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. 54   | 1 | 2 |

Section Scattering by one atom

J. Drentha*

aLaboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: Scattering by one atom

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Electrons in an atom are bound by the nucleus and are – in principle – not free electrons.

However, to a good approximation, they can be regarded as such if the frequency of the incident radiation ν is greater than the natural absorption frequencies, [\nu_{n}], at the absorption edges of the scattering atom, or the wavelength of the incident radiation is shorter than the absorption-edge wavelength (Section[link]. This is normally true for light atoms but not for heavy ones (Table[link]

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The position of the Kα edge of different elements

Atomic numberElementKα edge (Å)
6 C 43.68
16 S 5.018
26 Fe 1.743
34 Se 0.980
78 Pt 0.158

If the electrons in an atom can be regarded as free electrons, the scattering amplitude of the atom is a real quantity, because the electron cloud has a centrosymmetric distribution, i.e. [\rho ({\bf r})] [ = \rho (-{\bf r})].

A small volume, [\hbox{d}v_{r}], at r contains [\rho ({\bf r}) \times \hbox{d}v_{r}] electrons, and at −r there are [\rho (-{\bf r}) \times \hbox{d}v_{r}] electrons. The combined scattering of the two volume elements, in units of the scattering of a free electron, is[\rho ({\bf r}) \hbox{d}v_{r} \{ \exp (2\pi i{\bf r}\cdot {\bf S}) + \exp [2\pi i(-{\bf r})\cdot {\bf S}]\} = 2\rho ({\bf r}) \cos (2\pi {\bf r}\cdot {\bf S})\hbox{d}v_{r}\semi]this is a real quantity.

The scattering amplitude of an atom is called the atomic scattering factor f. It expresses the scattering of an atom in terms of the scattering of a single electron. f values are calculated for spherically averaged electron-density distributions and, therefore, do not depend on the scattering direction. They are tabulated in IT C (2004)[link] as a function of [\sin \theta /\lambda]. The f values decrease appreciably as a function of [\sin \theta /\lambda] (Fig.[link] This is due to interference effects between the scattering from the electrons in the cloud. In the direction [\theta = 0], all electrons scatter in phase and the atomic scattering factor is equal to the number of electrons in the atom.


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The atomic scattering factor f for carbon as a function of [\sin \theta /\lambda], expressed in units of the scattering by one electron. Reproduced with permission from Drenth (1999[link]). Copyright (1999) Springer-Verlag.


International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.

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