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On the Crystallization of 2D Hexagonal Lattices

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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.

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Authors and Affiliations

  1. Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Weinan E

  2. School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA

    Dong Li

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  1. Weinan E
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  2. Dong Li
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Correspondence to Dong Li.

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Communicated by G. Gallavotti

Dedicated with admiration to Professor Tom Spencer on occasion of his 60th birthday

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E, W., Li, D. On the Crystallization of 2D Hexagonal Lattices. Commun. Math. Phys. 286, 1099–1140 (2009). https://doi.org/10.1007/s00220-008-0586-2

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  • DOI: https://doi.org/10.1007/s00220-008-0586-2

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