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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 203-204   | 1 | 2 |

Section 2.1.8.1. General representations of p.d.f.'s of [|E|] by Fourier series

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

2.1.8.1. General representations of p.d.f.'s of [|E|] by Fourier series

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We assume, as before, that (i) the atomic phase factors [\vartheta_{j} = 2\pi {\bf h}^{T}{\bf r}_{j}] [cf. equation (2.1.1.2)[link]] are uniformly distributed on (0–2[\pi]) and (ii) the atomic contributions to the structure factor are independent. For a centrosymmetric space group, with the origin chosen at a centre of symmetry, the random variable is the (real) normalized structure factor E and its bounds are [-E_{M}] and [E_{M}], where [E_{M} = {\textstyle\sum\limits_{j = 1}^{N}}n_{j}, \hbox{ with } n_{j} = {{f_{j}}\over{\left({\textstyle\sum_{k = 1}^{N}}\;f_{k}^{2}\right)^{1/2}}}. \eqno(2.1.8.1)] Here, [E_{M}] is the maximum possible value of E and [f_{j}] is the conventional scattering factor of the jth atom, including its temperature factor. The p.d.f., [p(E)], can be non-zero in the range ([-E_{M},E_{M}]) only and can thus be expanded in the Fourier series [p(E) = (\alpha/2)\textstyle\sum\limits_{k = -\infty}^{\infty}C_{k}\exp(-\pi ik\alpha E), \eqno(2.1.8.2)] where [\alpha = 1/E_{M}]. Only the real part of [p(E)] is relevant. The Fourier coefficients can be obtained in the conventional manner by integrating over the range ([-E_{M},E_{M}]),[C_{k} = \textstyle\int\limits_{-E_{M}}^{E_{M}}p(E)\exp(\pi ik\alpha E)\;{\rm d}E. \eqno(2.1.8.3)] Since, however, [p(E) = 0] for [E \;\lt\; -E_{M}] and [E \;\gt\; E_{M}], it is possible and convenient to replace the limits of integration in equation (2.1.8.3)[link] by infinity. Thus [C_{k} = \textstyle\int\limits_{-\infty}^{\infty}p(E)\exp(\pi ik\alpha E)\;{\rm d}E = \langle \exp(\pi ik\alpha E) \rangle. \eqno(2.1.8.4)] Equation (2.1.8.4)[link] shows that [C_{k}] is a Fourier transform of the p.d.f. [p(E)] and, as such, it is the value of the corresponding characteristic function at the point [t_{k} = \pi\alpha k] [i.e., [C_{k} = C(\pi\alpha k)], where the characteristic function [C(t)] is defined by equation (2.1.4.1)[link]]. It is also seen that [C_{k}] is the expected value of the exponential [\exp(\pi ik\alpha E)]. It follows that the feasibility of the present approach depends on one's ability to evaluate the characteristic function in closed form without the knowledge of the p.d.f.; this is analogous to the problem of evaluating absolute moments of the structure factor for the correction-factor approach, discussed in Section 2.1.7[link]. Fortunately, in crystallographic applications these calculations are feasible, provided individual isotropic motion is assumed. The formal expression for the p.d.f. of [|E|], for any centrosymmetric space group, is therefore [p(|E|) = \alpha\left[1+2\textstyle\sum\limits_{k = 1}^{\infty}C_{k}\cos(\pi k\alpha |E|) \right], \eqno(2.1.8.5)] where use is made of the assumption that [p(E) = p(-E)], and the Fourier coefficients are evaluated from equation (2.1.8.4)[link].

The p.d.f. of [|E|] for a noncentrosymmetric space group is obtained by first deriving the joint p.d.f. of the real and imaginary parts of E and then integrating out its phase. The general expression for E is [E = A + iB = |E|\cos\varphi + i|E|\sin\varphi, \eqno(2.1.8.6)] where [\varphi] is the phase of E. The required joint p.d.f. is [p(A,B) = (\alpha^{2}/4)\textstyle\sum\limits_{m}\textstyle\sum\limits_{n}C_{mn}\exp[-\pi i\alpha(mA+nB)], \eqno(2.1.8.7)] and introducing polar coordinates [m = r\sin\Delta] and [n = r\cos\Delta], where [r = \sqrt{m^{2}+n^{2}}] and [\Delta = \tan^{-1}(m/n)], we have [\eqalignno{p(|E|,\varphi) &= (\alpha^{2}/4)|E|\textstyle\sum\limits_{m}\textstyle\sum\limits_{n}C_{mn}\exp[-\pi i \alpha |E| & \cr &\quad\times{}\sqrt{m^{2}+n^{2}}\sin(\varphi + \Delta)]. &(2.1.8.8) \cr}] Integrating out the phase [\varphi], we obtain [p(|E|) = (\pi\alpha^{2} |E|/2)\textstyle\sum\limits_{m}\textstyle\sum\limits_{n}C_{mn}J_{0}(\pi\alpha |E|\sqrt{m^{2}+n^{2}}), \eqno(2.1.8.9)] where [J_{0}(x)] is the Bessel function of the first kind (e.g. Abramowitz & Stegun, 1972[link]). This is a general form of the p.d.f. of [|E|] for a noncentrosymmetric space group. The Fourier coefficients are obtained, similarly to the above, as [C_{mn} = \langle \exp[\pi i\alpha(mA+nB)] \rangle \eqno(2.1.8.10)] and the average in equation (2.1.8.10)[link], just as that in equation (2.1.8.4)[link], is evaluated in terms of integrals over the appropriate trigonometric structure factors. In terms of the characteristic function for a joint p.d.f. of A and B, the Fourier coefficient in equation (2.1.8.10)[link] is given by [C_{mn} = C(\pi\alpha m,\pi\alpha n)].

We shall denote the characteristic function by [C(t_{1})] if it corresponds to a Fourier coefficient of a Fourier series for a centrosymmetric space group and by [C(t_{1},t_{2})] or by [C(t,\Delta)], where [t = (t_{1}^{2}+t_{2}^{2})^ {1/2}] and [\Delta = \tan^{-1}(t_{1}/t_{2})], if it corresponds to a Fourier series for a noncentrosymmetric space group.

References

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








































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