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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 63   | 1 | 2 |

Section 1.3.4.2.1.2. Structure factors in terms of form factors

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.1.2. Structure factors in terms of form factors

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In many cases, [\rho\llap{$-\!$}^{0}] is a sum of translates of atomic electron-density distributions. Assume there are n distinct chemical types of atoms, with [N_{j}] identical isotropic atoms of type j described by an electron distribution [\rho\llap{$-\!$}_{j}] about their centre of mass. According to quantum mechanics each [\rho\llap{$-\!$}_{j}] is a smooth rapidly decreasing function of x, i.e. [\rho\llap{$-\!$}_{j} \in {\scr S}], hence [\rho\llap{$-\!$}^{0} \in {\scr S}] and (ignoring the effect of thermal agitation)[\rho\llap{$-\!$}^{0}({\bf x}) = {\textstyle\sum\limits_{j=1}^{n}} \left[{\textstyle\sum\limits_{k_{j}=1}^{N_{j}}} \rho\llap{$-\!$}_{j} ({\bf x} - {\bf x}_{k_{j}})\right],]which may be written (Section 1.3.2.5.8[link])[\rho\llap{$-\!$}^{0} = {\textstyle\sum\limits_{j=1}^{n}} \left[\rho\llap{$-\!$}_{j} * \left({\textstyle\sum\limits_{k_{j}=1}^{N_{j}}} \delta_{({\bf x}_{k_{j}})}\right)\right].]By Fourier transformation:[F({\bf h}) = {\textstyle\sum\limits_{j=1}^{n}} \left\{\bar{\scr F}[\rho\llap{$-\!$}_{j}] ({\bf h}) \times \left[{\textstyle\sum\limits_{k_{j}=1}^{N_{j}}} \exp (2\pi i{\bf h} \cdot {\bf x}_{k_{j}})\right]\right\}.]Defining the form factor [f_{j}] of atom j as a function of h to be[f_{j}({\bf h}) = \bar{\scr F}[\rho\llap{$-\!$}_{j}] ({\bf h})]we have[F({\bf h}) = {\textstyle\sum\limits_{j=1}^{n}}\, f_{j}({\bf h}) \times \left[{\textstyle\sum\limits_{k_{j}=1}^{N_{j}}} \exp (2\pi i{\bf h} \cdot {\bf x}_{k_{j}})\right].]If [{\bf X} = {\bf Ax}] and [{\bf H} = ({\bf A}^{-1})^{T} {\bf h}] are the real- and reciprocal-space coordinates in Å and Å−1, and if [\rho_{j}(\|{\bf X}\|)] is the spherically symmetric electron-density function for atom type j, then[f_{j}({\bf H}) = \int\limits_{0}^{\infty} 4\pi \|{\bf X}\|^{2} \rho_{j} (\|{\bf X}\|) {\sin (2\pi \|{\bf H}\| \|{\bf X}\|) \over 2\pi \|{\bf H}\| \|{\bf X}\|} \,\hbox{d}\|{\bf X}\|.]

More complex expansions are used for electron-density studies (see Chapter 1.2[link] in this volume). Anisotropic Gaussian atoms may be dealt with through the formulae given in Section 1.3.2.4.4.2[link].








































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