The BCC lattice in a long range interaction system

TL;DR

This paper proves that in a three-dimensional geometric variational problem with specific long-range interactions, the body centered cubic lattice is the optimal structure, highlighting its significance in systems modeled by diblock copolymer theory.

Contribution

The paper demonstrates that the body centered cubic lattice minimizes energy in a 3D variational problem with a nonlocal operator, advancing understanding of lattice preferences in such systems.

Findings

BCC lattice is energetically favored in the model

Long-range interactions influence lattice selection

Results connect to diblock copolymer structures

Abstract

While the hexagonal lattice is ubiquitous in two dimensions, the body centered cubic lattice and the face centered lattice are both commonly observed in three dimensions. A geometric variational problem motivated by the diblock copolymer theory consists of a short range interaction energy and a long range interaction energy. In three dimensions, and when the long range interaction is given by the nonlocal operator $(-\Delta)^{-3/2}$, it is proved that the body centered cubic lattice is the preferred structure.