A Concept for Crystal Structure Determination without FOURIER Inversion: Some Steps towards Application*

Determination of a crystal structure without Fourier calculation of the scattering density (thus also avoiding the phase problem) is achieved in a fractional coordinate parameter space of dimension 3m where m is the number of independent atoms, reduced to equal point scatterers at rest. For demonstration of the basic ideas, two-dimensional parameter spaces (representing, e. g., one-dimensional two-atom structures) are used. "Central reciprocal lattice row" reflections allow for solving one-dimensional projections of the structure, each requiring less reflections and simultaneously providing better resolution than does a corresponding Fourier summation. The projection solution can be obtained either from the common intersection of the hyper-faces in the m-dimensional parameter space defined by the chosen scattering amplitudes or by exploring the permitted "solution region(s)" that follow from the mere ranking of these amplitudes. All possible solutions satisfying the data are found, including "false minima". The reconstruction of a hypothetical three-dimensional 11 atom structure from the solutions of one-dimensional projections is illustrated in an example based on "theoretical", i. e. error-free data. Since most of the theoretical background is laid down in two former, refereed publications, emphasis is put on different options to cope with the computing demands in practical applications. Advantages and shortcomings of the concept are discussed.