{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,19]],"date-time":"2026-10-19T19:26:56Z","timestamp":1792438016963,"version":"4.5.0"},"reference-count":31,"publisher":"International Union of Crystallography (IUCr)","issue":"4","license":[{"start":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T00:00:00Z","timestamp":1685664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"},{"start":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T00:00:00Z","timestamp":1685664000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["21-CE30-0029-04 G"],"award-info":[{"award-number":["21-CE30-0029-04 G"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["iucr.org","wiley.com","iucrj.org"],"crossmark-restriction":true},"short-container-title":["Acta Crystallogr A Found Adv","Acta Crystallogr A","Acta Crystallogr A Found Crystallogr","Acta Cryst A","Acta Cryst A Found Adv","Acta Cryst Sect A","Acta Cryst Sect A Found Adv","Acta Crystallogr Sect A","Acta Crystallogr Sect A Found Adv","Acta Crystallogr Sect A Found Crystallogr"],"accepted":{"date-parts":[[2023,4,22]]},"published-print":{"date-parts":[[2023,7,1]]},"abstract":"<jats:p>\n                    This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias &amp; Portier (1982).\n                    <jats:italic>J. Phys. Colloq.<\/jats:italic>\n                    <jats:bold>43<\/jats:bold>\n                    , C6-15\u2013C6-24; Pond &amp; Vlachavas (1983).\n                    <jats:italic>Proc. R. Soc. Lond. Ser. A<\/jats:italic>\n                    ,\n                    <jats:bold>386<\/jats:bold>\n                    , 95\u2013143]. The focus here is on the twist rotations such that the superimposition of the two layers generates a coincidence lattice. The set of such coincidence rotations plotted as a function of the lengths of their coincidence lattice unit-cell nodes exhibits remarkable arithmetic properties. The second part of the paper is devoted to determination of the space groups of the bilayers as a function of the rigid-body translation associated with the coincidence rotation. These general results are exemplified with a detailed study of graphene bilayers, showing that the possible symmetries of graphene bilayers with a coincidence lattice, whatever the rotation and the rigid-body translation, are distributed in only six distinct types of space groups. The appendix discusses some generalized cases of heterophase bilayers with coincidence lattices due to specific lattice constant ratios, and mechanical deformation by elongation and shear of a layer on top of an undeformed one.\n                  <\/jats:p>","DOI":"10.1107\/s2053273323003662","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T10:26:26Z","timestamp":1683887186000},"page":"301-317","update-policy":"https:\/\/doi.org\/10.1107\/cm_01","source":"Crossref","is-referenced-by-count":11,"title":["Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups"],"prefix":"10.1107","volume":"79","author":[{"given":"Denis","family":"Gratias","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02s6m8n84","id-type":"ROR","asserted-by":"publisher"}],"department":["CNRS UMR 8247"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marianne","family":"Quiquandon","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02s6m8n84","id-type":"ROR","asserted-by":"publisher"}],"department":["CNRS UMR 8247"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"329","published-online":{"date-parts":[[2023,6,2]]},"reference":[{"key":"nv5002_bb1","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1524\/zkri.2006.221.8.571","volume":"221","author":"Baake","year":"2006","journal-title":"Z. Kristallogr."},{"key":"nv5002_bb2","doi-asserted-by":"crossref","unstructured":"Baake, M. & Zeiner, P. (2017). Aperiodic Order, Vol. 2, Crystallography and Almost Periodicity, edited by M. Baake & U. Grimm, pp. 73-172. Cambridge University Press.","DOI":"10.1017\/9781139033862"},{"key":"nv5002_bb3","doi-asserted-by":"crossref","first-page":"12233","DOI":"10.1073\/pnas.1108174108","volume":"108","author":"Bistritzer","year":"2011","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"nv5002_bb4","doi-asserted-by":"crossref","first-page":"235449","DOI":"10.1103\/PhysRevB.75.235449","volume":"75","author":"Campanera","year":"2007","journal-title":"Phys. Rev. B"},{"key":"nv5002_bb6","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1038\/nature26154","volume":"556","author":"Cao","year":"2018","journal-title":"Nature"},{"key":"nv5002_bb5","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1038\/nature26160","volume":"556","author":"Cao","year":"2018","journal-title":"Nature"},{"key":"nv5002_bb7","unstructured":"Coxeter, H. S. M. (1963). Regular Polytopes, 2nd ed. New York: Macmillan."},{"key":"nv5002_bb8","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1107\/S2053273321007245","volume":"77","author":"Feuerbacher","year":"2021","journal-title":"Acta Cryst. A"},{"key":"nv5002_bb9","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1126\/science.1158877","volume":"324","author":"Geim","year":"2009","journal-title":"Science"},{"key":"nv5002_bb10","doi-asserted-by":"crossref","unstructured":"Gratias, D. & Portier, R. (1982). J. Phys. Colloq. 43, C6-15-C6-24.","DOI":"10.1051\/jphyscol:1982602"},{"key":"nv5002_bb11","doi-asserted-by":"crossref","first-page":"560","DOI":"10.3390\/cryst10070560","volume":"10","author":"Gratias","year":"2020","journal-title":"Crystals MDPI"},{"key":"nv5002_bb12","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1016\/0036-9748(73)90079-3","volume":"7","author":"Grimmer","year":"1973","journal-title":"Scr. Metall."