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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 19

Table 1.2.7.4 

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

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}}]