Structure determination without Fourier inversion. Part II: The use of intensity ratios and inequalities

Abstract

Abstract Using for a one-dimensional centrosymmetric structure of m equal point scatterers an m-dimensional coordinate parameter space P m and the (n – 1) independent ratios q(h, k) = [I′(h)/I′(k)]1/2 = g′(h)/g′(k) of n > 1 structure amplitudes observed on relative scale it is shown that the determination of the structure or possibly homometric structures is equivalent to finding the common intersection(s) of (n – 1) ≥ m independent isosurfaces Q[h, k; q] of dimension (m – 1).A fast and efficient reduction of the parameter space to the fraction(s) of P m that contain(s) the solution(s) can already be achieved on the basis of the qualitative inequalities between the observations. Therefore, emphasis is put on showing that each q(h, k) < 1, i.e. g′(h) < g′(k) (or vice versa) rules out all the regions in P m that contain point scatterer arrangements incompatible with q(h, k). For small structures, solution strategies are discussed and an estimate is given on the number of data necessary for solving even ‘tough’ problems.