Laszlo Tisza and the two-fluid model of superfluidity

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Laszlo Tisza and the two-fluid model of superfluidity

Sebastien Balibar

To cite this version:

Sebastien Balibar. Laszlo Tisza and the two-fluid model of superfluidity. Comptes Rendus. Physique,

Académie des sciences (Paris), 2017, 18 (9-10), pp.586-591. �10.1016/j.crhy.2017.10.016�. �hal-

01677664�

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Comptes Rendus Physique

www.sciencedirect.com

The Comptes rendus de l’Académie des sciencesthroughout history / Les Comptes rendus de l’Académie des sciencesà travers l’histoire

Laszlo Tisza and the two-fluid model of superfluidity

Laszlo Tisza et le modèle à deux fluides de la superfluidité

Sébastien Balibar

LaboratoirePierre-Aigrain,Départementdephysiquedel’Écolenormalesupérieure(ENS),PSLResearchUniversity,UniversitéParis-Diderot, SorbonneParisCité,SorbonneUniversités,UPMCUniversitéParis-6,CNRS,75005Paris,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Availableonline10November2017

Keywords:

Superfluidity Helium Tisza London Landau

Mots-clés : Superfluidité Hélium Tisza London Landau

The“two-fluid model”ofsuperfluiditywas firstintroduced byLaszlo Tisza in1938. On thatyear,Tiszapublishedthe principlesofhismodelas abrief noteinNatureand two articles inFrench inthe Comptesrendusdel’Académiedessciences,followed in 1940 by twootherarticlesinFrenchintheJournaldephysiqueetleRadium.In1941,thetwo-fluid modelwasreformulatedbyLevLandauonamorerigorousbasis.Successiveexperiments confirmed the revolutionary idea introduced by Tisza: superfluid helium is indeed a surprisingmixture oftwo fluidswith independentvelocity fields. His prediction ofthe existenceofheatwaves,aconsequenceofhismodel,was alsoconfirmed.Then,it took severaldecadesforthesuperfluidityofliquidheliumtobefullyunderstood.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ré s u m é

Le« modèleàdeuxfluides » delasuperfluidité aété introduiten1938par LaszloTisza.

Cetteannée-là,Tiszapublialesprincipesde sonmodèlesouslaformed’unenotebrève danslarevueNatureetdedeuxarticlesenfrançaisdanslesComptesrendusdel’Académie dessciences, qui furent eux-mêmes suivis en 1940 de deux autres articles en français dansleJournaldephysiqueetleRadium.En1941,lemodèleàdeuxfluides futreformulé parLevLandausurunebaseplusrigoureuse.Différentesexpériencesontconfirmél’idée révolutionnairedeTisza :l’héliumsuperfluideest,eneffet,unmélangesurprenantdedeux fluides,dontleschampsdevitessesontindépendants.Saprédictiondel’existenced’ondes dechaleur,quiestuneconséquencedecemodèle,futconfirmée,elleaussi.Ensuite,ilfallut plusieursdécenniespourquelasuperfluiditédel’héliumsoitcomplètementcomprise.

©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Historicalcontext:superfluidityin1938

LetmefirstsummarizethehistoricalcontextofTisza’swork.

E-mailaddress:[email protected].

https://doi.org/10.1016/j.crhy.2017.10.016

1631-0705/©^{2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

On10July1908,HeikeKamerlinghOnnesliquefiedheliumgasforthefirsttimeinhisLeidenlaboratory[1,2].Tenyears after the discovery of liquid hydrogenby JamesDewar, helium was the last so-called “permanent gas”, andKamerlingh Onnesdiscoveredthatitwasactuallynotpermanent,sinceitliquefiedat4 Kunderatmosphericpressure.Bypumpingon liquidhelium,KamerlinghOnneswasabletocoolitdownto1 K.Hedescribedhisdiscoveryintwocommunicationstothe NetherlandsAcademyofSciences(KNAW)[1]andina notetotheComptesrendusdel’Académiedessciences[3],whichwas receivedon17Augustandpublishedoneweeklater.Henoticedthatliquidheliumdidnotfreeze,andhedidnotobserve anyparticularchangeintheliquidpropertiesdownto1 K,theworldtemperaturerecordatthattime.Forthisdiscovery,he wasawardedtheNobelprizein1913.Inthemeantime,hehadalsodiscoveredsuperconductivity.Lowtemperaturephysics hadbegun.

