On the application of phase relationships to complex structures. XI. A theory of magic integers

Abstract

Magic integers enable several unknown phases to be expressed in terms of a single variable at the expense of introducing some error in the phases represented. The economy of variables is particularly useful in a multisolution direct-methods program like MULTAN where a successful solution may depend upon the ability to use a large number of phases at the beginning of phase determination and the computing time is proportional to the number of variables. Formulae are presented which give the phase errors in the magic-integer representation. A recipe is given for the generation of efficient magic-integer sequences in which the r.m.s. error is spread evenly over all the phases represented. These sequences minimize the overall phase error for a given maximum integer in the sequence. It is found that the errors are minimized when the differences between adjacent members of the magic-integer sequence form the terms of a geometric progression and the smallest integer is greater than half the largest.