# Sayre equation

### From Online Dictionary of Crystallography

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:

F_{hkl} = | ∑ | F_{h'k'l'}F_{h − h',k − k',l − l'} |

h'k'l' |

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:

where the signs S are positive if the phase is 0 and negative if it is π and the symbol indicates a certain degree of probability that the relationship is true, which becomes higher the stronger the reflections are.