Molecular-envelope transforms via Green's theorem

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 86 | 1 | 2 |

Section 1.3.4.4.3.5. Molecular-envelope transforms via Green's theorem

G. Bricogne^a

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

1.3.4.4.3.5. Molecular-envelope transforms via Green's theorem

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Green's theorem stated in terms of distributions (Section 1.3.2.3.9.1) is particularly well suited to the calculation of the Fourier transforms $[\bar{{\scr F}}[\chi_{U}]]$ of indicator functions. Let f be the indicator function $[\chi_{U}]$ and let S be the boundary of U (assumed to be a smooth surface). The jump $[\sigma_{0}]$ in the value of f across S along the outer normal vector is $[\sigma_{0} = -1]$ , the jump $[\sigma_{\nu}]$ in the normal derivative of f across S is $[\sigma_{\nu} = 0]$ , and the Laplacian of f as a function is (almost everywhere) 0 so that $[T_{\Delta f} = 0]$ . Green's theorem then reads: $[\eqalign{ \Delta (T_{f}) &= T_{\Delta f} + \sigma_{\nu}\delta_{(S)} + \partial_{\nu} [\sigma_{0} \delta_{(S)}]\cr &= -\partial_{\nu} [\delta_{(S)}].}]$

The function $[e_{{\bf H}}({\bf X}) = \exp (2\pi i{\bf H} \cdot {\bf X})]$ satisfies the identity $[\Delta e_{{\bf H}} = - 4\pi^{2} \|{\bf H}\|^{2} e_{{\bf H}}]$ . Therefore, in Cartesian coordinates: $[\eqalign{\bar{F} [\chi_{U}] ({\bf H}) &= \langle T_{\chi_{U}}, e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle T_{\chi_{U}}, \Delta e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle \Delta (T_{\chi_{U}}), e_{{\bf H}}\rangle \quad \quad [\hbox{Section } 1.3.2.3.9.1(a)]\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle - \partial_{\nu} [\delta_{(S)}], e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \int\limits_{S} \partial_{\nu} e_{{\bf H}} \;{\rm d}^{2} S \quad \quad [\hbox{Section } 1.3.2.3.9.1(c)]\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \int\limits_{S} 2\pi i{\bf H} \cdot {\bf n} \exp (2\pi i{\bf H} \cdot {\bf X}) \hbox{ d}^{2} S,}]$ i.e. $[\bar{{\scr F}}[\chi_{U}] ({\bf H}) = {1 \over 2\pi i\|{\bf H}\|^{2}} \int\limits_{S} {\bf H} \cdot {\bf n} \exp (2\pi i{\bf H} \cdot {\bf X}) \hbox{ d}^{2} S,]$ where n is the outer normal to S. This formula was used by von Laue (1936) for a different purpose, namely to calculate the transforms of crystal shapes (see also Ewald, 1940). If the surface S is given by a triangulation, the surface integral becomes a sum over all faces, since n is constant on each face. If U is a solid sphere with radius R, an integration by parts gives immediately: $[\eqalign{{1 \over \hbox{vol} (U)} \bar{{\scr F}}[\chi_{U}] ({\bf H}) &= {3 \over X^{3}} [\sin X - X \cos X]\cr &\phantom{[X - X \cos X]} \hbox{with } X = 2\pi \|{\bf H}\| R.}]$

References

Ewald, P. P. (1940). X-ray diffraction by finite and imperfect crystal lattices. Proc. Phys. Soc. London, 52, 167–174.Google Scholar

Laue, M. von (1936). Die aüßere Form der Kristalle in ihrem Einfluß auf die Interferenzerscheinungen an Raumgittern. Ann. Phys. (Leipzig), 26, 55–68.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 86