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Dislocations in a quantum crystal: Solid helium: A

model and an exception

Sébastien Balibar, John Beamish, Andrew Fefferman, Ariel Haziot, Xavier

Rojas, Fabien Souris

To cite this version:

Sébastien Balibar, John Beamish, Andrew Fefferman, Ariel Haziot, Xavier Rojas, et al.. Dislocations

in a quantum crystal: Solid helium: A model and an exception. Comptes Rendus Physique, Centre

Mersenne, 2016, Condensed matter physics in the 21st century: The legacy of Jacques Friedel, 17

(3-4), pp.264-275. �10.1016/j.crhy.2015.12.015�. �hal-01312392�

Condensed matter physics in the 21st century: The legacy of Jacques Friedel

Dislocations

in

a

quantum

crystal

Solid

helium:

A

model

and

an

exception

Dislocations dans un cristal quantique

L’hélium solide: un modèle et une exception

Sébastien Balibar

a

,

∗

,

John Beamish

b

,

Andrew Fefferman

c

,

Ariel Haziot

d

,

Xavier Rojas

e

,

Fabien Souris

b

a_{LaboratoirePierre-Aigrain,Départementdephysique,Écolenormalesupérieure,associéauCNRSetauxUniversitésPierre-et-Marie-Curieet}

Paris-Diderot,24,rueLhomond,75231 Pariscedex 05,France

b_{DepartmentofPhysics,UniversityofAlberta,Edmonton,T6G 2E1,Canada} c_{UniversitéGrenobleAlpes,InstitutNéel,BP 166,38042 Grenoblecedex 9,France} d_{DepartmentofPhysics,PennsylvaniaStateUniversity,UniversityPark,PA 16802,USA} e_{DepartmentofPhysics,RoyalHolloway,UniversityofLondon,Egham,Surrey TW20 0EX,UK}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline17December2015

Keywords: Dislocations Elasticity Plasticity Quantumcrystals Helium Mots-clés : Dislocations Élasticité Plasticité Crystauxquantiques Hélium

Solidheliumisparadoxical:itisbothamodelandanexception.Itisamodelforcrystal propertiesmainlybecauseofitsextremepuritywhichmakesuniversalphenomenasimpler and easier to identify. It is also exceptionalbecause the large quantum fluctuations of atomsaroundthenodesintheircrystallatticeallowthesephenomenatooccuratverylow temperaturewithalargeamplitude.AsnoticedbyJacquesFriedelin2013,theproperties ofhelium 4 crystals illustrate how the motionof dislocations may reduce their shear elasticmodulus,asit doesinallhexagonalclosepacked(hcp)crystalsincludingmetals. But thismotion takes place withoutany dissipation in the limit of T=0 and in the absenceofimpurities,whichisnowexceptionaland leadstoanelasticanomalyatlow temperature,whichwascalled“giantplasticity”byHaziotet al.in2013.Morerecently,we havediscovered that,in helium-4crystals,helium-3impurities arenot necessarilyfixed pinningcentersfordislocations.Evenatrelativelylargevelocities,dislocationsareableto movedressedwithimpuritiessomehowasanecklaceofatomicpearlsthroughtheperiodic lattice.Thisillustrateswhatisreallyquantuminthesecrystals:itismainlythedynamics oftheirdislocationsandthebehaviorofimpurities.

©2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

L’héliumsolide est paradoxal : c’est àla foisun cristal modèle etuneexception. C’est un modèle pour l’étude des propriétés cristallinesà cause de son extrême pureté, qui rendcertainsphénomènesuniversels plussimplesetplusfaciles àidentifier.C’est aussi unsystèmeexceptionnel,carlesfluctuationsquantiquesdesesatomesautourdesnœuds

*

Correspondingauthor.

E-mailaddress:sebastien.balibar@lpa.ens.fr(S. Balibar).

http://dx.doi.org/10.1016/j.crhy.2015.12.015

1631-0705/©2015Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

duréseau cristallin permettentàcesphénomènesd’avoirlieuàtrèsbassetempérature, avecuneamplitudeparticulièrementgrande.Commel’avait remarquéJacquesFriedelen 2013,lespropriétésdescristauxd’hélium4illustrentlamanièredontlemouvementdes dislocations peutréduireleur moduleélastiquede cisaillementtransverse, commedans toutcristalhexagonalcompact(hcp),ycompriscertainsmétaux.Maiscemouvementalieu sansdissipationlorsquelatempératuretendverszéro etenl’absencetotaled’impuretés, cequiest exceptionneletconduitàuneanomalieélastiquequiaété appelée« plasticité géante » parHaziotetal.en 2013.Plusrécemment,nousavonsdécouvertque,dansces cristaux d’hélium 4, les impuretés d’hélium 3 ne sont pas nécessairement des points d’ancragefixespourlesdislocations.Mêmeàrelativementgrandevitesse,lesdislocations sontcapablesdesedéplacerhabilléesd’hélium3,commeuncollierdeperlesatomiquesà traversleréseaupériodique.Celaillustrecequiestvraimentquantiquedanscescristaux : il s’agit principalementde la dynamiquede leursdislocationset ducomportement des impuretés.

