Introduction

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 246 | 1 | 2 |

Section 2.3.4.1. Introduction

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.4.1. Introduction

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The physical basis for anomalous dispersion has been well reviewed by Ramaseshan & Abrahams (1975), James (1965), Cromer (1974) and Bijvoet (1954). As the wavelength of radiation approaches the absorption edge of a particular element, then an atom will disperse X-rays in a manner that can be defined by the complex scattering factor $[f_{0} + \Delta f' + i\Delta f'',]$ where $[f_{0}]$ is the scattering factor of the atom without the anomalous absorption and re-scattering effect, $[\Delta f']$ is the real correction term (usually negative), and $[\Delta f'']$ is the imaginary component. The real term $[f_{0} + \Delta f']$ is often written as f′, so that the total scattering factor will be [f' + if''] . Values of $[\Delta f']$ and $[\Delta f'']$ are tabulated in IT IV (Cromer, 1974), although their precise values are dependent on the environment of the anomalous scatterer. Unlike $[f_{0}]$ , $[\Delta f']$ and $[\Delta f'']$ are almost independent of scattering angle as they are caused by absorption of energy in the innermost electron shells. Thus, the anomalous effect resembles scattering from a point atom.

The structure factor of index h can now be written as $[{\bf F}_{{\bf h}} = {\textstyle\sum\limits_{j = 1}^{N}}\; f'_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}) + i {\textstyle\sum\limits_{j = 1}^{N}}\; f''_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}). \eqno(2.3.4.1)]$ (Note that the only significant contributions to the second term are from those atoms that have a measurable anomalous effect at the chosen wavelength.)

Let us now write the first term as [A + iB] and the second as . Then, from (2.3.4.1), $[{\bf F} = (A + iB) + i(a + ib) = (A - b) + i(B + a). \eqno(2.3.4.2)]$ Therefore, $[|{\bf F}_{{\bf h}}|^{2} = (A - b)^{2} + (B + a)^{2}]$ and similarly $[|{\bf F}_{{{\bar {\bf h}}}}|^{2} = (A + b)^{2} + (- B + a)^{2},]$ demonstrating that Friedel's law breaks down in the presence of anomalous dispersion. However, it is only for noncentrosymmetric reflections that $[|{\bf F}_{{\bf h}}| \neq |{\bf F}_{{\bar{\bf h}}}|]$ .

Now, $[\rho ({\bf x}) = {1 \over V} {\sum\limits_{{\bf h}}^{\rm sphere}} {\bf F}_{{\bf h}} \exp (2\pi i{\bf h}\cdot {\bf x}).]$ Hence, by using (2.3.4.2) and simplifying, $[\eqalignno{\rho ({\bf x}) = &{2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} [(A \cos 2\pi {\bf h}\cdot {\bf x} - B \sin 2\pi {\bf h}\cdot {\bf x})\cr &+ i(a \cos 2\pi {\bf h}\cdot {\bf x} - b \sin 2\pi {\bf h}\cdot {\bf x})]. &(2.3.4.3)}]$ The first term in (2.3.4.3) is the usual real Fourier expression for electron density, while the second term is an imaginary component due to the anomalous scattering of a few atoms in the cell.

References

Bijvoet, J. M. (1954). Structure of optically active compounds in the solid state. Nature (London), 173, 888–891.Google Scholar

Cromer, D. T. (1974). Dispersion corrections for X-ray atomic scattering factors. In International tables for X-ray crystallography, Vol. IV, edited by J. A. Ibers & W. C. Hamilton, pp. 148–151. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar

James, R. W. (1965). The optical principles of the diffraction of X-rays. Ithaca: Cornell University Press.Google Scholar

Ramaseshan, S. & Abrahams, S. C. (1975). Editors. Anomalous scattering. Copenhagen: Munksgaard.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 246