International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.7, p. 719

Table 8.7.3.2 

P. Coppens,a Z. Sub and P. J. Beckerc

a732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

Table 8.7.3.2| top | pdf |
Expressions for the shape factors S for a parallelepiped with edges δx, δy, and δz (from Moss & Coppens, 1981[link])

j0 and j1 are the zero- and first-order spherical Bessel functions: j0(x) = sin x/x, j1(x) = sin x/x2 − cos x/x; VT is volume of integration.

[\hat y]Property[S[\hat y({\bf r}), {\bf h}]]
1 Charge [V_T\,j_0(2\pi h_x\delta_x)\, j_0(2\pi h_y\delta_y)\, j_0(2\pi h_z\delta_z)]
rα Dipole μα [\eqalign { -i&V_T\delta_\alpha\,j_1(2\pi h_\alpha\delta_\alpha) \cr & \times\, j_0(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_y\delta_y)}]
rαrβ Second moment
μαβ off-diagonal
[\eqalign { -V_T&\delta_\alpha\delta_\beta\,j_1 (2\pi h_\alpha\delta_\alpha) \cr \times\,& j_1(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_\gamma \delta_\gamma) }]
rαrα Second moment
μαα diagonal
[\eqalign { -V_T&\delta^2_\alpha\bigg\{\displaystyle{j_1(2\pi h_\alpha\delta_\alpha) \over \pi h_\alpha \delta_\alpha} - j_0(2\pi h_\alpha\delta_\alpha)\bigg\} \cr \quad \times\, &j_0(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_\gamma\delta_\gamma) }]