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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 83   | 1 | 2 |

Section 1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups

G. Bricognea

a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.3.6.4. Trigonal, tetragonal and hexagonal groups

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All the symmetries in this class of groups can be handled by the generalized Rader/Winograd algorithms of Section 1.3.4.3.4.3[link], but no implementation of these is yet available.

In groups containing axes of the form [n_{m}] with g.c.d. [(m, n) = 1\ (i.e.\ 3_{1}, 3_{2}, 4_{1}, 4_{3}, 6_{1}, 6_{5})] along the c direction, the following procedure may be used (Ten Eyck, 1973)[link]:

  • (i) to calculate electron densities, the unique structure factors indexed by [[\hbox{unique } (h, k)] \times (\hbox{all } l)] are transformed on l; the results are rearranged by the transposition formula of Section 1.3.4.3.3[link] so as to be indexed by [[\hbox{all } (h, k)] \times \left[\hbox{unique } \left({1 \over n}\right){\rm th} \hbox{ of } z\right]] and are finally transformed on (h, k) to produce an asymmetric unit. For a dihedral group, the extra twofold axis may be used in the transposition to produce a unique [(1/2n)]th of z.

  • (ii) to calculate structure factors, the unique densities in [(1/n)]th of z [or [(1/2n)]th for a dihedral group] are first transformed on x and y, then transposed by the formula of Section 1.3.4.3.3[link] to reindex the intermediate results by [[\hbox{unique } (h, k)] \times (\hbox{all } z)\hbox{;}] the last transform on z is then carried out.

References

First citation Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.Google Scholar








































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