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Study on the Effects of Anisotropic Disorder on Superfluid Helium Three in High Porosity Aerogel Using Longitudinal Ultr...

Permanent Link: http://ufdc.ufl.edu/UFE0041531/00001

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Title: Study on the Effects of Anisotropic Disorder on Superfluid Helium Three in High Porosity Aerogel Using Longitudinal Ultrasound
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Moon, Byoung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: aerogel, attenuation, gapless, helium, longitudinal, phase, pvdf, superfluid, ultrasound
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

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Abstract: STUDY ON THE EFFECTS OF ANISOTROPIC DISORDER ON SUPERFLUID HELIUM THREE IN HIGH POROSITY AEROGEL USING LONGITUDINAL ULTRASOUND Longitudinal sound attenuation measurements in superfluid 3He in 98% aerogel were conducted at pressures between 14 and 33 bar and in magnetic fields up to 4.44 kG. The temperature dependence of the ultrasound attenuation in the A-like phase was determined for the entire superfluid region by exploiting the field induced meta-stable A-like phase at the highest field. In lower fields, the A-B transition in aerogel was identified by a smooth jump in attenuation on both cooling and warming. Based on the transitions observed on warming, a phase diagram as a function of pressure (P), temperature (T) and magnetic field (B) is constructed. The transitions obtained by isothermal field sweeps are consistent with those by temperature sweeps at constant magnetic fields. The A-B phase boundary in aerogel recedes to the corner of zero temperature and melting pressure in response to an increasing magnetic field, which is drastically different from the bulk. The presence of elastic impurity scattering by aerogel limits the growth of the mean free path at low temperature. In this case, the dominance of temperature independent elastic scattering keeps the system from entering into collisionless limit on cooling. Therefore, it is expected that the sound attenuation obeys the omega^2-dependence. However, our result reveals that non-trivial frequency dependencies, departing from the omega^2-dependence appear as temperature lowers into the superfluid regime. This tendency is more evident at higher pressure and lower temperature. We attribute this property to the gapless behavior of superfluid 3He in aerogel.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Byoung Moon.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lee, Yoonseok.

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Source Institution: UFRGP
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Classification: lcc - LD1780 2010
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Permanent Link: http://ufdc.ufl.edu/UFE0041531/00001

Material Information

Title: Study on the Effects of Anisotropic Disorder on Superfluid Helium Three in High Porosity Aerogel Using Longitudinal Ultrasound
Physical Description: 1 online resource (108 p.)
Language: english
Creator: Moon, Byoung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: aerogel, attenuation, gapless, helium, longitudinal, phase, pvdf, superfluid, ultrasound
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: STUDY ON THE EFFECTS OF ANISOTROPIC DISORDER ON SUPERFLUID HELIUM THREE IN HIGH POROSITY AEROGEL USING LONGITUDINAL ULTRASOUND Longitudinal sound attenuation measurements in superfluid 3He in 98% aerogel were conducted at pressures between 14 and 33 bar and in magnetic fields up to 4.44 kG. The temperature dependence of the ultrasound attenuation in the A-like phase was determined for the entire superfluid region by exploiting the field induced meta-stable A-like phase at the highest field. In lower fields, the A-B transition in aerogel was identified by a smooth jump in attenuation on both cooling and warming. Based on the transitions observed on warming, a phase diagram as a function of pressure (P), temperature (T) and magnetic field (B) is constructed. The transitions obtained by isothermal field sweeps are consistent with those by temperature sweeps at constant magnetic fields. The A-B phase boundary in aerogel recedes to the corner of zero temperature and melting pressure in response to an increasing magnetic field, which is drastically different from the bulk. The presence of elastic impurity scattering by aerogel limits the growth of the mean free path at low temperature. In this case, the dominance of temperature independent elastic scattering keeps the system from entering into collisionless limit on cooling. Therefore, it is expected that the sound attenuation obeys the omega^2-dependence. However, our result reveals that non-trivial frequency dependencies, departing from the omega^2-dependence appear as temperature lowers into the superfluid regime. This tendency is more evident at higher pressure and lower temperature. We attribute this property to the gapless behavior of superfluid 3He in aerogel.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Byoung Moon.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lee, Yoonseok.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041531:00001


