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arXiv · 2107.14020

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

Laurent Bétermin·Ladislav Šamaj·Igor Travěnec
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Abstract

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones type potentials $f_{n,m}^{\rm{LJ}}(r):=a r^{-n}-b r^{-m}$, $n>m$ 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 with exponents $3<m<n$, 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 $\Delta$ between the inter-layer distance $d$ and the lattice parameter $a$ is studied as $V$ increases. In the critical region of exponents $0<m<n<3$, the SH phase with an extreme value of the anisotropy parameter $\Delta$ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For $-2<m<n<0$, the BCC lattice is the only energy minimizer. Choosing $-4<m<n<-2$, the FCC and SH latices become minimizers.

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Laurent Bétermin, Ladislav Šamaj, Igor Travěnec. 2021-07-29. Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies. https://doi.org/10.1111/sapm.12533

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