InternationalSpace-group symmetryTables for Crystallography Volume A Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 125-126
## Section 1.6.5.1.2. Status of centrosymmetry and resonant scattering H. D. Flack
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The basic starting point in this analysis is the following linear transformation of and , applicable to both observed and model values, to give the average (*A*) and difference (*D*) intensities:In equation (1.6.2.1), was denoted by . The expression for corresponding to that for given in equation (1.6.2.1) and using the same nomenclature isIn general is small compared to . A compound with an appreciable resonant-scattering contribution has , whereas a compound with a small resonant-scattering contribution has . For centric reflections, , and so the values of of these are entirely due to random uncertainties and systematic errors in the intensity measurements. of acentric reflections contains contributions both from the random uncertainties and the systematic errors of the data measurements, and from the differences between and which arise through the effect of resonant scattering. A slight experimental limitation is that a data set of intensities needs to contain both reflections *hkl* and in order to obtain and .

The Bijvoet ratio, defined byis the ratio of the root-mean-square value of *D* to the mean value of *A*. In a structure analysis, two independent estimates of the Bijvoet ratio are available and their comparison leads to useful information as to whether the crystal structure is centrosymmetric or not.

The first estimate arises from considerations of intensity statistics leading to the definition of the Bijvoet ratio as a value called Friedif_{stat}, whose functional form was derived by Flack & Shmueli (2007) and Shmueli & Flack (2009). One needs only to know the chemical composition of the compound and the wavelength of the X-radiation to calculate Friedif_{stat} using various available software.

The second estimate of the Bijvoet ratio, Friedif_{obs}, is obtained from the observed diffraction intensities. One problematic point in the evaluation of Friedif_{obs} arises because *A* and *D* do not have the same dependence on and it is necessary to eliminate this difference as far as possible. A second problematic point in the calculation is to make sure that only acentric reflections of any of the noncentrosymmetric point groups in the chosen Laue class are selected for the calculation of Friedif_{obs}. In this way one is sure that if the point group of the crystal is centrosymmetric, all of the chosen reflections are centric, and if the point group of the crystal is noncentrosymmetric, all of the chosen reflections are acentric. The necessary selection is achieved by taking only those reflections that are general in the Laue group. To date (2015), the calculation of Friedif_{obs} is not available in distributed software. On comparison of Friedif_{stat} with Friedif_{obs}, one is able to state with some confidence that:

*Example 1*

The crystal of compound Ex1 (Udupa & Krebs, 1979) is known to be centrosymmetric (space group ) and has a significant resonant-scattering contribution, Friedif_{stat} = 498 and Friedif_{obs} = 164. The comparison of Friedif_{stat} and Friedif_{obs} indicates that the crystal structure is centrosymmetric.

*Example 2*

The crystal of compound Ex2, potassium hydrogen (2*R*,3*R*) tartrate, is known to be enantiomerically pure and appears in space group . The value of Friedif_{obs} is 217 compared to a Friedif_{stat} value of 174. The agreement is good and allows the deduction that the crystal is neither centrosymmetric, nor twinned by inversion in a proportion near to 50:50, nor that the data set is unsatisfactorily dominated by random uncertainty and systematic error.

*Example 3*

The crystals of compound Ex3 (Zhu & Jiang, 2007) occur in Laue group . One finds Friedif_{stat} = 70 and Friedif_{obs} = 499. The huge discrepency between the two shows that the observed values of *D* are dominated by random uncertainty and systematic error.

### References

Flack, H. D. & Shmueli, U. (2007). The mean-square Friedel intensity difference in*P*1 with a centrosymmetric substructure.

*Acta Cryst.*A

**63**, 257–265.

Shmueli, U. & Flack, H. D. (2009). Concise intensity statistics of Friedel opposites and classification of the reflections.

*Acta Cryst.*A

**65**, 322–325.

Udupa, M. R. & Krebs, B. (1979). Crystal and molecular structure of creatininium tetrachlorocuprate(II).

*Inorg. Chim. Acta*,

**33**, 241–244.

Zhu, H.-Y. & Jiang, S.-D. (2007). 1,3,4,6-Tetra-

*O*-acetyl-2-(trifluoromethylsulfonyl)-β-D-mannopyranose.

*Acta Cryst.*E

**63**, o2833.