©2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. MeetingJacquesFriedel

Sébastien Balibarwishesto dedicatethisreview articletothememoryofProfessorJacquesFriedel,withthefollowing personaltestimony:

“ThefirsttimeImentionedourstudyofdislocationmotioninheliumcrystalstoJacquesFriedel,hesaidsomethinglike “all thathasalreadybeenmeasured inhexagonal metals”.Afterafewminutes ofsurprise, Ifeltquitehappywiththat.It meantthatourcrystalscouldbeofgeneralinterestfarbeyondthelittlefieldofquantumsolids.So,Iinsistedandstarted showingsomedetailsofwhatweweremeasuring.TogetherwithmyvisitorJohnBeamish(Edmonton,Alberta,Canada)and agroupofbrightstudentsandpostdocswhoworkedonthissubjectwithmeatthe‘Laboratoiredephysiquestatistiquede l’ENS’andareco-authorsofthisreview,wewereprogressivelydiscoveringaseriesofratherspectacularpropertiesofthese dislocations, andIsucceededincapturingJacques Friedel’sattention. From2011to thevery lastmoments ofhislife, we exchangedcommentsandideas onthemotionofdislocationsincrystals.Everymeetingwasan opportunitytostrengthen ourgrowingfriendship.Thesediscussionshavebeenextraordinaryusefultome.IamextremelygratefultoJacquesFriedel forhisinterest inourwork. Itisa greathonor andagreatpleasure forme to summarizeour resultsonthemechanical propertiesofheliumcrystalsinhismemory.”

2. Quantum,ultrapure,visible

Whyissolid helium-4consideredasaquantumcrystal?Above25 barandbelow1.4 K,4_{Hecrystallizes inahexagonal} closepacked(hcp)structurewherethe atomshavelarge quantumfluctuations.Thisis dueto theHeisenberguncertainty relationbetweenmomentumandposition.Ifanatom ofmassm anddiameterd is confinedbyits neighborsinaregion of typical size a, it acquires a fluctuating momentum p

= ¯

h

/(

a

−

d

)

and consequently a quantum kinetic energy Eq of orderh

¯

2

/

[

2m

(

a

−

d

)

2

]

. Eq islargeinthe caseof4Hebecauseits massis lowandthehard cored

=

2

.

6 Å iscomparable tothelatticespacinga

=

3

.

7 Å.Thisroughargumentleads to Eq

≈

15 K,whichiscomparabletothedepthoftheHe–He interatomicpotentialV

≈

11 K[1].V isduetoavanderWaalsinteractionwhichisrathersmallbecauseofthelowvalueof thepolarizabilityofHeatoms.ThelargevalueofquantumfluctuationshadalreadybeennoticedbyF.Londonin1936[2].It hasbeenrecentlycalculatedbyE.J.Rugelesetal.[3]whofoundthattheprecisemagnitudeofthekineticenergyassociated withquantumfluctuationsvariesfrom10to20 Kintheliquid,andfrom30to40 Kinthesolid,dependingondensity.

The large quantumfluctuationshave aseries ofconsequences. Firstof all, 4_{He remains liquiddown to T}

₌

_0,witha largemolarvolume(28 cm3_/mol).In4_{Hegas,thenaturalconcentrationofthelightisotope}3_{Heisbetween300and25 ppb,} depending on its precise origin, forexample Texasor Qatar wells[4].4_{He gas can be purifieddown to the 10}−12 _level. 3_{He impuritiescould even be totallyremoved from}4_{He crystalsby usingan adapted zone meltingmethod[5].Since all} chemicalimpuritiescan befilteredout,4Hecrystalsarethepurestcrystalsonecanstudy.But,aswe shallseebelow,the elasticpropertiesof4_{Hecrystalsareextremelysensitivetotinytracesof}3_{Heimpurities.}

3_{Hehavingasmallermass,itsmolar volumeintheliquidstateisevenlarger:37 cm}3_/mol.4_{Hecrystallizesonlyabove} 25 bar(3_{Heabove 30to35 bar, butthe crystalsunderconsiderationinthepresentreview are}4_{Hecrystals). Thereisno} triplepointinthephasediagramwheretheliquid,thegasandthesolid phasewouldmeetasinusualmaterials.Helium crystalsarealwaysgrownfromtheir liquidphase.Thesolid-vaporinterfacedoesnotexist.Furthermore,aroundthenodes oftheircrystallinestructure,theatomsfluctuatemorethaninordinaryclassicalcrystals.Forexample,the“Lindemann cri-terion”saysthat,inmostclassicalcrystalsontheirmeltingcurve,theamplitudeofthethermalfluctuationsistypically10% oftheinteratomicdistance[6].In4_{Hecrystals,BurnsandIsaacs[7]measureditandfoundarootmeansquaredisplacement} of26%.Inotherwords,theHeatomsareweaklylocalizedintheirnetwork.

