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

International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 204-205   | 1 | 2 |

Section 2.1.8.2. Fourier–Bessel 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

2.1.8.2. Fourier–Bessel series

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Equations (2.1.8.5)[link] and (2.1.8.9)[link] are the exact counterparts of equations (2.1.7.5)[link] and (2.1.7.6)[link], respectively. The computational effort required to evaluate equation (2.1.8.9)[link] is somewhat greater than that for (2.1.8.5)[link], 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 [t = (t_{1}^{2}+t_{2}^{2})^{1/2}] alone, rather than on [t_{1}] and [t_{2}] separately. In such cases the p.d.f. of [|E|] for a noncentrosymmetric space group can be expanded in a single Fourier–Bessel series (Barakat, 1974[link]; Weiss & Kiefer, 1983[link]; Shmueli et al., 1984[link]). The general form of this expansion is [p(|E|) = 2\alpha^{2}|E|\textstyle\sum\limits_{u = 1}^{\infty}D_{u}J_{0}(\alpha\lambda_{u}|E|), \eqno(2.1.8.11)] where [D_{u} = {{C(\alpha\lambda_{u})}\over{J_{1}^{2}(\lambda_{u})}} \eqno(2.1.8.12)] and [C(\alpha\lambda_{u}) = \textstyle\prod\limits_{j = 1}^{N/g}C_{ju}, \eqno(2.1.8.13)] where [J_{1}(x)] is the Bessel function of the first kind, and [\lambda_{u}] is the uth root of the equation [J_{0}(x) = 0]; the atomic contribution [C_{ju}] to equation (2.1.8.13)[link] is computed as [C_{ju} = C(\alpha n_{j}\lambda_{u}). \eqno(2.1.8.14)] The roots [\lambda_{u}] are tabulated in the literature (e.g. Abramowitz & Stegun, 1972[link]), but can be most conveniently computed as follows. The first five roots are given by [\eqalign{ \lambda_1& = 2.4048255577\cr \lambda_2& = 5.5200781103 \cr \lambda_3& = 8.6537279129\cr \lambda_4& = 11.7915344390\cr \lambda_5& = 14.9309177085 }] and the higher ones can be obtained from McMahon's approximation (cf. Abramowitz & Stegun, 1972[link]) [{\lambda_{u} = \beta+{{1}\over{8\beta}}-{{124}\over{3(8\beta)^{3}}}+{{120928}\over{15(8\lambda)^{5}}}-{{401743168}\over{105(8\lambda)^{7}}}+\ldots,} \eqno(2.1.8.15)] where [\beta = (u-{\textstyle {1\over4}})\pi]. For [u \;\gt\; 5] the values given by equation (2.1.8.15)[link] have a relative error less than 10−11 so that no refinement of roots of higher orders is needed (Shmueli et al., 1984[link]). Numerical computations of single Fourier–Bessel series are of course faster than those of the double Fourier series, but both representations converge fairly rapidly.

References

First citationAbramowitz, M. & Stegun, I. A. (1972). Handbook of mathematical functions. New York: Dover.Google Scholar
First citationBarakat, R. (1974). First-order statistics of combined random sinusoidal waves with application to laser speckle patterns. Opt. Acta, 21, 903–921.Google Scholar
First citationShmueli, U., Weiss, G. H., Kiefer, J. E. & Wilson, A. J. C. (1984). Exact random-walk models in crystallographic statistics. I. Space groups [P\bar{1}] and [P1]. Acta Cryst. A40, 651–660.Google Scholar
First citationWeiss, G. H. & Kiefer, J. E. (1983). The Pearson random walk with unequal step sizes. J. Phys. A, 16, 489–495.Google Scholar








































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