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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 15   | 1 | 2 |

Section 1.2.7.2. Reciprocal-space description of aspherical atoms

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

1.2.7.2. Reciprocal-space description of aspherical atoms

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The aspherical-atom form factor is obtained by substitution of (1.2.7.6)[link] in expression (1.2.4.3a)[link]: [f_{j}({\bf S}) = {\textstyle\int} \rho_{j}({\bf r}) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}{\bf r}. \eqno(1.2.4.3a)] In order to evaluate the integral, the scattering operator [{\exp (2\pi i{\bf S} \cdot {\bf r})}] must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959[link]; Cohen-Tannoudji et al., 1977[link]) [{\exp (2\pi i{\bf S} \cdot {\bf r}) = 4\pi {\textstyle\sum\limits_{l = 0}^{\infty}}\; {\textstyle\sum\limits_{m = -l}^{l}} i^{l} j_{l} (2\pi Sr) Y_{lm} (\theta, \varphi) Y_{lm}^{*} (\beta, \gamma).} \eqno(1.2.7.7a)]

The Fourier transform of the product of a complex spherical harmonic function with normalization [{\textstyle\int} |Y_{lm}|^{2}\ \hbox{d}\Omega = 1] and an arbitrary radial function [R_{l}(r)] follows from the orthonormality properties of the spherical harmonic functions, and is given by [{{\textstyle\int} Y_{lm} R_{l}(r) \exp (2\pi i {\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}{\textstyle\int}j_{l} (2\pi {S}r) R_{l}(r) r^{2}\ \hbox{d}r Y_{lm} (\beta, \gamma),} \eqno(1.2.7.8a)] where [j_{l}] is the lth-order spherical Bessel function (Arfken, 1970[link]), and θ and ϕ, β and γ are the angular coordinates of r and S, respectively.

For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics: [\eqalignno{\exp (2\pi i{\bf S} \cdot {\bf r}) &= {\sum\limits_{l = 0}^{\infty}} i^{l} j_{l} (2\pi Sr) (2 - \delta_{m0}) (2l + 1) {\sum\limits_{m = 0}^{l}} {(l - m)! \over (l + m)!}\cr &\quad \times P_{l}^{m} (\cos \theta) P_{l}^{m} (\cos \beta) \cos [m(\phi - \gamma)], &(1.2.7.7b)}] which leads to [{\textstyle\int} y_{lmp} (\theta, \varphi) R_{l}(r) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}\langle j_{l}\rangle y_{lmp} (\beta, \gamma). \eqno(1.2.7.8b)] Since [y_{lmp}] occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions [d_{lmp}] [{\textstyle\int} d_{lmp} (\theta, \varphi) R_{l}(r) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}\langle j_{l}\rangle d_{lmp} (\beta, \gamma). \eqno(1.2.7.8c)]

In (1.2.7.8b)[link] and (1.2.7.8c)[link], [\langle j_{l}\rangle], the Fourier–Bessel transform, is the radial integral defined as [\langle j_{l}\rangle = {\textstyle\int} j_{l}(2\pi Sr) R_{l}(r) r^{2}\ \hbox{d}r \eqno(1.2.7.9)] of which [\langle j_{0}\rangle] in expression (1.2.4.3)[link] is a special case. The functions [\langle j_{l}\rangle] for Hartree–Fock valence shells of the atoms are tabulated in scattering-factor tables (IT IV, 1974[link]). Expressions for the evaluation of [\langle j_{l}\rangle] using the radial function (1.2.7.5a[link][link]c[link]) have been given by Stewart (1980)[link] and in closed form for (1.2.7.5a)[link] by Avery & Watson (1977)[link] and Su & Coppens (1990)[link]. The closed-form expressions are listed in Table 1.2.7.4[link].

Table 1.2.7.4| top | pdf |
Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977[link]; Su & Coppens, 1990[link])

[\langle j_{k}\rangle \equiv {\textstyle\int_{0}^{\infty}} r^{N} \exp(-Zr)j_{k}(Kr)\;\hbox{d}r, K = 4\pi \sin \theta/\lambda.]