Fig. 1. Thegrowthshapeofheliumcrystalsshowsevidencefortheirhexagonalstructure.Thispicturewastakenin1994at0.1 Kintheiropticalrefrigerator byS.Balibar,E.RolleyandC.Guthman.Fromtheanglebetweenthebasalfacetandthe6prismaticones,theorientationofthesix-foldsymmetryaxisc

canbedetermined.Thiscrystalis1 cmwide.Thecolorsareobtainedbyilluminatingitwithwhitelightdispersedthroughaprism[8].

Fig. 2. Theexperimentalcellthatwasusedtomeasurethemechanicalpropertiesof4_{HecrystalsatENSin2013–2014.Singlecrystalsaregrownfromthe} bottomupinsidethe0.7-mmslitbetweentwoverticaltransducersinthecenterofthecell.Thecrystalorientationwasobtainedfromphotographsofthe growthshapeinthebottompart.

Giventhesepeculiarities,someauthorsconsideredthatquantumfluctuationscoulddestroythesurfaceorderleading to theexistence offacetsonthecrystalshape.Thiswasprovedwrong [8].Inrealitytheexistenceoffacetsisa consequence of the existence of a periodic potential due to the crystalline structure, which localizesthe crystal surfaces nearcrystal planes at low enough temperature. This potential is reduced by quantum fluctuations butnot to zero.The existence of facetshasproved essential inallourstudies because,thanksto thewindows ofourrefrigerator,we coulddetermine the crystalorientationfromphotographsoftheirgrowthshapesuchasshowninFig. 1 [9].

Some other authors proposed that 4_{He crystals could be superfluid, not only liquid} 4_{He. A coexistence of crystalline} orderinrealspaceandsuperfluidorderinmomentumspaceiscalled“supersolidity”andisdefinitely paradoxicalbutnot impossible.AnexperimentbyKimandChan[10,11]hadraisedalotofinterestforthisquestionin2004.Theyhadobserved an anomaly ina “torsionaloscillator”(TO)filledwithsolid 4_{He,whichthey interpretedasaconsequenceofsupersolidity} appearingbelow0.1 K.However,theresonanceperiodofaTOisproportionaltothesquarerootofamomentumofinertia

I divided by an elasticconstant K .Supersoliditywould indeedimplyan anomalous reductionin I,butan increase in K

couldhavethesameeffect.DayandBeamish[12]showedin2007that4Hecrystalshadalargershearmodulusbelow0.1 K than above.Afterseveralyears ofstudies, itwasshownby severalgroups[13–15] that,inmostexperimentstherotation anomalyobservedinTOswasduetoachangeinelasticshearmodulus,nottoanysuperfluidflowreducingthemomentum ofinertia.Thediscoveryofsupersoliditywouldhavebeenamajorevent,butourcarefulstudyofthemechanicalproperties of 4_{Hecrystals ledusto thediscovery ofa “giantplasticity”[16],which isnot lessspectacularnorlessinteresting. This} plasticityisthesubjectofthepresentreview.

3. Agiantplasticity

Tomeasuretheshearmodulusofvarious4Hecrystals,weusedsuccessiveexperimentalcells.Thelastone isshownin Fig. 2.Itismadeofaroughlyhexagonalholeinacopperplatethatisattachedtoadilutionrefrigerator,whosetemperature canreach15 mKevenwhenitslargewindowsareopenforopticalmeasurements.Thecellisclosedbytwosapphire win-dowssothatanordinarycameracanrecord growthshapesofthecrystalsunderstudy[9,16].Itcontainstwopiezoelectric transducers madefromPZT material witha transverse polarization. Oriented single crystalsof 4He are grown along the verticalslit(0.7 mmthicknessinthiscase)betweenthetwotransducers.Onetransducerproducesaverticaldisplacement (0.95 Å/V), whilethe othermeasures thestress that istransmittedbythe 4_{Hecrystal. Thesensitivityandthestability of}

