Scattering by one atom

Drenth, J.

doi:10.1107/97809553602060000658

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. 2.1, pp. 53-54 | 1 | 2 |

Section 2.1.4.3.1. Scattering by one atom

J. Drenth^a ^*

^a Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: j.drenth@chem.rug.nl

2.1.4.3.1. 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 2.1.4.4). This is normally true for light atoms but not for heavy ones (Table 2.1.4.1).

Table 2.1.4.1 | top | pdf |
The position of the Kα edge of different elements

Atomic number	Element	K α 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 $[\rho ({\bf r}) \hbox{d}v_{r} \{ \exp (2\pi i{\bf r}\cdot {\bf S}) + \exp [2\pi i(-{\bf r})\cdot {\bf S}]\} = 2\rho ({\bf r}) \cos (2\pi {\bf r}\cdot {\bf S})\hbox{d}v_{r}\hbox{;}]$ 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) as a function of $[\sin \theta /\lambda]$ . The f values decrease appreciably as a function of $[\sin \theta /\lambda]$ (Fig. 2.1.4.5). 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.

Figure 2.1.4.5 | top | pdf |

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). Copyright (1999) Springer-Verlag.

References

International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 53-54