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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
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High-Fugacity Expansion and Crystallization in Non-sliding Hard-Core Lattice Particle Models Without a Tiling

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|October 24, 2024
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Summary

This study proves crystallization transitions in hard-core particle models on graphs. A new criterion, applicable to non-tiling particle configurations, is established using Pirogov-Sinai theory and local density definitions.

Keywords:
CrystallizationDiscrete Voronoi diagramHard-core modelsLocal densityPirogov-Sinai theory

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Area of Science:

  • Statistical Mechanics
  • Mathematical Physics
  • Materials Science

Background:

  • Crystallization transitions are fundamental in statistical mechanics.
  • Previous criteria for crystallization in hard-core models had limitations.

Purpose of the Study:

  • To establish a general criterion for crystallization in hard-core particle models on periodic graphs.
  • To extend existing theories to include models with non-tiling close-packing configurations.

Main Methods:

  • Application of Pirogov-Sinai theory to prove pressure analyticity.
  • Development of a local density definition using discrete Voronoi cells.
  • Analysis of hard-core particle models on periodic graphs in dimension d.

Main Results:

  • Existence of a crystallization transition for a family of hard-core particle models.
  • A new, more general criterion for crystallization is established.
  • The criterion applies to models where particles do not tile space, like discrete hard-disk models.

Conclusions:

  • The developed criterion is illustrated with staircase, hard-disk, and heptacube models.
  • The Pirogov-Sinai theory, enhanced by local density, provides a powerful tool for studying phase transitions.
  • This work advances the understanding of crystallization in diverse particle systems.