On Born’s Conjecture about Optimal Distribution of Charges for an Infinite Ionic Crystal
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.
KeywordsCalculus of variations Lattice energy Theta functions Electrostatic energy Ewald summation
Mathematics Subject ClassificationPrimary 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.
- Bétermin, L.: Local optimality of cubic lattices for interaction energies. Anal. Math. Phys. (2017). https://doi.org/10.1007/s13324-017-0205-5
- Bétermin, L., Knüpfer, H.: Optimal lattice configurations for interacting spatially extended particles. Lett. Math. Phys. (2018). https://doi.org/10.1007/s11005-018-1077-9
- Borwein, J., Glasser, M., McPhedran, R., Wan, J., Zucker, I.: Lattice Sums Then and Now (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2013). https://doi.org/10.1017/CBO9781139626804
- Coulangeon, R., Schürmann, A.: Local energy optimality of periodic sets. (Preprint) arXiv:1802.02072 (2018)
- E, W., Li, D.: On the crystallization of 2D hexagonal lattices. Commun. Math. Phys. 286, 1099–1140 (2009)Google Scholar
- Henn, A.: The hexagonal lattice and the Epstein zeta function. In: Dynamical Systems, Number Theory and Applications, pp. 127–140 (2016). https://doi.org/10.1142/9789814699877_0007
- Schiff, J.L.: The Laplace transform: theory and applications. Springer, Berlin (2013)Google Scholar