An approximation for organic compounds

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 191 | 1 | 2 |

Section 2.1.2.3. An approximation for organic compounds

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School 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

2.1.2.3. An approximation for organic compounds

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In organic compounds there are very many interatomic distances of about 1.5 or 1.4 Å. Adoption of the preceding criterion would mean that the inner portion of the region of reciprocal space accessible by the use of copper $[K\alpha]$ radiation is not within the sphere of intensity statistics based on fixed-index (first process) averaging. No substantial results are available for fixed-parameter (second process) averaging, and very few from the approximation to it (third process).

To the extent to which the third process is acceptable, an approximation to the variation of $[\langle I \rangle]$ with $[\sin\theta]$ is obtainable. The exponent in equation (2.1.2.2) can be written as $[2\pi i sr_{jk} \cos\psi, \eqno(2.1.2.9)]$ where s is the radial distance in reciprocal space, $[r_{jk}]$ is the distance from the jth to the kth atom and $[\psi]$ is the angle between the vectors s and r. Averaging over a sphere of radius s, with $[\psi]$ treated as the co-latitude, gives $[\langle I \rangle = \sum_{j}\sum_{k}f_{j}\;f_{k}{{\sin 2\pi sr_{jk}}\over {2\pi sr_{jk}}}. \eqno(2.1.2.10)]$ This is the familiar Debye expression. It has the correct limits for s zero and s large, and is in accord with the argument from resolution.

References

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 191