Barlow Packings All Have Maximum Packing Fraction: Why it is so Hard to Find Barlow Structures Other than fcc or hcp?

Abstract

We apply lattice theory to periodic Barlow packings $(X_{1}X_{2}\dots X_{N})_{\infty}$ where $X_{i}\in\{A,B,C\}$ and $X_{i}\neq X_{i+1}$, formed by stacking two-dimensional hexagonal close-packed layers, interacting via generalized $(n,m)$-Lennard-Jones potentials. The cohesive energies of all packings lie between those of hexagonal close-packed (hcp, $(AB)$) and face-centered cubic (fcc, $(ABC)$) structure and vary almost linearly with the fcc stacking fraction. The equilibrium lattice constant, bulk modulus, Einstein frequency, and vibrational entropy show similarly near-linear trends. For the standard $(12,6)$ potential, fcc has the highest vibrational entropy and hcp the lowest, with other Barlow packings falling between them, consistent with previous sticky-hard-sphere Monte Carlo simulations and supported by phonon dispersion calculations. These trends imply an approximately linear dependence of free energy on fcc stacking fraction, with hcp and fcc as endpoints. This provides a simple thermodynamic explanation for why intermediate Barlow stackings are rare in nature: near-linearity offers little driving force to favor them. Exceptions may arise when hcp and fcc are nearly degenerate in energy or when many-body interactions, absent from the present model, favor another stacking. Density-functional calculations including dispersion forces for argon Barlow packings with periods $N=2,3,4$, and $12$ corroborate the predicted near-linear cohesive energy dependence.