The firstindication ofa changein theproperties ofliquid heliumwas observed in1932 by JohnC. McLennan inhis Toronto laboratory. At that time, cryostats andrefrigerators were made oftransparent glass and McLennan noticed that when pumping to cool it down, liquid helium stopped boiling below 2.2 K[4], asif some mysterious order took place below2.2 K inthisextremelycold liquid.WillemH.Keesomwas a formerstudent ofKamerlingh Onnesworkinginthe sameLeidenlaboratorywithhisdaughterAnna,andtheystudiedtogetherthetemperaturedependenceofthespecificheat ofliquid helium.They founda singularity at2.2 K, another indication that a transitiontosome orderedstate happened there[5].Itmusthavebeenratherastonishingtofindtwodifferentstatesinaverypureliquidmadeofsmallroundatoms withveryweakinteractionsandnochemicalproperties.Anyhow,thissingularityinthetemperaturevariationofthespecific heat havingtheshape oftheGreek letterλ, theycalled2.2 Kthe “lambdatemperature T_λ”or“λ-point”andthe change occurringatTλthe“λ-transition”.AboveTλ,theheliumliquidwascalled“heliumI”and“heliumII”below.

Soon afterwards (1936–1937), the same Willem H. Keesom with his daughterAnna showed that liquid helium was conductingheat surprisingly well below the same2.2 K [6]. Afew months later, J.F.Allen, R. Peierls, and Z.Uddin also observeda surprisinglyhighthermalconductivityinCambridge[7].Lateron, oneunderstoodthat thisvery highthermal conductivitywas thereasonwhyliquidhelium didnot boilbelow T_λ. Indeed,ifthermalconductivityisvery high,there arenosignificanttemperaturedifferencesintheliquid,consequentlynoregionswheretheprobabilityofbubblenucleation ishigherthan elsewhere,sothat theliquidevaporates fromits freesurface, butnoboiling occurs.It happensthatmany yearslater IworkedformyPhDon thequantumevaporationofliquidheliumanddiscovered thatit isanalogoustothe photoelectriceffect[8,9].Butletuscomebacktothe1930s.

Atthattime,someresearchersprobablythoughtthatthelargethermalconductivitymightbeaconsequenceofconvec- tionbeingveryefficientiftheliquidviscositywasverysmall[10].Indeed,anundergraduatestudent–A.D.Misener–and two technicians–J.O.WilhelmandA.R.Clark –workinginTorontounderthe authorityofE.F.Burton[11] measuredthe dampingofarotatingcylinderto deducetheviscosityofliquidhelium.Oncemore,they observedtheabsenceofboiling below the λ-point, but, mostimportantly, they found that viscosity apparently decreased sharply below Tλ [10]. Liquid heliumwasthusattractingmoreandmoreattentionattheendof1937,andthehistoryofsuperfluiditystarted[10].

It startedwithtwo famous notes published side by side by Nature inits January 8th(1938) issue.The first one [12]

–“Viscosity ofliquidheliumbelowthelambda point”,receivedon 3December1937, by P.Kapitza (InstituteforPhysical Problems, Moscow)– appearedon page74.Thesecond one [13] –“Flowofliquidhelium II ”,receivedon 22December 1937,byJ.F.AllenandA.D.Misener–appearedonpage75.A.D.MisenerhadmovedfromTorontotoCambridge,wherehe wanted tograduate underthesupervision ofJ.F.Allen.Fora detailedcomparisonbetweenthescientific contentofthese two notes, andfor commentson priority in relationwith the dates of receipt,see ref. [10]. Let usnow summarizethe scientificcontentofthesetwonotes.

P. Kapitza hadobservedthe flow ofliquidhelium througha slit betweentwowell-polishedcylinders pressed against eachother.Thisslitwas verythin(0.5micrometers)andheobservedthat“theflow ofliquidabove theλ-point couldbe onlyjustdetectedoverseveralminutes,whilebelowtheλ-pointtheliquidheliumflowedquiteeasily,andthelevelinthe tubesettleddowninafewseconds.FromthemeasurementswecanconcludethattheviscosityofheliumIIisatleast1500 timessmallerthan thatof heliumIatnormalpressure.”Aftersome considerations onturbulence,heconcludedthat “by analogywithsuperconductors,. . . theheliumbelowtheλ-pointentersaspecialstatewhichmightbecalledsuperfluid.”This isafamoussentencewhereKapitzaintroducedtheword“superfluid”forthefirsttime.Hisintuitionwasquiteremarkable becausesuperfluidsandsuperconductorsareindeedanalogousstatesofmatter,butKapitzawrotethissentencelongbefore theBCStheoryofsuperconductivitywasestablished,afortioribeforeanydemonstrationofsuchananalogy.

