Local optimality of cubic lattices for interaction energies

Local optimality of cubic lattices for interaction energies

We study the local optimality of Simple Cubic, Body-Centred-Cubic and Face-Centred-Cubic lattices among Bravais lattices of fixed density for some finite energy per point. Following the work of Ennola [Math. Proc. Cambridge, 60:855–875, 1964], we prove that these lattices are critical points of all the energies, we write the second derivatives in a simple way and we investigate the local optima...

Electrostatic interaction energies of homogeneous cubic charge distributions

The starting point is the problem of finding the interaction energy of two coinciding homogeneous cubic charge distributions. The brute force method of subdividing the cube into N sub-cubes and doing the sums results in slow convergence because of the Coulomb singularity. Using symmetry and algebra the Coulomb singularities can be eliminated. This leads to an accurate numerical algorithm as wel...

An equivalence functor between local vector lattices and vector lattices

We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-...

Extended Watson integrals for the cubic lattices.

The known exact expressions for extended Watson integrals relating to various cubic and modified cubic lattices are summarized. A new closed form expression for Watson's result on the simple cubic lattice is given in terms of gamma functions.

Interfacial energies on Penrose lattices