Letters in Mathematical Physics

, Volume 108, Issue 10, pp 2213–2228 | Cite as

Optimal lattice configurations for interacting spatially extended particles

  • Laurent Bétermin
  • Hans Knüpfer


We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.


Calculus of variations Lattice energy Triangular lattice Crystal 

Mathematics Subject Classification

Primary 74G65 Secondary 82B20 



LB is grateful for the support of the Mathematics Center Heidelberg (MATCH) during his stay in Heidelberg. He also acknowledges support from ERC advanced Grant Mathematics of the Structure of Matter (Project No. 321029) and from VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). HK is grateful about support from DFG Grant 392124319. The authors also thank the referees for their interesting suggestions and comments.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Institute of Applied Mathematics and IWRUniversity of HeidelbergHeidelbergGermany

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