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Iwouldliketothankmyadvisor,YoonseokLeeforhisinvaluableassistanceandguidance.Thisresearchwouldnothavebeenpossiblewithouthisenthusiasticcoordination.IthankMarkW.Meisel.Hewasalwayswillingtohelpandsupportme.Isincerelyappreciateit.IwishtoexpressmygratitudetoNaotoMasuhara.Hisknowledgeonthecryostatandphysicswereessentialtonishmyexperiment.Wewentthroughlotsofproblemstogetherforseveralyears.TherearemanypeopleIwouldliketoconveymythanksto.GregLabbeandJohnGrahamfromcryogenicservices,andMarkLink,EdStorchandBillMalphursfromphysicsmachineshop,andLarryPhelpsandPeteAxsonfromtheelectronicshop.Theirsupportswerealwaysgreatlyappreciableandtheirnejobswereamazing.IwouldliketothankGaryIhasforthespectrometers,andNobertMuldersinUniversityofDelawareforprovidingusaerogel.Specialthanksgotomycommitteemembers,PradeepKumar,YasumasaTakano,AmlanBiswasandCliffordR.Bowers.Ithankmycolleagues,Hyunchang,Pradeep,MiguelandPanfortheirsupportandcooperation.Mostofall,Icannotsaythankenoughtomybelovedparentsandsisters.TheyhavebeenalwayssupportiveandnevershownanyquestiononwhatIhavebeendoing.Iamverygladbecausetheyaresohappyformygraduation.Finally,mywifeHyerin!Thankyouforyourunderstandingandlove.Icannotthinkofanylifewithoutyou.Iloveyou!.Chaeun!Iamsomissingyou,mydaughter. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2BASICPROPERTIESOFLIQUIDHELIUMTHREE ............... 16 2.1NormalLiquid3He ............................... 16 2.1.1FermiLiquidTheory .......................... 16 2.1.2CollectiveModes ............................ 17 2.2Superuid3He ................................. 20 2.2.1SuperuidPhasesof3He ....................... 21 2.2.2OrientationalEffects .......................... 24 3SUPERFLUIDHELIUMTHREEIN98%AEROGEL ............... 27 3.1AerogelandScatteringModels ........................ 27 3.2SuperuidityandSuperuidPhases ..................... 30 3.3PhaseDiagram ................................. 33 3.4GaplessSuperuidity ............................. 36 3.5LongitudinalSound ............................... 38 4LONGITUDINALSOUNDATTENUATIONINSUPERFLUIDHELIUMTHREEIN98%AEROGELANDPHASEDIAGRAM .................... 42 4.1Experiments .................................. 42 4.2UltrasoundAttenuationandaP-B-TPhaseDiagram ............ 48 4.2.1Overview ................................ 48 4.2.2ResultsandDiscussion ........................ 50 4.2.2.1LongitudinalSoundAttenuationandtheA-BTransitioninAerogel ........................... 50 4.2.2.2TheABTransitioninAerogelbyIsothermalFieldSweeps ............................ 59 4.2.2.3PhaseDiagram ........................ 66 4.2.2.4AttenuationPropertiesofA-likePhase ........... 70 4.3FrequencyDependentUltrasoundAttenuation ................ 71 4.3.1Overview ................................ 71 4.3.2ResultsandDiscussion ........................ 73 5

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................. 85 5.1Overview .................................... 85 5.2PropertiesofPVDFTransducers ....................... 86 5.3AcousticCellandExperiment ......................... 88 5.4ResultsandDiscussion ............................ 90 6CONCLUSION .................................... 100 REFERENCES ....................................... 102 BIOGRAPHICALSKETCH ................................ 108 6

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Table page 2-1Healinglengthsforvariousorientingforces .................... 26 4-1Parametersfordeterminingg(). ......................... 57 4-2Importantparametersestimatedforthreepressuresusedinthiswork. ..... 79 5-1Fittingparameters .................................. 93 5-2ValuesofT2byseveralgroups .......................... 98 7

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Figure page 1-1Phasediagramofliquid3Heintheabsenceofmagneticeld. .......... 14 2-1Phasediagramofliquid3Heinthemagneticeld. ................ 23 2-2Gapdistortionaccordingtoanincreasingmagneticeld. ............ 24 3-1Superuidtransitionof3Heinaerogel ....................... 29 3-2Superuiddensityatvariouspressureobtainedbytortionaloscillator ...... 31 3-3Suppressionofthetransitiontemperatureandtheorderparameter ....... 32 3-4FigureofindicatingthecoexistenceoftheA-andtheB-likephases. ...... 34 3-5FigureofillustratingtheshiftupofPCPduetotheanisotropicscatteringandthedecreaseofstrongcouplingeffects ....................... 35 3-6ThetemperaturedependenceofshearviscosityandthedensityofstatesatT=0 37 3-7Thetemperaturedependenceofthefrictionalrelaxationtimef 41 4-1Schematicdiagramoftheexperimentalcell. .................... 43 4-2Picturesoftheacousticcavityandtheassembledcell. .............. 43 4-3Resonancetestforselectingbest-matchedpair. ................. 44 4-4Pictureoftheexperimentalregionofthecryostat. ................. 45 4-5Aschematicdiagramofthemeasurementscheme(MATECpulsedspectrometer). 46 4-6Atypicalreceivedsignalandanintegrationscheme. ............... 47 4-7Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat29bar. ............................ 51 4-8TheA-Btransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat29bar. ....................................... 52 4-9Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat19.5bar. .......................... 53 4-10TheA-Btransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat19.5bar. ...................................... 54 4-11Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat33bar. ............................ 55 8

