Crystallization of one-dimensional alternating two-component systems

Laurent Bétermin (1) , Hans Knüpfer (2) , Florian Nolte (2)

1 Department of Mathematical Sciences [Copenhagen]
2 Universität Heidelberg [Heidelberg] = Heidelberg University

We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration – also called crystallization – is shown in various settings, at any scale and at high density. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also prove the crystallization at high density for Lennard-Jones type interactions and ionic screened potentials involving inverse power laws and Yukawa potentials. These high density results are derived from a general sufficient condition based on a convexity argument. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

Mots clés: Optimisation; Crystallization; Ionic crystals; Lennard-Jones potentials; Inverse power laws potentials; Two-component systems; Convexity

Domaines: Optimisation et contrôle [math.OC]; Physique mathématique [math-ph]

Laurent Bétermin, Hans Knüpfer, Florian Nolte. Crystallization of one-dimensional alternating two-component systems. 2018. ⟨hal-01761439v1⟩