In crystallography, the Sayre equation, named after David Sayre who introduced it in 1952, is a mathematical relationship that allows one to calculate probable values for the phases of some diffracted beams. It is used when employing direct methods to solve a structure. Its formulation is the following:
F h k l = ∑ h ′ k ′ l ′ F h ′ k ′ l ′ F h − h ′ , k − k ′ , l − l ′ {\displaystyle F_{hkl}=\sum _{h'k'l'}F_{h'k'l'}F_{h-h',k-k',l-l'}}
which states how the structure factor for a beam can be calculated as the sum of the products of pairs of structure factors whose indices sum to the desired values of h , k , l {\displaystyle h,k,l} . Since weak diffracted beams will contribute a little to the sum, this method can be a powerful way of finding the phase of related beams, if some of the initial phases are already known by other methods. In particular, for three such related beams in a centrosymmetric structure, the phases can only be 0 or π {\displaystyle \pi } and the Sayre equation reduces to the triplet relationship:
S h ≈ S h ′ S h − h ′ {\displaystyle S_{h}\approx S_{h'}S_{h-h'}}
where the S {\displaystyle S} indicates the sign of the structure factor (positive if the phase is 0 and negative if it is π {\displaystyle \pi } ) and the ≈ {\displaystyle \approx } sign indicates that there is a certain degree of probability that the relationship is true, which becomes higher the stronger the beams are.
References
