Crystallization in the hexagonal lattice for ionic dimers

Friedrich, Manuel; Kreutz, Leonard

doi:10.1142/s0218202519500362

Cited by 18 publications

(31 citation statements)

References 33 publications

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“…Answering this question in a rigorous mathematical way is known to be extremely challenging, even though the interactions between atoms or molecules are assumed to be a sum of pairwise potentials. Whereas the one-dimensional version of this problem is wellunderstood [14,35,65,66,67], only few results have been proved in dimensions 2 and 3 for models consisting of short-range interactions [34,38,43,44,45], perturbations of the hard-sphere potential [30,33,63] and oscillating functions [61].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Minimizing lattice structures for Morse potential energy in two and three dimensions

Bétermin

2019

Journal of Mathematical Physics

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We investigate the local and global optimality of the triangular, square, simple cubic, facecentred-cubic (FCC), body-centred-cubic (BCC) lattices and the hexagonal-close-packing (HCP) structure for a potential energy per point generated by a Morse potential with parameters (α, r 0 ). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all twodimensional Bravais lattices with respect to their density. The behaviour of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, FCC and BCC lattices are checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, FCC and BCC lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities, as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from BCC, FCC to HCP, as α increases, that we partially and heuristically explain from the lattice theta functions properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Minimizing lattice structures for Morse potential energy in two and three dimensions

Bétermin

2019

Journal of Mathematical Physics

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“…Sharp constants for this law were then established in [17] and the uniqueness of ground states was characterized in [19]. We also mention extensions to other crystals [16,40,42] and dimers [26,27]. By way of contrast, in dimension three or higher the recent results [11,39,41] characterize optimal energy configurations within classes of lattices and are in this sense conditional to crystallization.…”

Section: Introductionmentioning

confidence: 95%

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Friedrich

Kreutz

Schmidt

2021

Arch Rational Mech Anal

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We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the ‘sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of $$\Gamma $$ Γ -convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, that is, on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. We further provide a fine analysis of the surface energies at grain boundaries both for vacuum–solid and solid–solid phase transitions. The latter relies fundamentally on a structure result for grain boundaries showing that, due to the extremely brittle setup, interpolating boundary layers near cracks are energetically not favorable.

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“…Using three different radially symmetric short-range interaction potentials, he proved the minimality of a two-dimensional binary quasiperiodic configuration. Furthermore, Friedrich and Kreutz [ 16 , 17 ] have shown the optimality of two-dimensional structures with alternating charges. They explored the possibilities to obtain a rock-salt structure or an alternate honeycomb lattice as minimizer of an interaction energy involving only short-range radially symmetric potentials.…”

Section: Introductionmentioning

confidence: 99%

Note on Crystallization for Alternating Particle Chains

2020

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We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

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Crystallization in the hexagonal lattice for ionic dimers

Cited by 18 publications

References 33 publications

Minimizing lattice structures for Morse potential energy in two and three dimensions

Minimizing lattice structures for Morse potential energy in two and three dimensions

Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy

Note on Crystallization for Alternating Particle Chains

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