{"id":"https://openalex.org/W1998406998","doi":"https://doi.org/10.1098/rspa.1950.0139","title":"On the Bose-Einstein condensation","display_name":"On the Bose-Einstein condensation","publication_year":1950,"publication_date":"1950-09-22","ids":{"openalex":"https://openalex.org/W1998406998","doi":"https://doi.org/10.1098/rspa.1950.0139","mag":"1998406998"},"language":"en","primary_location":{"id":"doi:10.1098/rspa.1950.0139","doi":"https://doi.org/10.1098/rspa.1950.0139","is_oa":false,"landing_page_url":"https://doi.org/10.1098/rspa.1950.0139","pdf_url":null,"source":{"id":"https://openalex.org/S4210212881","display_name":"Proceedings of the Royal Society of London A Mathematical and Physical Sciences","issn_l":"0080-4630","issn":["0080-4630","2053-9169"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310319787","host_organization_name":"Royal Society","host_organization_lineage":["https://openalex.org/P4310319787"],"host_organization_lineage_names":["Royal Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019436339","display_name":"S.R. DE GROOT","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"S.r. De Groot","raw_affiliation_strings":["Institute for Theoretical Physics, The University, Utrecht, Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Theoretical Physics, The University, Utrecht, Netherlands","institution_ids":["https://openalex.org/I193662353"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067457346","display_name":"G.J. Hooyman","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I193662353","display_name":"Utrecht University","ror":"https://ror.org/04pp8hn57","country_code":"NL","type":"education","lineage":["https://openalex.org/I193662353"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"G. J. Hooyman","raw_affiliation_strings":["Institute for Theoretical Physics, The University, Utrecht, Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Theoretical Physics, The University, Utrecht, Netherlands","institution_ids":["https://openalex.org/I193662353"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016478686","display_name":"Cornelis A. Ten Seldam","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I887064364","display_name":"University of Amsterdam","ror":"https://ror.org/04dkp9463","country_code":"NL","type":"education","lineage":["https://openalex.org/I887064364"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"C. A. Ten Seldam","raw_affiliation_strings":["Van der Waals Laboratory, The University, Amsterdam, Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Van der Waals Laboratory, The University, Amsterdam, Netherlands","institution_ids":["https://openalex.org/I887064364"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6853,"has_fulltext":false,"cited_by_count":136,"citation_normalized_percentile":{"value":0.63663664,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"203","issue":"1073","first_page":"266","last_page":"286"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10425","display_name":"Cold Atom Physics and Bose-Einstein Condensates","score":0.5112000107765198,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10425","display_name":"Cold Atom Physics and Bose-Einstein Condensates","score":0.5112000107765198,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.19820000231266022,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12612","display_name":"Strong Light-Matter Interactions","score":0.06920000165700912,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bose-einstein-condensation","display_name":"Bose-Einstein condensation","score":1.0},{"id":"https://openalex.org/keywords/bose-statistics","display_name":"Bose statistics","score":0.7900000214576721},{"id":"https://openalex.org/keywords/specific-heat-discontinuity","display_name":"specific heat discontinuity","score":0.8289999961853027},{"id":"https://openalex.org/keywords/lambda-point","display_name":"lambda point","score":0.597000002861023},{"id":"https://openalex.org/keywords/harmonic-potential","display_name":"harmonic potential","score":0.9570000171661377},{"id":"https://openalex.org/keywords/generalized-dimensions","display_name":"generalized dimensions","score":0.4000000059604645},{"id":"https://openalex.org/keywords/finite-size-effects","display_name":"finite-size