Face-Centered Cubic Crystallization of Atomistic Configurations

We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance as it presents a significant milestone towards understanding, on the atomistic level, phenomena such as melting or plastic behavior. We characterize a set of rotation- and translation-invariant two- and three-body potentials V2, V3 such that the energy minimum of 1#YE(Y)=1#Y2∑{y,y′}⊂YV2(y,y′)+6∑{y,y′,y′′}⊂YV3(y,y′,y′′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\#Y}E(Y) = \frac{1}{\# Y} \left(2\sum_{\{y,y'\} \subset Y}V_2(y, y') + 6\sum_{\{y,y',y''\} \subset Y} V_3(y,y',y'')\right)$$\end{document}over all Y⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Y \subset \mathbb{R}^3}$$\end{document}, #Y = n, converges to the energy per particle in the face-centered cubic (fcc) lattice as n tends to infinity. The proof involves a careful analysis of the symmetry properties of the fcc lattice.