Emergence of Wulff-Crystals from Atomistic Systems on the FCC and HCP Lattices

We consider a system of N hard spheres sitting on the nodes of either the FCC or HCP lattice and interacting via a sticky-disk potential. As N tends to infinity (continuum limit), assuming the interaction energy does not exceed that of the ground-state by more than N^{2 / 3} (surface scaling), we obtain the variational coarse grained model by Γ -convergence. More precisely, we prove that the continuum limit energies are of perimeter type and we compute explicitly their Wulff shapes. Our analysis shows that crystallization on FCC is preferred to that on HCP for N large enough. The method is based on integral representation and concentration-compactness results that we prove for general periodic lattices in any dimension.