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....................................... 56 4-13Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat25bar. ............................ 57 4-14TheABtransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat25bar. ....................................... 58 4-15Temperaturedependenceofattenuationat33barusing6.22MHzexcitation. 59 4-16MagneticelddependenceofthewidthoftheA-likephase. ........... 60 4-17MagneticelddependenceofthewidthoftheA-likephasescaledbyB2. ... 61 4-18Pressuredependenceofg(). ........................... 62 4-19Resultsoftheisothermaleldsweep(IFS)at0.3mKandP=25bar. ..... 63 4-20Resultsoftheisothermaleldsweepat14bar. .................. 64 4-21Resultsoftheisothermaleldsweepat29barandT0:86mK. ........ 64 4-22Resultsoftheisothermaleldsweep(rampuponly)at29barandT1:38mK. 65 4-23Phasediagramofsuperuid3Hein98%aerogel. ................. 67 4-24Relativeattenuationat6.22MHzforzeromagneticeldoncoolingandwarmingand4.44kGonwarming. .............................. 70 4-25Absoluteattenuationsforpressuresfrom8to34barasafunctionoftemperatureat9.5MHz(onwarmingexcept8bar). ....................... 72 4-26Aratioofthezerotemperatureattenuation0totheoneatthesuperuidtransitiontemperaturec. .................................... 73 4-27Temperaturedependenceofrelativeattenuationat3.69,6.22,and11.30MHztakenat33bar. .................................... 75 4-28Temperaturedependenceofabsoluteattenuationforthreefrequenciesalongwiththepreviousmeasurementat9.5MHzfor25bar(a)and14bar(b)ofsamplepressures. .................................. 76 4-29Soundattenuationasafunctionoffrequencyforselectreducedtemperaturesat33(a),25(b),and14(c)bar. ........................... 78 4-30Schematicdiagramofresonantscatterings. .................... 80 4-31Temperaturedependenceofrelativeattenuationat3.69,6.22,8.73,9.50and11.30MHztakenat29bar. ............................. 83 9

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....................................... 84 5-1(a)Quarterwavelengthforheavybackingmaterials(b)Halfwavelengthforpolymerbackingmaterials. ............................. 87 5-2Drawingofcopperbacking. ............................. 88 5-3Pictureofacousticcavitieswithaerogel(a)andwithoutaerogel(b). ...... 89 5-4Pictureofcellwithtwoacousticcavitiesinit:(a)Bottomview(b)Topview. ... 90 5-5Pictureofexperimentalcellset-up. ......................... 91 5-6Integratedmagnitude(inarbitraryunits)versusexcitationfrequencyforliquid3Heat28.1barand12.1mK. ............................ 92 5-7Integratedmagnitude(inarbitraryunits)versusfrequencyresponsemeasuredbyGranrothetal.inliquid4Heat1barand30mK. ............... 93 5-8Linearitytestfor6MHzpulse. ............................ 94 5-9Linearitytestfor20MHzpulse. ........................... 94 5-10Soundattenuationinnormalandsuperuid3Heatfourdifferentsoundfrequency.Inset:MagniedviewnearTc: 95 5-11Datattingfor24MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. ...................... 96 5-12Datattingfor17MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. ...................... 96 5-13Datattingfor11MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. ...................... 97 5-14Datattingfor6MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. ...................... 97 10

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Longitudinalsoundattenuationmeasurementsinsuperuid3Hein98%aerogelwereconductedatpressuresbetween14and33barandinmagneticeldsupto4.44kG.ThetemperaturedependenceoftheultrasoundattenuationintheA-likephasewasdeterminedfortheentiresuperuidregionbyexploitingtheeldinducedmeta-stableA-likephaseatthehighesteld.Inlowerelds,theABtransitioninaerogelwasidentiedbyasmoothjumpinattenuationonbothcoolingandwarming.Basedonthetransitionsobservedonwarming,aphasediagramasafunctionofpressure(P),temperature(T)andmagneticeld(B)isconstructed.Thetransitionsobtainedbyisothermaleldsweepsareconsistentwiththosebytemperaturesweepsatconstantmagneticelds.TheABphaseboundaryinaerogelrecedestothecornerofzerotemperatureandmeltingpressureinresponsetoanincreasingmagneticeld,whichisdrasticallydifferentfromthebulk. Thepresenceofelasticimpurityscatteringbyaerogellimitsthegrowthofthemeanfreepathatlowtemperature.Inthiscase,thedominanceoftemperatureindependentelasticscatteringkeepsthesystemfromenteringintocollisionlesslimitoncooling.Therefore,itisexpectedthatthesoundattenuationobeysthe!2-dependence.However,ourresultrevealsthatnon-trivialfrequencydependencies,departingfromthe!2-dependenceappearastemperaturelowersintothesuperuidregime.Thistendency 11

