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

International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 216-221   | 1 | 2 |

Section 2.2.4. Normalized structure factors

C. Giacovazzoa*

aDipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy
Correspondence e-mail: carmelo.giacovazzo@ic.cnr.it

2.2.4. Normalized structure factors

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2.2.4.1. Definition of normalized structure factor

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The normalized structure factors E (see also Chapter 2.1[link] ) are calculated according to (Hauptman & Karle, 1953[link]) [|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[link] ).

  • (b) P atomic groups having a known configuration but with unknown orientation and position (Main, 1976[link]). 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[link]). 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[link]) 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 [[x0z]]). 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[link]; Giacovazzo, 1983a[link]) [\langle |F_{\bf h}|^{2} \rangle = |F_{{\rm p}, \, {\bf h}}|^{2} + \varepsilon_{\bf h} \textstyle\sum\nolimits_{q},]where [F_{{\rm 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[link],b[link]) [\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[link]; Cascarano et al., 1988[link]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.

2.2.4.2. Definition of quasi-normalized structure factor

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When probability theory is not used, the quasi-normalized structure factors [ \hbox{\scr E}_{\bf h}] and the unitary structure factors [U_{\bf h}] are often used. [ \hbox{\scr E}_{\bf h}] and [U_{\bf h}] are defined according to [ \eqalign{|\hbox{\scr E}_{\bf h}|^{2} &= \varepsilon_{\bf h}|E_{\bf h}|^{2}\cr U_{\bf h} &= F_{\bf h} \Big/ \left(\textstyle\sum\limits_{j=1}^{N} f_{j}\right).}]Since [\sum_{j=1}^{N} f_{j}] is the largest possible value for [F_{\bf h}, U_{\bf h}] represents the fraction of [F_{\bf h}] with respect to its largest possible value. Therefore [0 \leq |U_{\bf h}| \leq 1.]If atoms are equal, then [ U_{\bf h} = \hbox{\scr E}_{\bf h} / N^{1/2}].

2.2.4.3. The calculation of normalized structure factors

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N.s.f.'s cannot be calculated by applying (2.2.4.1)[link] to observed s.f.'s because: (a) the observed magnitudes [I_{\bf h}] (already corrected for Lp factor, absorption, …) are on a relative scale; (b) [\langle |F_{\bf h}|^{2} \rangle] cannot be calculated without having estimated the vibrational motion of the atoms.

This is usually obtained by the well known Wilson plot (Wilson, 1942[link]), according to which observed data are divided into ranges of [s^{2} = \sin^{2} \theta / \lambda^{2}] and averages of intensity [\langle I_{\bf h} \rangle] are taken in each shell. Reflection multiplicities and other effects of space-group symmetry on intensities must be taken into account when such averages are calculated. The shells are symmetrically overlapped in order to reduce statistical fluctuations and are restricted so that the number of reflections in each shell is reasonably large. For each shell [K \langle I \rangle = \langle |F|^{2} \rangle = \langle |F^{\rm o}|^{2} \rangle \exp (- 2 Bs^{2}) \eqno(2.2.4.3)]should be obtained, where K is the scale factor needed to place X-ray intensities on the absolute scale, B is the overall thermal parameter and [\langle |F^{\rm o}|^{2} \rangle] is the expected value of [|F|^{2}] in which it is assumed that all the atoms are at rest. [\langle |F^{\rm o}|^{2} \rangle] depends upon the structural information that is available (see Section 2.2.4.1[link] for some examples).

Equation (2.2.4.3)[link] may be rewritten as [\ln \left\{{\langle I \rangle\over \langle |F^{\rm o}|^{2} \rangle}\right\} = - \ln K - 2Bs^{2},]which plotted at various [s^{2}] should be a straight line of which the slope (2B) and intercept (ln K) on the logarithmic axis can be obtained by applying a linear least-squares procedure.

Very often molecular geometries produce perceptible departures from linearity in the logarithmic Wilson plot. However, the more extensive the available a priori information on the structure is, the closer, on the average, are the Wilson-plot curves to their least-squares straight lines.

Accurate estimates of B and K require good strategies (Rogers & Wilson, 1953[link]) for:

  • (1) treatment of weak measured data. If weak data are set to zero, there will be bias in the statistics. Methods are, however, available (French & Wilson, 1978[link]) that provide an a posteriori estimate of weak (even negative) intensities by means of Bayesian statistics.

  • (2) treatment of missing weak data (Rogers et al., 1955[link]; Vicković & Viterbo, 1979[link]). All unobserved reflections may assume [\eqalign{\mu &= |F_{\rm o\min}|^{2}/3 \hbox{ for cs. space groups}\cr \mu &= |F_{\rm o\min}|^{2}/2 \hbox{ for ncs. space groups,}}]where the subscript `o min' refers to the minimum observed intensity.

Once K and B have been estimated, [E_{\bf h}] values can be obtained from experimental data by [|E_{\bf h}|^{2} = {KI_{\bf h}\over \langle |F_{\bf h}^{\rm o}|^{2} \rangle \exp (- 2Bs^{2})},]where [\langle |F_{\bf h}^{\rm o}|^{2} \rangle] is the expected value of [|F_{\bf h}^{\rm o}|^{2}] for the reflection h on the basis of the available a priori information.

2.2.4.4. Probability distributions of normalized structure factors

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Under some fairly general assumptions (see Chapter 2.1[link] ) probability distribution functions for the variable [|E|] for cs. and ncs. structures are (see Fig. 2.2.4.1[link]) [{}_{\bar{1}}P(|E|) \, \hbox{d}|E| = \sqrt{{2\over \pi}} \exp \left(- {E^{2}\over 2}\right) \, \hbox{d}|E| \eqno (2.2.4.4)]and[{}_{1}P(|E|) \,\hbox{d}|E| = 2|E| \exp (- |E|^{2}) \,\hbox{d}|E|, \eqno (2.2.4.5)]respectively. Corresponding cumulative functions are (see Fig. 2.2.4.2[link]) [\eqalign{_{\bar{1}}N(|E|) &= \sqrt{{2\over \pi}} \int\limits_{0}^{|E|} \exp \left(- {t^{2}\over 2}\right) \,\hbox{d}t = \hbox{erf} \left({|E|\over \sqrt{2}}\right),\cr {}_{1}N(|E|) &= \int\limits_{0}^{|E|} 2t \exp (- t^{2}) \,\hbox{d}t = 1 - \exp (- |E|^{2}).}]

[Figure 2.2.4.1]

Figure 2.2.4.1 | top | pdf |

Probability density functions for cs. and ncs. crystals.

[Figure 2.2.4.2]

Figure 2.2.4.2 | top | pdf |

Cumulative distribution functions for cs. and ncs. crystals.

Some moments of the distributions (2.2.4.4)[link] and (2.2.4.5)[link] are listed in Table 2.2.4.1[link]. In the absence of other indications for a given crystal structure, a cs. or an ncs. space group will be preferred according to whether the statistical tests yield values closer to column 2 or to column 3 of Table 2.2.4.1[link].

Table 2.2.4.1| top | pdf |
Moments of the distributions (2.2.4.4)[link] and (2.2.4.5)[link]

[R(E_{s})] is the percentage of n.s.f.'s with amplitude greater than the threshold [E_{s}].

CriterionCentrosymmetric distributionNoncentrosymmetric distribution
[\langle |E|\rangle] 0.798 0.886
[\langle |E|^{2}\rangle] 1.000 1.000
[\langle |E|^{3}\rangle] 1.596 1.329
[\langle |E|^{4}\rangle] 3.000 2.000
[\langle |E|^{5}\rangle] 6.383 3.323
[\langle |E|^{6}\rangle] 15.000 6.000
[\langle |E^{2} - 1|\rangle] 0.968 0.736
[\langle (E^{2} - 1)^{2}\rangle] 2.000 1.000
[\langle (E^{2} - 1)^{3}\rangle] 8.000 2.000
[\langle |E^{2} - 1|^{3}\rangle] 8.691 2.415
R(1) 0.320 0.368
R(2) 0.050 0.018
R(3) 0.003 0.0001

For further details about the distribution of intensities see Chapter 2.1[link] .

References

Cascarano, G., Giacovazzo, C. & Luić, M. (1985a). Non-crystallographic translational symmetry: effects on diffraction-intensity statistics. In Structure and Statistics in Crystallography, edited by A. J. C. Wilson, pp. 67–77. Guilderland, USA: Adenine Press.
Cascarano, G., Giacovazzo, C. & Luić, M. (1985b). Direct methods and superstructures. I. Effects of the pseudotranslation on the reciprocal space. Acta Cryst. A41, 544–551.
Cascarano, G., Giacovazzo, C. & Luić, M. (1988a). Direct methods and structures showing superstructure effects. III. A general mathematical model. Acta Cryst. A44, 176–183.
French, S. & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Cryst. A34, 517–525.
Giacovazzo, C. (1983a). From a partial to the complete crystal structure. Acta Cryst. A39, 685–692.
Giacovazzo, C. (1988). New probabilistic formulas for finding the positions of correctly oriented atomic groups. Acta Cryst. A44, 294–300.
Hauptman, H. & Karle, J. (1953). Solution of the Phase Problem. I. The Centrosymmetric Crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service.
Mackay, A. L. (1953). A statistical treatment of superlattice reflexions. Acta Cryst. 6, 214–215.
Main, P. (1976). Recent developments in the MULTAN system. The use of molecular structure. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 97–105. Copenhagen: Munksgaard.
Rogers, D., Stanley, E. & Wilson, A. J. C. (1955). The probability distribution of intensities. VI. The influence of intensity errors on the statistical tests. Acta Cryst. 8, 383–393.
Rogers, D. & Wilson, A. J. C. (1953). The probability distribution of X-ray intensities. V. A note on some hypersymmetric distributions. Acta Cryst. 6, 439–449.
Vicković, I. & Viterbo, D. (1979). A simple statistical treatment of unobserved reflexions. Application to two organic substances. Acta Cryst. A35, 500–501.
Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152.








































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