The ambiguity factor in implication theory

The implication is a transformation of a Harker synthesis that has the property of indicating the possible positions of atoms in the crystal structure. The location of the atoms is subject to a certain ambiguity which can be defined by an ambiguity factor, m = M/q. M is a basic ambiguity characteristic of the symmetry axis and arising from group products, while q is a degeneracy factor expressing a particular kind of equivalence of these products. The ambiguity factor can also be determined more easily with the aid of missing spectra. It is shown that M is the modulus of the transformation from the implication cell to the crystal cell, and 1/q is the fraction of spectra present. The ambiguity factor of any implication synthesis can, therefore, be easily determined from the symmetry and characteristic missing spectra of the space group. The ambiguity of an implication can be resolved with the aid of m Patterson line syntheses. The implication synthesis, therefore, provides a theoretical device for untangling the Patterson synthesis. This suggests that there should exist a simple relation between the Fourier coefficients in the implication function and the corresponding coefficients in the Fourier expression of the projected electron density. The details of this relation are dealt with in another paper.