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. 2.1, pp. 207212
Section 2.1.8. Nonideal distributions: the Fourier method^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England 
The starting point of the method described in the previous section is the centrallimit theorem approximation, and the method consists of finding correction factors which result in better approximations to the actual p.d.f. Conceptually, this is equivalent to improving the approximation of the characteristic function [cf. equation (2.1.4.10)] over that which led to the centrallimit theorem result.
The method to be described in this section does not depend on any initial approximation and will be shown to utilize the dependence of the exact value of the characteristic function on the spacegroup symmetry, atomic composition and other factors. This approach has its origin in a simple but ingenious observation by Barakat (1974), who noted that if a random variable has lower and upper bounds then the corresponding p.d.f. can be nonzero only within these bounds and can therefore be expanded in an ordinary Fourier series and set to zero (identically) outside the bounded interval. Barakat's (1974) work dealt with intensity statistics of laser speckle, where sinusoidal waves are involved, as in the present problem. This method was applied by Weiss & Kiefer (1983) to testing the accuracy of a steepestdescents approximation to the exact solution of the problem of random walk, and its first application to crystallographic intensity statistics soon followed (Shmueli et al., 1984). Crystallographic (e.g. Shmueli & Weiss, 1987; Rabinovich et al., 1991a,b) and noncrystallographic (Shmueli et al., 1985; Shmueli & Weiss, 1985a; Shmueli, Weiss & Wilson, 1989; Shmueli et al., 1990) symmetry was found to be tractable by this approach, as well as joint conditional p.d.f.'s of several structure factors (Shmueli & Weiss, 1985b, 1986; Shmueli, Rabinovich & Weiss, 1989). The Fourier method is illustrated below by deriving the exact counterparts of equations (2.1.7.5) and (2.1.7.6) and specifying them for some simple symmetries. We shall then indicate a method of treating higher symmetries and present results which will suffice for evaluation of Fourier p.d.f.'s of for a wide range of space groups.
We assume, as before, that (i) the atomic phase factors [cf. equation (2.1.1.2)] are uniformly distributed on (0–2π) and (ii) the atomic contributions to the structure factor are independent. For a centrosymmetric space group, with the origin chosen at a centre of symmetry, the random variable is the (real) normalized structure factor E and its bounds are and , where Here, is the maximum possible value of E and is the conventional scattering factor of the jth atom, including its temperature factor. The p.d.f., , can be nonzero in the range () only and can thus be expanded in the Fourier series where . Only the real part of is relevant. The Fourier coefficients can be obtained in the conventional manner by integrating over the range ,Since, however, for and , it is possible and convenient to replace the limits of integration in equation (2.1.8.3) by infinity. Thus Equation (2.1.8.4) shows that is a Fourier transform of the p.d.f. and, as such, it is the value of the corresponding characteristic function at the point [i.e., , where the characteristic function is defined by equation (2.1.4.1)]. It is also seen that is the expected value of the exponential . It follows that the feasibility of the present approach depends on one's ability to evaluate the characteristic function in closed form without the knowledge of the p.d.f.; this is analogous to the problem of evaluating absolute moments of the structure factor for the correctionfactor approach, discussed in Section 2.1.7. Fortunately, in crystallographic applications these calculations are feasible, provided individual isotropic motion is assumed. The formal expression for the p.d.f. of , for any centrosymmetric space group, is therefore where use is made of the assumption that , and the Fourier coefficients are evaluated from equation (2.1.8.4).
The p.d.f. of for a noncentrosymmetric space group is obtained by first deriving the joint p.d.f. of the real and imaginary parts of E and then integrating out its phase. The general expression for E is where is the phase of E. The required joint p.d.f. is and introducing polar coordinates and , where and , we have Integrating out the phase , we obtain where is the Bessel function of the first kind (e.g. Abramowitz & Stegun, 1972). This is a general form of the p.d.f. of for a noncentrosymmetric space group. The Fourier coefficients are obtained, similarly to the above, as and the average in equation (2.1.8.10), just as that in equation (2.1.8.4), is evaluated in terms of integrals over the appropriate trigonometric structure factors. In terms of the characteristic function for a joint p.d.f. of A and B, the Fourier coefficient in equation (2.1.8.10) is given by .
