Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

Bétermin, Laurent

doi:10.48550/arxiv.2104.09795

Cited by 4 publications

(6 citation statements)

References 32 publications

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“…The same holds for the Lennard-Jones type energy E LJ n,m . Indeed, the two-dimensional ground state of E LJ n,m in L 2 has been proven to be a triangular lattice for many exponents (n, m) by the first author [3,8]. Furthermore, we have observed and partially proved a phase transition of type "triangular-rhombic-squarerectangular" for the minimizer of E LJ n,m in L 2 (V ) as V increases, see [4,64].…”

Section: Introduction Setting and Main Resultsmentioning

confidence: 57%

“…To compute numerically the minimizers, we have used the optimization tool FindMinimum in Mathematica, which includes ConjugateGradient, PrincipalAxis, LevenbergMarquardt, Newton, QuasiNewton, InteriorPoint, and LinearProgramming methods. It has to be noticed that, as in [59], we can systematically restrict our study to a compact domain of G 3 since our energy diverges or goes to zero as the parameters (u, v) go to infinity (see also [8] for such example in two dimensions).…”

Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

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Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Bétermin¹,

Šamaj²,

Travěnec³

2021

Preprint

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The Riesz potential fs(r) = r −s is known to be an important building block of many interactions, including Lennard-Jones type potentials f LJ n,m (r) := ar −n − br −m , n > m that are widely used in Molecular Simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices (SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s < 0 and the HCP lattice is computed to have higher energy than the FCC (for s > 3/2) and BCC (for s < 3/2) lattices. In the Lennard-Jones case, the ground state among lattices is confirmed to be a FCC lattice whereas a HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type "FCC-SH" and "FCC-HCP-SH" (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents (n, m). In the SH phase, the variation of the ratio ∆ between the inter-layer distance d and the lattice parameter a is studied as V increases.

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Section: Introduction Setting and Main Resultsmentioning

confidence: 57%

Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Bétermin¹,

Šamaj²,

Travěnec³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many physical, chemical and number theoritic problems can be formulated to the following minimization problem on lattices: (1.1) See e.g. [2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,28,29,38,39,36,34,40,42,43,44,45,46,47,4]. The summation ranges over all the lattice points except for the origin 0 and the function f denotes the potential of the system.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On minima of difference of theta functions and application to hexagonal crystallization

Luo

Wei

2022

Math. Ann.

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Let L = 1 Im(z) Z ⊕ zZ where z ∈ H = {z = x + iy or (x, y) ∈ C : y > 0} be the two dimensional lattices with unit density. Assuming that α ≥ 1, we prove that min L P∈L,|L|=1|P| 2 e −πα|P| 2 is achieved at hexagonal lattice. More generally we prove that for α ≥ 1 minis achieved at hexagonal lattice for b ≤ 1 2π and does not exist for b > 1 2π . As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in [1,7,10,14] and extend the main results in [34] on minima of difference of two theta functions.

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“…Indeed, the two-dimensional ground state of 𝐸 LJ 𝑛,𝑚 in  2 has been proven to be a triangular lattice for many exponents (𝑛, 𝑚) by the first author. 59,60 We aim to present a complete picture of the lattice ground states of 𝐿 ↦ 𝜁 𝐿 (𝑠) and 𝐸 LJ 𝑛,𝑚 in  3 and  3 (𝑉). In particular, we want to show minimality properties of the following important threedimensional lattices (given here with unit density):…”

Section: Minimization Among Lattices and Settingmentioning

confidence: 99%

Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

2022

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The Riesz potential fs(r)=r−s$f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard‐Jones–type potentials fn,mLJ(r):=ar−n−br−m$f_{n,m}^{\rm {LJ}}(r):=a r^{-n}-b r^{-m}$, n>m$n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three‐dimensional lattices of Riesz and Lennard‐Jones energies. We discuss the minimality of the body‐centered‐cubic (BCC) lattice, face‐centered‐cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close‐packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s<0$s<0$ and the HCP lattice is computed to have higher energy than the FCC (for s>3/2$s>3/2$) and BCC (for s<3/2$s<3/2$) lattices. In the Lennard‐Jones case with exponents 3

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Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

Cited by 4 publications

References 32 publications

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

On minima of difference of theta functions and application to hexagonal crystallization

Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

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