Whole Pattern-Fitting Quantitative Analysis I. Principles

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Principles

We now consider a very different approach to quantitative analysis. The methods described above, involving one or just a few peaks in the powder diffraction pattern, have served well over an era in which powder diffraction equipment has changed technically to the point where now computational analysis is accepted as a fairly standard supplement within the powder diffraction package. There still may be highly specialised industrial situations where an extremely rapid quantitative analysis response is important, requiring the type of "one-peak estimate" that the above methods can deliver. However, most powder diffraction practitioners these days have easy access to powerful computers, and this sits alongside one of the other revolutions in powder diffraction - whole pattern fitting. This concept has already been dealt with at length within this powder diffraction course, mainly via discussion of the Rietveld structure refinement method. The main reasons for the introduction of whole pattern fitting into the field of quantitative analysis are very similar to those for Rietveld structure refinement: Powder Diffraction patterns can sometimes be very congested. If a diagnostic peak overlaps another peak then extraction of its intensity can be uncertain; if the whole quantitative analysis hangs on this peak intensity then the uncertainty propagates. On the other hand, if the whole pattern is fitted, then all the available information is being used and success does not rest on the measurement of just one or two peaks.

You will already be aware that during the process of Rietveld structure refinement, a scale factor is refined. This scale factor takes responsibility for many parameters such as the conditions of the experiment (X-ray beam intensity etc.) and the amount of material present. If two structural phases are present in the sample and structure-refined then two scale factors must be assigned to the sample and, since only one (constant) experiment and sample (and indeed only one sample absorption) is involved, these two scale factors will simply represent the different amounts of the components present. This is much as with the single peak methods discussed earlier but with the normalisation process being effectively handled via the known unit cell structures of each phase. A general formula has been given by Hill and Howard (see footnote ¹ ) for the weight fraction, w_A, of phase A in a multi-component sample:

where S, Z, M, V, are, respectively, the scale factors, number of formula units per unit cell, the mass of the formula unit, the unit cell volume, and the summation is over all components in the mixture. The method does not depend on the choice of unit cell (i.e. choice of Z and/or V) since the scale factors automatically adjust to compensate.

This procedure is now increasingly used as an alternative to the previous single peak (or few peaks) methods of quantitative analysis. In addition to the advantages already outlined, the method also has distinct advantages in cases where one or more phase in a mixture occurs as a slightly variable structure: for example one phase might be part of a solid solution series (e.g. the cement phase Ca₂Fe_xAl_1− xO₅ where x varies depending on the amount of Fe and Al in the source materials). Since x can become another variable in the Rietveld refinement, then the chosen structure will automatically adjust to optimise the fit with the whole pattern data. Of course this would not happen with the single peak methods.

Another important development has been the use of whole pattern fitting methods for quantitative analysis even when the structure is, for whatever reason, completely unknown. In these cases it is the actual measured patterns of pure phase components, I_{pure i} (2θ), rather than the predicted patterns from crystal structures, that are manipulated in the least squares minimisation process. Quantities such as:

are minimised in total over all 2θ data points of the pattern (see footnote ² ) the summation being taken over likely candidates, i, from a library of pure reference materials whose diffraction patterns, I_pure i (2θ), will have already been collected and normalised via the constant c_i in an analogous fashion to that for single peak methods (Reference Intensity Ratio etc.); the w_i values are essentially the required output, the best-fit weight fractions for each phase.

Yet another development has been to combine both known structures and reference patterns simultaneously in whole pattern fitting quantitative analysis; that is, for a given mixture, to use crystal structures for those phases where they are known and reliable, and to use measured pure phase patterns in the other cases. An interesting twist to this is that the measured patterns do not have to be manipulated as "fixed patterns" but can be varied within the whole pattern fitting construct: That is, rather than using a set of I_pure i (2θ) values, an "artificial" F_hkl set can be created for each pure phase i that would yield the same I_pure i (2θ) intensities within the refinement process. The individual atom coordinates are not needed yet by calculating from a working F_hkl set many of the usual refinement parameters (unit cell size, peak widths) are allowed to vary and be refined just as though they belonged to a known structure. These features have been included within a commercial software package for whole pattern quantitative analysis, known as SIROQUANT (see footnote ³  and links page).

Prior to looking at an example, we note the following features of whole pattern fitting quantitative analysis:

• Whole-pattern quantitative analysis utilises all the diffraction information, and is not tied to one or a few peaks;
• Preferred orientation effects, although not necessarily removed, are less severe with whole pattern methods since the intensity errors are spread over many peaks;
• The problem of peak overlap is effectively minimised by using whole pattern fitting;
• Crystal structures can vary during refinement - this aids situations where the real candidate phase is slightly different from the model reference phase;
• Fitted patterns can also vary away from the pure measured pattern, again allowing for reality where the candidate phase is slightly different to the pure phase (e.g. the unit cell is allowed to vary to accommodate small mismatches between the peak positions of the pure phase and of the actual sample);
• Imperfectly known structures can in principle be used by permitting some re-apportioning of intensity between different hkl reflections but so as to conserve the total scattering over the whole pattern;
• The peak widths relating to each phase can vary independently, for example to reflect likely situations where the crystallite size/strain of one phase is very different to that of another phase in a mixture;
• One is not tied to wholly measured pure reference patterns or to wholly known structures.

¹ R.J.Hill & C.J.Howard, J. Applied Crystallography 20, 467 (1987)
² D.K.Smith et al., Powder Diffraction 2, 73 (1987)
³ J.C.Taylor, Powder Diffraction 6, 2, (1991); J.C.Taylor & Z.Rui, ibid 7, 152, (1992)

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