Line profile analysis and the Rietveld method: crossing paths?

Is a combination of the Rietveld Method (RM) and Line Profile Analysis (LPA) possible? Or, in other words, is it possible to obtain meaningful microstructural information from pattern refinement? Opinions in the scientific community are divided, but the answer is probably "Yes, provided we are aware of what we are doing", with the additional proviso that, at least for the time being, sample imperfections are not too complicated. However, in all cases the physical meaning and validity of a solution have to be carefully checked against one's experience and intuition and, if practicable, information from other techniques. While the traditional Rietveld recipe for pattern generation and refinement is mostly unequivocal, the correspondence between an observed broadened peak and a set of physically sound microstructural parameters is not: a given line-profile shape can be compatible with different broadening models. The question of profile shape is of course central to obtaining precise and meaningful results from the structure refinement, but empirical shape and shape-trend functions have mostly been used since the original work of Rietveld. From the technical point of view it is quite easy to embed in the refinement procedure a routine for the generation of profile shapes based on microstructural parameters. More-or-less independently, some authors (Scardi and Lutterotti, Delhez and Mittemeijer, Cheary and Coelho, von Dreele, Bergmann, for example) have gone some way towards accomplishing this in recent years by using different approaches and models, but rarely has a convincing analysis of the validity of the results been carried out. Clearly both the RM and LPA are defined mathematically over a wide range of parameters, but, owing to the phenomenological nature of the models which often underlie the mathematical analysis, and also to various approximations, the results can be of limited physical significance. This in turn means they must be taken cum grano salis, unless a validation a priori carried out with traditional methods is undertaken. For example, extremely high values for crystallite size or microstrain derived from Rietveld refinement have been reported in the literature. These cannot be justified by any model since the criteria on which the models are based have not been met. It must be stressed that LPA is limited in its application and has not yet been developed as a global tool. It merely gives an explanation for line-profile shapes in somewhat restricted cases. There are various models, applied singly or simultaneously, which are only valid when certain sources of broadening are present, such as crystallite size, stacking faults, dislocations, microstrain or composition gradients, etc.. A definitive microstructural refinement tool, able to handle all of these has not yet been established, though some proposed procedures perhaps represent a move in the right direction; those derived from integral breadth methods (English and Dutch schools) and single-peak Fourier approaches (Italian school) seem the most reasonable to be included in Rietveld codes. Due to their nature, and since they can account for anisotropic broadening, they can permit a direct comparison to be made with traditional profile analysis techniques, such the Warren-Averbach method. However, even in this case, validity is restricted by the limitations of the underlying theories, i.e. limited crystallite size and non- specific microstrain, both presenting simple distributions. Improvements in the field may well be reported in the future, but the best solution is likely to be a comparative one. A few words should be added on purely quantitative aspects, related to the instrumental contribution to the profile shape. Recent additions to the literature show the possibility, in limited cases, of calculating the instrumental profile shape for a given geometry; actual measurements are, in fact, conducted on suitable standard materials that are not guaranteed to be defect-free (even the NBS line-profile standard has been proven to be slightly defective). On the other hand, since it is impossible to be certain that calculations take into account all instrumental effects, quantitative values have to be considered good estimates within the reliability of the instrumental profile. (This is valid also for traditional LPA.) Even this field is still open and a definitive solution is still awaited. In conclusion, the state-of-the-art allows for a microstructural refinement in the Rietveld method, but clearly the nature of any structural imperfections present in the sample must be ascertained by ‘traditional’ methods beforehand; characterization of microstructure is more than simply reporting values of ‘size’ and ‘strain’. Also, prior to attaching any physical significance to the results, a check on the range of validity of the model employed and, whenever possible, a cross-check of the results, should be carried out. However, even when the results are not physically meaningful, a profile shape model which adequately accounts for anisotropic sample broadening can enhance the quality of pattern fitting and hence the quality of the refined structural data. Since the literature on the subject is vast and growing continuously, a dedicated Website, intended to expand with further contributions from the scientific community, is being set-up by the authors under the URL http://bragg.ing.unitn.it/sizestrain/