International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 109110

Most structuresolving software packages contain a section dedicated to several probabilistic methods based on the Wilson (1949) paper on the probability distribution of structurefactor magnitudes. These statistics sometimes correctly indicate whether the intensity data set was collected from a centrosymmetric or noncentrosymmetric crystal. However, not infrequently these indications are erroneous. The reasons for this may be many, but outstandingly important are (i) the presence of a few very heavy atoms amongst a host of lighter ones, and (ii) a very small number of nearly equal atoms. Omission of weak reflections from the data set also contributes to failures of Wilson (1949) statistics. These erroneous indications are also rather strongly spacegroup dependent.
The well known probability density functions (hereafter p.d.f.'s) of the magnitude of the normalized structure factor , also known as ideal p.d.f.'s, arewhere it is assumed that all the atoms are of the same chemical element. Let us see their graphical representations.
It is seen from Fig. 1.6.2.1 that the two p.d.f.'s are significantly different, but usually they are not presented as such by the software. What is usually shown are the cumulative distributions of , the moments: for n = 1, 2, 3, 4, 5, 6, and the averages of low powers of for ideal centric and acentric distributions, based on equation (1.6.2.3). Table 1.6.2.2 shows the numerical values of several loworder moments of and that of the lowest power of . The higher the value of n the greater is the difference between their values for centric and acentric cases. However, it is most important to remember that the influence of measurement uncertainties also increases with n and therefore the higher the moment the less reliable it tends to be.


Ideal p.d.f.'s for the equalatom case. The dashed line is the centric, and the solid line the acentric ideal p.d.f. 
There are several ideal indicators of the status of centrosymmetry of a crystal structure. The most frequently used are: (i) the N(z) test (Howells et al., 1950), a cumulative distribution of , based on equation (1.6.2.3), and (ii) the loworder moments of , also based on equation (1.6.2.3). Equation (1.6.2.3), however, is very seldom used as an indicator of the status of centrosymmetry of a crystal stucture.
Let us now briefly consider p.d.f.'s that are valid for any atomic composition as well as any spacegroup symmetry, and exemplify their performance by comparing a histogram derived from observed intensities from a structure with theoretical p.d.f.'s for the space groups P1 and . The p.d.f.'s considered presume that all the atoms are in general positions and that the reflections considered are general (see, e.g., Section 1.6.3). A general treatment of the problem is given in the literature and summarized in the book Introduction to Crystallographic Statistics (Shmueli & Weiss, 1995).
The basics of the exact p.d.f.'s are conveniently illustrated in the following. The normalized structure factor for the space group , assuming that all the atoms occupy general positions and resonant scattering is neglected, is given bywhere is the normalized scattering factor. The maximum possible value of is and the minimum possible value of is . Therefore, must be confined to the range. The probability of finding E outside this range is of course zero. Such a probability density function can be expanded in a Fourier series within this range (cf. Shmueli et al., 1984). This is the basis of the derivation, the details of which are well documented (e.g. Shmueli et al., 1984; Shmueli & Weiss, 1995; Shmueli, 2007). Exact p.d.f.'s for any centrosymmetric space group have the formwhere , and exact p.d.f.'s for any noncentrosymmetric space group can be computed as the double Fourier serieswhere is a Bessel function of the first kind and of order zero. Expressions for the coefficients and are given by Rabinovich et al. (1991) and by Shmueli & Wilson (2008) for all the space groups up to and including .
The following example deals with a very high sensitivity to atomic heterogeneity. Consider the crystal structure of [(Z)ethyl NisopropylthiocarbamatoκS](tricyclohexylphosphineκP)gold(I), published as with Z = 2, the content of its asymmetric unit being AuSPONC_{24}H_{45} (Tadbuppa & Tiekink, 2010). Let us construct a histogram from the data computed from all the observed reflections with nonnegative reduced intensities and compare the histogram with the p.d.f.'s for the space groups P1 and , computed from equations (1.6.2.5) and (1.6.2.4), respectively. The histogram and the p.d.f.'s were put on the same scale. The result is shown in Fig. 1.6.2.2.
A visual comparison strongly indicates that the spacegroup assignment as was correct, since the recalculated histogram agrees rather well with the p.d.f. (1.6.2.4) and much less with (1.6.2.5). The ideal Wilsontype statistics incorrectly indicated that this crystal is noncentrosymmetric. It is seen that the ideal p.d.f. breaks down in the presence of strong atomic heterogeneity (gold among many lighter atoms) in the space group . Other space groups behave differently, as shown in the literature (e.g. Rabinovich et al., 1991; Shmueli & Weiss, 1995).
Additional examples of applications of structurefactor statistics and some relevant computing considerations and software can be found in Shmueli (2012) and Shmueli (2013).
References
Howells, E. R., Phillips, D. C. & Rogers, D. (1950). The probability distribution of Xray intensities. II. Experimental investigation and the Xray detection of centres of symmetry. Acta Cryst. 3, 210–214.Rabinovich, S., Shmueli, U., Stein, Z., Shashua, R. & Weiss, G. H. (1991). Exact randomwalk models in crystallographic statistics. VI. P.d.f.'s of E for all plane groups and most space groups. Acta Cryst. A47, 328–335.
Shmueli, U. (2007). Theories and Techniques of Crystal Structure Determination. Oxford University Press.
Shmueli, U. (2012). Structurefactor statistics and crystal symmetry. J. Appl. Cryst. 45, 389–392.
Shmueli, U. (2013). INSTAT: a program for computing nonideal probability density functions of E. J. Appl. Cryst. 46, 1521–1522.
Shmueli, U. & Weiss, G. H. (1995). Introduction to Crystallographic Statistics. Oxford University Press.
Shmueli, U., Weiss, G. H., Kiefer, J. E. & Wilson, A. J. C. (1984). Exact randomwalk models in crystallographic statistics. I. Space groups and P1. Acta Cryst. A40, 651–660.
Shmueli, U. & Wilson, A. J. C. (2008). Statistical properties of the weighted reciprocal lattice. In International Tables for Crystallography, Volume B, Reciprocal Space, edited by U. Shmueli, ch. 2.1. Dordrecht: Springer.
Tadbuppa, P. P. & Tiekink, E. R. T. (2010). [(Z)Ethyl NisopropylthiocarbamatoκS](tricyclohexylphosphineκP)gold(I). Acta Cryst. E66, m615.
Wilson, A. J. C. (1949). The probability distribution of Xray intensities. Acta Cryst. 2, 318–321.