Peter Stephens, a pioneer in synchrotron diffraction and ab initio structure determination and also the developer of the model we use for anisotropic microstrain broadening, once said something along the lines of “A Rietveld refinement is never done, one merely gives up.” This is because there seems to be an endless range of structural variations and fitting options one can try. As was noted in Chapter 9, the Reduced \(\chi ^2\) should ideally be close 1 to but never less than 1. Equivalently, \(R_{wp}\) should be close to \(R_{exp}\) but always greater than \(R_{exp}\). When the Reduced \(\chi ^2\) is close to 1, there is no reason to look for models that improve the fit by adding more complexity. However, one never knows if there might be a more simple model that fits the data just as well, unless one looks for it.
There are many fits where small mismatches in the peak shape or inaccuracies in fitting the background will prevent the Reduced \(\chi ^2\) value from ever getting close to 1. How then do we know that there is no point in trying to improve the fit to the peak intensities? The suggestion that I have is to make a copy of the GSAS-II project and in that copy perform a Le Bail or Pawley fit (see sections 20.4 and 20.3, where this type of fitting is discussed in detail.) These methods optimize the reflection intensities directly rather than try to fit the structure and this provide a lower limit for the best \(\chi ^2\) or \(R_{wp}\) that one can hope to obtain by fitting the peak intensities via a structure refinement. Either of these fits will represent more reasonable lower limits when refining peak intensities than use of 1 for \(\chi ^2\) or \(R_{exp}\)..