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

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 84   | 1 | 2 |

Section 1.3.4.4.3.1. Squaring

G. Bricognea

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

1.3.4.4.3.1. Squaring

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Sayre (1952a)[link] derived his `squaring method equation' for structures consisting of equal, resolved and spherically symmetric atoms by observing that squaring such an electron density is equivalent merely to sharpening each atom into its square. Thus [F_{{\bf h}} = \theta_{{\bf h}} {\textstyle\sum\limits_{{\bf k}}}\; F_{{\bf k}} F_{{\bf h}-{\bf k}},] where [\theta_{{\bf h}} = f({\bf h})/f^{\rm sq} ({\bf h})] is the ratio between the form factor [f({\bf h})] common to all the atoms and the form factor [f^{\rm sq} ({\bf h})] for the squared version of that atom.

Most of the central results of direct methods, such as the tangent formula, are an immediate consequence of Sayre's equation. Phase refinement for a macromolecule by enforcement of the squaring method equation was demonstrated by Sayre (1972[link], 1974[link]).

References

Sayre, D. (1952a). The squaring method: a new method for phase determination. Acta Cryst. 5, 60–65.
Sayre, D. (1972). On least-squares refinement of the phases of crystallographic structure factors. Acta Cryst. A28, 210–212.
Sayre, D. (1974). Least-squares phase refinement. II. High-resolution phasing of a small protein. Acta Cryst. A30, 180–184.








































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