  N
k 1 2 3 4 5 6 7 8
0 [\displaystyle{1 \over K^{2} + Z^{2}}] [\displaystyle{2Z \over (K^{2} + Z^{2})^{2}}] [\displaystyle{2(3Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{3}}] [\displaystyle{24Z(Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{4}}] [\displaystyle{24(5Z^{2} - 10K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{240Z(K^{2} - 3Z^{2}) (3K^{2} - Z^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{720(7Z^{6} - 35K^{2}Z^{4} + 21K^{4}Z^{2} - K^{6}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{40320(Z^{7} - 7K^{2}Z^{5} + 7K^{4}Z^{3} - K^{6}Z) \over (K^{2} + Z^{2})^{8}}]
1   [\displaystyle{2K \over (K^{2} + Z^{2})^{2}}] [\displaystyle{8KZ \over (K^{2} + Z^{2})^{3}}] [\displaystyle{8K(5Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{4}}] [\displaystyle{48KZ(5Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{48K(35Z^{4} - 42K^{2}Z^{2} + 3K^{4}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{1920KZ(7Z^{4} - 14K^{2}Z^{2} + 3K^{4}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{5760K(21Z^{6} - 63K^{2}Z^{4} + 27K^{4}Z^{2} - K^{6}) \over (K^{2} + Z^{2})^{8}}]
2     [\displaystyle{8K^{2} \over (K^{2} + Z^{2})^{3}}] [\displaystyle{48K^{2}Z \over (K^{2} + Z^{2})^{4}}] [\displaystyle{48K^{2}(7Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{384K^{2}Z(7Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{1152K^{2}(21Z^{4} - 18K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{11520K^{2}Z(21Z^{4} - 30K^{2}Z^{2} + 5K^{4}) \over (K^{2} + Z^{2})^{8}}]
3       [\displaystyle{48K^{3} \over (K^{2} + Z^{2})^{4}}] [\displaystyle{384K^{3}Z \over (K^{2} + Z^{2})^{5}}] [\displaystyle{384K^{3}(9Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{11520K^{3}Z(3Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{11520K^{3}(33Z^{4} - 22K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{8}}]
4         [\displaystyle{384K^{4} \over (K^{2} + Z^{2})^{5}}] [\displaystyle{3840K^{4}Z \over (K^{2} + Z^{2})^{6}}] [\displaystyle{3840K^{4}(11Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{46080K^{4}Z(11Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{8}}]
5           [\displaystyle{3840K^{5} \over (K^{2} + Z^{2})^{6}}] [\displaystyle{46080K^{5}Z \over (K^{2} + Z^{2})^{7}}] [\displaystyle{40680K^{5}(13Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{8}}]
6             [\displaystyle{46080K^{6} \over (K^{2} + Z^{2})^{7}}] [\displaystyle{645120K^{6}Z \over (K^{2} + Z^{2})^{8}}]
7               [\displaystyle{645120K^{7} \over (K^{2} + Z^{2})^{8}}]

Expressions (1.2.7.8)[link] [link] show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant.

The scattering factors [f_{lmp}({\bf S})] of the aspherical density functions [R_{l}(r)d_{lmp}(\theta, \phi)] in the multipole expansion (1.2.7.6)[link] are thus given by [f_{lmp}({\bf S}) = 4\pi i^{l} \langle j_{l}\rangle d_{lmp} (\beta, \gamma). \eqno(1.2.7.8d)]

The reciprocal-space spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1[link], except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S.

References

First citationInternational Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar
First citationArfken, G. (1970). Mathematical models for physicists, 2nd ed. New York, London: Academic Press.Google Scholar
First citationAvery, J. & Watson, K. J. (1977). Generalized X-ray scattering factors. Simple closed-form expressions for the one-centre case with Slater-type orbitals. Acta Cryst. A33, 679–680.Google Scholar
First citationCohen-Tannoudji, C., Diu, B. & Laloe, F. (1977). Quantum mechanics. New York: John Wiley and Paris: Hermann.Google Scholar
First citationStewart, R. F. (1980). Electron and magnetization densities in molecules and solids, edited by P. J. Becker, pp. 439–442. New York: Plenum.Google Scholar
First citationSu, Z. & Coppens, P. (1990). Closed-form expressions for Fourier– Bessel transforms of Slater-type functions. J. Appl. Cryst. 23, 71–73.Google Scholar
First citationWeiss, R. J. & Freeman, A. J. (1959). X-ray and neutron scattering for electrons in a crystalline field and the determination of outer electron configurations in iron and nickel. J. Phys. Chem. Solids, 10, 147–161.Google Scholar








































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