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Physical systems are frequently modeled as sets of points in space, eachrepresenting the position of an atom, molecule, or mesoscale particle. As manyproperties of such systems depend on the underlying ordering of theirconstituent particles, understanding that structure is a primary objective ofcondensed matter research. Although perfect crystals are fully described by aset of translation and basis vectors, real-world materials are never perfect,as thermal vibrations and defects introduce significant deviation from idealorder. Meanwhile, liquids and glasses present yet more complexity. A completeunderstanding of structure thus remains a central, open problem. Here wepropose a unified mathematical framework, based on the topology of the Voronoicell of a particle, for classifying local structure in ordered and disorderedsystems that is powerful and practical. We explain the underlying reason whythis topological description of local structure is better suited for structuralanalysis than continuous descriptions. We demonstrate the connection of thisapproach to the behavior of physical systems and explore how crystallinestructure is compromised at elevated temperatures. We also illustrate potentialapplications to identifying defects in plastically deformed polycrystals athigh temperatures, automating analysis of complex structures, andcharacterizing general disordered systems.

A Topological Framework for Local Structure Analysis in Condensed Matter