  # The Triplet Relation from Sayre's Equation

The earliest formulation of the triplet-relation (10) for the centrosymmetric case was via Sayre's equation (Sayre, 1952). This equation can be derived from Fourier theory as follows.

The electron density can be written as (17)
and upon squaring this function becomes (18)
(18) is rewritten by setting and to (19)
Because is also a periodic function it can be written, by analogy with (17), as (20)
in which GH is the structure factor of the squared structure. Comparing (19) and (20) it follows that (21)
The structure factor GH is: (22)
in which gj is the form factor of the squared atoms. For equal atoms (22) reduces to (23)
The normal structure factor for equal atoms is (24)
Thus from (23) and (24) we obtain (25)
Finally from (21) and (25) it follows that (26)
which is known as Sayre's Equation. It is emphasised that, given an equal-atom structure, Sayre's equation is exact. The summation (26) contains a large number of terms; however, in general it will be dominated by a smaller number of large |FKFH-K|. Rewriting (26) to (27)
and considering a reflection with large |FH| it can therefore be assumed that the terms with large |FKFH-K| have their angular part approximately equal to the angular part of |FH| itself, illustrated in Fig. 8. For one strong |FKFH-K| this leads to: (28)
or or (29)
Relation (29) is identical to (16), the triplet relation. Thus by introducing the obvious argument that the most important terms in Sayre's equation (27) must reflect the phase the triplet relation is found.

In the event that only a number of larger terms in (27) are available the scaling constant f/gV has no meaning. Nevertheless most likely the phase information included in these terms is correct and thus an expression such as (30)
in which K ranges over a limited number of terms may be very helpful.

The so called tangent formula (Karle and Hauptman, 1956) (31)
in which the signs of numerator and denominator are used to determine the quadrant of the phase , is closely related to (30). This formula is used in almost all direct method procedures.   Next: The Positive Quartet Relation Up: An Introduction to Direct Methods. The Previous: Large |EH|, |EK| and |E-H-K|: The