Theory of anomalous scattering

Matthews, B. W.

doi:10.1107/97809553602060000685

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 14.1, pp. 295-296 | 1 | 2 |

Section 14.1.7. Theory of anomalous scattering

B. W. Matthews^a ^*

^aInstitute of Molecular Biology, Howard Hughes Medical Institute and Department of Physics, University of Oregon, Eugene, OR 97403, USA
Correspondence e-mail: brian@uoxray.uoregon.edu

14.1.7. Theory of anomalous scattering

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Suppose that two isomorphous crystals are differentiated by N heavy atoms of position $[{\bf r}_{n}]$ and scattering factor $[({\bf f}_{n}' + i{\bf f}_{n}'')]$ . Then, for the reflection hkl, the calculated structure factor of the N atoms is $[\eqalignno{F_{H}({\bf h}) + iF_{H}''({\bf h}) &= {\textstyle\sum\limits_{n=1}^{N}}\ f_{n}' ({\bf h}) \exp (2\pi i{\bf h} \cdot {\bf r}_{n})\cr &\quad + i {\textstyle\sum\limits_{n=1}^{N}}\ f_{n}'' ({\bf h}) \exp (2\pi i{\bf h} \cdot {\bf r}_{n}). &(14.1.7.1)}]$ If the heavy atoms are all of the same type, i.e. they all have the same ratio of $[f_{n}'/f_{n}''\ (= k)]$ , then $[F_{H}]$ and $[F_{H}'']$ are orthogonal, and $[F_{H}'' = F_{H}/k]$ .

The relation between the structure factors of the reflection hkl and its Friedel mate $[\bar{h}\bar{k}\bar{l}]$ is illustrated in Fig. 14.1.7.1(a). The situation can be conveniently represented (Fig. 14.1.7.1b) by reflecting the $[\bar{h}\bar{k}\bar{l}]$ diagram through the real axis onto the hkl diagram. In cases such as this, where Friedel's law breaks down, we shall refer to the difference $[\Delta_{PH} = (F_{PH+} - F_{PH-})]$ as the Bijvoet difference, or simply the anomalous-scattering difference. The Harker phase circles corresponding to Fig. 14.1.7.1(b) are shown in Fig. 14.1.7.2. It will be seen that, as in the case of single isomorphous replacement, and similarly with the anomalous-scattering data alone, there is an ambiguous phase determination. In the absence of error, the three phase circles (Fig. 14.1.7.2) will meet at a point, resolving the phase ambiguity and giving a unique solution for the phase of $[{\bf F}_{P}]$ . The isomorphous-replacement method gives phase information symmetrical about the vector $[{\bf F}_{H}]$ , whereas the anomalous-scattering phase information for $[{\bf F}_{PH}]$ is symmetrical about $[{\bf F}_{H}'']$ , which, for heavy atoms of the same type, is at right angles to $[{\bf F}_{H}]$ . In other words, the two methods complement each other, one method providing exactly that information which is not given by the other.

Figure 14.1.7.1| top | pdf |

(a) Vector diagrams illustrating anomalous scattering for the reflections hkl and $[\overline{hkl}]$ . (b) Combined vector diagram for reflections hkl and $[\overline{hkl}]$ .

Figure 14.1.7.2| top | pdf |

Harker construction for a single isomorphous replacement with anomalous scattering, in the absence of errors.

On average, the experimentally measured isomorphous-replacement difference, $[(F_{PH} - F_{P})]$ , will be larger than the anomalous-scattering difference, $[(F_{PH+} - F_{PH-})]$ . The former, however, relies on measurements from different crystals and is also susceptible to errors due to non-isomorphism between the parent and derivative crystals. The latter can be obtained from measurements on the same crystal, under closely similar experimental conditions, and is not affected by non-isomorphism. Therefore, it is desirable to use methods that take into account the different sources of error in the respective measurements (Blow & Rossmann, 1961; North, 1965; Matthews, 1966b). One method is as follows.

References

Blow, D. M. & Rossmann, M. G. (1961). The single isomorphous replacement method. Acta Cryst. 14, 1195–1202.Google Scholar

Matthews, B. W. (1966b). The extension of the isomorphous replacement method to include anomalous scattering measurements. Acta Cryst. 20, 82–86.Google Scholar

North, A. C. T. (1965). The combination of isomorphous replacement and anomalous scattering data in phase determination of non-centrosymmetric reflexions. Acta Cryst. 18, 212–216.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 14.1, pp. 295-296