A Crystallization Result in Two Dimensions for a Soft Disc Affine Potential

We prove finite crystallization for particles in the plane interacting through a soft disc potential, as originally shown by Radin [9]. We give an alternative proof that relies on the geometric decomposition of the energy proved in [6], and that extends the range of validity from 25/24 to the optimal value \(1/(2\sin \tfrac{\pi }{7})\) . The strategy is based on showing that any minimizer has at least as many boundary points as the canonical “spiral” configuration.