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Liquidheliumisafascinatingcondensedmattersystemthathasattractedavastamountofinterestowingtoitsuniquelowtemperatureproperties.Itistheonlymaterialinnaturethatexistsasliquidevenatabsolutezerotemperatureduetothelargezeropointenergyandtheweakattractiveinteratomicinteraction.Theboilingpointsare4.21Kand3.19Kfor4Heand3He,respectively.Thisuniquepropertyallowsphysiciststoinvestigatequantumphenomenainliquidstate. Themostintriguingpartisthatbothliquidsundergophasetransitionstosuperuidstates.However,differentquantumstatisticsapplicableto3Heand4Hemakestrikingdifferencesinthephysicalbehaviorinsuperuidaswellasinnormaluid.Incontrastto4He,whichisacompositebosonwithspin0,3HeobeysFermistatisticswithspin1/2.Asaresult,3HebecomesasuperuidthroughBCSpairing.Becauseofitschargeneutralityandtheabsenceofthelatticestructure,theattractivepairinteractionismediatedthroughthespinpolarization,givingrisetothespintripletpairing(S=1)ratherthanthespinsingletpairing(S=0)asinconventionalsuperconductors.Ithasbeenexperimentallyknownthatthreedistinctstablesuperuidphasesexistinbulk3He,referredtoastheA-,theB-andtheA1-phases.Figure 1-1 showstheTPphasediagramofliquid3Heintheabsenceofmagneticeld.Sincethediscoveryofthesuperuidityin3He[ 1 ],atremendousamountoftheoreticalandexperimentalworkshavebeenperformedandrevealedrichquantumphenomenaassociatedwithspontaneoussymmetrybreaking.Moreover,itsexceptionalpurityhasofferedanopportunitytotesttheoreticalideassuchasthegeneralizedBCStheoryandFermiliquidtheory. Inadditiontotheexceptionalpurity,sincethestructuresoforderparametersofsuperuid3Hewerewellknown,itwasexpectedthatthissystemwouldprovideamuchbetterunderstandingfortheimpurityeffectsonunconventionalCooperpairingsystems. 13

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Phasediagramofliquid3Heintheabsenceofmagneticeld. Unlikethecaseofconventionalsuperconductors,alltypesofimpuritiesaredetrimentaltoCooperpairswithanon-zeroangularmomentum.Therefore,superuid3Hewithp-wavepairingisexpectedtobestronglyinuencedbyanytypesofimpurity.However,asystematicinvestigationoneffectsofimpurityordisorderhadnotbeenachieveduntilhighporositysilicaaerogelwasemployedinsuperuid3HeforthersttimebyPortoandParpia[ 2 ].Sincethen,aerogelhasplayedamajorroleasanimpurityinsuperuid3He.Mostphysicalquantitiessuchastransitiontemperature,orderparameter,andtransportparametersarerescaledorinuenceddramaticallyinthissystemduetothepair-breakingbyscatteringofftheaerogel.Inaddition,thecorrelationofaerogelstructureandtheirunusuallengthscalesmakethesystemmoreinterestingandalsounique. Ultrasound(MHz)hasbeenoneofthemostusefultoolstoprobethepropertiesofliquid3He.Thereexistseveralsoundmodesbothinpureandimpure(aerogel) 14

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Inthiswork,theeffectsofhighporosityaerogelonsuperuidaswellasnormal3Hewereinvestigatedusingvariousultrasoundtechniques.Inchapter2,someofbasicpropertiesofnormalandsuperuid3Hearediscussed.Inchapter3,theimpurityeffectsofaerogelonsuperuid3Hearediscussed.Wewillsurveythecurrentstatusoftheeldbyreviewingsomeoftheimportantexperimentalresultsandtheoreticalideas.Chapter4describesourlongitudinalultrasoundexperimentconductedinsuperuid3Hein98%porosityaerogel.Thesoundattenuationmeasurementsinbulkliquid3Heusingbroadbandtransducersarediscussedinchapter5.Finally,intheconclusion,asummaryandafewsuggestionsaregiveninconnectionwiththedirectionforthefutureworks. 15

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3 5 ].Liquid3HeisaperfectexampleofthisFermiliquidtheoryandhasservedastheparadigmformanydecades. wheretheinteractionfunctioncanbeparametrizedforisotropicsystembyf~k~k00=1 16

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whereisthegyromagneticratio.Since,fromtheGalileaninvariancem 3Fs1; 6 ],indicatingtheexistenceofstronginteractions. 7 ]: 17

