Sayre equation

In crystallography, the Sayre equation, named after David Sayre who introduced it in 1952, is a mathematical relationship that allows 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 ′

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 . 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 and the Sayre equation reduces to the triplet relationship:

S h ≈ S h ′ S h − h ′

where the S indicates the sign of the structure factor (positive if the phase is 0 and negative if it is π ) and the ≈ sign indicates that there is a certain degree of probability that the relationship is true, which becomes higher the stronger the beams are.