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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 69   | 1 | 2 |

Section 1.3.4.2.2.7. Electron-density calculations

G. Bricognea

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

1.3.4.2.2.7. Electron-density calculations

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A formula for the Fourier synthesis of electron-density maps from symmetry-unique structure factors is readily obtained by orbit decomposition: where L is a subset of such that contains exactly one point of each orbit for the action of G on . The physical electron density per cubic ångström is then with V in Å3.

In the absence of anomalous scatterers in the crystal and of a centre of inversion −I in Γ, the spectrum has an extra symmetry, namely the Hermitian symmetry expressing Friedel's law (Section 1.3.4.2.1.4 ). The action of a centre of inversion may be added to that of Γ to obtain further simplification in the above formula: under this extra action, an orbit with is either mapped into itself or into the disjoint orbit ; the terms corresponding to and may then be grouped within the common orbit in the first case, and between the two orbits in the second case.

 Case 1: is centric. The cosets in may be partitioned into two disjoint classes by picking one coset in each of the two-coset orbits of the action of −I. Let denote one such class: then the reduced orbit contains exactly once the Friedel-unique half of the full orbit , and thus Grouping the summands for and yields a real-valued summand Case 2: is acentric. The two orbits are then disjoint, and the summands corresponding to and may be grouped together into a single real-valued summand In order to reindex the collection of all summands of , put where labels the Friedel-unique centric reflections in L and the acentric ones, and let stand for a subset of containing a unique element of each pair for . Then 