Topological framework for local structure analysis in condensed matter

Significance Richard Feynman famously described the hypothesis “All things are made of atoms” as among the most significant of all scientific knowledge. How atoms are arranged in “things” is an interesting and natural question. However, aside from perfect crystals and ideal gases, understanding these arrangements in an insightful yet tractable manner is challenging. We introduce a unified mathematical framework for classifying and identifying local structure in imperfect condensed matter systems using Voronoi topology. This versatile approach enables visualization and analysis of a wide range of complex atomic systems, including highly defected solids and glass-forming liquids. The proposed framework presents a new perspective into the structure of discrete systems of particles, ordered and disordered alike. Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent particles, understanding that structure is a primary objective of condensed matter research. Although perfect crystals are fully described by a set of translation and basis vectors, real-world materials are never perfect, as thermal vibrations and defects introduce significant deviation from ideal order. Meanwhile, liquids and glasses present yet more complexity. A complete understanding of structure thus remains a central, open problem. Here we propose a unified mathematical framework, based on the topology of the Voronoi cell of a particle, for classifying local structure in ordered and disordered systems that is powerful and practical. We explain the underlying reason why this topological description of local structure is better suited for structural analysis than continuous descriptions. We demonstrate the connection of this approach to the behavior of physical systems and explore how crystalline structure is compromised at elevated temperatures. We also illustrate potential applications to identifying defects in plastically deformed polycrystals at high temperatures, automating analysis of complex structures, and characterizing general disordered systems.

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