Fig. 3. Inatemperaturedomainaround0.2 K,thisultrapurecrystalshowsa“giantplasticity”:itsshearmodulusishighlyreducedwithrespecttoits intrinsicvalue(127 bar),whichisduetotheelasticityofthelatticeonly.Thisplasticityisduetothelargemobilityofdislocationsinthistemperature domain.Atlowertemperature,tracesof3_{Heimpuritiesstartbindingtothedislocationsandpreventingtheirdisplacement.Athighertemperaturethe} motionisdampedbycollisionswiththermalphonons.

thissetuparesuchthatonecouldapplyverysmallstrainsintherangefrom10−10_to10−6_{andmeasurestresses downto} 10−9 _{bar.Thestresstostrainratiogivesanabsolutemeasurementoftheshearmodulusinoneparticulardirectionthanks} toacarefulcalibration.Strainsareappliedatafrequencybetween1 Hzand20 kHzsothat,thankstoalock-inamplifier, boththeamplitudeoftheelasticshearmodulusandthedissipationcouldbemeasured.

Fig. 3showsoneparticularrecordingoftheshearmodulusofacrystalthatwasorientedwithitssix-foldsymmetryaxis (thec axisofthehcpstructure)nearlyvertical.InthisparticularcrystalnamedX15byHaziotetal.[16],theconcentration of 3_{Heimpurities had beenreduced to 4}

_×

₁₀−10 _{only. In a temperatureregion around 0.2 K, its shearmodulus is 43%} smaller than theintrinsic modulus (127 bar)that is knownfrom soundvelocity measurements at higherfrequency and higher temperature. By intrinsic we mean what is only due to the elasticity of the crystal lattice. But, as explained in all necessary details by Jacques Friedelin his famous book [17], realcrystals contain linear defects called“dislocations” whosemotionmayproduce an additionalstrainunder theactionofan applied stress.It meansa reductionin theshear modulus.Wehavedemonstratedthat,inthecaseof4He,highlymobiledislocationsstronglysoftenthecrystalsbutonlyina temperaturedomainbetweentwootherregionswheretwodifferentmechanismspreventthedislocationlinesfrommoving. Itdependsontheirconcentrationbuttypicallybelow0.1 K,3Heimpuritiesbindtodislocationsandeitherpinorslowdown theirmotion.Asfortheregionabove0.3 K,itisthecollisionswiththermalphonons,whichslowdownthismotion.

Beforeconsideringthiselasticanomalyinmoredetails,letusmakeaquickcomparisonwithclassicalcrystals.Inclassical crystals,dislocationsmoveonlyathighenoughtemperatureandundersufficientlylargestress.Thisisbecausedislocations aredefectlinesmovinginaperiodiclattice,somethingwhichmayonlyhappenbythermalactivationaboveenergybarriers called“Peierl’sbarriers”[17].Applyingastressreducesthesebarriers.Highthermalfluctuationsallowpointdefects(named “kinks”or “jogs”)on thedislocation linesto passabove the barriersandproduce adisplacement ofthe line. Inclassical crystals,the dislocationmotion induces asmall softeningthat is highly dependenton temperatureandstress amplitude, whilein4Heitislargeandindependentoftemperatureifimpuritiesareremoved.Weshalldiscussbelowtherelevanceof theterm“plasticity”whichsomespecialistsinMaterial’sscienceconsidermisleading,butwhichJ.Friedelaccepted[18].

Fig. 3 suggeststhat, atlow temperature, the pinningofdislocations byimpurities is not yetcompleted atthe lowest temperature we could reach. If due to 3_{He impurities, using naturalhelium with a larger} 3_{He concentration X}

3 should displace the pinning to higher temperatureand allow to reach the complete pinning at reachable temperatures.This is indeed what we observed with various crystals of naturalpurity containing X3

=

2

.

5

×

10−8 3He, asshown on Fig. 4. ContrarytotheultrapurecrystalX15ofFig. 3,allthecrystalswithnaturalpurity(X2,X3,X5,X6,X20,X21andapolycrystal) reachedtheirintrinsicshearmodulusbelow60 mK.Thisintrinsicvalueisindicatedwithticsontheverticalaxis.Itstrongly dependsonorientation.NotetheparticularcrystalX3,whosec axiswastiltedby45degreesfromthevertical:itshowsno temperaturedependenceatall,whichallowedustocalibrateourtransducers.

Theelasticitytensorofhexagonalclosepackedcrystalscontains only5elasticcoefficients.Amongthem,thecoefficient

c44 determines thestress that is necessaryto makebasal planes glideagainst each other (see casesb and b on Fig. 5). The coefficient c66 determines the stress thatis necessaryto deformthe hexagonal symmetryin thesebasalplanes (see casea onFig. 5).What happensintheparticularcaseofX3isthattheresponseisindependentofbothc44 andc66.Our measurements showed that the elasticanomaly isa softening from theintrinsic value when dislocationsare pinned (at

Fig. 4. Themagnitudeofthechangeinshearmodulusishighlysensitivetotheorientationsofcrystalswithrespecttotheverticalshearthatisusedto measurethisshearmodulus.Allcrystalsaremadeoutofnatural4Hegasthatcontains2.5×10−8 3Heimpurities,exceptX15(4×10−10).

