A Method for Obtaining a Possibly Unique (Unambiguous) Crystal Structure Solution using Multiple Resonant Scattering Data

Abstract

A sequence of steps for determining a crystal structure, possibly without ambiguities, is presented. The prerequisites are: centrosymmetry (at present) and two different anomalous scatterers, a~, a~. Their partial structure amplitudes [F,t(hkl) I and IF,2(hkl)l are separated by multiple-wavelength measurements (MAD). The core part of the method is a recursive algebraic technique applied to the geometrical part of these structure amplitudes from central reciprocal-lattice rows. At least m + 1 reflections are necessary at each row if 2m atoms of e.g.

Solving Crystal Structures without Fourier Mapping. I. Centrosymmetric Case

“…[In another paper (Fischer & Pilz, 1997, Section 5), we have described a method that may lead from onedimensional projections of a monoatomic structure to an unambiguous geometrical ('point') model via a three-dimensional reconstruction: The yj, coordinates, for example, independently found from the g (OkO), are allocated to the xj or -xj by using 'diagonal' coordinates obtained from g (hhO). Similarly, the zj,, can be allocated to the two-dimensional projection.…”

“…Another aspect is offered for a simple case m = 2, which may also be considered in connection with the experimental example described in Fischer & Pilz (1997): R(x) permits direct calculation of the coordinates (without 'root-finding') from Consequently, even small errors in g cause either a 'forbidden' (and thus impossible) region or lead to a small I Ox/Ogl, which results in a high spatial resolution. Fig.…”

“…How robust the algebraic method will behave with respect to experimental uncertainties can best be investigated by actual experiments. So far, we have performed only one study on the partial structures of Sb and Se in Cu3SbSe3 (Fischer & Pilz, 1997). For determining the partial structures of Se and Sb, we used 22 and 26 gh,obs, respectively, from six central reciprocallattice rows each.…”

“…After having successfully applied the recursive algebra to a centrosymmetric three-dimensional structure and being able to show that the solution was unique [at least for the main part (Fischer & Pilz, 1997)], we are now trying to develop the theory towards also solving noncentrosymmetric structures.…”

Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

“…In addition, Fischer & Pilz (1995) and were able to demonstrate in theory and by experiment that a unique (unambiguous) solution is obtained or all possible structures compatible with the data can be presented. Also in the case of centrosymmetric partial structures, the three-dimensional atomic positions are reconstructed (Fischer & Pilz, 1997) from three independent one-dimensional projections (e.g. onto the lattice rows a, b, c from h00, 0k0 and 00l data: class 1' reflections) plus`diagonals' (e.g.…”