International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 205
Section 2.1.8.3. Simple examples^{a}School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^{b}St John's College, Cambridge, England |
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 right-hand 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 .
References
Abramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar