International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 216221
Section 2.2.4. Normalized structure factors ^{a}Dipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy 
The normalized structure factors E (see also Chapter 2.1 ) are calculated according to (Hauptman & Karle, 1953) where is the squared observed structurefactor magnitude on the absolute scale and is the expected value of .
depends on the available a priori information. Often, but not always, this may be considered as a combination of several typical situations. We mention:
When probability theory is not used, the quasinormalized structure factors and the unitary structure factors are often used. and are defined according to Since is the largest possible value for represents the fraction of with respect to its largest possible value. Therefore If atoms are equal, then .
N.s.f.'s cannot be calculated by applying (2.2.4.1) to observed s.f.'s because: (a) the observed magnitudes (already corrected for Lp factor, absorption, …) are on a relative scale; (b) cannot be calculated without having estimated the vibrational motion of the atoms.
This is usually obtained by the well known Wilson plot (Wilson, 1942), according to which observed data are divided into ranges of and averages of intensity are taken in each shell. Reflection multiplicities and other effects of spacegroup 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 should be obtained, where K is the scale factor needed to place Xray intensities on the absolute scale, B is the overall thermal parameter and is the expected value of in which it is assumed that all the atoms are at rest. depends upon the structural information that is available (see Section 2.2.4.1 for some examples).
Equation (2.2.4.3) may be rewritten as which plotted at various 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 leastsquares 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 Wilsonplot curves to their leastsquares straight lines.
Accurate estimates of B and K require good strategies (Rogers & Wilson, 1953) for:
Once K and B have been estimated, values can be obtained from experimental data by where is the expected value of for the reflection h on the basis of the available a priori information.
Under some fairly general assumptions (see Chapter 2.1 ) probability distribution functions for the variable for cs. and ncs. structures are (see Fig. 2.2.4.1) andrespectively. Corresponding cumulative functions are (see Fig. 2.2.4.2)
Some moments of the distributions (2.2.4.4) and (2.2.4.5) are listed in Table 2.2.4.1. 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.

For further details about the distribution of intensities see Chapter 2.1 .
References
Cascarano, G., Giacovazzo, C. & Luić, M. (1985a). Noncrystallographic translational symmetry: effects on diffractionintensity 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 Xray 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 Xray intensity data. Nature (London), 150, 151–152.