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Theory of dynamical stability for two- and three-dimensional Lennard-Jones crystals

The dynamical stability of three-dimensional (3D) Lennard-Jones (LJ) crystals has been studied for many years. The face-centered-cubic and hexagonal close packed structures are dynamically stable, while the body-centered cubic structure is stable only for long range LJ potentials that are characterized by relatively small integer pairs $(m,n)$. Here, we study the dynamical stability of two-dimensional (2D) LJ crystals, where the planar hexagonal, the buckled honeycomb, and the buckled square structures are assumed. We demonstrate that the stability property of 2D and 3D LJ crystals can be classified into four groups depending on $(m,n)$. The instabilities of the planar hexagonal, the buckled square, and the body-centered cubic structures are investigated within analytical expressions. The structure-stability relationship between the LJ crystals and the elemental metals in the periodic table is also discussed.