Crystallinity in two dimensions: A note on a paper of C. Radin

Wagner, Heinz-Jürgen

doi:10.1007/bf01018831

Cited by 12 publications

(7 citation statements)

References 3 publications

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“…+ 6s = 3s 2 + 3s + 1 , whereas for general N ∈ (N s , N s+1 ) they are obtained by nestling the remaining points around the boundary of the regular hexagon constructed for N s . Our analysis extends [9] (see also [12]), where the above result was proved for δ = 1 24 using an ansatz on the value of the minimal energy (coinciding with the energy of the canonical configurations), previously found by Harborth in [7]. Such an approach had been previously exploited in [8] to treat the sticky disc model, corresponding to the choice δ = 0 in (1.1).…”

Section: Introductionsupporting

confidence: 60%

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

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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.

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Section: Introductionsupporting

confidence: 60%

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

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“…which, together with relation (25), entails (23). By (22) and (23) we finally obtain that −( −E +1) −2n + 2 2( −E +1) − 3n + 4 + 4.…”

Section: Ground States Are Squarementioning

confidence: 83%

“…by adapting an argument from [16,22]. To this end, let h j be the number of elementary j -gons in the bond graph and h be the total number of elementary polygons.…”

Section: Ground States Are Squarementioning

confidence: 99%

“…More precisely, in [13] the authors considered the crystallization of an ensemble of hard discs which maximize relative tangencies (see also [11]). In [16] and [22] the results were refined for firstneighbour interacting soft discs which allow interaction at distance and have been extended to quasicrystals in [17]. Instead, the emergence of a macroscopic Wulff shape for shortrange two-body interaction potentials has recently been investigated in [24].…”

Section: Introductionmentioning

confidence: 99%

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Finite crystallization in the square lattice

2014

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This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two-and three-body shortranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.

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“…Results on this "crystal problem" [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] have concentrated on classical mechanical models, mostly lattice gas models. The problem is essentially solved for one dimension, both for lattice gas [19] and continuum [20] models.…”

Section: Introductionmentioning

confidence: 99%