The Cambridgenote by Allen andMisenerpresented a detailedstudyof theflow through two micro-capillaries with different sectionsat two different temperatures (1.07and 2.17 K) andten different pressures. Theirmain findings were that,contrarytoPoiseuille’slaw,whichdescribeslaminarsituations,thevelocitywas nearlyindependentofpressure,also independentofthecapillarysection.Theyconcludedthat“theobservedtypeofflow. . . inwhichthevelocitybecomesalmost independent of pressure, most certainly cannot be treated as laminar oreven asordinary turbulent flow. Consequently anyknownformulacannot, fromourdata,give avalue oftheviscosity whichwouldhavemuchmeaning.” Atthat time, everyoneelsekeptconsideringthatliquidheliumwasaliquidwithasmallviscosity.ButAllenandMisenerdiscoveredthat, belowTλ,thehydrodynamicsofheliumrequiredanentirelynewinterpretation.Hereistherealexperimentalbreakthrough thattriggeredallthefollowingdevelopments.

2. FritzLondonandBose–Einsteincondensation

Before readingthesetwo noteson viscosity,that is,in 1936, FritzLondonhadalreadytried to findthe type oforder takingplace inthisliquidbelow T_λ. F.Simon[14] andF. London[15] hadrealizedthat,in liquidhelium,themagnitude of quantum fluctuationsmust be rather large. The kinetic energy associated with these quantum fluctuations, which is usually called“zero-pointenergy”,is intheorderof E_k= ¯^h2/[^2m(a−^d)²]^{,that is,itisinversely} proportionalto asmall confinementlength(a−^d)whereaistheinteratomicdistanceanddtheatomdiameter.Itisalsoinverselyproportionalto theatommassm,whichissmallagain.SincethevanderWaalsinteractionEiisweakbecausethepolarizabilityoftheHe atomsisweak,F. SimonandF. Londonfoundthat Ek iscomparableto Ei,whichexplainswhythemolarvolume ofliquid helium isanomalouslylarge,alsowhyliquidheliumdoesnot freeze,evenwhenapproaching theabsolutezero,exceptat pressureslargerthan25 bar.

F. London [15],followed byH.Frölich[16],hadconsideredastrange possibility:liquidheliumII couldbea bodycen- teredcubiccrystalwithonlyhalfofthenetworksitesoccupiedbyatoms[17].Equivalently,itcould beconsideredastwo entangledcrystallinestructures,respectivelyoneofatomsandtheother ofvacancies,whichareholes,absent atoms.¹ But, in1938,F. Londonunderstoodthatthistentativemodelcouldnotbe“maintained”andthat“anentirelydifferentinterpre- tationwasneeded”[18]. IthappenedafteronemorenotewassenttoNaturebyJ.F.Allen,thistimewithH.Jones,ayoung theorist[19].

Entitled “New phenomena connected with heat flow in helium II”, the new letter by Allen and Jones described the discovery ofwhatisnowknownasthefountaineffect.Theydiscoveredthat,whenheatwasappliedtoliquidheliumIIon one sideofamicro-capillary,thepressureincreasedproportionallytotheheatcurrentsothatthelevelofthefreesurface went up onthewarm side.A liquidjet couldeven occur ifthe pressurewas highenough. ForLondon,itwas no longer possibletodoubtthatthisliquidhadtotallyanomalouspropertiesforwhicharadicallynewinterpretationwasneeded.On 5March1938,LondonsentanotetoNature,whichwaspublishedon9April.There,heexplainedthatliquidheliumIIwas notcrystalline,beforeproposingthatitwasundergoingatransitionrelatedtotheBose–EinsteincondensationatT_λ [18].

ByextendingacalculationbySatiendraNathBose[20]toasetofatoms,AlbertEinsteinhadfoundthat,belowacritical temperatureTc,asetofatomsobeyingthe“Bose–Einsteinstatistics”shouldaccumulateinasinglequantumstate[21].This

“Bose–Einsteincondensation”(BEC)wastheresultofordertakingplaceinmomentumspace,not inrealspace.Onecould also say that, below Tc, all atoms startedbehaving the sameway becausethey formed a macroscopic wave of indistin- guishableatoms. ForEinstein,thisstrange “condensation”was therathersurprisingresultofacalculation,buthedidnot reallybelieveinit.Asiswellknown,hesaid:“DieTheorieisthübschaberobauchetwaswahresdranist?”(“Thetheoryis pretty,butisthereanythingtrueinit?”)Furthermore,Uhlenbeck hadraisedan objectiontoit.ButinNovember1937,an internationalconferencetookplaceinAmsterdamtocelebratethe100thanniversaryofvanderWaals.Londonparticipated thereinadiscussionbetweenUhlenbeck,Ehrenfest,Kramers,andEinstein,whereUhlenbeckwithdrewhisobjectiontothe Bose–Einsteintheory[22].

