International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, p. 126

Section 1.6.5.1.3. Resolution of noncentrosymmetric ambiguities

H. D. Flackb

1.6.5.1.3. Resolution of noncentrosymmetric ambiguities

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It was shown in Section 1.6.5.1.2[link] that under certain circumstances it is possible to determine whether or not the space group of the crystal investigated is centrosymmetric. Suppose that the space group was found to be noncentrosymmetric. In each Laue class, there is one centrosymmetric point group and one or more noncentrosymmetric point groups. For example, in the Laue class mmm we need to distinguish between the point groups 222, 2mm, m2m and mm2, and of course between the space groups based on them. We shall show that it is possible in practice to distinguish between these noncentrosymmetric point groups using intensity differences between Friedel opposites caused by resonant scattering.

An excellent intensity data set from a crystal (Ex2 above) of potassium hydrogen (2R, 3R) tartrate, measured with a wavelength of 0.7469 Å at 100 K, was used. The Laue group was assumed to be mmm. The raw data set was initially merged and averaged in point group 1 and all special reflections of the Laue group mmm (i.e. 0kl, h0l, hk0, h00, 0k0, 00l) were set aside. The remaining data were organized into sets of reflections symmetry-equivalent under the Laue group mmm, and only those sets (589 in all) containing all 8 of the mmm-symmetry-equivalent reflections were retained. Each of these sets provides 4 [A_{\rm obs}] and 4 [D_{\rm obs}] values which can be used to calculate Rmerge values appropriate to the five point groups in the Laue class mmm. The results are given in Table 1.6.5.1[link]. The value of 100% for Rmerge in a centrosymmetric point group, such as mmm or 2/m, arises by definition and not by coincidence. The [R_{D}] of the true point group has the lowest value, which is noticeably different from the other choices of point group.

Table 1.6.5.1| top | pdf |
Rmerge values for Ex2 for the 589 sets of general reflections of mmm which have all eight measurements in the set

Rmerge (%)mmm2mmm2mmm2222
[R_{A}] 1.30 1.30 1.30 1.30 1.30
[R_{D}] 100.0 254.4 235.7 258.1 82.9

The crystal of Ex1 above (space group [P2_{1}/c]) was treated in a similar manner. Table 1.6.5.2[link] shows that [R_{D}] values display no preference between the three point groups in Laue class 2/m.

Table 1.6.5.2| top | pdf |
Rmerge values for Ex1 for the 724 sets of general reflections of 2/m which have all four measurements in the set

Rmerge (%)2/mm2
[R_{A}] 1.29 1.29 1.29
[R_{D}] 100.0 98.3 101.7

Intensity measurements comprising a full sphere of reflections are essential to the success of the Rmerge tests described in this section.








































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