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

International Tables for Crystallography (2012). Vol. F, ch. 2.1, p. 55   | 1 | 2 |

Section Anomalous dispersion

J. Drentha*

aLaboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: Anomalous dispersion

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In classical dispersion theory, the scattering power of an atom is derived by supposing that the atom contains dipole oscillators. In units of the scattering of a free electron, the scattering of an oscillator with eigen frequency [\nu_{n}] and moderate damping factor [\kappa_{n}] was found to be a complex quantity:[f_{n} = \nu^{2}/(\nu^{2} - \nu_{n}^{2} - i \kappa_{n} \nu), \eqno(]where [\nu] is the frequency of the incident radiation [James, 1965[link]; see also IT C (2004)[link], equation ([link] ]. When [\nu \gg \nu_{n}] in equation (,[link] [f_{n}] approaches unity, as is the case for scattering by a free electron; when [\nu\ll\nu_{n}], [f_{n}] approaches zero, demonstrating the lack of scattering from a fixed electron. Only for [\nu \cong \nu_{n}] does the imaginary part have an appreciable value.

Fortunately, quantum mechanics arrives at the same result by adding a rational meaning to the damping factors and interpreting [\nu_{n}] as absorption frequencies of the atom (Hönl, 1933[link]). For heavy atoms, the most important transitions are to a continuum of energy states, with [\nu_{n} \geq \nu_{K}] or [\nu_{n} \geq \nu_{L}] etc., where [\nu_{K}] and [\nu_{L}] are the frequencies of the K and L absorption edges.

In practice, the complex atomic scattering factor, [f_{\rm anomalous}], is separated into three parts: [f_{\rm anomalous} = f + f' + if'']. f is the contribution to the scattering if the electrons are free electrons and it is a real number (Section[link]. f′ is the real part of the correction to be applied and f″ is the imaginary correction; f″ is always [\pi/2] in phase ahead of f (Fig.[link] [f + f'] is the total real part of the atomic scattering factor.


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The atomic scattering factor as a vector in the Argand diagram. (a) When the electrons in the atom can be regarded as free. (b) When they are not completely free and the scattering becomes anomalous with a real anomalous contribution [f'] and an imaginary contribution [if'']. Reproduced with permission from Drenth (1999[link]). Copyright (1999) Springer-Verlag.

The imaginary correction [if''] is connected with absorption by oscillators having [\nu_{n} \cong \nu]. It can be calculated from the atomic absorption coefficient of the anomalously scattering element. For each of the K, L etc. absorption edges, [f''] is virtually zero for frequencies below the edge, but it rises steeply at the edge and decreases gradually at higher frequencies.

The real correction [f'] can be derived from [f''] by means of the Kramers–Kronig transform [IT C (2004)[link], Section[link] ]. For frequencies close to an absorption edge, [f'] becomes strongly negative.

Values for f, [f'] and [f''] are always given in units equal to the scattering by one free electron. f values are tabulated in IT C (2004)[link] as a function of [\sin \theta/\lambda], and the anomalous-scattering corrections for forward scattering as a function of the wavelength. Because the anomalous contribution to the atomic scattering factor is mainly due to the electrons close to the nucleus, the value of the corrections diminishes much more slowly than f as a function of the scattering angle.


International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.
Hönl, H. (1933). Atomfaktor für Röntgenstrahlen als Problem der Dispersionstheorie (K-Schale). Ann. Phys. 18, 625–655.
James, R. W. (1965). The Optical Principles of the Diffraction of X-rays, p. 135. London: G. Bell and Sons Ltd.

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