Madsen et al. (2011).
This leads to the question: which crystal structure should be selected for QPA? Databases contain multiple entries for the same phase with the structures determined using different methods. While ADPs and site-occupation factors determined using neutron diffraction and single-crystal analysis should be favoured over those determined using X-ray powder data, many database entries do not have refined ADPs for all (and in some cases, any) atoms. Often, arbitrarily chosen default values of 0.5 or 1.0 Å^{2} for B_{eq} are entered for all atoms, but this should be viewed or used with great caution. There is clearly a need to carefully evaluate the crystal-structure data used for QPA. This is particularly worth mentioning in view of the advent of new `user-friendly' software that automatically assigns crystal structures after having performed the phase identification.
Empirical profile-shape models contribute significantly to the complexity (and correlations) of whole-powder-pattern fitting for QPA because of the large number of phases and multiple parameters required to model the profile shape of each phase. The use of convolution-based profile fitting [in, for example, BGMN (Bergmann et al., 1998, 2000) and TOPAS (Bruker AXS, 2013)] greatly reduces the number of parameters, because the instrument-resolution function (which is constant for a given setup) can be separated from sample-related peak broadening. The instrument component can be refined using a standard and then fixed for subsequent analysis. The sample contribution to peak width and shape can then be related directly to crystallite size and microstrain using a minimal number of parameters. The reduction of the total number of parameters reduces the refinement complexity and the chance of parameter correlation.
The choice of the function used to model the pattern background may also have a strong influence on amorphous content (Gualtieri, 2000; Madsen et al., 2011). Given that the intensities of both the background and the amorphous contribution vary slowly as a function of 2θ, it is inevitable that there will be a high degree of correlation between them. Hence, any errors in determining the true background will result in errors in amorphous phase determination. A simple approach is to use a background function with a minimal number of parameters. A more exact approach requires the separation of the amorphous contribution from background components such as Compton scattering and parasitic scattering by the sample environment and air in the beam path. This is routinely done in pair distribution function (PDF) analysis; details can be found in Chapter 5.7 in this volume and in Egami & Billinge (2003).
Another parameter that correlates with the pattern background is the width of broad peaks for phases of low concentration. If allowed to refine to very large width values, the peaks are `smeared' over a broad range of the pattern with no clear distinction between peaks and background. The same issue applies when there is a high degree of peak overlap, particularly at high angles, leading to severe under- or over-estimation of the phase. The careful use of limits for either crystallite size or corresponding parameters in empirical peak-shape modelling assists in minimizing this effect.
There can be a subtle interplay between the profile-shape function and the pattern background that has an impact on whole-pattern fitting (Hill, 1992). The data in Fig. 3.9.21, collected using a Cu tube and an Ni Kβ filter, exhibit low-angle truncation of the peak tails at the β-filter absorption edge. On the high-angle side, the anatase peak displays a wide tail which extends to the position of the strongest rutile peak at about 27.5° 2θ. In this case, rutile is present as a minor phase and the error in the background determination using conventional peak-profile modelling (Fig. 3.9.21a) introduces about 0.5% bias in the rutile QPA. The use of a more accurate profile model that incorporates the effect of the β-filter absorption edge (Fig. 3.9.21b) serves to improve the accuracy (Bruker AXS, 2013).
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