On Born's conjecture about optimal distribution of charges for an infinite ionic crystal
arXiv:1704.02887 · doi:10.1007/s00332-018-9460-3
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 holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.
32 pages. 3 Figures. To appear in Journal of Nonlinear Science. DOI :10.1007/s00332-018-9460-3
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- A nonlocal isoperimetric problem with dipolar repulsion
- Optimal lattice configurations for interacting spatially extended particles
- Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies
- Finite crystallization and Wulff shape emergence for ionic compounds in the square lattice
- A variational principle for Gaussian lattice sums
- The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators