Sayre equation

From Online Dictionary of Crystallography

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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.