Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Abstract

Abstract. We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential V (r) = r −6 − 2r −12 , where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

Minimizing lattice structures for Morse potential energy in two and three dimensions

“…Answering this question in a rigorous mathematical way is known to be extremely challenging, even though the interactions between atoms or molecules are assumed to be a sum of pairwise potentials. Whereas the one-dimensional version of this problem is wellunderstood [14,35,65,66,67], only few results have been proved in dimensions 2 and 3 for models consisting of short-range interactions [34,38,43,44,45], perturbations of the hard-sphere potential [30,33,63] and oscillating functions [61].…”

Crystallization in the hexagonal lattice for ionic dimers

“…(8). This already gives the upper bound in (9). Moreover, we will construct explicitly configurations C n with net charge (see (6)) of the order n 1/4 which establishes Theorem 2.5(ii).…”

“…(ii) The ground states for n ≤ 29 can also be characterized, but due to the smallness of the structures, more degeneracies can occur. In particular, for n = 8, 9,12,15,18,21,29 there might be one octagon at the boundary. (We refer to Remark 7.11 for more details.)…”

“…For n = 10, 11, the presence of an octagon is excluded by Proposition 7.9 and the fact that each hexagon can share at most 2 atoms with an octagon (see Lemma 7.5 and Lemma 7.6). Consequently, for n ≤ 29, ground states may contain a boundary octagon only for n = 8,9,12,15,18,21,29. This is indeed possible as shown in Fig.…”

“…In two dimensions for a finite number of identical particles, ground states have been proved to be subsets of the triangular lattice by Heitman, Radin, and Wagner [20,27,35] for sticky and soft, purely two-body interaction energies. Recently, a new perspective on this problem was given by De Luca and Friesecke [9] using a discrete Gauss-Bonnet method from discrete differential geometry. Including angular potentials favoring 2π/3 bond-angles, which is typically the case for graphene, crystallization in the hexagonal lattice has been proved by Mainini and Stefanelli [25], see also [14] for a related classification of ground states.…”