Journal of Nonlinear Science

, Volume 28, Issue 5, pp 1629–1656 | Cite as

On Born’s Conjecture about Optimal Distribution of Charges for an Infinite Ionic Crystal

  • Laurent Bétermin
  • Hans Knüpfer


We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born’s conjecture about the optimality of the rock salt alternate distribution of charges on a cubic lattice (and more generally on a d-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results hold for a class of completely monotone interaction potentials which includes Coulomb-type interactions for \(d\ge 3\). In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.


Calculus of variations Lattice energy Theta functions Electrostatic energy Ewald summation 

Mathematics Subject Classification

Primary 49S99 Secondary 82B20 



LB is grateful for the support of MATCH during his stay in Heidelberg. Both authors would like to thank Florian Nolte for interesting discussions.


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Copyright information

© Springer Science+Business Media, LLC, 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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