},{"key":"nv5002_bb13","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1016\/0036-9748(74)90334-2","volume":"8","author":"Grimmer","year":"1974","journal-title":"Scr. Metall."},{"key":"nv5002_bb14","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1107\/S0108767384000246","volume":"40","author":"Grimmer","year":"1984","journal-title":"Acta Cryst. A"},{"key":"nv5002_bb18","doi-asserted-by":"crossref","unstructured":"Hahn, T. (2005). Editor. International Tables for Crystallography, Vol. A, Space-Group Symmetry, 5th ed. Heidelberg: Springer.","DOI":"10.1107\/97809553602060000502"},{"key":"nv5002_bb15","doi-asserted-by":"crossref","first-page":"170","DOI":"10.24033\/bsmf.118","volume":"5","author":"Halphn","year":"1877","journal-title":"Bull. Soc. Math. Fr."},{"key":"nv5002_bb16","unstructured":"Hardy, G. H. & Wright, E. M. (1979). An Introduction to the Theory of Numbers, 5th ed. Oxford University Press."},{"key":"nv5002_bb17","doi-asserted-by":"crossref","first-page":"125504","DOI":"10.1103\/PhysRevLett.100.125504","volume":"100","author":"Hass","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"nv5002_bb20","doi-asserted-by":"crossref","first-page":"266401","DOI":"10.1103\/PhysRevLett.121.266401","volume":"121","author":"Naik","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"nv5002_bb23","unstructured":"Philippon, P. (2008). Un oeil et Farey, https:\/\/hal.science\/hal-00488471."},{"key":"nv5002_bb21","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1063\/1.531424","volume":"37","author":"Pleasants","year":"1996","journal-title":"J. Math. Phys."},{"key":"nv5002_bb22","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1098\/rspa.1983.0028","volume":"386","author":"Pond","year":"1983","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"nv5002_bb24","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1107\/S0365110X66002615","volume":"21","author":"Ranganathan","year":"1966","journal-title":"Acta Cryst."},{"key":"nv5002_bb25","first-page":"52","volume":"01","author":"Romeu","year":"2012","journal-title":"Cryst. Struct. Theory Appl."},{"key":"nv5002_bb27","doi-asserted-by":"crossref","first-page":"474001","DOI":"10.1088\/1361-6463\/abaa15","volume":"53","author":"Soriano","year":"2020","journal-title":"J. Phys. D Appl. Phys."},{"key":"nv5002_bb28","doi-asserted-by":"crossref","first-page":"121407","DOI":"10.1103\/PhysRevB.82.121407","volume":"82","author":"Su\u00e1rez Morell","year":"2010","journal-title":"Phys. Rev. B"},{"key":"nv5002_bb29","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1021\/nl902948m","volume":"10","author":"Trambly de Laissardi\u00e8re","year":"2010","journal-title":"Nano Lett."},{"key":"nv5002_bb30","doi-asserted-by":"crossref","first-page":"125413","DOI":"10.1103\/PhysRevB.86.125413","volume":"86","author":"Trambly de Laissardi\u00e8re","year":"2012","journal-title":"Phys. Rev. B"},{"key":"nv5002_bb31","doi-asserted-by":"crossref","first-page":"165415","DOI":"10.1103\/PhysRevB.77.165415","volume":"77","author":"Varchon","year":"2008","journal-title":"Phys. Rev. B"},{"key":"nv5002_bb32","doi-asserted-by":"crossref","first-page":"081103","DOI":"10.1103\/PhysRevB.102.081103","volume":"102","author":"Venkateswarlu","year":"2020","journal-title":"Phys. Rev. B"},{"key":"nv5002_bb33","doi-asserted-by":"crossref","first-page":"086402","DOI":"10.1103\/PhysRevLett.122.086402","volume":"122","author":"Wu","year":"2019","journal-title":"Phys. Rev. Lett."}],"container-title":["Acta Crystallographica Section A Foundations and Advances"],"original-title":[],"link":[{"URL":"https:\/\/journals.iucr.org\/a\/issues\/2023\/04\/00\/nv5002\/nv5002.pdf","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.iucr.org\/a\/issues\/2023\/04\/00\/nv5002\/nv5002.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T06:23:56Z","timestamp":1736749436000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.iucr.org\/paper?S2053273323003662"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,2]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["S2053273323003662"],"URL":"https:\/\/doi.org\/10.1107\/s2053273323003662","relation":{},"ISSN":["2053-2733"],"issn-type":[{"value":"2053-2733","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,2]]},"assertion":[{"value":"Acta Crystallographica Section A: Foundations and Advances","order":0,"name":"journal","label":"Publication","explanation":{"URL":"https:\/\/journals.iucr.org\/a"},"group":{"name":"general","label":"General"}},{"value":"research papers","order":1,"name":"content_type","label":"Content type","explanation":{"URL":"https:\/\/journals.iucr.org\/a\/services\/notesforauthors.html"},"group":{"name":"general","label":"General"}},{"value":"Available","URL":"https:\/\/scripts.iucr.org\/cgi-bin\/citedin?nv5002","order":2,"name":"article_metrics","label":"Article metrics","group":{"name":"general","label":"General"}},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"14 July 2022","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication history"}},{"value":"22 April 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication history"}},{"value":"2 June 2023","order":2,"name":"published_online","label":"Published online","group":{"name":"publication_history","label":"Publication history"}},{"value":"\u00a9 2023 Gratias and Quiquandon","order":0,"name":"copyright","label":"Copyright","explanation":{"URL":"https:\/\/journals.iucr.org\/services\/copyrightpolicy.html"},"group":{"name":"copyright_and_licencing","label":"Copyright and licencing"}},{"value":"Creative Commons Attribution (CC-BY)","order":1,"name":"license","label":"Licence","explanation":{"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"},"group":{"name":"copyright_and_licencing","label":"Copyright and licencing"}}]}}