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

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Abstract

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.

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Notes

  1. “Ein endliches Stück eines einfachen kubischen Raumgitters soll so mit gleich vielen positiven und negativen Ladungen von gleichem absoluten Betrage besetzt werden, daß die elektrostatische Energie des Systems möglichst klein wird.”

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Acknowledgements

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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Authors and Affiliations

  1. QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Laurent Bétermin

  2. Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Hans Knüpfer

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  1. Laurent Bétermin
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  2. Hans Knüpfer
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Correspondence to Laurent Bétermin.

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Communicated by Michael Ward.

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Bétermin, L., Knüpfer, H. On Born’s Conjecture about Optimal Distribution of Charges for an Infinite Ionic Crystal. J Nonlinear Sci 28, 1629–1656 (2018). https://doi.org/10.1007/s00332-018-9460-3

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