# Sayre equation

### From Online Dictionary of Crystallography

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In [[direct methods]], the '''Sayre equation''' is used to calculate probable values for the phases of some reflections. Its formulation is the following: | 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> | + | |

+ | <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''. | 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 | + | 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> | <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 | + | 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. |

[[Category: Structure determination]] | [[Category: Structure determination]] |

## Latest revision as of 11:06, 15 December 2017

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:

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.