R factors in Rietveld analysis: How good is good enough?

People mastering Rietveld refinement techniques commonly ask the same questions: What do the various Rietveld discrepancy values, i.e., goodness-of-fit, !2, and R factors mean? Also, which ones are most important? Finally, what values allow one to distinguish good refinements from poor ones? These questions are also important to people who review Rietveld results, as well as individuals trying to decide if the results in a paper are likely to be trustworthy. These discrepancy values are only one criterion for judging the quality of Rietveld fits; of greater importance is the “chemical reasonableness” of the model. Also, as will be discussed further, graphical analysis of a fit is very valuable. In this article, I will explain how several of the most important of these discrepancy terms arise, what they mean, and what they measure, as well as slipping in a few of my own opinions—which may not be universally held in the field. But to start with the last question, there is no simple way to distinguish a good fit from one that is just plain wrong based on R factors or other discrepancy values. A large number of Rietveld indices have been proposed, but I have yet to see one that can be used as an absolute measure of refinement quality. The reason for this should be clear by the end of this article, but to get started, let’s define the concepts needed for this discussion. In the following paragraphs, when a term is first defined, it is presented in bold face to make the definition easier to see. Diffraction data are a set of intensity values measured at a set of specific momentum transfer !Q" values, which are usually expressed as 2" settings. It should be noted that diffraction measurements can also be made with fixed 2" while the wavelength varies, for example, in time-of-flight or energy-dispersive diffraction. However, for convenience, I will assume that data are collected as a function of 2" for this paper. By convention, the intensity values are labeled yO,i, where O indicates these are observed values and i indicates the intensity was measured at 2" value 2"i. To perform Rietveld analysis, we must have an uncertainty estimate for yO,i, which I will label ![yO,i]. In the past, this was called the estimated standard deviation !esd", but crystallographic convention now uses the term standard uncertainty !s.u." for this !Schwartzenbach et al., 1995, 1996". The meaning of ![yO,i] is that if we knew the “true” value for this intensity, which I will label yT,i, say, by measuring it an infinite number of times, then on average yO,i will be ±##yO,i$ of yT,i. Another way to express this is that %!yO,i− %yO,i&"& =##yO,i$, where % & indicates the expected value. When intensities are measured by directly counting individual photons or neutrons arriving at the detector, e.g., pulse counting, then yO,i=##yO,i$. In cases where intensity values incorporate implicit scaling factors, the s.u. must be computed from the number of counts and then be scaled by the same factor as the intensity. !If yO,i=sIO,i, where IO,i is the actual number of counts, then sIO,i=##yO,i$." Examples where this is needed include the use of variable counting times or scaling by a monitor detector or from instruments that report counts per second. Estimation of experimental uncertainties can be quite difficult for detectors that do not directly count quanta, e.g., charge coupled detectors, image plates, or energydispersive detectors that automatically correct for detector dead time.