Scaling of X-ray diffraction data

one might think that this should not be a serious problem, the scaling of reflections, but it is, and furthermore it often serves as one of the most serious sources of error in the creation of a data set when it is done improperly or carelessly. If we consider, for example, the hkl and —hk — l symmetry equivalent reflections from a monoclinic crystal, or the hkl and —h — k — l reflections from any crystal (again ignoring anomalous dispersion) we would expect any measured intensity hu to be exactly equal to its symmetry or Friedel mate. They seldom are, however, and we may consider the reasons why. 

A crystal is never a perfect sphere, it has some geometric shape. As it rotates in the X-ray beam, different volumes of the whole crystal may be exposed. As a consequence hki will not be equal to I-h-k-i as we supposed. The same effect is produced by absorption because the diffracted X-rays must pass through different volumes of the crystal, or different volumes of the glass capillary or solvent, liquid, or ice, that surrounds the crystal. 

Given all of these effects, it is a wonder that the data sets we obtain are as precise as they are. It is a testament to the patience and persistence of generations of crystallographers that these problems have been dealt with successfully. Current software normally addresses all, or most of these factors, and their effects are now more or less transparent to most practicing crystallographers. Nonetheless, it is wise to know that all these problems are present in the background, and they constantly present a danger to a structure determination if they are mishandled. 

Scaling diffraction data from separate crystals, or scaling observed and calculated intensities or structure amplitudes is even more fraught with problems. Consider two separate crystals whose data were collected in two separate experiments, and remember that there are frequently several crystals. First of all, the independent data set from each crystal is subject to all the factors recounted above, and it is unlikely that any two will be identically affected. The two crystals may have had different geometrical properties, different amounts of liquid around them, different rates or allowances of radiation damage, or they 

FIGURE 7.9 In (a) and (b) are the resolution dependencies of two X-ray data sets recorded from two different but isomorphous crystals that show a clear difference in the rates at which average intensity declines with sin 9. The difference may be due to differences in crystal quality, decay rates, or many other factors. If the two data sets are scaled to one another using only a linear scale factor, the result in (c) is obtained, in which the crosshatched area represents the scaling mismatch, or residual. If a linear plus exponential scale factor is used, the falloff of the second data set is made to more closely approximate that of the first set, and the result in (d) is obtained in which the residual is substantially reduced. 

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