Optimal and non-optimal lattices for non-completely monotone interaction potentials

Bétermin, Laurent; Petrache, Mircea

doi:10.1007/s13324-019-00299-6

Cited by 26 publications

(80 citation statements)

References 72 publications

(129 reference statements)

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“…We observe that the Morse potential V M , for r 0 = 1 fixed, converges to −δ(x − 1) as α → +∞ (see Figure 1). A similar proof as the one we have done for the Lennard-Jones type potential in [16,Thm 1.13] shows the global optimality of a lattice achieving the kissing number with balls of radius 1/2, for sufficiently large α. In particular, in dimension 2 (resp.…”

Section: Confirmation Of Our Conjecture In Dimensionsupporting

confidence: 70%

“…This also holds for the Lennard-Jones potential for large exponents (see e.g. [3,16]). Even though P 3 hcp ∈ L 3 , using formula (1.10) we have computed the numerical values of E α,r 0 [λ hcp].…”

Section: Introduction and Main Resultsmentioning

confidence: 73%

“…The two most important observations are the following: 1. Global minimizer: for all α, r 0 , the global minimizer of E α,r 0 in L 2 seems to be a triangular lattice, as it seems to be the case for Lennard-Jones type potentials (see [8,10,16]).…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…It is a natural first step for keeping or rejecting periodic structures which could be good candidates for the Crystal Problem associated to the interaction potential. We have already studied the Lennard-Jones type potential case in [8,9,10,16,17] and our goal is to give the same kind of quantitative results for the Morse potential.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Furthermore, the non-optimality of the triangular lattice at low density (i.e. large area) can be shown as a direct consequence of [16,Thm 1.5], which is itself a consequence of Montgomery Theorem [49, Thm 1] and the functional equation for the lattice theta function [49,Eq. (2)] defined by…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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: Confirmation Of Our Conjecture In Dimensionsupporting

confidence: 70%

Section: Introduction and Main Resultsmentioning

confidence: 73%

Section: Introduction and Main Resultsmentioning

confidence: 97%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Bétermin

2019

Journal of Mathematical Physics

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Effect of Periodic Arrays of Defects on Lattice Energy Minimizers

Bétermin

2021

Ann. Henri Poincaré

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We consider interaction energies $$E_f[L]$$ E f [ L ] between a point $$O\in {\mathbb {R}}^d$$ O ∈ R d , $$d\ge 2$$ d ≥ 2 , and a lattice L containing O, where the interaction potential f is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for $$E_f$$ E f when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodic arrays of substitutional defects). We consider separately the non-shifted ($$O\in k L$$ O ∈ k L ) and shifted ($$O\not \in k L$$ O ∉ k L ) cases and we derive several general conditions ensuring the (non-)optimality of a universal optimizer among lattices for the new energy including defects. Furthermore, in the case of inverse power laws and Lennard-Jones-type potentials, we give necessary and sufficient conditions on non-shifted periodic vacancies or substitutional defects for the conservation of minimality results at fixed density. Different examples of applications are presented, including optimality results for the Kagome lattice and energy comparisons of certain ionic-like structures.

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