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3v2F(1+Fs0)1+Fs1 whereistherelaxationtimeforviscosity.Comparingthisattenuationwithaclassicalexpression,1=2!2 onecanndtheexpressionfortheshearviscosityofaFermiliquidas=1 5v2F1+Fs1 Therefore,when!1,therstsoundmodeexperiencesincreasingdampingandeventuallyceasestopropagate.LandauarguedthatanewmodeofsoundcouldemergeinthiscollisionlesslimitinaFermiliquidinwhichtheessentialrestroingforcewouldarisefromthemoleculareld.Wole[ 8 ]calculatedthezerosoundvelocityandattenuationusingakineticequationandacollisionintegral, 5m 5Fs2vF 45m where0NisthequasiparticlelifetimeontheFermisurfaceandrelatedtoby 18

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42(12)10N; wheretheparameter2isanangularaverageofthescatteringcrosssectioncorrespondingtol=1inLegendrepolynomial,Pl(cos).20:72,giving=2:750N[ 8 ].Infact,histheorycanbeappliedforanarbitraryvalueof!.Inthe!1limit,thetheoryrecoversexactlythesameresultsasEq. 2 andEq. 2 whenignoringFs2. Since0N/1 2 and 2 ,11=T2and0T2.Therefore,theunmistakablecrossoverbetweentwosoundregimesshouldappearastemperaturechanges,whichwasconclusivelyconrmedexperimentallybyAbeletal.[ 9 ]. Later,Rudnick[ 10 ]developedatheoryonzerosoundbasedonthefactthatzerosoundisaviscoelasticeffect,aswasrecognizedbyLea[ 11 ].Hewasabletoexpressthesoundvelocityandattenuationinoneequationforeachinsteadofhavingtwoexpressionsforthetwolimits,respectively.TheviscoelasticmodelprovidedexcellenttstothemeasurementsofAbeletal.[ 9 ]atlowpressuresandKettersonetal.[ 12 ]athighpressuresforawidetemperaturerangeincludingthecrossoverregionwithcl=c1+(c0c1)!22 NotethattheseareclassicalresultswithouttheconsiderationofaFermiliquid. Inarealexperimentalsituation,thepresenceofconningwallcausesanadditionalattenuationandacorrectiontothevelocitybecausetheuidstartstoslipatthewallswhenthemeanfreepathiscomparabletothesamplesize.NagaiandWole[ 13 ]calculatedthesoundvelocityandattenuationusingasetofhydrodynamicequations, 19

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!1=2issmallerthanthedimensionoftheresonatorandthewavelengthofthesound.Theyobtainedthegeneralexpressionforthesoundvelocityandattenuation,c=c11+2! whereZ(!)isthecomplexsurfaceimpedance,andRandLaretheradiusandthelengthofthecylindricalacousticresonator. OneofthemostfascinatingaspectsofzerosoundinaFermiliquidisthepossibilityofpropagatingtransversezerosound(TZS)mode[ 3 5 ].Ingeneral,thetransversewaveinliquidsdecayswithinalengthcomparabletothewavelength.Therefore,thepredictionbyLandauontransversezerosoundmodeinaFermiliquidisextremelyinteresting.However,therelevantFermiliquidinteractionforTZSinliquid3Heismarginallystrongtosupportthismode.ThisresultsinthespeedofTZSveryclosetovFcausingstrongLandaudamping.AlthoughthereisnounequivocalexperimentalevidenceofTZSinthenormalstateof3He,theexistenceofpropagatingTZSintheB-phaseof3HewasbeautifullydemonstratedbyLeeetal.[ 14 ]. 20

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where^kisaunitvectorinmomentumspace.Thevector~d(^k)canbeexpandedin^kjwithatensorquantitydj,a33matrixofcomplexcomponents,whereandjcorrespondtothevaluesof-1,0,+1ofthequantumnumbersSzandLz,respectively,d(~k)=3Xjdj^kj: Intheabsenceofamagneticeld,thefreeenergyminimumisobtainedbythefollowingsphericallysymmetricorderparameter,dj=eij!~k=ei(^kx+i^ky) whereisanisotropicenergygap.ThisstatewasrstdiscussedbyBalianandWerthamer[ 15 ]andisreferredtoasthe`BWstate',whichcorrespondstotheB-phaseofsuperuid3He.Thisstateconsistsofthesuperpositionofallspintripletstatesandhasanisotropicgap.Sincethefreeenergyisinvariantundertherotationofthespinspacerelativetotheorbitalspace,thegeneralformoftheBWstatecanbewrittenbydj=eiRj(^n;); whereRj(^n;)describesarelativerotationofspinandorbitalspacesaround^nbyanangle.Therefore,theBWstateisinvariantundersimultaneousrotationinspinandorbitalspaceandonlytherelativespin-orbitsymmetryisbroken. 21

