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Two-dimensional equal atom structures or structure projections are determined applying the parameter space concept, either directly or supported by coordinate indications obtained from one or two separate runs on 1-d projection(s). By-passing the “phase problem” a grid-search based trial and error technique is used for optimizing model vs. experimental quasi-normalized structure amplitudes (or intensities) of a small number of reflections. The concept can also be applied to multi-wavelengths X-ray data exhibiting resonant scattering contrasts as well as to neutron data from structures including elements with differing signs of scattering powers. Unique results (or all possible combinations of atom sites meeting the data) are obtained from rather few reflections. They provide remarkable spatial resolution, but much more computing power is needed than for standard techniques. Considering the numbers of test models to be checked as a relative measure of computing time the computational efforts for different solution options and increasing numbers of independent atoms are assessed. In the examples given for symmetry   p1  , where one atom defines the origin, up to 8 additional atoms were located within Δ( x ,  y ) ≤ 0.025 from, e.g., 12 reflections with randomly generated maximum relative intensity errors of up to 20 %. All calculations were carried out with a standard PC.

Solving surface and embedded structures without Fourier inversion: a 2-dimensional direct space approach