The determination of structure invariants. I. Quadrupoles and their uses

The triple-phase relationships (tpr) involving a particular set of reflexions can, in general, be related to each other since they form an interdependent set. A particular form of relationship, called a quadrupole, involves the sum of four tpr's and this sum can either be 0 or some other value, usually π, if there is translational symmetry. A tpr involved in a number of quadrupoles with zero sum has an enhanced probability of being close to zero whereas if it is involved in many quadrupoles with sum π then it may be expected that this tpr will deviate significantly from zero. An analysis of the quadrupoles can lead to a selection of tpr's which is more reliable than one selected on the basis of the three values of |E| alone. Such tpr's can be used in the early stages of phase determination to increase the probability of getting through the first critical stages without serious error. A few tpr's which deviate significantly from zero may also be detected from the quadrupoles.