The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets

The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which `intensity only' measurements as well as `higher-order invariants' can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of `cyclotomic sets', corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into N equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.