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16 ].Later,itwasrealizedbyAndersonandBrinkman[ 17 ]thattheexperimentallyobservedA-phaseathighpressuresisconsistentwiththetheoreticalaxialphasewhichisnowcalledtheABMstate.UnliketheBWstate,theABMstateishighlyanisotropicandhasCooperpairswithonlySz=1.Thisiswhythisstateisreferedasanequal-spinpairingstate.TheorderparameterdiscussedbyAndersonandMorel[ 16 ]isdj=0~d(^mj+i^nj); where^l=^m^nindicatestheorbitalangularmomentumoftheCooperpairand^mand^naremutuallyorthogonalunitvectorsinorbitalspace. Inthepresenceofamagneticeld,thephasediagramchangesdrastically.SincethemagneticsusceptibilityoftheBWstateislowerthanthatoftheABMstate(duetotheSz=0component),themagneticenergyislowerintheABMstatethanintheBWstate.Therefore,itisreasonabletothinkthattheABMstatewouldgainitsgroundagainsttheBWstateasthestrengthofmagneticeldincreasesandwouldeventuallybecomemorestable.However,theeffectofamagneticeldonthephasediagramismoresubtleandprofound[ 18 ].EvenaninnitesimallyweakmagneticeldopensupasliveroftheA-phaseregionbelowthesuperuidtransitionatallpressures.TheA-phaseregioncontinuouslygrowsasthestrengthofmagneticeldincreasesandeventuallypushestheB-phaseoutofthephasediagramaroundB0:6T.ThephasediagramshowninFig. 2-1 clearlydemonstratesthisbehavior.Theprofoundeffectofmagneticeldwillbediscussedindetailinthefollowingchapter. Thethirdphaseofsuperuid3He,theA1-phase,appearsonlyinthepresenceofamagneticeldbysplittingthesuperuidtransitionintotwosecondordertransitions. 22

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Phasediagramofliquid3Heinthemagneticeld. TheZeemanenergyseparatesthetransitiontemperatureforthespinupanddowncomponentsbecauseoftheminuteparticle-holeasymmetry.Asaresult,intheA1-phaseonlythespinupcomponentformsCooperpairsintofullypolarizedsuperuidstate.ThewidthoftheA1-phaseisrathersmallandalmostproportionaltothestrengthofmagneticeld,60K/Tatthemeltingpressure. ThepresenceofamagneticeldcausesadistortionoftheotherwiseisotropicBWstategap,thesocalledgapdistortion.TewordtandSchopohl[ 19 ]studiedthiseffectandshowedthatthegapcomponentperpendiculartomagneticeld,?,increaseswithincreasingmagneticeld,whiletheparallelcomponent,k,decreasesandsuddenlyfallstozeroatacertaincriticalvalueofmagneticeld.AprospectivedistortionprocessisillustratedinFig. 2-2 23

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Gapdistortionaccordingtoanincreasingmagneticeld. 2 and 2 )revealtheiranisotropicnatureintheformof^n,^d,and^l.Theappearanceofthesevectorsindicatesthatthesystemwillchooseapreferreddirection.However,thedegeneracyofaspecicdirectionstillremains.Severalexternalorinternalperturbationswhichcoupletotheorderparametercanliftthisdegeneracy:magneticeld,electriceld,wall,superow,anddipole-dipoleinteractions.Whenonlyoneperturbationisconsidered,theorderparameteralignsuniformlythroughoutthesystemshowingauniformtexture.Inreality,multiplesourcesofperturbationcompeteeachother.Inthiscase,thesystemwillndthelowestenergycongurationincorporatingspatialvariationsofthepreferreddirection,producinganon-uniformtexture. Forexample,thedipoleenergydensityintheA-andtheB-phasesarefoundas[ 20 ]fAD=3 5gD(T)(^d^l)2; 5gD(T)cos+1 42; wheregD(T)isthedipolecouplingconstantdenedbygD(T)DD(F)2(T);D=3 (D(F)jVlj)2: 24

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4)104fortheB-phasewhereLiscalledLeggettangle.Inmagneticelds,theLeggettangleismodiedbythegapdistortion,0(H)=cos11 4k Theeffectofamagneticeldisobtainedbyconsideringthemagneticenergydensityas[ 20 ]fAH/(^d~H)2; Therefore,thepreferredorientationswouldbe^d?~Hand^nk~H. Thewallalsohasasignicanteffectonthetexture.Theorderparameter,component,?,isstronglysuppressedwithinalayerofthicknessofthecoherencelength,whichrendersbothvectors,~lintheA-phaseand^nintheB-phase,alignedtothesurfacenormal(^s).Ingeneral,althoughtheorderparameterisrestoredtoitsbulkvaluewithinafewcoherencelength,theorderparameterpertainsthedirectiondeterminedbythesurfaceforamuchlargerlengthscalethehealinglength[ 21 ]. Itisusefultodenethehealinglengthtohaveanideaofhowacontinuouscongurationoftheorderparametereld,calledatexture,formsinvarioussituations.Thehealinglengtharedenedforseveralorientingforcessuchasdipole,magneticeld,wall,andetc.byequatingtheircorrespondingenergygainstothebendingenergycost.ThehealinglengthsforthreeorientingforcesaresummarizedinTable 2-1 .NotethatthesurfacehealinglengthintheB-phaseisquitelongsothatinatypicalexperimental 25

