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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 205   | 1 | 2 |

## Section 2.1.8.3. Simple examples

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

#### 2.1.8.3. Simple examples

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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