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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 205   | 1 | 2 |

Section 2.1.8.3. Simple examples

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

2.1.8.3. Simple examples

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Consider the Fourier coefficient of the p.d.f. of [|E|] for the centrosymmetric space group [P\bar{1}]. The normalized structure factor is given by [E = 2\textstyle\sum\limits_{j = 1}^{N/2}n_{j}\cos\vartheta_{j}, \quad{\rm with}\quad \vartheta_{j} = 2\pi {\bf h}^{T}\cdot{\bf r}_{j}, \eqno(2.1.8.16)] and the Fourier coefficient is [\eqalignno{C_{k} & = \langle \exp(\pi ik\alpha E) \rangle &(2.1.8.17) \cr& = \left\langle \exp\left[2\pi ik\alpha\textstyle\sum\limits_{j = 1}^{N/2}n_{j}\cos\vartheta_{j}\right] \right\rangle &(2.1.8.18) \cr & = \left\langle \textstyle\prod\limits_{j = 1}^{N/2} \exp(2\pi ik\alpha n_{j}\cos\vartheta_{j}) \right\rangle &(2.1.8.19) \cr & = \textstyle \prod\limits_{j = 1}^{N/2} \langle \exp(2\pi ik\alpha n_{j}\cos\vartheta_{j}) \rangle &(2.1.8.20) \cr & = \textstyle\prod\limits_{j = 1}^{N/2}\left\{{({1}/{2\pi})}\int\limits_{-\pi}^{\pi}\exp(2\pi ik\alpha n_{j}\cos\vartheta)\;{\rm d}\vartheta \right\}\cr &&(2.1.8.21) \cr & = \textstyle\prod\limits_{j = 1}^{N/2}J_{0}(2\pi k\alpha n_{j}). &(2.1.8.22)}%2.1.8.22] Equation (2.1.8.20)[link] is obtained from equation (2.1.8.19)[link] if we make use of the assumption of independence, the assumption of uniformity allows us to rewrite equation (2.1.8.20)[link] as (2.1.8.21)[link], and the expression in the braces in the latter equation is just a definition of the Bessel function [J_{0}(2\pi k\alpha n_{j})] (e.g. Abramowitz & Stegun, 1972[link]).

Let us now consider the Fourier coefficient of the p.d.f. of [|E|] for the noncentrosymmetric space group [P1]. We have [A = \textstyle\sum\limits_{j = 1}^{N}n_{j}\cos\vartheta_{j} \quad {\rm and}\quad B = \textstyle\sum\limits_{j = 1}^{N}n_{j}\sin\vartheta_{j}. \eqno(2.1.8.23)] These expressions for A and B are substituted in equation (2.1.8.10)[link], resulting in [\eqalignno{C_{mn} & = \left\langle \textstyle\prod\limits_{j = 1}^{N} \exp[\pi i\alpha n_{j}(m\cos\vartheta_{j} + n\sin\vartheta_{j})] \right\rangle & \cr & & (2.1.8.24) \cr & = \left\langle \textstyle\prod\limits_{j = 1}^{N} \exp[\pi i\alpha n_{j}\sqrt{m^{2}+n^{2}} \sin(\vartheta_{j}+\Delta)] \right\rangle & \cr & & (2.1.8.25)\cr &= \textstyle \prod\limits_{j = 1}^{N} J_{0}(\pi\alpha n_{j}\sqrt{m^{2}+n^{2}}). &(2.1.8.26)}] Equation (2.1.8.24)[link] leads to (2.1.8.25)[link] by introducing polar coordinates analogous to those leading to equation (2.1.8.8)[link], and equation (2.1.8.26)[link] is then obtained by making use of the assumptions of independence and uniformity in an analogous manner to that detailed in equations (2.1.8.12)[link]–(2.1.8.22)[link] above.

The right-hand side of equation (2.1.8.26)[link] is to be used as a Fourier coefficient of the double Fourier series given by (2.1.8.9)[link]. Since, however, this coefficient depends on [(m^{2}+n^{2})^{1/2}] alone rather than on m and n separately, the p.d.f. of [|E|] for P1 can also be represented by a Fourier–Bessel series [cf. equation (2.1.8.11)[link]] with coefficient [D_{u} = {{1}\over{J_{1}^{2}(\lambda_{u})}}\prod_{j = 1}^{N}J_{0}(\alpha n_{j}\lambda_ {u}), \eqno(2.1.8.27)] where [\lambda_{u}] is the uth root of the equation [J_{0}(x) = 0].

References

First citationAbramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar








































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