International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 268
Section 2.4.3.6. Anomalous scattering without phase change^{a}Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^{b}Raman Research Institute, Bangalore 560 080, India |
The phase determination, and hence the structure solution, outlined above relies on the imaginary component of the dispersion correction. Variation in the real component can also be used in structure analysis. In early applications of anomalous scattering, the real component of the dispersion correction was made use of to distinguish between atoms of nearly the same atomic numbers (Mark & Szillard, 1925; Bradley & Rodgers, 1934). For example, copper and manganese, with atomic numbers 29 and 25, respectively, are not easily distinguishable under normal X-ray scattering. However, the real components of the dispersion correction for the two elements are −1.129 and −3.367, respectively, when Fe radiation is used (IT IV, 1974). Therefore, the difference between the scattering factors of the two elements is accentuated when this radiation is used. The difference is more pronounced at high angles as the normal scattering factor falls off comparatively rapidly with increasing scattering angle whereas the dispersion-correction term does not.
The structure determination of KMnO_{4} provides a typical example for the use of anomalous scattering without phase change in the determination of a centrosymmetric structure (Ramaseshan et al., 1957; Ramaseshan & Venkatesan, 1957). f′ and f″ for manganese for Cu radiation are and 2.808, respectively. The corresponding values for Fe radiation are and 0.481, respectively (IT IV, 1974). The data sets collected using the two radiations can now be treated as those arising from two perfectly isomorphous crystals. The intensity differences between a reflection in one set and the corresponding reflection in the other are obviously caused by the differences in the dispersion-correction terms. They can, however, be considered formally as intensity differences involving data from two perfectly isomorphous crystals. They can be used, as indeed they were, to determine the position of the manganese ion through an appropriate Patterson synthesis (see Section 2.4.4.2) and then to evaluate the signs of structure factors using (2.4.2.6) when the structure is centrosymmetric. When the structure is noncentrosymmetric, a twofold ambiguity exists in the phase angles in a manner analogous to that in the isomorphous replacement method. This ambiguity can be removed if the structure contains two different subsets of atoms Q1 and Q2 which, respectively, scatter radiations and anomalously. Data sets can then be collected with , which is scattered normally by all atoms, and . The three sets can be formally treated as those from three perfectly isomorphous structures and the phase determination effected using (2.4.2.7) (Ramaseshan, 1963).
References
International Tables for X-ray Crystallography (1974). Vol. IV, pp. 148–151. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google ScholarBradley, A. J. & Rodgers, J. W. (1934). The crystal structure of the Heusler alloys. Proc. R. Soc. London Ser. A, 144, 340–359.Google Scholar
Mark, H. & Szillard, L. (1925). Ein einfacher versuch zur auffinclung eines selectiven effecktes bei der zerstrenung von Röntgenstrahlen. Z. Phys. 33, 688–691.Google Scholar
Ramaseshan, S. (1963). The use of anomalous scattering in crystal structure analysis. In Advanced methods of crystallography, edited by G. N. Ramachandran, pp. 67–95. London and New York: Academic Press.Google Scholar
Ramaseshan, S. & Venkatesan, K. (1957). The use of anomalous scattering without phase change in crystal structure analysis. Curr. Sci. 26, 352–353.Google Scholar
Ramaseshan, S., Venkatesan, K. & Mani, N. V. (1957). The use of anomalous scattering for the determination of crystal structures – KMnO_{4}. Proc. Indian Acad. Sci. 46, 95–111.Google Scholar