Computational techniques for nanostructure determination of substances that resist standard crystallographic methods are often laborious processes starting from initial guess solutions not derived from experimental data. The Liga algorithm can create nanostructures using only lists of lengths or distances between atom pairs, providing an experimental basis for starting structures. These distance lists may be extracted from a variety of experimental probes and we illustrate the procedure with distances determined from the pair distribution function. Candidate subclusters that are a subset of a structure's atoms compete based on adherence to the length list. Atoms are added to well performing candidates and removed from poor ones, until a complete structure with sufficient agreement to the length list emerges. The Liga algorithm is shown to reliably recreate Lennard-Jones clusters from ideal length lists and the C60 structure from neutron-scattering data. The correct fullerene structure was obtained with experimental data which missed several distances and had loosened constraints on distance multiplicity. This suggests that the Liga algorithm may have robust applicability for a wide range of nanostructures even in the absence of ideal data.
An extension of the Liga algorithm for structure solution from atomic pair distribution functions (PDFs), to handle periodic crystal structures with multiple elements in the unit cell, is described. The procedure is performed in three separate steps. First, pair distances are extracted from the experimental PDF. In the second step the Liga algorithm is used to find unit‐cell sites consistent with these pair distances. Finally, the atom species are assigned over the cell sites by minimizing the overlap of their empirical atomic radii. The procedure has been demonstrated on synchrotron X‐ray PDF data from 16 test samples. The structure solution was successful for 14 samples, including cases with enlarged supercells. The algorithm success rate and the reasons for the failed cases are discussed, together with enhancements that should improve its convergence and usability.
Reconstruction of complex structures is an inverse problem arising in virtually all areas of science and technology, from protein structure determination to bulk heterostructure solar cells and the structure of nanoparticles. We cast this problem as a complex network problem where the edges in a network have weights equal to the Euclidean distance between their endpoints. We present a method for reconstruction of the locations of the nodes of the network given only the edge weights of the Euclidean network. The theoretical foundations of the method are based on rigidity theory, which enables derivation of a polynomial bound on its efficiency. An efficient implementation of the method is discussed and timing results indicate that the run time of the algorithm is polynomial in the number of nodes in the network. We have reconstructed Euclidean networks of about 1000 nodes in approximately 24 h on a desktop computer using this implementation. We also reconstruct Euclidean networks corresponding to polymer chains in two dimensions and planar graphene nanoparticles. We have also modified our base algorithm so that it can successfully solve random point sets when the input data are less precise.
The study presents an algorithm, ParSCAPE, for model-independent extraction of peak positions and intensities from atomic pair distribution functions (PDFs). It provides a statistically motivated method for determining parsimony of extracted peak models using the information-theoretic Akaike information criterion (AIC) applied to plausible models generated within an iterative framework of clustering and chi-square fitting. All parameters the algorithm uses are in principle known or estimable from experiment, though careful judgment must be applied when estimating the PDF baseline of nanostructured materials. ParSCAPE has been implemented in the Python program SrMise. Algorithm performance is examined on synchrotron X-ray PDFs of 16 bulk crystals and two nanoparticles using AIC-based multimodeling techniques, and particularly the impact of experimental uncertainties on extracted models. It is quite resistant to misidentification of spurious peaks coming from noise and termination effects, even in the absence of a constraining structural model. Structure solution from automatically extracted peaks using the Liga algorithm is demonstrated for 14 crystals and for C60. Special attention is given to the information content of the PDF, theory and practice of the AIC, as well as the algorithm's limitations.
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