AfterreadingthenewnotebyAllenandJones[19],LondonthoughtagaintotheBose–Einsteincondensation(BEC)[22], and he madea rough estimate of the BEC temperature Tc by considering liquid helium asan idealgas with the same density.Hefound T_c=³.09 K,avalueclosetoT_λ=².2 K.Healsocalculatedthetemperaturevariationofthespecificheat neartheBECtransitionpoint,andfoundagainabehaviorclosetotheoneobservedinliquidhelium.Afterall,aliquidwith a largemolarvolumecouldperhapsbenotsodifferentfromagas[22].Londonhesitatedinpublishinghisresults,buthe finallysentanotetoNatureon5March1938,wherehewrote:“...thedegeneracyoftheBose–Einsteingashasrathergotthe reputation ofhavingapurely imaginaryexistence[...]amodelwhichissofarfromrealitythat itsimplifiesliquidhelium toanidealgas[...][but]itseemsdifficultnottoimagineaconnectionwiththecondensationphenomenonofBose–Einstein statistics.”Thisnoteisfamousbecauseitsignalsthehistoricaljumpofquantummechanicsfromthemicroscopicphysicsof atomstomacroscopicsystemslikealiterofliquidhelium,whichisvisibletothenakedeye.Inreality,Londonhadalready thoughtwithhisbrotherHeinzaboutmacroscopicwavefunctionstodescribesuperconductors(see,forexample,[23]).

Fritz London hadalso discussed his revolutionary idea with Laszlo Tisza, who had joinedhim a few months before (September 1937).BothLondon andTiszawhererefugees flyingaway fromnazism inEasternEurope,andtheyhadbeen attractedto Parisby Paul Langevinandhiscollaborators[10,24].LaszloTiszaimmediatelyagreed withLondon’s tentative theoryandaddedtoitnewideasinordertointerpretmoreexperimentalresults,especiallythefountaineffect.

3. ThefivearticleswritteninParisbyLaszloTisza

AfterdiscussingwithFritzLondon,LaszloTiszahadonemorerevolutionaryidea.StartingfromLondon’swork, hepro- posedthat,foranidealBosegasbelowitsBECtransitiontemperatureT_c,amacroscopicnumberofatomsnofthehelium atoms should be “condensed” on the ground state, while the rest (n0−ⁿ) should be distributed on all excited states.

Supposingthatthisassumptionwascorrect,thefractionn/n0 shoulddecreasewithtemperatureaccordingto

1 Eightyyears, laterthisassumptionlooksnotsofarfrom whatisnowcalled“supersolidity”,whichwasconsideredforafewyearsasapossible explanationoftheanomalousmechanicalpropertiesofsolidhelium,beforebeingrejected[17].

n

n₀=(T/T₀)^s (1)

wheretheexponents=³/2 foranidealgas,butitshouldbelargerforaliquidwithstronginteractionsbetweenatoms.It meantthat,belowtheBECtemperatureTc,theidealgasshouldbeamixtureoftwocomponents:a“superfluidcomponent”

madeofcondensedatomsanda“normalcomponent”madeofnon-condensedatoms.Theconcentration ofthe“superfluid component”shouldvarycontinuouslyfrom0atTc to1atT=^0.Asaconsequence,atanytemperaturebetweenTc and0, theliquidshouldbemadeofadefinite mixtureofthesuperfluidcomponentwiththenormalcomponent.Nowadays,this viewisconsideredtoosimpleforaliquidbecauseatom–atominteractionsmustplayaroleonthenatureofboththeground stateandtheexcitedstates,andtheatomslosetheir individualcharacter.Tiszawasawareofsomeofthesedifficulties,but hecontinued.Thecondensedatomsbeingonthegroundstate,theycouldnotcontributetoanymomentumexchange,con- sequentlytoanydissipation,contrarytothenon-condensed atoms,whichshouldbe responsiblefordissipation,thatisfor viscousbehavior,whateveritis.Itallowedhimtounderstandtheapparentcontradictionwhichhadbeennoticedbetween theTorontoexperimentusinga macroscopiccylinderandtheexperiments byKapitza andby Allen,wherea microscopic slitoramicro-capillarywere used.Indeed,intheTorontoexperiment,boththecondensed andthenon-condensedatoms shouldbemoving,whileinthetwoothersonlyasuperflowofcondensedatomsshouldtakeplace.Hethenpredictedthat this superflow should produce a gradient inthe concentration ofcondensed atoms, which should be accompanied by a temperaturegradientaccordingtoEq.(1).