Fig. 5. Usualnotationsforstressesandcorrespondingstrainsinhexagonalcrystals.Thecasea (topline),correspondstoadeformationofthehexagonsin thebasalplanes,wheretheappliedstressσ isproportionaltotheelasticcoefficientc66.Thetwocasesb andbbelowcorrespondtotheglidingofbasal planeswithrespecttoeachother,wheretheappliedstressσisproportionaltotheelasticcoefficientc44.Wearegratefultotheanonymousrefereewho allowedustoreproducethisfigure.

low T),andthatthissofteningishighlyanisotropic.Itsuggestedthatitiseitherduetoareductioninc44orinc66,andwe haveshownthatitisc44,notc66.Indeed,fromthecrystalorientationitispossibletoexpressthemeasuredshearmodulus intermsofall theelasticcoefficients.Then wecould eitherextract theT-dependentanomalousvalue ofc44byassuming thatc66andthethreeothersareconstant,ordotheopposite,thatisassumethatc44isconstantandextractaT-dependent value ofc66.Fig. 6showsthattheassumptionofaconstantc66iscorrect,sinceitgivessimilarvaluesforthereductionof

c44(

−

60%),whiletheoppositeassumptionwithaconstantc44wouldleadtoresultsthatareabsurd(changesinc66bymore than 1000%forX21andby300%forX6).Now,whatistheoriginofthisreduction?It isduetothefactthat dislocations

Fig. 6. Thesefourcrystalshavedifferentorientations,buttheyshowthesamevariationoftheirelasticcoefficientc44.IthasbeenunderstoodbyHaziotet al.asaconsequenceoftheglidingofdislocationsthatoccursonlyparalleltothebasalplanesofthehcpstructure(seetext).

Fig. 7. Therelativeamplitudeoftheshearmodulusc44for4differentcrystalsat20 mKasafunctionofthestressprojectedonthebasalplane(“resolved”). Atathresholdstressofafewμbar,dislocationsbreakawayfrom3_{Heimpurities.Thisthresholdislargerwhenincreasingthestressthanwhendecreasing} it.Beingfreeofimpurities,thecrystalX4showsareductionofc44by80%,independentofstress.

havepreferential glidingdirections.Areductioninc44meansthatdislocationsglideparalleltothebasalplanes.Thanksto discussionswithJ.FriedelandO.Hardouin-Duparc,we foundareferencebyB.Legrand[19] whoexplainsthat thisisdue tothesplittingofedgedislocationsintotwo“partial” dislocationswitha “stackingfault”inbetween, ifthestacking fault energyissmallenough.Realdislocationsareoftenatomicribbons, notreally1D-lines,inwhichcasetheyglideparallelto theribbonplane.Inmanyhexagonalmetals(Be,Mg,Co,Zn),glidingisalsoparalleltobasalplanes,butinsomeothers(Zr, Ti)itisalongprismaticplanes.Weseethat4Hecrystalsshowphenomenawhichexistinclassicalmaterialsbutperhapsnot asclearly.Indeed,notonlytheamplitudeofthesofteningisverylargein4_{He(wehaveobservedupto90%reductioninc}

44

[20])but,inclassicalcrystalsathightemperature,theglidingofdislocationsisusuallymixedwithmanyotherphenomena. Itappearedalsopossibletoremoveallimpuritiesfrom4Hecrystalsbyamethodthatisreminiscentoftheclassical“zone melting”.Itisbasedon thefactthat impuritiesare usuallymoresolubleintheliquidthan inthesolid,wherethe strain field around eachimpurity costs elastic energy.Inthe caseof4_{He, thedifference inpotential energybetweentheliquid} andthesolidhasbeencalculatedandfoundtobe

−

1.359 Kper3Heatom[5,22].Asaconsequence,whencoolingdowna crystalinthepresenceofitsliquidphase(rememberthatthereisnotriplepointinthephasediagramof4He),allthe3He impuritiesaretrappedintheliquidphase.Ithelpsshakinggentlythedislocationsby applyinganoscillating stresssothat 3_{Heimpuritiesdonotbindtodislocationsandarefreetomovetowardstheliquid,asweshallseelater.Fig. 7showsthat}

Fig. 8. Thestress–straindiagramofcrystalX4showalinearbehaviorwithareducedslope(samedataasinFig. 7).Onthecontrary,theplasticityof classicalcrystalsisanon-linearphenomenon[21].

