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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 203-204   | 1 | 2 |

## Section 2.1.8.1. General representations of p.d.f.'s of by Fourier series

U. Shmuelia* and A. J. C. Wilsonb

aSchool of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

#### 2.1.8.1. General representations of p.d.f.'s of by Fourier series

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We assume, as before, that (i) the atomic phase factors [cf. equation (2.1.1.2) ] are uniformly distributed on (0–2 ) 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 non-zero 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 correction-factor 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.

### References Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar