InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 230-231
## Section 2.2.6. Direct methods in real and reciprocal space: Sayre's equation |

The statistical treatment suggested by Wilson for scaling observed intensities corresponds, in direct space, to the origin peak of the Patterson function, so it is not surprising that a general correspondence exists between probabilistic formulation in reciprocal space and algebraic properties in direct space.

For a structure containing atoms which are fully resolved from one another, the operation of raising to the *n*th power retains the condition of resolved atoms but changes the shape of each atom. Let where is an atomic function and is the coordinate of the `centre' of the atom. Then the Fourier transform of the electron density can be written as If the atoms do not overlap and its Fourier transform gives is the scattering factor for the *j*th peak of :

We now introduce the condition that all atoms are equal, so that and for any *j*. From (2.2.6.1) and (2.2.6.2) we may write where is a function which corrects for the difference of shape of the atoms with electron distributions and . Since the Fourier transform of both sides gives from which the following relation arises: For , equation (2.2.6.4) reduces to Sayre's (1952) equation [but see also Hughes (1953)] If the structure contains resolved isotropic atoms of two types, *P* and *Q*, it is impossible to find a factor such that the relation holds, since this would imply values of such that and simultaneously. However, the following relationship can be stated (Woolfson, 1958): where and are adjustable parameters of . Equation (2.2.6.6) can easily be generalized to the case of structures containing resolved atoms of more than two types (von Eller, 1973).

Besides the algebraic properties of the electron density, Patterson methods also can be developed so that they provide phase indications. For example, it is possible to find the reciprocal counterpart of the function For the function (2.2.6.7) coincides with the usual Patterson function ; for , (2.2.6.7) reduces to the double Patterson function introduced by Sayre (1953). Expansion of as a Fourier series yields *Vice versa,* the value of a triplet invariant may be considered as the Fourier transform of the double Patterson.

Among the main results relating direct- and reciprocal-space properties it may be remembered:

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