atlowtemperaturewhenimpuritiesareboundtodislocations(crystalsX2,X5andX6),applyinganoscillatingstrainlarger thanafewmicrobarsdetachestheimpuritiessothattheshearmodulusisreduced.ForcrystalX4whereallimpuritieshave beendetached beforecooling,theshearmodulus isreducedby 80%fromitsintrinsicvalue andit staysatthislow value whencyclingtheappliedstress.Fig. 8showsthatthisimpurity freecrystalhasareducedelasticitythatisindependentof stress,contrarytoclassicalcrystalsthatwouldshowtheirintrinsicelasticityatlowstressandareducedoneathighstress, whichistheclassicalnon-linearplasticbehavior.Inthecaseof4_{Hecrystalstheelasticityisreducedbythehighmobilityof} dislocations. Thatiswhywecalledita“giantplasticity”.Butthisplasticityislinear,whichmaybeconfusingforMaterials scientistswhoaremorefamiliarwithclassicalplasticity,whichisalsoduetothemotionofdislocationsbutisanon-linear phenomenon.

4. Thedissipationassociatedwithdislocationmotion

In order to understandthe dislocation motion, we havemeasured the dissipation from thephase shift ofthe crystal responsetoadrivingstrain.Therearetwoseparate domainsintemperature.Athightemperature,i.e.above0.2 K,Granato and Lucke[23] hadpredictedin 1956 that dislocationsshould interact withthermal phonons.These interactions would transfermomentumfromthedislocationstothephonongaswherethecorrespondingenergyshouldthermalize.Thistheory waslaterimprovedbyNinomiya[24].Iftrue,thismechanismshouldleadtoamaximumchange

δ

c44intheshearmodulus

c0₄₄thatisgivenby

δ

c44 c0₄₄

=

α



L2 1

+

α



L2 (1) andtoadissipation1

/

Q 1 Q

=

α



L2 1

+

α



L2B L 2

_ω

_T3 ₍₂₎

where



isthedensityofdislocationlinesperunitvolume,L isatypicallengthbetweennodesinthedislocationnetwork,

α

=

0

.

019 andB

=

905 s m−2K−3isathermalphonondampingparametercalculated[4]byusingNinomiya’swork[24]. Haziot etal. [26] measured the dissipation 1

/

Q . Fig. 9showsa remarkable agreement withtheory.The initial slope is perfectlyconsistent withthe predicted

ω

T3 _{behavior. Witha more precisecalculation takingcare ofthe existence of} a distribution in lengths L, Feffermanetal. [25] could fit thewhole curve, not only the initialslope. But therough ap-proximationwithasinglelength L alreadyleadsanimportantconclusion.Indeed,fittingourresultswitheqs.(1)and(2), allowstodeterminetwoimportantquantities,thelinedensity



andtheirmeanlength L.Ifdislocationsformedaregular cubic lattice, thesetwo quantitieswouldbe relatedby the simplerelation



L2

₌

_{3. Foranythree dimensionallattice of} dislocations,thedensity



hastobeoforder1

/

L2.Wehavefoundthatitisnotthecase.Intheir2013experiment,Haziot etal.[26]founddensities



between104and106 percm2,whichisrathersmall,andlengthsL between100and230 μm, whichisverylarge.Thelattervaluesaremacroscopicand,mostinterestingly,muchlargerthanfora3-dimensionalnetwork ofdislocations.Theproduct



L2_{rangesfrom17to57.Inanattempttomakeevenbetterqualitycrystals,Sourisetal.[20]}

Fig. 9. Above0.3 K,thedissipation1/Q associatedwiththedislocationoptionisproportionaltoωT3_{asexpectedifthisdissipationisduetocollisions} withthermalphonons[23].Thedeparturefromlinearityhasbeen understoodbyFeffermanetal.asaconsequence ofthedistributionofdislocation lengths[25].

found



L2 _{valuesupto471!Itmeansthatthedislocationsdonotforma3D-network.Theyavoidcrossingbyforming2D} arraysofparallellinescalled“sub-boundaries”,andtheyglidetogetherparalleltothebasalplanesinacooperativeway.

Ascanbefoundoncemoreinhisbook,JacquesFriedelexplainsthatsuchaformationofsub-boundariesimpliesamuch largersoftening (Sourisetal.[20] foundupto 90%)than inthecaseofa 3Dnetwork whereitisoforder10%.Thistype ofcollective motionhadbeenpredictedin1940by W.L. Bragg[33] andobservedin1952 byParker andWashburn[34]. JacquesFriedelshowedin1955that,inaluminumcrystals,dislocationsarealsoarrangedinsub-boundariesandcouldmove enough nearthe meltingtemperaturethatthe shearmodulusishighly reduced[27].Heremarks thatwhen approaching themeltingpointat933 K,thedissipationassociatedwiththedislocationbecomesvery largeandthedislocation motion mixeswithgrainboundarymigration.

