|
International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 274
Section A13.2.1.6. Expansion of the interference function
aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France |
![[\eqalignno{{\cal \chi}_{b}({\bf h - hR}^{-1}) &= (3/4\pi b^{3}) {\textstyle\int\limits_{0}^{b}} {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2 \pi}} \exp[2\pi i({\bf h - kR}^{-1}){\bf r}] r^{2} \sin (\theta)\ \hbox{d}r\ \hbox{d}\theta\ \hbox{d}\varphi\cr &= {\textstyle\sum\limits_{\ell, \, \ell' = 0}^{\infty}}\ {\textstyle\sum\limits_{m = -\ell}^{\ell}}\ {\textstyle\sum\limits_{m'= -\ell'}^{\ell'} i^{\ell - \ell'}} \overline{Y_{\ell, \, m}({\hat {\bf h}})} Y_{\ell', \, m'}({\hat {\bf k}})\cr &\quad \times (12\pi/b^{3}) {\textstyle\int\limits_{0}^{b}} j_{\ell} (2\pi hr)\ j_{\ell'} (2\pi kr)r^{2}\ \hbox{d}r\cr &\quad \times {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2\pi}} Y_{\ell, \, m} (\hat{{\bf r}}) \overline{Y_{\ell', \, m'} ({\bf R}^{-1} \hat{{\bf r}})} \sin (\theta)\ \hbox{d}\theta\ \hbox{d}\varphi\cr &= {\textstyle\sum\limits_{\ell = 0}^{\infty}}\ {\textstyle\sum\limits_{m=-\ell}^{\ell}}\ {\textstyle\sum\limits_{\ell'=0}^{\infty}}\ {\textstyle\sum\limits_{m'=-\ell'}^{\ell'} i^{\ell - \ell'}} \overline{Y_{\ell, \, m} (\hat{{\bf h}})} Y_{\ell', \, m'} (\hat{{\bf k}})\cr &\quad \times (12\pi/b^{3}) {\textstyle\int\limits_{0}^{b}} j_{\ell} (2\pi hr)\ j_{\ell'} (2\pi kr) r^{2}\ \hbox{d}r\cr &\quad \times {\textstyle\sum\limits_{m''=-\ell'}^{\ell'}} {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2\pi}} Y_{\ell, \, m} (\hat{{\bf r}}) \overline{Y_{\ell', \, m''} (\hat{{\bf r}})} \sin (\theta)\ \hbox{d}\theta\ \hbox{d}\varphi \ D_{m'', \, m'}^{\ell'} ({\bf R})\cr &= {\textstyle\sum\limits_{\ell = 0}^{\infty}}\ {\textstyle\sum\limits_{m'=-\ell}^{\ell}} \overline{Y_{\ell, \, m} (\hat{{\bf h}})} Y_{\ell, \, m'} (\hat{{\bf k}}) 12\pi U^{\ell} (2\pi hb, 2\pi kb) D_{m, \, m'}^{\ell} ({\bf R}).\cr &&(\rm A13.2.1.16)}]](/teximages/fach13o2/fach13o2fd61.svg)