We shall denote the characteristic function by if it corresponds to a Fourier coefficient of a Fourier series for a centrosymmetric space group and by or by , where and , if it corresponds to a Fourier series for a noncentrosymmetric space group.
Equations (2.1.8.5) and (2.1.8.9) are the exact counterparts of equations (2.1.7.5) and (2.1.7.6), respectively. The computational effort required to evaluate equation (2.1.8.9) is somewhat greater than that for (2.1.8.5), because a double Fourier series has to be summed. The p.d.f. for any noncentrosymmetric space group can be expressed by a double Fourier series, but this can be simplified if the characteristic function depends on alone, rather than on and separately. In such cases the p.d.f. of for a noncentrosymmetric space group can be expanded in a single Fourier–Bessel series (Barakat, 1974; Weiss & Kiefer, 1983; Shmueli et al., 1984). The general form of this expansion is where and where is the Bessel function of the first kind, and is the uth root of the equation ; the atomic contribution to equation (2.1.8.13) is computed as The roots are tabulated in the literature (e.g. Abramowitz & Stegun, 1972), but can be most conveniently computed as follows. The first five roots are given by and the higher ones can be obtained from McMahon's approximation (cf. Abramowitz & Stegun, 1972) where . For the values given by equation (2.1.8.15) have a relative error less than 10^{−11} so that no refinement of roots of higher orders is needed (Shmueli et al., 1984). Numerical computations of single Fourier–Bessel series are of course faster than those of the double Fourier series, but both representations converge fairly rapidly.
Consider the Fourier coefficient of the p.d.f. of for the centrosymmetric space group . The normalized structure factor is given by and the Fourier coefficient is Equation (2.1.8.20) is obtained from equation (2.1.8.19) if we make use of the assumption of independence, the assumption of uniformity allows us to rewrite equation (2.1.8.20) as (2.1.8.21), and the expression in the braces in the latter equation is just a definition of the Bessel function (e.g. Abramowitz & Stegun, 1972).
Let us now consider the Fourier coefficient of the p.d.f. of for the noncentrosymmetric space group . We have These expressions for A and B are substituted in equation (2.1.8.10), resulting in Equation (2.1.8.24) leads to (2.1.8.25) by introducing polar coordinates analogous to those leading to equation (2.1.8.8), and equation (2.1.8.26) is then obtained by making use of the assumptions of independence and uniformity in an analogous manner to that detailed in equations (2.1.8.12)–(2.1.8.22) above.
The righthand side of equation (2.1.8.26) is to be used as a Fourier coefficient of the double Fourier series given by (2.1.8.9). Since, however, this coefficient depends on alone rather than on m and n separately, the p.d.f. of for P1 can also be represented by a Fourier–Bessel series [cf. equation (2.1.8.11)] with coefficient where is the uth root of the equation .
We now illustrate the methodology of deriving characteristic functions for space groups of higher symmetries, following the method of Rabinovich et al. (1991a,b). The derivation is performed for the space group (No. 174). According to Table A1.4.3.6 , the real and imaginary parts of the normalized structure factor are given by and whereNote that , i.e., one of these contributions depends on the other two; this is a recurring problem in calculations pertaining to trigonal and hexagonal systems. For brevity, we write directly the general form of the characteristic function from which the functional form of the Fourier coefficient can be readily obtained. The characteristic function is given by where , and the assumption of independence was used. If we further employ the assumption of uniformity, while remembering that the angular variables are not independent, the characteristic function can be written as where is the Fourier representation of the periodic delta function. Equation (2.1.8.34) then becomes If we change the variable to , becomes and . Hence The imaginary part of the summation, involving Bessel functions of odd orders, vanishes upon integration and the latter is restricted to the positive quadrant in . Thus, upon replacing cosines by sines (this is permissible at this stage) the atomic contribution to the characteristic function becomes and a double Fourier series must be used for the p.d.f.
