# Sayre equation

### From Online Dictionary of Crystallography

##### Revision as of 18:33, 1 September 2017 by MassimoNespolo (talk | contribs) (Created page with "<font color="blue">Équation de Sayre</font> (''Fr''); <font color="black">Equazione di Sayre</font> (''It''). In direct methods, the '''Sayre equation''' is used to calcul...")

Équation de Sayre (*Fr*); Equazione di Sayre (*It*).

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 π 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 π 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.