Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 73-74   | 1 | 2 |

Section Structure-factor calculation

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 Structure-factor calculation

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Structure factors may be calculated from a list of symmetry-unique atoms by Fourier transformation of the orbit decomposition formula for the motif [\rho\llap{$-\!$}^{0}] given in Section[link]:[\eqalignno{F({\bf h}) &= \bar{{\scr F}}[\rho\llap{$-\!$}^{0}] ({\bf h})&\cr &= \bar{{\scr F}}\left[{\textstyle\sum\limits_{j \in J}} \left({\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} S_{\gamma_{j}}^{\#} (\tau_{{\bf x}_{j}} \rho\llap{$-\!$}_{j})\right)\right] ({\bf h})&\cr &= {\textstyle\sum\limits_{j \in J}} \,{\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \bar{{\scr F}}[\tau_{{\bf t}_{\gamma_{j}}} {\bf R}_{\gamma_{j}}^{\#} \tau_{{\bf x}_{j}} \rho\llap{$-\!$}_{j}] ({\bf h})&\cr &= {\textstyle\sum\limits_{j \in J}} \,{\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp (2\pi i {\bf h} \cdot {\bf t}_{\gamma_{j}})&\cr &\quad \times [({\bf R}_{\gamma_{j}}^{-1})^{T_{\#}} [\exp (2\pi i{\boldxi} \cdot {\bf x}_{j}) \bar{{\scr F}}[\rho\llap{$-\!$}_{j}]_{{\boldxi}}]] ({\bf h}) &\cr &= {\textstyle\sum\limits_{j \in J}}\, {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp (2\pi i{\bf h} \cdot {\bf t}_{\gamma_{j}})&\cr &\quad \times \exp [2\pi i({\bf R}_{\gamma_{j}}^{T} {\bf h}) \cdot {\bf x}_{j}] \bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h})\hbox{\semi}&}]i.e. finally:[F({\bf h}) = {\textstyle\sum\limits_{j \in J}}\, {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\} \bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h}).]

In the case of Gaussian atoms, the atomic transforms are[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp [- {\textstyle{1 \over 2}} {\bf h}^{T} (4\pi^{2} {\bf Q}_{j}) {\bf h}]]or equivalently[\bar{{\scr F}}[\rho_{j}] ({\bf H}) = Z_{j} \exp [-{\textstyle{1 \over 2}} {\bf H}^{T} (4\pi^{2} {\bf U}_{j}) {\bf H}].]

Two common forms of equivalent temperature factors (incorporating both atomic form and thermal motion) are

  • (i) isotropic B:[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp (-{\textstyle{1 \over 4}} B_{j} {\bf H}^{T} {\bf H}),]so that [{\bf U}_{j} = (B_{j}/8\pi^{2}) {\bf I}], or [{\bf Q}_{j} = (B_{j}/8\pi^{2}) {\bf A}^{-1} ({\bf A}^{-1})^{T}];

  • (ii) anisotropic β's:[\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf h}) = Z_{j} \exp (-{\bf h}^{T} {\boldbeta}_{j} {\bf h}),]so that [{\boldbeta}_{j} = 2\pi^{2} {\bf Q}_{j} = 2\pi^{2} {\bf A}^{-1} {\bf U}_{j} ({\bf A}^{-1})^{T}], or [{\bf U}_{j} = (1/2\pi^{2}) {\bf A}\beta_{j} {\bf A}^{T}].

In the first case, [\bar{{\scr F}}[\rho\llap{$-\!$}_{j}] ({\bf R}_{\gamma_{j}}^{T} {\bf h})] does not depend on [\gamma_{j}], and therefore:[\eqalign{ F({\bf h}) &= {\textstyle\sum\limits_{j \in J}} \,Z_{j} \exp \{-{\textstyle{1 \over 4}} B_{j} {\bf h}^{T} [{\bf A}^{-1} ({\bf A}^{-1})^{T}] {\bf h}\}\cr &\quad \times {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\}.}]In the second case, however, no such simplification can occur:[\eqalign{ F({\bf h}) &= {\textstyle\sum\limits_{j \in J}} \,Z_{j} {\textstyle\sum\limits_{\gamma_{j} \in G/G_{{\bf x}_{j}}}} \exp [-{\bf h}^{T} ({\bf R}_{\gamma_{j}} {\boldbeta}_{j} {\bf R}_{\gamma_{j}}^{T}) {\bf h}]\cr &\quad \times \exp \{2\pi i{\bf h} \cdot [S_{\gamma_{j}} ({\bf x}_{j})]\}.}]These formulae, or special cases of them, were derived by Rollett & Davies (1955)[link], Waser (1955b)[link], and Trueblood (1956)[link].

The computation of structure factors by applying the discrete Fourier transform to a set of electron-density values calculated on a grid will be examined in Section[link].


Rollett, J. S. & Davies, D. R. (1955). The calculation of structure factors for centrosymmetric monoclinic systems with anisotropic atomic vibration. Acta Cryst. 8, 125–128.
Trueblood, K. N. (1956). Symmetry transformations of general anisotropic temperature factors. Acta Cryst. 9, 359–361.
Waser, J. (1955b). The anisotropic temperature factor in triclinic coordinates. Acta Cryst. 8, 731.

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