Sayre equation

From Online Dictionary of Crystallography

Équation de Sayre (Fr). Sayre-Gleichung (Ge). Equazione di Sayre (It). Ecuación de Sayre (Sp).

In direct methods, the Sayre equation is used to calculate probable values for the phases of some reflections. Its formulation is the following:

[math]F_{hkl} = \sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'}[/math]

and states that the structure factor of a reflection hkl can be calculated as a function of structure factors whose Laue indices sum to the desired values of hkl. In particular, in a centrosymmetric structure, the phases of three reflections satisfying the above relation of Laue indices can only be 0 or [math]\pi[/math] and the Sayre equation reduces to a relation between signs of structure factors:

[math]S_{h} \approx S_{h'} S_{h-h'} [/math]

where the signs S are positive if the phase is 0 and negative if it is [math]\pi[/math] and the [math]\approx[/math] symbol indicates a certain degree of probability that the relationship is true, which becomes higher the stronger the reflections are.