SOLVING CRYSTAL-STRUCTURES WITHOUT FOURIER MAPPING - I - CENTROSYMMETRIC CASE

A recursive algebraic procedure for solving one-dimensional monoatomic crystal structures is presented. (If applied to projections, also a t hree-dimensional atom arrangement may be reconstructed.) Moduli of the geometrical parts of the corresponding structure factors serve as exp erimental input. The atom coordinates are found from the roots of a po lynomial. For space group <P(1)over bar> with m atoms in the asymmetri c 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 uncerta inties of the derived coordinates is simulated. In a similar way, hint s for discriminating between sign variations are obtained. gh The reso lution in direct space is better than that of a one-dimensional Fourie r summation over the same number of reflections. Error-free data provi de 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 fur ther reduced by employing reflection orders higher than m + 1. Some ap plications of the method are discussed.