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Table2-1. Healinglengthsforvariousorientingforces OrientingforcesHealinglengthNotes DipoleAD8m,BD7mintheG-LregimeatmeltingPMagneticeldAH/1

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Observationofsuperuidtransitioninliquid3Heimpregnatedinhighporositysilicaaerogelhasopenedawaytointroducingstaticdisorder/impuritiesinthissystemandtriggeredimmediatetheoreticalandexperimentalactivities.TheuniquestructureofaerogelformedbyanentanglednetworkofnanometersizedSiO2strandspresentsmorethanconventionalrandomlydistributedisotropicscatteringcenters.Thenetworkoftheimpurityscatteringcanbealteredbymodifyingthecompositionofthesurfacelayersfrommagnetictopurelypotentialscattering.Furthermore,thecorrelatedstrand-likestructureinevitablyintroduceslocalrandomanisotropy.Therefore,theeffectofdisorderisnotsimplylimitedtothesuppressionofsuperuidbypair-breaking.Therearenumerousinterestingphenomenaobservedandexpectedinthissystem.Inthischapter,wewillsurveythecurrentstatusoftheeldbyreviewingsomeoftheimportantexperimentalresultsandtheoreticalideas. Mostoftheexperimentsincludingthisworkinliquid3Heused98%porosityaerogels.Foratypicalaerogelsamplewith98%porosity,theaveragedistancebetweenthestrands,whichisessentiallythecorrelationlengthoftheaerogel(a),isintherangeof30-40nm.Thegeometricmeanpathis`a120150nm.Anotherimportantlengthisthecoherencelengthofsuperuid,o,denedbyo=~vF=2kBTc,whereTcisthe 27

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22 ].ItisexpectedthatthechangesintheFermiliquidparametersandthedipole-dipoleinteractionconstant(gD)duetothescatteringofftheaerogelarenegligiblysmallsincetheFermiwavelength,F=2=kF0:7nm,ismuchsmallerthanthelengthscalesofaerogel[ 23 ]. Consideringalloftheseaerogelproperties,Thunebergetal.[ 22 ]andThuneberg[ 23 ]discussedvariousscatteringmodelsbasedonthequasiclassicaltheory.Homogeneousscatteringmodel(HSM)isthesimplestamongthoseandassumesthatthemediumisisotropic,i.e.,themeanfreepath(`)isindependentofthequasiparticlemomentumdirection,andthescatteringcenterdistributionisuniformandrandom.ThismodelconvenientlygivesthesameformalismasinbulkfortheGinzburg-LandautheoryandLeggett'stheoryonNMR[ 24 ]withrenormalizedparameters.AlthoughHSMpredictssuppressionofTcandsuperuiddensity(s)intherightdirection,itfellshortinexplainingtheexperimentalresultsfortheentirepressurerange.Thesuperuidtransitioninaerogel(Tca)accordingtothismodelisgivenbylnTc 2n11 2n1+x; wherex=o=`istheAbrikosov-Gorkovdepairingparameter[ 22 ]. Whenaiscomparabletoo,asisathighpressures,itisimportanttoconsiderinhomogenietyandanisotropyofaerogel.HanninenandThuneberg[ 25 ]studiedinhomogeneousbutisotropicscatteringmodel(IISM)extensively.Thismodelgivesbetteragreementwithexperimentsbutpredictsasignicanttemperaturedependenceofsuppressionfactorfortheorderparameter. SaulsandSharma[ 28 ]proposedaphenomenologicalIISmodelbyredeningthedepairingparameterx=^x=(1+2a=^x),wherea=a=`;^x=o=`.Thismodelproducedanexcellenttofthesuperuidtransitionin98%aerogeltotheexperimentallydetermined 28

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Superuidtransitionof3Heinaerogel.ThebluedatapointsarefromGervaisetal.[ 26 ]andtheredonesarefromMatsumotoetal.[ 27 ].ThebluelineisthetheoreticalcalculationbySaulsandSharma[ 28 ].TwosolidcirclesareA-BlikephasetransitionfromVicenteetal.[ 29 ].[FigurereproducedwithpermissionfromJ.A.SaulsandPriyasharma,Phys.Rev.B68,224502(2003).Copyright(2003)bytheAmericanPhsicalSociety.] transitiontemperaturesbytheCornellgroup[ 27 ]andtheNorthwesterngroup[ 26 ](seeFig. 3-1 ). TheanisotropicHSMemphasizestheimportanceofanisotropicnatureofaerogelstrands.ThemaineffectofanisotropycanbeincorporatedintotheGinzburg-Landaufreeenergythroughthequadraticorderparametertermwhichshiftsthetransitiontemperature[ 23 ].Inthelimitofao,thislocalanisotropyisaveragedoutrecoveringHISMwiththemodiedGinzberg-Landaucoefcients,i[ 22 ].ItisbelievedthatanisotropicscatteringstabilizestheA-phase.Recently,theeffectsofanisotropicscatteringhaveattractedattentionfollowingtheobservationoftheexistenceoftheA-phaseatlowpressureinaerogel[ 29 ].Thissubjectwillbediscussedindetaillater. 29

