Optimal triangular lattice crystallization and exponential decay energy minimization problem for the attractive-repulsive model

In this paper, we discuss lattice energy minimization problems for various interacting potentials, including Born-Mayer, Coulomb, Lennard-Jones, and Rydberg potentials. The main methods and techniques used in this paper are derived from Bétermin, SIAM J. Math. Anal. 48, 3236–3269 (2016). Based on this, we derive a more comprehensive attractive-repulsive model that can be understood in the sense of exponential decay. Our research leads to a new lower bound for the inverse density of the triangular lattice when the Lennard-Jones potential with exponential decay is minimal, which is a significant improvement over previous results. More specifically, in Bétermin, SIAM J. Math. Anal. 48, 3236–3269 (2016), the degree of the lowest order term of the Lennard-Jones potential is to satisfy x1 > 3/2, but our result covers the interval (1, 3/2], making it the optimal case with a lower bound of 1. We extend the results for Morse potential to derive the optimal attractive-repulsive correlation coefficient conditions, i.e., ln(b2: b1) ≥ 2.3853 and ln(b2: b1) ≥ 7.3069. Also, our findings provide the first-ever inverse density bound when the triangular lattice is optimal for the Coulomb and Rydberg potentials, while the former gives global optimality without density constraints, and the latter gives upper and lower bounds on the density interval for optimality. Finally, we present numerical estimates of the inverse density, and these findings have critical applications in improving numerical simulations of radiation defects in crystals, especially in the α-iron lattice, and better understanding the properties of rare gases. Our solutions and improved estimates will enable researchers to better understand lattice energy minimization.