{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"4ff7f97e1fa1743143d76b974bd2cdf4e07f26868a3e3037f6512a47de8532ed","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"b112451b1c759a4ca37af0dc2b87a467ac79eb26a9e90b2d0116a4fb916dcd8a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystallization.jellium-bcc","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Point charges in a uniform background of opposite charge are expected to form a body-centered cubic crystal, the Wigner crystal. No proof exists that this arrangement has the lowest energy.","posed_since":"","precise":"For $N$ point charges $e$ in a neutralizing uniform background of density $\\rho0$ in $R^{3}$ (Coulomb $e^{2}/r$), prove that in the thermodynamic limit the minimal energy per particle equals the bcc value, about $-0.8959\\,e^{2}/r_{s}$ with $r_{s}=(3/(4\\pi \\rho0))^{1/3}$, and that minimizers are bcc up to $o(N)$ particles. The quantum version asks the same for the electron gas as $r_{s}\\to\\infty$.","problem_ref":null,"references":"","settled_by":"A proof that no configuration beats the bcc Madelung energy per particle in the thermodynamic limit.","status_note":"Open in both the classical and the quantum low-density version.","title":"Body-centered cubic ground state of classical 3D jellium","topic_ref":"00bca35d0adb193ca8c103df758e6ebf929de8f1f684bf0377fdce9923d82670"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"c5d8280c7688f3caf879b2efe1eb7346fe8a69d31ee6ba7dab6d9796893b8cc5","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"9156181a0a818ff0aad9ed36fa8b7b9312d0359499430541cf7af6e4b9ace809","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"t_nm7a1qF6o3MCLFRSq5XRw_Hhk4ToI_8g16VG5IdkE_EzzckR3E8GNpEBfLc65qjVTluFw31MbGAQjuc7UsDg"},"schema":"pubphys.envelope/1"},"record_hash":"c5d8280c7688f3caf879b2efe1eb7346fe8a69d31ee6ba7dab6d9796893b8cc5","leaf_index":1608}