Local optimality of cubic lattices for interaction energies

Journal of Mathematical Physics

2019

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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.

Section: Lemma 212 (Local Optimality Of the Triangular Lattice At Hisupporting

confidence: 92%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Confirmation Of Our Conjecture In Dimensionmentioning

confidence: 99%

See 2 more Smart Citations

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

Journal of Mathematical Physics

2019

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“…II.2]. Then, as already discussed in [9], Corollary 5.1 is of great interest for the understanding of BCC and FCC stability in the smeared out particle case, using dimension reduction techniques as in [13].…”

Section: Let Zmentioning

confidence: 91%

Minimal Soft Lattice Theta Functions

2020

Constr Approx

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We study the minimality properties of a new type of "soft" theta functions. For a lattice L ⊂ R d , a L-periodic distribution of mass µ L and an other mass ν z centred at z ∈ R d , we define, for all scaling parameter α > 0, the translated lattice theta function θ µ L +νz (α) as the Gaussian interaction energy between ν z and µ L . We show that any strict local or global minimality result that is true in the point case µ = ν = δ 0 also holds for L → θ µ L +ν0 (α) and z → θ µ L +νz (α) when the measures are radially symmetric with respect to the points of L∪{z} and sufficiently rescaled around them (i.e. at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies and an approximation argument. Furthermore, for the honeycomb lattice H, the center of any primitive honeycomb is shown to minimize z → θ µ H +νz (α) and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centred-cubic and face-centred-cubic lattices.

Effect of Periodic Arrays of Defects on Lattice Energy Minimizers

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.