Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.2, p. 17   | 1 | 2 |

Section 1.2.9. The atomic temperature factor

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA
Correspondence e-mail:

1.2.9. The atomic temperature factor

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Since the crystal is subject to vibrational oscillations, the observed elastic scattering intensity is an average over all normal modes of the crystal. Within the Born–Oppenheimer approximation, the theoretical electron density should be calculated for each set of nuclear coordinates. An average can be obtained by taking into account the statistical weight of each nuclear configuration, which may be expressed by the probability distribution function [P({\bf u}_{1}, \ldots , {\bf u}_{N})] for a set of displacement coordinates [{\bf u}_{1}, \ldots , {\bf u}_{N}].

In general, if [\rho ({\bf r},{\bf u}_{1}, \ldots , {\bf u}_{N})] is the electron density corresponding to the geometry defined by [{\bf u}_{1}, \ldots , {\bf u}_{N}], the time-averaged electron density is given by [\langle \rho ({\bf r})\rangle = {\textstyle\int} \rho ({\bf r}, {\bf u}_{1}, \ldots , {\bf u}_{N}) P ({\bf u}_{1}, \ldots , {\bf u}_{N})\,{\rm d} {\bf u}_{1} \ldots \hbox{d} {\bf u}_{N}. \eqno(]

When the crystal can be considered as consisting of perfectly following rigid entities, which may be molecules or atoms, expression ([link] simplifies: [{\langle \rho_{\rm rigid\ group} ({\bf r})\rangle = {\textstyle\int} \rho_{\rm r.g.,\ static} ({\bf r} - {\bf u}) P ({\bf u})\ \hbox{d} {\bf u} = \rho_{\rm r.g.,\ static} * P ({\bf u}).} \eqno(]

In the approximation that the atomic electrons perfectly follow the nuclear motion, one obtains [\langle \rho_{\rm atom} ({\bf r})\rangle = \rho_{\rm atom, \, static} ({\bf r}) * P ({\bf u}). \eqno(]The Fourier transform of this convolution is the product of the Fourier transforms of the individual functions: [\langle f ({\bf H})\rangle = f({\bf H})T({\bf H}). \eqno(]Thus [T({\bf H})], the atomic temperature factor, is the Fourier transform of the probability distribution [P({\bf u})].

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