Orthorhombic groups

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 82-83 | 1 | 2 |

Section 1.3.4.3.6.3. Orthorhombic groups

G. Bricogne^a

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

1.3.4.3.6.3. Orthorhombic groups

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Almost all orthorhombic space groups are generated by two monoclinic transformations $[g_{1}]$ and $[g_{2}]$ of the type described in Section 1.3.4.3.6.2, with the addition of a centre of inversion [-e] for centrosymmetric groups. The only exceptions are Fdd2 and Fddd which contain diamond glides, in which some non-primitive translations are `square roots' not of primitive lattice translations, but of centring translations. The generic case will be examined first.

To calculate electron densities, the unique octant of data may first be transformed on $[{\bf h}^{+}]$ (respectively $[{\bf h}^{-}]$ ) as in Section 1.3.4.3.6.2 using the symmetry pertaining to generator $[g_{1}]$ . These intermediate results may then be expanded by generator $[g_{2}]$ by the formula of Section 1.3.4.3.3 prior to the final transform on $[{\bf h}^{-}]$ (respectively $[{\bf h}^{+}]$ ). To calculate structure factors, the reverse operations are applied in the reverse order.

The two exceptional groups Fdd2 and Fddd only require a small modification. The F-centring causes the systematic absence of parity classes with mixed parities, leaving only (000) and (111). For the former, the phase factors $[\exp [2 \pi i ({\bf h}^{+} \cdot {\bf t}_{g}^{+} + {\bf h}^{-} \cdot {\bf t}_{g}^{-})]]$ in the symmetry relations of Section 1.3.4.3.6.2 become powers of (−1) so that one is back to the generic case. For the latter, these phase factors are odd powers of i which it is a simple matter to incorporate into a modified multiplexing/demultiplexing procedure.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 82-83