Definition of normalized structure factor

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 215-216 | 1 | 2 |

Section 2.2.4.1. Definition of normalized structure factor

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.4.1. Definition of normalized structure factor

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The normalized structure factors E (see also Chapter 2.1 ) are calculated according to (Hauptman & Karle, 1953) $[|E_{\bf h}|^{2} = |F_{\bf h}|^{2}/\langle |F_{\bf h}|^{2}\rangle, \eqno(2.2.4.1)]$ where $[|F_{\bf h}|^{2}]$ is the squared observed structure-factor magnitude on the absolute scale and $[\langle |F_{\bf h}|^{2}\rangle]$ is the expected value of $[|F_{\bf h}|^{2}]$ .

$[\langle |F_{\bf h}|^{2}\rangle]$ depends on the available a priori information. Often, but not always, this may be considered as a combination of several typical situations. We mention:

(a) No structural information. The atomic positions are considered random variables. Then $[\langle |F_{\bf h}|^{2}\rangle = \varepsilon_{\bf h} \textstyle\sum\limits_{j = 1}^{N} f_{j}^{2} = \varepsilon_{\bf h} \textstyle\sum\nolimits_{N}]$ so that $[E_{\bf h} = {F_{\bf h} \over (\varepsilon_{\bf h} \sum\nolimits_{N})^{1/2}}. \eqno(2.2.4.2)]$ $[\varepsilon_{\bf h}]$ takes account of the effect of space-group symmetry (see Chapter 2.1 ).
(b) P atomic groups having a known configuration but with unknown orientation and position (Main, 1976). Then a certain number of interatomic distances $[r_{j_{1}j_{2}}]$ are known and $[\langle |F_{\bf h}|^{2}\rangle = \varepsilon_{\bf h} \left(\sum\nolimits_{N} + \sum\limits_{i = 1}^{P} \sum\limits_{j_{1} \neq j_{2} = 1}^{M_{i}} f_{j_{1}} f_{j_{2}} {\sin 2 \pi qr_{j_{1}j_{2}} \over 2 \pi qr_{j_{1}j_{2}}}\right),]$ where $[M_{i}]$ is the number of atoms in the ith molecular fragment and $[q = |{\bf h}|]$ .
(c) P atomic groups with a known configuration, correctly oriented, but with unknown position (Main, 1976). Then a certain group of interatomic vectors $[{\bf r}_{j_{1} j_{2}}]$ is fixed and $[\langle |F_{{\bf h}}|^{2} \rangle = \varepsilon_{{\bf h}} \left(\textstyle\sum\nolimits_{N} + \textstyle\sum\limits_{i=1}^{P} \textstyle\sum\limits_{j_{1} \neq j_{2} = 1}^{M_{i}} f_{j_{1}} f_{j_{2}} \exp 2\pi i{\bf h} \cdot {\bf r}_{j_{1} j_{2}}\right).]$ The above formula has been derived on the assumption that primitive positional random variables are uniformly distributed over the unit cell. Such an assumption may be considered unfavourable (Giacovazzo, 1988) in space groups for which the allowed shifts of origin, consistent with the chosen algebraic form for the symmetry operators $[{\bf C}_{s}]$ , are arbitrary displacements along any polar axes. Thanks to the indeterminacy in the choice of origin, the first of the shifts $[\boldtau _{i}]$ (to be applied to the ith fragment in order to translate atoms in the correct positions) may be restricted to a region which is smaller than the unit cell (e.g. in P2 we are free to specify the origin along the diad axis by restricting $[\boldtau _{1}]$ to the family of vectors $[\{\boldtau _{1}\}]$ of type ). The practical consequence is that $[\langle |F_{{\bf h}}|^{2} \rangle]$ is significantly modified in polar space groups if h satisfies $[{\bf h} \cdot \boldtau _{1} = 0,]$ where $[\boldtau _{1}]$ belongs to the family of restricted vectors $[\{\boldtau _{1}\}]$ .
(d) Atomic groups correctly positioned. Then (Main, 1976; Giacovazzo, 1983a) $[\langle |F_{\bf h}|^{2} \rangle = |F_{p, \, {\bf h}}|^{2} + \varepsilon_{\bf h} \textstyle\sum\nolimits_{q},]$ where $[F_{p, {\bf h}}]$ is the structure factor of the partial known structure and q are the atoms with unknown positions.
(e) A pseudotranslational symmetry is present. Let $[{\bf u}_{1}, {\bf u}_{2}, {\bf u}_{3}, \ldots]$ be the pseudotranslation vectors of order $[n_{1}, n_{2}, n_{3}, \ldots]$ , respectively. Furthermore, let p be the number of atoms (symmetry equivalents included) whose positions are related by pseudotranslational symmetry and q the number of atoms (symmetry equivalents included) whose positions are not related by any pseudotranslation. Then (Cascarano et al., 1985a,b) $[\langle |F_{\bf h}|^{2} \rangle = \varepsilon_{\bf h} (\zeta_{\bf h} \textstyle\sum\nolimits_{p} + \textstyle\sum\nolimits_{q}),]$ where $[\zeta_{\bf h} = {(n_{1} n_{2} n_{3} \ldots) \gamma_{\bf h}\over m}]$ and $[\gamma_{\bf h}]$ is the number of times for which algebraic congruences $[{\bf h} \cdot {\bf R}_{s} {\bf u}_{i} \equiv 0\ (\hbox{mod 1})\quad \hbox{for } i = 1, 2, 3, \ldots]$ are simultaneously satisfied when s varies from 1 to m. If $[\gamma_{\bf h} = 0]$ then $[F_{\bf h}]$ is said to be a superstructure reflection, otherwise it is a substructure reflection.

Often substructures are not ideal: e.g. atoms related by pseudotranslational symmetry are ideally located but of different type (replacive deviations from ideality); or they are equal but not ideally located (displacive deviations); or a combination of the two situations occurs. In these cases a correlation exists between the substructure and the superstructure. It has been shown (Mackay, 1953; Cascarano et al., 1988 a) that the scattering power of the substructural part may be estimated via a statistical analysis of diffraction data for ideal pseudotranslational symmetry or for displacive deviations from it, while it is not estimable in the case of replacive deviations.

References

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 215-216