The R Factor and Crystallographic Refinement

If the model derived from the diffraction data were correct in every way, and the measured data were perfect, then the agreement would be exact Fhkt-obs would always equal Fhki-caic and R = 0. But this is never the case. The data, of course, contain measurement errors, the atomic positions may be accurate and precise, but still not perfect, temperature factors and the ellipsoids of vibration may be only approximate, and so on. In general, even for a very well and correctly determined structure, R will commonly be in the range of 0.05 to 0.10 for a conventional crystal having an asymmetric unit of 50 or so atoms. For macro-molecular structure determinations, R is normally in the range of 0.15 to 0.25. The R factor is in most cases the ultimate criterion of model quality at the resolution it was determined. 

It so happens that the R factor is useful not only in telling us when we have the correct solution, and how good it is. The R factor is also invaluable in directing us from an imperfect model, or a very approximate structure solution, to the best model that we can obtain. This occurs in the process known as crystallographic refinement (see Chapter 10). 

Refinement also may be accomplished in real space by calculating difference Fourier syntheses using as coefficients A F = F i,s — Fcaic and phases j caic derived from the trial structure. In general, if an atom has been incorrectly placed near its true location, a negative peak will appear at its assigned position and a positive peak will appear at its proper location. That is, in the difference Fourier we have subtracted electron density from where the atom isn t, and not subtracted density from where it is. The atom is then shifted by altering x, y, z, improved atomic parameters are included in a new round of structure factor calculations, and another difference Fourier computed. The process is repeated until the map is devoid of significant features. 

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