Solving Crystal Structures without Fourier Mapping. I. Centrosymmetric Case
Abstract
A recursive algebraic procedure for solving one-dimensional monoatomic crystal structures is presented. (If applied to projections, also a three-dimensional atom arrangement may be reconstructed.) Moduli of the geometrical parts of the corresponding structure factors serve as experimental input. The atom coordinates are found from the roots of a polynomial. For space group P1 with m atoms in the asymmetric unit, the first m + 1 reflections are needed for finding their signs by means of a determinant technique. Using Monte Carlo calculations, the influence of the standard uncertainties of the data on the uncertainties of the derived coordinates is simulated. In a similar way, hints for discriminating between sign variations are obtained. The resolution in direct space is better than that of a one-dimensional Fourier summation over the same number of reflections. Error-free data provide a unique solution (if homometries are excluded). For data affected by experimental uncertainties, all possible solutions (compatible with the data) are found. Their number is always finite, and it may be further reduced by employing reflection orders higher than m + 1. Some applications of the method are discussed.