The BCC lattice in a long range interaction system

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 $(-\mathrm{\Delta }{)}^{-3\mathrm{/}2}$, it is proved that the body centered cubic lattice is the preferred structure.