{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"4d77d65f7db702f86cd588a32ee9b671f19af9b6910ff7a378911fbcc007ab99","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"28378b3b047f4787ae6f328e4e1e5ed879615fa1e6e17d04a641151392edf98d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystallization.lj-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Prove that many particles interacting through the standard Lennard-Jones force (repulsive at short range, attractive at long range) have a lowest-energy arrangement that is a periodic crystal. Computations suggest the hexagonal close-packed crystal.","posed_since":"","precise":"For $V(r)=4\\epsilon ((s/r)^{12}-(s/r)^6)$, let $x_1..x_N \\text{ in } R^3$ minimize $E_N=\\sum_{i<j} V(\\mid x_i-x_j\\mid)$. Prove that $\\min E_N/N$ converges to the energy per particle of one periodic configuration (expected hcp, nearly degenerate with fcc) and that minimizers, after a rigid motion, coincide with it outside a boundary layer of $o(N)$ particles. A proof for any pair potential with a repulsive core and an attractive tail in 3D would also answer the qualitative question.","problem_ref":null,"references":"","settled_by":"A proof of convergence of $N$-particle minimizers to a periodic lattice for Lennard-Jones or a comparable two-body potential in $R^{3}$.","status_note":"Proved in 3D only for a tailored potential with a three-body term and fcc ground state (Flatley, Theil, ARMA 2015).","title":"Periodic ground states for the Lennard-Jones potential in 3D","topic_ref":"00bca35d0adb193ca8c103df758e6ebf929de8f1f684bf0377fdce9923d82670"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"10d56200699267ad15a619ab84fe229ec72c6c949a61ee6b17d7ba46cefef59d","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"e7301c86cd87f65afbdc5475fd83c0d94529a6b38f372a85b77f91b0eb4f4d18","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"ZTK3P9jsNV9S2nHh7Fv3d53q8fBYndSQbuNpRx4Jlt_mHM9kOEgE3VEa0ycMk6YOw9JhrS7uiJfuO0tQdNoZAw"},"schema":"pubphys.envelope/1"},"record_hash":"10d56200699267ad15a619ab84fe229ec72c6c949a61ee6b17d7ba46cefef59d","leaf_index":1610}