Linearization routines for the parameter space concept to determine crystal structures without Fourier inversion

We present the elaboration and first generally applicable linearization routines of the parameter space concept (PSC) for determining one-dimensionally projected structures of <i>m</i> independent scatterers. This crystal determination approach does not rely on Fourier inversion but rather considers all structure parameter combinations consistent with available diffraction data in a parameter space of dimension <i>m</i>. The method utilizes <i>m</i> structure-factor amplitudes or intensities represented by piecewise analytic hyper-surfaces to define the acceptable parameter regions. The coordinates of the point scatterers are obtained through the intersection of multiple isosurfaces. This approach allows for the detection of all possible solutions for the given structure-factor amplitudes in a single derivation. Taking the resonant contrast into account, the spatial resolution achieved by the presented method may exceed that of traditional Fourier inversion, and the algorithms can be significantly optimized by exploiting the symmetry properties of the isosurfaces. The applied one-dimensional projection demonstrates the efficiency of the PSC linearization approach based on fewer reflections than Fourier sums. Monte Carlo simulations, using the projections of various random two- and three-atom structure examples, are presented to illustrate the universal applicability of the proposed method. Furthermore, ongoing efforts aim to enhance the efficiency of data handling and to overcome current constraints, promising further advancements in the capabilities and accuracy of the PSC framework.