{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1f9e8d36ac44fb8f721f33cfeafcb8e5767b1ee7a794c50b1ecdf81237d790a3","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"cb227b6f0f7e97d9b6eedee1fe299684c2ecf0ca722cab5c62e94a238432d043","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystallization.lj-2d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"In the plane, particles with Lennard-Jones forces should arrange into the triangular lattice. Existing proofs work only for forces with a much narrower attractive well.","posed_since":"","precise":"For the Lennard-Jones pair potential in $R^2$, prove that $N$-particle minimizers of $E_N$ converge (energy per particle and structure away from an $o(N)$ boundary layer) to the triangular lattice with the spacing that minimizes the lattice energy. Theil's 2006 proof and its extensions (E, Li 2009) need a narrow potential well and steep core, which excludes Lennard-Jones.","problem_ref":null,"references":"","settled_by":"An extension of the 2D crystallization proof to the Lennard-Jones potential itself.","status_note":"Optimality of the triangular lattice for Lennard-Jones is known only within the restricted class of Bravais lattices and for some densities.","title":"Triangular-lattice crystallization for the 2D Lennard-Jones potential","topic_ref":"00bca35d0adb193ca8c103df758e6ebf929de8f1f684bf0377fdce9923d82670"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"57e837ab2b857dd5662b451587626bba44ea4bd7d6465a00df48d97e91e56191","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"2e2981079d5d3c109c974cc31080d15fec440aa563ff02ed88338e5cf51b8259","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"UdZXItJbAHhtLGdeUjWJEAvcLztNlPcay3dzhcRZ-lLIV4ca2e8_GsjCUhWCWjgG7YhNauD1PGC3FaNCTdGkBg"},"schema":"pubphys.envelope/1"},"record_hash":"57e837ab2b857dd5662b451587626bba44ea4bd7d6465a00df48d97e91e56191","leaf_index":1609}