Conditional ambiguity of one-dimensional crystal structures determined from a minimum of diffraction intensity data

Abstract

When the number of intensities greatly exceeds the number of unknown atomic coordinates, the problem of obtaining a crystal structure from the intensities is overdetermined and, for a sufficiently small structure, a chemically meaningful solution can be found by direct methods. A difficulty in determining a structure has been historically attributed to the non-uniqueness of such a structure owing to multiple, or homometric, structures that yield the same set of intensities. The number of homometric structures has not been rigorously analyzed owing to the complexity of this problem. By using the method of elementary symmetric polynomials with a new origin definition, one-dimensional crystal structures of a small number of identical atoms (N < 5), determined from a minimum (N - 1) of the lowest-resolution intensities, are enumerated. It is demonstrated that such a structure is unique for N ≤ 3. Interestingly, for N = 4, the structure can be determined either uniquely or twofold ambiguously, depending on the intensity values. These results suggest that, even for larger structures, a minimum set of (or not many more) accurately measured intensities can yield a unique structure.