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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 101   | 1 | 2 |

Section 1.3.4.5.1.3. The transform of an axially periodic fibre

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

1.3.4.5.1.3. The transform of an axially periodic fibre

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Let ρ be the electron-density distribution in a fibre, which is assumed to have translational periodicity with period 1 along z, and to have compact support with respect to the (x, y) coordinates. Thus ρ may be written[\rho = \left[\delta_{x} \otimes \delta_{y} \otimes \left({\textstyle\sum\limits_{k \in {\bb Z}}} \delta_{(k)}\right)_{z}\right] * \rho^{0},]where [\rho^{0} = \rho^{0} (x, y, z)] is the motif.

By the tensor product property, the inverse Fourier transform [F = \bar{{\scr F}}_{xyz} [\rho]] may be written[F = \left[1_{\xi} \otimes 1_{\eta} \otimes \left({\textstyle\sum\limits_{l \in {\bb Z}}} \delta_{(l)}\right)_{\zeta}\right] \times \bar{{\scr F}}[\rho^{0}]]and hence consists of `layers' labelled by l:[F = {\textstyle\sum\limits_{l \in {\bb Z}}} F(\xi, \eta, l) (\delta_{(l)})_{\zeta}]with[F(\xi, \eta, l) = {\textstyle\int\limits_{0}^{1}} \bar{{\scr F}}_{xy} [\rho^{0}] (\xi, \eta, z) \exp (2 \pi i l z) \hbox{ d}z.]

Changing to polar coordinates in the (x, y) and [(\xi, \eta)] planes decomposes the calculation of F from ρ into the following steps:[\eqalign{g_{nl} (r) &= {1 \over 2\pi} {\textstyle\int\limits_{0}^{2\pi}} {\textstyle\int\limits_{0}^{1}} \rho (r, \varphi, z) \exp [i (-n \varphi + 2 \pi l z)] \hbox{ d}\varphi \hbox{ d}z \cr G_{nl} (R) &= {\textstyle\int\limits_{0}^{\infty}} g_{nl} (r) \,J_{n} (2 \pi Rr) 2 \pi r \hbox{ d}r\cr F (R, \psi, l) &= {\textstyle\sum\limits_{n \in {\bb Z}}} i^{n} G_{nl} (R) \exp (in\psi)}]and the calculation of ρ from F into:[\eqalign{G_{nl} (R) &= {1 \over 2\pi i^{n}} {\textstyle\int\limits_{0}^{2\pi}} F(R, \psi, l) \exp (-in \psi) \hbox{ d}\psi\cr g_{nl} (r) &= {\textstyle\int\limits_{0}^{\infty}} G_{nl} (R) \,J_{n} (2 \pi rR) 2 \pi R \hbox{ d}R\cr \rho (r, \varphi, z) &= {\textstyle\sum\limits_{n \in {\bb Z}}}\, {\textstyle\sum\limits_{l \in {\bb Z}}} g_{nl} (r) \exp [i (n\varphi - 2 \pi l z)].}]

These formulae are seen to involve a 2D Fourier series with respect to the two periodic coordinates ϕ and z, and Hankel transforms along the radial coordinates. The two periodicities in ϕ and z are independent, so that all combinations of indices (n, l) occur in the Fourier summations.








































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