1.3 What is Rietveld Analysis?

So what is Rietveld analysis about? As mentioned, it is the fitting of an as-measured powder diffraction pattern. To contrast to single-crystal crystallographic fitting, there one only needs to fit to the observed structure factor intensities (which, by the way, are not directly measured, but must be extracted as part of the single-crystal experimental process). In both single-crystal and Rietveld analysis, one optimizes details of the atomistic structure of the phase to best-fit the observed intensities. However, with powder diffraction, much more must be done. Note that it now may be necessary to fit more than one structure, as a material may contain many phases. Additionally, in Rietveld fitting, there are three more major aspects in the as-measured powder diffraction pattern that must be fit:

• There will always be non-Bragg scattering intensity present. We call this background. In the single-crystal measurement, this is automatically removed (one hopes – well), but in powder diffraction the background levels are more substantial, can vary very significantly across the pattern, and must be modeled point by point.
• Peak positions are determined by the lattice parameters, so these need to be fit as well. Fortunately, powder diffraction is both extremely accurate and precise in this measurement – frequently an order of magnitude more precise than single crystal measurements.
• In order to fit the as-measured powder diffraction pattern point-by-point, we must account not only for the intensity of the reflections from the structure, but also the way that these reflections are broadened (in single-crystal work, one just integrates over the breadth.) This broadening arises from both the sample and the instrument and is modeled by the peak shape function. When multiple data types are combined, the breadths need to be treated differently for each phase, as well as for each measurement when more than one set of experimental data is used.

In later sections of this book I will discuss how each of the four aspects of this modeling are done, but it should be clear that any metrics for how well the data are fitted will combine the overall fit quality with respect to all four. In Chapter 9, I will discuss these metrics in detail, but for now let us concentrate on the quantity called the R-factor. Powder diffraction R-factors are poor at reflecting the quality of a structural fit since that single R-factor value indicates any problems in fitting the structure, background, peak shape and/or cell dimensions. If a powder diffraction pattern contains a large amount of background, and that is well-fit, but the structure is not, the R-factor may be low. On the other hand, if minor aspects of the background or peak shape are poorly fit, a high R-factor may result, even though the structural fit is excellent. This is in contrast to a single-crystal R-factor, which only measures how well the crystal structure reproduces the observed intensities. Thus, the single-crystal R-factor is a more clear indication of quality – though it is often overemphasized.