Expressions for the atomic contributions to the characteristic functions were obtained by Rabinovich et al. (1991a) for a wide range of space groups, by methods similar to those described above. These expressions are collected in Table 2.1.8.1 in terms of symbols which are defined below. The following abbreviations are used in the subsequent definitions of the symbols: and the symbols appearing in Table 2.1.8.1 are given below:where and
^{†}And the enantiomorphous space group.

The averages appearing in the above summary are, in general, computed asexcept and which are computed as where is any of the atomic characteristic functions indicated above. The superscripts preceding the symbols in the above summary are appended to the corresponding symbols in Table 2.1.8.1 on their first occurrence.
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) obtained the following Fourier coefficient for the bicentric distribution in the space group :to be used with equation (2.1.8.5). Furthermore, if we use the important property of the characteristic function as outlined in Section 2.1.4.1, it is easy to write down the Fourier coefficient for a 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) and (2.1.8.41), with the appropriate number of atoms in each factor (Shmueli & Weiss, 1985a). While the purely bicentric p.d.f. obtained by using (2.1.8.41) with (2.1.8.5) is significantly different from the ideal bicentric p.d.f. given by equation (2.1.5.13) only when the atomic composition is sufficiently heterogeneous, the above partially bicentric p.d.f. appears to be a useful development even for an equalatom structure.
The problem of the coexistence of several noncrystallographic centres of symmetry within the asymmetric unit of , and its effect on the p.d.f. of , was examined by Shmueli, Weiss & Wilson (1989) 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. A Fourierseries representation of a nonideal p.d.f. corresponding to this case was developed by Shmueli, Rabinovich & Weiss (1989), 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 nonideal p.d.f.'s occur when heavy atoms are present in special positions (Shmueli & Weiss, 1988). 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 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 is here utilized.
The need for theoretical nonideal distributions was exemplified by Fig. 2.1.7.1, referred to above, and the performance of the two approaches described above, for this particular example, is shown in Fig. 2.1.8.1. Briefly, the Fourier p.d.f. shows an excellent agreement with the histogram of recalculated values, while the agreement attained by the Hermite correction factor is much less satisfactory, even for the (longest available to us) fiveterm expansion. It must be pointed out that (i) the inadequacy of `short' correction factors, in the example shown, is due to the large deviation from the ideal behaviour and (ii) the number of terms used there in the Fourier summation is twenty, whereafter the summation is terminated. Obviously, the computation of twenty (or more) Fourier coefficients is easier than that of five terms in the correction factor. The convergence of the Fourier series is very satisfactory. It appears that the (analytically) exact Fourier approach is the preferred one in cases of large or intermediate deviations, while the correctionfactor approach may cope well with small ones. As far as the availability of symmetrydependent centric and acentric p.d.f.'s is concerned, correction factors are available for all the space groups (see Table 2.1.7.1), while Fourier coefficients of p.d.f.'s are available for the first 206 space groups (see Table 2.1.8.1). It should be pointed out that p.d.f.'s based on the correctionfactor method cope very well with cubic symmetries higher than , even if the asymmetric unit of the space group is strongly heterogeneous (Rabinovich et al., 1991b).
Both approaches described in this section are related to the characteristic function of the required p.d.f. The correctionfactor p.d.f.'s (2.1.7.5) and (2.1.7.6) can be obtained by expanding the logarithm of the appropriate characteristic function in a series of cumulants [e.g. equation (2.1.4.13); see also Shmueli & Wilson (1982)], truncating the series and performing its termbyterm Fourier inversion. The Fourier p.d.f., on the other hand, is computed by forming a Fourier series whose coefficients are exact analytical forms of the characteristic function at points related to the summation indices [e.g. equations (2.1.8.5), (2.1.8.9) and (2.1.8.11), and Table 2.1.8.1] and truncating the series when the terms become small enough.
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