Comingback toour4_{Hecrystals,ourresultscouldbe thoughtofinanalogy withapileofpapersheetsthatisstiff in} all directionsexceptparalleltothe sheetswerethe pileisvery soft. Butitisnot exactlythewholeatomicplaneswhich move,it ispartofthemneareach dislocation.Furthermore,thereisa non-negligiblefriction betweenpapersheets orin classicalcrystalssothatthestress/strainrelationisnon-linear,while,asshowninFigs. 7–9,thedissipationassociatedwith theshear motionin4Hetends tozeroatlow temperatureintheabsence ofimpurities.Thisisan evidenceforquantum behavior.Onepossibilityisthatquantumfluctuationsmakethekinkenergyvanishsothatdislocationlinesaretotallyfree tomove inaperiodic latticethat hasnoinfluenceonthem.Anotherpossibilityisthat kinkshavea non-zeroenergybut dislocation linescontainsome kinksforgeometric reasons,butthese“geometric kinks”wouldhavetomove byquantum tunneling throughverysmallPeierl’s barriers,a situationthatwouldbe hard todistinguishexperimentally fromthefirst one.

5. Thebindingofimpuritiestomovingdislocations

Intheabove sectionwe mainlyfocusedon thedissipationabove 0.3 Kwhereithasbeenshownto beaconsequence ofaninteractionwiththermalphonons.Below0.2 K,thedissipationmechanismisdifferent. Whenthe3Heconcentration

X3 isnot negligible,theseimpuritiesprogressivelybindtodislocationsasT decreases.Adefinitedissipationisassociated withthisbinding,asshownbyFig. 10.Whenimpuritiesstartbinding,thedislocationmotiondecreasesandthedissipation increases.Itreachesa peak atsometemperature Tp beforevanishing atlow T whenthe dislocationsarefullyanchored. WestudiedthefrequencydependenceofTp.

Knowing thetypical length anddensityofdislocationsinthe crystalsunderstudy, Haziotetal.[35] coulddetermine theirtypicaldisplacementandthespeedoftheirmiddlepointforagivenstrain.Theresultisshownonthesemi-logplot ofFig. 11 wheretwo differentregimesappear. Athighspeed, thepeak temperatureisindependent ofspeed. Thisisthe expectedregimewhereimpuritiesactaspinningpoints:theycannotmovefastenoughandtheyanchordislocations.Butat lowspeed,morepreciselybelow45 μm/s,theconstantslopedemonstratesathermallyactivatedregimeoffrictionwhere the speed decays exponentially asthe temperature is reducedand more impurities bind.It means that the dislocations move dressed withimpurities attachedto them.Assuming that thefriction is proportional tothe densityofbound 3He, the slopeinthissemi-log plotallows todetermine thebinding energy E3 of 3Heimpurities to thedislocation lines.We foundE3

=

0

.

67 K forthisparticularcrystal.Inamoreprecisestudy[4],wefoundadistributionofbindingenergiesaround

Fig. 10. Thetemperaturevariationoftheshearmodulus(a)andthedissipationassociatedwithdislocationmotion(b).Recordingsatdifferentfrequencies ofthedrivingstrainshowthatthetransitionfromstiff(atlowT)tosoft(athigherT)dependsonfrequencyinthelow-frequencyrangeonly.

Fig. 11. Knowingthedensityofdislocationlinesandtheirmeanlengthbetweenthenodesintheirnetwork,Haziotetal.[21]couldcalculatethemean dislocationspeedandstudyitsdependenceonthetemperatureTp ofthepeakdissipationinthemiddleofthetransition.Tworegimesappear:below 45 μm/s,3_{Heimpuritiesmoveattachedtothedislocationswhile,abovethiscriticalspeed,dislocationsarereallypinnedby}3_{Heimpurities(seetext).}

Fig. 12. Semilogplotoftherelaxationtime τ ofdislocationsversustheinversetemperature1/T forcrystalswithdifferentorientationsandimpurity concentrations.SlightvariationsinslopeshowthatthereisanarrowdistributioninthebindingenergyE3of3Heimpuritiestodislocations(seetext).

Fig. 13. ThreerecordingsoftheshearmodulusmeasuredbyFeffermanetal.[25]whiledecreasingthedrivingstrainnear25 mK.Byanalyzingtheshape ofthetransitiontowardtheintrinsicshearmodulusatlowstrain,Feffermandeterminedthewidthofthedistributionoflengthsbetweenthenodesofthe dislocationnetwork.

