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Communications in Mathematical Physics

, Volume 286, Issue 3, pp 1099–1140 | Cite as

On the Crystallization of 2D Hexagonal Lattices

  • Weinan E
  • Dong Li
Article

Abstract

It is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects. In this paper we focus on the zero temperature case and consider a class of atomic potentials V = V 2 + V 3, where V 2 is a pair potential of Lennard-Jones type and V 3 is a three-body potential of Stillinger-Weber type. For this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity. This value is given by the corresponding value for a optimal hexagonal lattice, optimized with respect to the lattice spacing. Furthermore, under suitable periodic or Dirichlet boundary condition, we show that the minimizers do form a hexagonal lattice.

Keywords

Bond Angle Side Length Hexagonal Lattice Triangular Lattice Quadratic Variation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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