Finally,Tiszaunderstoodtheoriginofthefountaineffectastheinverseprocess oftheabove-mentionedsuperflow:“If one maintainsa temperature differencebetween theends ofa capillary,a gradient ofdensity ofthe excited atomsand, thus,ofpressureisproduced.”Henoticedasimilaritybetweenthispressuregradientandtheosmoticpressuredifference acrossasemi-permeablemembrane.Andhefinishedhisarticlebywriting:

AdetaileddiscussionoftheproblemwillbegivenintheJournaldephysique.IamgreatlyindebtedtoDr.F.Londonfortheopportunity ofseeinghispaperbeforepublication.

L. Tisza,Laboratoiredephysiqueexpérimentale,CollègedeFrance,Paris,16April.

Thishistoricalnote[25]toNatureisconsideredasthebirthofthe“two-fluidmodel”,but,atthattime,Londoncouldnot agreewithTisza’snewideas[26].ToimaginethatliquidheliumIIwasmadeoftwofluidcomponentsthataremixedbut thatcouldflowindependentlyofeachother wasimpossibleforhim.ButTiszadevelopedhismodelinthetwoothernotes hesenttotheComptesrendusdel’Académiedessciencesafewmonthslater.In1940,Tiszapublishedanothertwoarticlesin theJournaldephysiqueetleRadium[27,28],ashehadannounced.Evenlater(1941), Landaureformulated Tisza’stwo-fluid modelonamorerigorousbasis[29,30].

InhisfirstnotetotheComptesrendus[31],whichwaswritteninFrenchandpresentedbyPaulLangevinon14November 1938, Tiszafirstconsideredthenature ofthetwocomponentsinhis two-fluidmodel.Hederived thetwo equationsthat describe thehydrodynamicsofliquidheliumbelow T_λ andarrived atadiscovery: insteadofdiffusingasinanyclassical liquid,heatpropagatesasawave,andhecalculatedthevelocityoftheseheatwaves.Thispredictionhadtobecomeacrucial testofhismodel.Healsodescribedhowhismodelagreeswiththemeasurementsoftheanomalousthermalconductivity ofliquidhelium[6,7].

As for Tisza’s second note to the Comptesrendus[32], it describesthe viscosity of the normalcomponent, which he calculatedwithin hisapproximations.Tiszaexplainedwhyhismodelwas inagreement,notonly withtheTorontoexper- iment [4]anda later one by W.H. Keesom andG.E. Mac Wood[33] where the two components were moving, butalso withthe two experiments by Kapitza and by AllenandMisener, where onlythe superfluid component was moving. He alsoexplained why,in aquantum liquidlike heliumat low temperature, viscositywas disappearing atlow temperature, contrarytoclassicalliquidswhereviscosityincreasesastemperaturedecreases.Hethenexplainedthefountaineffectasa thermomechanical effectwithsome resemblancetothe osmoticpressureacross asemi-permeablemembrane.Finally,he explainedthe“Rollincreeping”[48],whichiswhy,inabeaker, superfluidheliumwasabletocreepupalong thesidesso thatthebeakeremptiesspontaneously.Itwasaconsequenceoftheexistenceofathinsuperfluidfilmcoveringthebeaker walls.

InOctober1939, despitethe growingstress followingthe invasionofFranceby thenazi army,whichsoonforcedthe LangevinlaboratorytomovetoToulouse[26],TiszasentacoupleofarticlestotheJournaldephysiqueetleRadium[27,28].In thisjournal,hecouldfurtherdevelop histwo-fluidmodelwithoutthelengthlimitationthatwas imposedbytheComptes rendus[27].HethusaddedagraphshowingtheagreementbetweenhistheoryandtheviscositymeasurementsbyKeesom andMacWood[33].Healsoaddedaparagraphexplainingwhy,contrarytoBoseliquids,Fermiliquidscouldnotbesuper- fluid.Healsoaskedquestionsaboutturbulenceinasuperfluid,buthewas farfromthe recentdevelopmentsofthisfield thatshowitscomplexity.Inordertogetmoreinformationonhisthermomechanicaleffect,hefinallyproposedtostudythe propagationofsmalltemperaturevariations below0.8 K.Thisproposaldemonstratedhisremarkableintuition,buthehad probablynotanticipatedthat,afewyearslater,suchexperimentswouldsupportsomeofLandau’scriticismonhiswork.

4. LevLandauandlaterdevelopments

ThisisnottherightplacetodescribethelifeofLevLandauwhohadwrittenaleafletcomparingStalintoHitlerinmay 1938andwhohadbeenconsequentlythrowninitssadlyknownLubyankaprisonbytheNKVD.Landausurvivedthanksto