Communications in Mathematical Physics

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

On the Crystallization of 2D Hexagonal Lattices

  • Weinan E
  • Dong Li


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.


Bond Angle Side Length Hexagonal Lattice Triangular Lattice Quadratic Variation 
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© 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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