0.7 K witha typical width of0.1 K. This can be seen onFig. 12 wherethe relaxation time

τ

= (

1

/

ω

)

√

1

+

α



L2 _show slightvariationsinslopefordifferentcrystals.Ourassumptionofadissipationproportionaltotheconcentration X3 of3He impuritieswas verifiedbySourisetal.[4]by comparingthreetypesofcrystalswith X3 respectivelyequalto2

.

5

×

10−8, 3

.

8

×

10−7_and2

_.

₃₂

_×

₁₀−6_.

Adistribution ofbinding energies was expectedbecausewe know thatdislocations arerarely purely edgenor purely screw type. Dependingon their orientation inthelattice, they havea mixedcharacter sothat the bindingenergy varies slightlyfrom a value corresponding to a pure screw type to a pure edge type where it should be slightlylarger. As we shallseenow,wehavealsofound someevidencefortheexistenceofthisenergydistributionfromotherfitswiththeory. Indeed,wefoundevidenceforadistributioninthe“networklength”betweennodesinthedislocationnetworkbystudying thestrain dependenceoftheshearmodulus atlow temperature.Ifone appliesalarge oscillatingstrain amplitude(10−6) at0.5 Kandcooldown,thelargeoscillationsofthedislocationlinespreventtheimpuritiesfrombinding.Whenreducing then the strain amplitude, 3He impurities start binding andFefferman etal. [25] showed that the shear modulus starts increasing(seeFig. 12).Iftherewereasinglepinninglength,thereshouldbeaprecisevalueoftheappliedstrainatwhich alldislocationswouldgetpinnedandtheshearmodulusshouldjumptotheintrinsicvalue.Thesmoothtransitiononesees onFig. 13isanevidencethatitisnotthecase.Infacttheshapeofthetransitionfromsoftathighstraintostiffatsmall strainallowedFeffermantodeterminethelengthdistribution.Thisisbecauseshortdislocationsmovelessthanlongones, 3_{Heimpuritiesbindtotheshortonesbeforelongonesasthedrivingstrainisreduced.Thenetworklengthdistributionwas} foundtoberatherbroad,extendingtypicallyfrom30to300 μminthecaseofthisparticularcrystal.

WhentryingthentofitdatasetssimilartothoseshowninFig. 10,thatisrecordingsofthestiffnessandofthe dissipa-tionasafunctionoftemperature,itappearednecessarytoaccountnotonlyforadistributionofnetworklengthsbutalso ofbindingenergiesof3_{He.Indoingthis,Feffermanetal.[25]foundthatthewidthofthisenergydistributionhadtobeof} order0.1 Karound0.7 K,inverygoodagreementwiththestraindependenceinFig. 13.

is strained so that quantum exchange with4He is at leastperturbed. As forthe mechanism that explains the observed dissipation, we proposed withthe help ofH.J. Maris [4]that it is an emissionof transversewaves along thedislocation lines.

Ourseriesofmeasurementsofstiffnessanddissipationof4Hecrystalsasafunctionoforientation,temperature,driving strain, and3He concentration allowed usto characterize nearly all the staticand dynamic propertiesof the dislocation network.Suchasetofresultsandthewayhowtheywereobtainedisclearly particulartothisquantumcrystal.Thereare afewmorequestionswhichwoulddeservemorestudy.Oneofthemconcernstheexactvalueofthekinkenergy:isitzero ornot?ArelatedoneisthevalueofthePeierl’sbarrierforthemotionofexistingkinks:oncemoreisitzeroornot?

It wouldbe nicealsotodeterminethewidthofsplit dislocation.Itisrelatedtotheenergyofstacking faultswhichis notyetpreciselyknown.

Thecaseofpure3Hecrystalsisobviouslyinteresting.Theyexistintwodifferentstructures:bodycenteredorhexagonal compact.Dislocationpropertiesshould beratherdifferentinthetwo structures.The tunnelingofkinksandofimpurities should be rather differentfromthe 4_{He casebecause}3_{He atomshave a non-zeronuclear spin that isdisordered except} below1 mK,anultralowtemperaturethatwouldmakethestudyofmechanicalpropertiesverydifficult.Furthermore,3He crystalsaremuchmoredifficulttopurifythan4_Heones.

Eventually, thereremainssomecontroversyaboutapossiblemassflowalongthedislocation cores.Experimentsatthe MassachusettsUniversity (Amherst)[31]andinEdmonton (Canada)[32]areinprogress,whosesetsofresultsare notyet fullyunderstood.

Acknowledgements

WeacknowledgesupportfromNSERC(Canada)andfromtheEuropeanResearchCouncilgrantAdG247258-SUPERSOLID.

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