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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 61   | 1 | 2 |

Section 1.3.4.2.1.5. Parseval's identity and other [L^{2}] theorems

G. Bricognea

a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.1.5. Parseval's identity and other [L^{2}] theorems

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By Section 1.3.2.4.3.3[link] and Section 1.3.2.6.10.2[link], [{\textstyle\sum\limits_{{\bf h}\in {\bb Z}^{3}}} |F({\bf h})|^{2} = {\textstyle\int\limits_{{\bb R}^{3} / {\bb Z}^{3}}} |\rho\llap{$-\!$} ({\bf x})|^{2} \;\hbox{d}^{3} {\bf x} = V {\textstyle\int\limits_{{\bb R}^{3} / {\Lambda}}} |\rho ({\bf X})|^{2} \;\hbox{d}^{3} {\bf X}.] Usually [\rho\llap{$-\!$} ({\bf x})] is real and positive, hence [|\rho\llap{$-\!$} ({\bf x})| = \rho\llap{$-\!$} ({\bf x})], but the identity remains valid even when [\rho\llap{$-\!$} ({\bf x})] is made complex-valued by the presence of anomalous scatterers.

If [\{G_{\bf h}\}] is the collection of structure factors belonging to another electron density [\sigma = A^{\#} \sigma\llap{$-$}] with the same period lattice as ρ, then [\eqalign{{\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} \overline{F({\bf h})}G({\bf h}) &= {\textstyle\int\limits_{{\bb R}^{3} / {\bb Z}^{3}}} \overline{\rho\llap{$-\!$} ({\bf x})} \sigma\llap{$-$} ({\bf x}) \;\hbox{d}^{3} {\bf x} \cr &= V {\textstyle\int\limits_{{\bb R}^{3} / {\Lambda}}} \rho ({\bf X}) \sigma ({\bf X}) \;\hbox{d}^{3} {\bf X}.}] Thus, norms and inner products may be evaluated either from structure factors or from `maps'.








































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