## The application of phase relationships to complex structures. VlI. Magic integers

**Abstract**

With phases expressed in cycles so that 0 < 1 it is possible with a single symbol x, in the range 0 to 1, to represent several phases, say m, by = nix mod (1) where i runs from 1 to m and the integers, ni, are referred to as 'magic integers'. A starting set of phases may consist of some which fix the origin and enantiomorph, some known by 1 relationships for example and others given magic-integer representation in terms of x, y and z. Relationships between the starting-set phases then appear in the form Hx + Ky + Lz + b 0, and maxima of the function, = |E1rE2rE3r| cos {2(Hrx + Kry + Lrz + b)} , lead to sets of possible values of the unknown phases in the starting set of reflexions. By means of the magic-integer process complex structures requiring very large starting sets may be tackled. Examples of the application of the method are given.