Ambiguities in the X-Ray Analysis of Crystal Structures

The intensities of the x-ray reflections from a given crystal of a known substance depend on the vector distances between the atoms and not directly on the coordinates of the atoms. The problem of uniqueness in the x-ray analysis of a crystal structure thus depends on the uniqueness of the determination of the arrangement of a periodic set of points in space by its vector distance set. In this paper a large number of cases is presented in which two, and sometimes three or four, non-congruent sets of points are homometric, i.e., have the same vector distance set. Although the investigation is largely based on a discussion of one-dimensional cyclotomic sets, i.e., those in which the points divide the period on the line into rational fractions, it is shown that there are many families of non-cyclotomic pairs and multiplets and that each of these families has its counterpart in two and three dimensions. The significance of these results for practical crystal analysis is discussed.