Topological framework for local structure analysis in condensed matter

Abstract

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. structure classification | Voronoi topology | atomic systems visualization C ondensed matter systems are often abstracted as large sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. Two challenges frequently encountered when studying systems at this scale are classifying and identifying local structure. Simulation studies of nucleation, crystallization, and melting, for example, as well as those of defect migration and transformation, require a precise understanding of which particles are associated with which phases, and which are associated with defects. As these systems are abstracted as large point sets, these dual challenges of classifying and identifying structure reduce to ones of understanding arrangements of points in space.A primary difficulty in classifying structure in spatial point sets arises from a tension between a desire for completeness and the necessity for practicality. The local neighborhood of a particle within an ensemble of particles can be completely described by a list of relative positions of each of its neighbors. However, although such a list of coordinates is complete in some sense, this raw data provides little direct insight, leaving us wanting for a practical and more illuminating description. This tension is often mediated by the choice of an "order parameter," which distills structural data into a single number or vector, and which is constructed to be both informative and computationally tractable (1).A central limitation of conventional order parameters is exhibited in degeneracies that arise in describing neighborhoods that are structurally...

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