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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 208   | 1 | 2 |

Section 2.1.8.6. Other non-ideal Fourier p.d.f.'s

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

2.1.8.6. Other non-ideal Fourier p.d.f.'s

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As pointed out above, the representation of the p.d.f.'s by Fourier series is also applicable to effects of noncrystallographic symmetry. Thus, Shmueli et al. (1985[link]) obtained the following Fourier coefficient for the bicentric distribution in the space group [P\bar{1}] [C_{k} = (2/\pi)\textstyle\int\limits_{0}^{\pi/2}\left[\textstyle\prod\limits_{j = 1}^{N/4}J_{0}(4\pi k\alpha n_{j}\cos\vartheta)\right]\;{\rm d}\vartheta \eqno(2.1.8.41)] to be used with equation (2.1.8.5)[link]. Furthermore, if we use the important property of the characteristic function as outlined in Section 2.1.4.1[link], it is easy to write down the Fourier coefficient for a [P\bar{1}] asymmetric unit containing a centrosymmetric fragment centred at a noncrystallographic centre and a number of atoms not related by symmetry. This Fourier for the above partially bicentric arrangement is a product of expressions (2.1.8.17)[link] and (2.1.8.41)[link], with the appropriate number of atoms in each factor (Shmueli & Weiss, 1985a[link]). While the purely bicentric p.d.f. obtained by using (2.1.8.41)[link] with (2.1.8.5)[link] is significantly different from the ideal bicentric p.d.f. given by equation (2.1.5.13)[link] only when the atomic composition is sufficiently heterogeneous, the above partially bicentric p.d.f. appears to be a useful development even for an equal-atom structure.

The problem of the coexistence of several noncrystallographic centres of symmetry within the asymmetric unit of P [\bar{1}], and its effect on the p.d.f. of [|E|], was examined by Shmueli, Weiss & Wilson (1989[link]) by the Fourier method. The latter study indicates that the strongest effect is produced by the presence of a single noncrystallographic centre.

Another kind of noncrystallographic symmetry is that arising from the presence of centrosymmetric fragments in a noncentrosymmetric structure – the subcentric arrangement already discussed in Section 2.1.5.4[link]. A Fourier-series representation of a non-ideal p.d.f. corresponding to this case was developed by Shmueli, Rabinovich & Weiss (1989[link]), and was also applied to the mathematically equivalent effects of dispersion and presence of heavy scatterers in centrosymmetric special positions in a noncentrosymmetric space group.

A variety of other non-ideal p.d.f.'s occur when heavy atoms are present in special positions (Shmueli & Weiss, 1988[link]). Without going into the details of this development, it can be noted that if the atoms are distributed among k types of Wyckoff positions, the characteristic function corresponding to the p.d.f. of [|E|] is a product of the k characteristic functions, each of which is related to one of these special positions; the same property of the characteristic function as that in Section 2.1.4.1[link] is here utilized.

References

First citationShmueli, U., Rabinovich, S. & Weiss, G. H. (1989). Exact conditional distribution of a three-phase invariant in the space group P1. I. Derivation and simplification of the Fourier series. Acta Cryst. A45, 361–367.Google Scholar
First citationShmueli, U. & Weiss, G. H. (1985a). Centric, bicentric and partially bicentric intensity statistics. Structure and statistics in crystallography, edited by A. J. C. Wilson, pp. 53–66. Guilderland: Adenine Press.Google Scholar
First citationShmueli, U. & Weiss, G. H. (1988). Exact random-walk models in crystallographic statistics. IV. P.d.f.'s of [|E|] allowing for atoms in special positions. Acta Cryst. A44, 413–417.Google Scholar
First citationShmueli, U., Weiss, G. H. & Kiefer, J. E. (1985). Exact random-walk models in crystallographic statistics. II. The bicentric distribution in the space group [P\bar{1}]. Acta Cryst. A41, 55–59.Google Scholar
First citationShmueli, U., Weiss, G. H. & Wilson, A. J. C. (1989). Explicit Fourier representations of non-ideal hypercentric p.d.f.'s of [|E|]. Acta Cryst. A45, 213–217.Google Scholar








































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