International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.4, pp. 271-272
Section 3.4.2.1. Figures of merit^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
An important task is the introduction of a figure of merit (FOM) that is able to (a) describe the physical plausibility of a trial cell and its agreement with the observed pattern, and (b) select the best cell among different possible ones. de Wolff (1968) made an important contribution in this direction. He developed the M_{20} figure of merit defined bywhere Q_{20} is the Q value corresponding to the 20th observed and indexed peak, N_{20} is the number of different calculated Q values up to Q_{20}, and is the average absolute discrepancy between the observed and the calculated Q values for the 20 indexed peaks; the factor 2 is a result of statistics, explained by the larger chance for an observed line to sit in a large interval as compared with sitting in a small interval. The rationale behind M_{20} is as follows: the better the agreement between the calculated and the observed peak positions (the smaller the value) and the smaller the volume of the unit cell (the smaller the N_{20} value), the larger the M_{20} value and consequently the confidence in the proposed unit cell. A rule of thumb for M_{20} is that if the number of unindexed peaks whose Q values are less than Q_{20} is not larger than 2 and if M_{20} > 10, then the indexing process is physically reasonable (de Wolff, 1968; Werner, 2002). This rule is often valid, but exceptions occur. The use of the first 20 peaks is a compromise (coming from experience) between introducing a quite large number of observed peaks (depending on the number of parameters of the unit cell) and avoiding the use of high-angle peak positions, which are more affected by errors. M_{20} is statistically expected to be 1 in case of completely arbitrary indexing. It has no upper limit (it can be very large when is very small).
Smith & Snyder (1979) proposed the F_{N} criterion in order to overcome the limits of M_{20} with respect to its dependence on the 20 lines and on crystal class and space group. The F_{N} figure of merit is given bywhere is the average absolute discrepancy between the observed and calculated 2θ peak position values and N_{poss} is the number of possible diffraction lines up to the Nth observed line. The values of and N_{poss}, (, N_{poss}), are usually given with F_{N}. With respect to M_{20}, F_{N} is more suitable for ranking the trial solutions and less for indicating their physical plausibility (Werner, 2002).
Both M_{20} and F_{N}, being based on the discrepancies between observed and calculated lines, are less reliable if there are impurity peaks; if the information about the unindexed lines is not taken into account, the risk of obtaining false solutions increases. Alternative FOMs based on joint probability have also been proposed (Ishida & Watanabe, 1967, 1971). Among the recently developed FOMs, we mention:
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