Structure determination without Fourier inversion. Part I. Unique results for centrosymmetric examples

A concept is presented for determining a one-dimensional structure of m independent point scatterers by mapping into an m-dimensional space P m at least m observations as (m - 1)-dimensional so-called isosurfaces defined by s(h) g(h) or g(h) alone, where s(h) and g(h) are sign and modulus, respectively, of the geometrical structure factor. Values of g(h) are derived from the observed |F(h)|. The solution vector(s), (x 1 ,..., x m ), representing the coordinates x j of the point scatterers is (are) found from the common intersection(s) of n ≥ m different isosurfaces. Homometric and quasi-homometric structures can thus safely be detected from multiple solutions, the latter mainly arising from the experimental uncertainties of the g(h). Spatial resolution is by far higher than that offered by a corresponding Fourier transform. Computer time can be drastically reduced upon consideration of both the symmetry of P m and the anti-symmetry and self-similarity properties of the isosurfaces, which allow for taylor-ing of the intersection search routines and/or applying linear approximations. The basic principles of the method are illustrated by various two- and three-atom structure examples and discussed in view of the application potential to real structure problems.