Local optimality of cubic lattices for interaction energies
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Local optimality of cubic lattices for interaction energies. / Bétermin, Laurent.
I: Analysis and Mathematical Physics, Bind 9, 2019, s. 403–426.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Local optimality of cubic lattices for interaction energies
AU - Bétermin, Laurent
PY - 2019
Y1 - 2019
N2 - We study the local optimality of simple cubic, body-centred-cubic and face-centred-cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola (Math Proc Camb 60:855–875, 1964), we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optimality of these lattices for the theta function and the Lennard–Jones-type energies. In particular, we prove the local minimality of the FCC lattice (resp. BCC lattice) for large enough (resp. small enough) values of its scaling parameter and we also prove the fact that the simple cubic lattice is a saddle point of the energy. Furthermore, we prove the local minimality of the FCC and the BCC lattices at high density (with an optimal explicit bound) and its local maximality at low density in the Lennard–Jones-type potential case. We then show the local minimality of FCC and BCC lattices among all the Bravais lattices (without a density constraint). The largest possible open interval of density’s values where the simple cubic lattice is a local minimizer is also computed.
AB - We study the local optimality of simple cubic, body-centred-cubic and face-centred-cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola (Math Proc Camb 60:855–875, 1964), we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optimality of these lattices for the theta function and the Lennard–Jones-type energies. In particular, we prove the local minimality of the FCC lattice (resp. BCC lattice) for large enough (resp. small enough) values of its scaling parameter and we also prove the fact that the simple cubic lattice is a saddle point of the energy. Furthermore, we prove the local minimality of the FCC and the BCC lattices at high density (with an optimal explicit bound) and its local maximality at low density in the Lennard–Jones-type potential case. We then show the local minimality of FCC and BCC lattices among all the Bravais lattices (without a density constraint). The largest possible open interval of density’s values where the simple cubic lattice is a local minimizer is also computed.
U2 - 10.1007/s13324-017-0205-5
DO - 10.1007/s13324-017-0205-5
M3 - Journal article
VL - 9
SP - 403
EP - 426
JO - Analysis and Mathematical Physics
JF - Analysis and Mathematical Physics
SN - 1664-2368
ER -
ID: 187631482