# 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.