### 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.

Original language | English (US) |
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Pages (from-to) | 1099-1140 |

Number of pages | 42 |

Journal | Communications In Mathematical Physics |

Volume | 286 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2009 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*286*(3), 1099-1140. https://doi.org/10.1007/s00220-008-0586-2