{"paper":{"title":"Crystal planes and reciprocal space in Clifford geometric algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA","physics.chem-ph"],"primary_cat":"cond-mat.mtrl-sci","authors_text":"Eckhard Hitzer","submitted_at":"2013-06-07T08:03:27Z","abstract_excerpt":"This paper discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space $\\R^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra $Cl(\\R^{n+1})$ (see Grassmann H., edited by Engel F., Die Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner: Leipzig, 1894.) and in the conformal algebra $Cl(\\R^{n+1,1})$. The crystallographic notions of $d$-spacing, phase angle, structure factors, condit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1646","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}