Close-packed ground states of sticky hard spheres · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

Title
Close-packed ground states of sticky hard spheres
Plain
Model atoms as hard balls that gain energy only when they touch. Prove that the best arrangement of many such balls stacks flat triangular layers, as in fcc or hcp crystals.
Precise · changed · core
Revision 1: For $V(r)=+\infty \text{ for } r<1, -1 \text{ for } r=1, 0 \text{ for } r>1 \text{ in } R^3$, minimizing $E_N$ is maximizing the contact number $c(N)$ of $N$ non-overlapping unit-diameter spheres. Prove that for large N every minimizer coincides, except for $o(N)$ spheres, with a subset of a close packing (a stacking of triangular layers), and determine whether $\operatorname{lim}(6N-c(N))/N^{2/3}$ exists and, if so, its value (expected to be set by the surface energy of the optimal close-packed shape).
Revision 2: For $V(r)=+\infty \text{ for } r<1, -1 \text{ for } r=1, 0 \text{ for } r>1 \text{ in } R^3$, minimizing $E_N$ is maximizing the contact number $c(N)$ of $N$ non-overlapping unit-diameter spheres. Prove that for large N every minimizer coincides, except for $o(N)$ spheres, with a subset of a close packing (a stacking of triangular layers), and determine whether $\lim(6N-c(N))/N^{2/3}$ exists and, if so, its value (expected to be set by the surface energy of the optimal close-packed shape).
Settled by
A structure theorem for maximal-contact configurations of N spheres together with the sharp surface constant.
Kind
well-posed
Answer type
proof
References
(empty)
Literature status
open
Status note
The 2D sticky-disk problem is solved (Heitmann, Radin 1980); in 3D only bounds on $c(N)$ of the form $6N - C N^{2/3}$ are known.
Posed since
(empty)

Decisions