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:
which states how the
structure factor
In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
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
. 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
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point g ...
structure, the phases can only be 0 or
and the Sayre equation reduces to the triplet relationship:
where the
indicates the sign of the structure factor (positive if the phase is 0 and negative if it is
) and the
sign indicates that there is a certain degree of
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
that the relationship is true, which becomes higher the stronger the beams are.
References
*
*{{cite book , title=Crystal Structure Determination , last= Werner , first=Massa , year= 2004, publisher=Springer, isbn=3540206442 , page= 102
Crystallography