On the crystallization of 2D hexagonal lattices. (English) Zbl 1180.82191
From the authors’ abstract: It is a fundamental problem to understand why solids form crystals 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 expressed as the sum of a pair potential of Lennard-Jones type and 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 an 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.
Reviewer: Erdoǧan S. Şuhubi (İstanbul)
References:
| [1] | Blanc X., Le Bris C.: Periodicity of the infinite-volume ground state of a one-dimensional quantum model. Nonlinear Anal. Ser. A: Theory Methods 48(6), 791–803 (2002) · Zbl 0992.82043 · doi:10.1016/S0362-546X(00)00215-7 |
| [2] | Friesecke G., James R., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55, 1461–1506 (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048 |
| [3] | Gardner C.S., Radin C.: The infinite-volume ground state of the Lennard-Jones potential. J. Stat. Phys. 20(6), 719–724 (1979) · doi:10.1007/BF01009521 |
| [4] | Hamrick G.C., Radin C.: The symmetry of ground states under perturbation. J. Stat. Phys. 21, 601–607 (1979) · doi:10.1007/BF01011171 |
| [5] | Heitman R., Radin C.: Ground states for sticky disks. J. Stat. Phys. 22, 281–287 (1980) · doi:10.1007/BF01014644 |
| [6] | John F.: Rotation and strain. Comm. Pure. Appl. Math. 14, 391–413 (1961) · Zbl 0102.17404 · doi:10.1002/cpa.3160140316 |
| [7] | Kohn R.V.: New integral estimates for deformations in terms of their nonlinear strain. Arch. Mech. Anal. 78, 131–172 (1982) · Zbl 0491.73023 · doi:10.1007/BF00250837 |
| [8] | Müller S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Part. Differ. Eqs. 1(2), 169–204 (1993) · Zbl 0821.49015 · doi:10.1007/BF01191616 |
| [9] | Radin C.: The ground state for soft disks. J. Stat. Phys. 26(2), 367–372 (1981) · doi:10.1007/BF01013177 |
| [10] | Radin C.: Classical ground states in one dimension. J. Stat. Phys. 35, 109–117 (1983) · doi:10.1007/BF01017368 |
| [11] | Radin C., Schulmann L.S.: Periodicity of classical ground states, Phys. Rev. Lett. 51(8), 621–622 (1983) · doi:10.1103/PhysRevLett.51.621 |
| [12] | Radin C.: Low temperature and the origin of crystalline symmetry. Int. J. Mod. Phys. B 1, 1157–1191 (1987) · doi:10.1142/S0217979287001675 |
| [13] | Rickman, S.: Quasiregular mappings. Berlin Heidelberg-New York Springer-Verlag, 1993 · Zbl 0816.30017 |
| [14] | Theil F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2005) · Zbl 1113.82016 · doi:10.1007/s00220-005-1458-7 |
| [15] | Ventevogel W.J.: On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Phys. A. 92, 343–361 (1978) · doi:10.1016/0378-4371(78)90136-X |
| [16] | Ventevogel W.J., Nijboer B.R.A.: On the configuration of systems of interacting particle with minimum potential energy per particle. Phys. A. 98, 274–288 (1979) · doi:10.1016/0378-4371(79)90178-X |
| [17] | Ventevogel W.J., Nijboer B.R.A.: On the configuration of systems of interacting particles with minimum potential energy per particle. Phys. A. 99, 565–580 (1979) · doi:10.1016/0378-4371(79)90072-4 |
| [18] | Nijboer B.R.A., Ruijgrok Th.W.: On the minimum-energy configuration of a one-dimensional system of particles interacting with the potential \({\phi(x)=(1+x^4)^{-1}}\) . Phys. A. 133, 319–329 (1985) · doi:10.1016/0378-4371(85)90071-8 |
| [19] | Yedder, A.B.H., Blanc, X., Le Bris, C.: A numerical investigation of the 2-dimensional crystal problem. Preprint CERMICS (2003), available at http://www.ann.jussieu.fr/publications/2003/R03003.html , 2003 |
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