One long-standing problem back in the 1960’s, when Hugo was busy developing Rietveld refinement, was how to come up with an estimate for the “observed” structure factors, \(|F_{obs, hkl}|\), from a powder diffraction pattern. These estimated observed structure factors would allow one of the more common crystallographic computations, Fourier maps, for structure completion as will be discussed in §20.2. While I don’t find them usually very useful from powder diffraction, with “observed” structure factors one can compute Patterson maps which might allow structure solution for “heavy atom” powder diffraction patterns.
Estimating the area under a peak was not a problem, but how to apportion out that intensity when a peak was composed of overlapped reflections posed a challenge. This could be done by use of deconvolution for peaks that were separated by a reasonable fraction of the FWHM, but even so deconvolution, but that was slow back in those days. However, when reflections are exactly or very nearly overlapped, the peak intensity can be divided amongst the reflections in any manner.
As part of his original software, Hugo Rietveld came up with a method for estimating “observed” structure factors, \(|F_{obs, hkl}|\) which has also become universal in the field. I have referred to this as “Hugo Rietveld’s other breakthrough” and it is a quite nice technique. In order to compute the powder diffraction pattern, Hugo’s software first had to compute structure factors from the model, \(F_{calc, hkl}\), these then multiplied by the profile function, the multiplicity and finally other scaling constants, such as the Lorentz-Polarization correction, the scale factor and the phase fraction and that product determined how much intensity to add to a given point in the calculated powder diffraction profile. This was done for every reflection and every data point in the pattern. In the end, that meant for every point in the computed diffraction pattern it could be tracked how much of that intensity came from each reflection. The beauty of Hugo’s intensity extraction algorithm was that he just turned that computation around. For every point in the observed powder diffraction he used the same factor applied to the \(F_{calc, hkl}\) and used that to divide each data point’s actual intensity and added that to the variable that would in the end contain the estimate for \(|F_{obs, hkl}|\). For convenience, the peak shape function is normalized to unit area.
This may not do a perfect job with two overlapping peaks, where \(|F_{obs, hkl}|\) and \(|F_{calc, hkl}|\) are closely matched for one reflection and not the other, but will put most of the extra intensity into the right reflection. With completely overlapped peaks, it will divvy up the extra intensity weighted by the multiplicity. Perhaps not correct, but there are no better ad hoc choices. I don’t believe anyone has improved over this algorithm in newer implementations of Rietveld fitting. As it turns out, for many years when I presented this intensity extraction method in my introduction to Rietveld analysis, I did this with a slide titled, “Hugo Rietveld’s other breakthrough.” I wonder if that slide prompted the e-mail that I very much prize that is reproduced in Fig. 20.1. Note that 20 years later, none of the links in the e-mail still work, but a version of my slides are still online1.
