Classification of Particle Numbers with Unique Heitmann–Radin Minimizer


We show that minimizers of the Heitmann–Radin energy (Heitmann and Radin in J Stat Phys 22(3):281–287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in De Luca and Friesecke (Crystallization in two dimensions and a discrete Gauss–Bonnet Theorem, 2016).

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We are indebted to Michael Baake for pointing out to us the website [12] after attending a lecture on the work reported here. The research of LDL was funded, and that of GF partially supported, by the DFG Collaborative Research Center CRC 109 “Discretization in Geometry and Dynamics”.

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Correspondence to Lucia De Luca.

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De Luca, L., Friesecke, G. Classification of Particle Numbers with Unique Heitmann–Radin Minimizer. J Stat Phys 167, 1586–1592 (2017).

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  • Crystallization
  • Wulff shape
  • Heitmann–Radin potential
  • Discrete differential geometry
  • Energy minimization