effects","score":0.40700000524520874}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28929054737091064},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17694976925849915}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1098/rspa.1950.0139","doi":"https://doi.org/10.1098/rspa.1950.0139","is_oa":false,"landing_page_url":"https://doi.org/10.1098/rspa.1950.0139","pdf_url":null,"source":{"id":"https://openalex.org/S4210212881","display_name":"Proceedings of the Royal Society of London A Mathematical and Physical Sciences","issn_l":"0080-4630","issn":["0080-4630","2053-9169"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310319787","host_organization_name":"Royal Society","host_organization_lineage":["https://openalex.org/P4310319787"],"host_organization_lineage_names":["Royal Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W282945202","https://openalex.org/W2146579064","https://openalex.org/W2324535796"],"related_works":["https://openalex.org/W2142655063","https://openalex.org/W2060365385","https://openalex.org/W2027676463","https://openalex.org/W760628398","https://openalex.org/W2040698636","https://openalex.org/W1967386090","https://openalex.org/W1972362141","https://openalex.org/W2008641866","https://openalex.org/W2050049415","https://openalex.org/W2060441579"],"abstract_inverted_index":{"Abstract":[0],"The":[1,49],"Bose-Einstein":[2],"condensation":[3],"of":[4,26,33,41,83,126,203,230,271,295],"a":[5,24,144,186,196,220,266],"gas":[6,35],"is":[7,81,103,115,135,177,206,224,265,292],"investigated.":[8],"Starting":[9],"from":[10],"the":[11,17,31,34,38,57,78,84,94,118,124,153,160,174,210,227,231,252,272,281,289],"well-known":[12],"formulae":[13],"for":[14,88,140,195,288],"Bose":[15],"statistics,":[16],"problem":[18],"has":[19,43],"been":[20,45],"generalized":[21],"to":[22,47,54,106,185,226,236,247],"include":[23],"variety":[25],"potential":[27,79],"fields":[28],"in":[29,280],"which":[30],"particles":[32],"move,":[36],"and":[37,66,159,257],"number":[39,119,201],"w":[40],"dimensions":[42],"not":[44],"restricted":[46],"three.":[48],"energy":[50,155],"levels":[51],"are":[52,164,301],"taken":[53],"be":[55],"\u03b5i\u2261\u03b5s1,....,s10=constanth2ms1\u03b1\u22121a12+...+sw\u03b1aw2(1\u2264\u03b1\u22642)":[56],"quantum":[58],"numbers":[59],"being":[60,70],"s1":[61],",w=":[62],"1,":[63,91,256],"2,":[64,77,152,173],"...,":[65,68],"a1,":[67],"aw":[69],"certain":[71],"characteristic":[72],"lengths.":[73],"(For":[74],"\u03b1":[75,89],"=":[76,90,121,168,181,213,278],"field":[80],"that":[82,107,117],"w-dimensional":[85,95],"rectangular":[86,221],"box;":[87],"we":[92],"obtain":[93],"harmonic":[96],"oscillator":[97],"field.)":[98],"A":[99,187,189],"direct":[100],"rigorous":[101],"method":[102],"used":[104],"similar":[105],"proposed":[108],"by":[109,275,307],"Fowler":[110],"&":[111],"Jones":[112],"(1938).":[113],"It":[114],"shown":[116],"q":[120,141,171,254],"w/\u03b1":[122],"determines":[123],"appearance":[125],"an":[127,293],"Einstein":[128],"transition":[129,145,191],"temperature":[130],"T0\u00b7For":[131],"q\u2264":[132,151],"1":[133,143,149],"there":[134,264],"no":[136],"such":[137],"point,":[138],"while":[139],">":[142,172,255],"point":[146,192],"exists.":[147],"For":[148,170,219],"<":[150,261],"mean":[154,228],"\u03f5-":[156],"per":[157],"particle":[158],"specific":[161,175],"heat":[162,176],"d\u03f5-/dT":[163],"continuous":[165,309],"at":[166,179,258],"T":[167,180,260,262],"T0\u00b7":[169],"discontinuous":[178],"T0,":[182],"giving":[183],"rise":[184],"\u03bb-point.":[188],"well-defined":[190],"only":[193,208],"appears":[194],"very":[197],"large":[198],"(theoretically":[199],"infinite)":[200],"N":[202,276],"particles.":[204],"T0":[205,244],"finite":[207,267],"if":[209],"quantity":[211],"v":[212,223,234],"N/(a1":[214],"....":[215],"aw)2/":[216],"\u03b1\u00afis":[217],"finite.":[218],"box,":[222],"equal":[225],"density":[229],"gas.":[232],"If":[233],"tends":[235,246],"zero":[237,248],"or":[238,249],"infinity":[239],"as":[240],"N\u2192":[241],"\u221e,":[242],"then":[243],"likewise":[245],"infinity.":[250],"In":[251],"case":[253],"temperatures":[259],"0'":[263],"fraction":[268],"N0":[269],"/N":[270],"particles,":[273],"given":[274,306],"0/N":[277],"1-(T/T0)q,":[279],"lowest":[282],"state.":[283],"London\u2019s":[284,308],"formula":[285],"(1938":[286],"b)":[287],"three-dimensional":[290],"box":[291],"example":[294],"this":[296],"equation.":[297],"Some":[298],"further":[299],"results":[300],"also":[302],"compared":[303],"with":[304],"those":[305],"spectrum":[310],"approximation.":[311]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":4}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}