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2 ]usingatortionaloscillator.Theymeasuredtheresonancefrequencyoftheoscillatorandfoundthatitshowedanabruptincreaseatatemperatureslightlylowerthanthebulksuperuidtransition,indicatingdecouplingofmassinsidetheoscillator.Thesuperuidfractionobtainedfromtheshiftinresonancefrequencyrevealedstrikingfeatures.Unlikebulk,thesuperuidfractionwasfoundtoreachavaluemuchlessthanunityinthezerotemperaturelimit,whichmonotonicallydecreaseswithpressure(seeFig. 3-2 ).Inthesameyear,Spragueetal.[ 30 ]alsoreportedsuperuidtransitionsin98%aerogelusingapulsedNMRtechnique.Basedonthefrequencyshiftandmagnetization,theyconcludedthattheobservedsuperuidphasewasanequalspinpairingstate.Ayearlater,Spragueetal.[ 31 ]observedatransitionfromanESPtoanon-ESPstate.Allesetal.[ 32 ]showedthatthisnon-ESPstatecouldbeidentiedastheB-phaseofthebulk,basedontheanalysesoftheirNMRspectra. Alongwiththesystematicsuppressionofthesuperuidtransitiontemperature(Tca)in98%aerogel,thegapsuppressionisalsoexpected.AccordingtoHSM,theratioofthegapsuppressionisthesameastheratioofsuppressionofthesuperuidtemperature.However,severalexperimentsestimatedmuchseverergapsuppressionthanpredictedbyHSM.Forexample,Barkeretal.[ 33 ]estimatedabout50%ofgapsuppressionfromtheNMRfrequencyshiftat32bar,andHalperinetal.[ 34 ]showedsimilarsuppressionfactorsutilizingthedataofNMRandspecicheatbySpragueetal.[ 30 31 ]andChoietal.[ 35 ],respectively(seeFig. 3-3 ),demonstratingthatHSMisnotasuitablemodelforsuperuid3Heinaerogel. Twodistinctsuperuidphaseshavebeenobservedin3Heinaerogelintheabsenceofamagneticeld,calledtheA-like(ESP)andtheB-like(nonESP)phases.Incontrasttothespinstructures,theidenticationoftheorbitalstructuresforboththeA-likeandtheB-likephasesarestillinconclusive.Presently,itisbelievedthattheB-likephasehas 30

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Superuiddensityatvariouspressureobtainedbytortionaloscillator[ 2 ].Thepressuresare3.4,4.0,5.0,6.1,7.0,8.5,10,13,15,20,25,and29barfromtheleft.Theinsetshowsthesuperuiddensityinthebulkfor0,5,10,15,and20barfromtheleft.[FigurereproducedwithpermissionfromJ.V.PortoandJ.M.Parpia,Phys.Rev.Lett.74,4667(1995).Copyright(1995)bytheAmericanPhsicalSociety.] thesameorderparameterasthatoftheB-phaseinbulk.InadditiontotheworkofAllesetal.[ 32 ],Dmitrievetal.providedthemostconvincingevidenceforthisidenticationbyobservingthesharpNMRfrequencyshiftatLeggettangle(L104)[ 36 ]aswellasthehomogeneousspinprecessiondomain(HPD)intheB-likephase[ 37 ].Bycontrast,theidenticationoftheorbitalstructureoftheA-likephaseisfarfromconclusive. AccordingtoImryandMa[ 38 ],anarbitrarilyweakelddestroyslong-rangeordersinceitisenergeticallyfavorabletohavethesystembreakintodomainsatlargedistances.Basedonthisargument,Volovik[ 39 ]pointedoutthattheA-likephaseofsuperuid3Heinaerogelisagloballyisotropicstatewithoutalongrangeorder,calledglassorLIM(Larkin-Imry-Ma)state.ThisLIMstatehasnosuperuidity,inotherwordssuperuiddensity(s)iszero.However,thesuperuiditycanberestoredbyan 31

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Suppressionofthetransitiontemperatureandtheorderparameter.TheamplitudeoforderparameterisdeterminedfromtheNMRfrequencyshifts[ 30 31 ]andthespecicheatjumpmeasurements[ 35 ].[FigurereproducedwithpermissionfromW.P.Halperinetal.,J.Phys.Soc.Jap.77,111002(2008).Copyright(2008)bythePhysicalSocietyofJapan.] applicationofasmallmagneticeld(30G)inthedipolelockedcaseinwhichL0>Dorlargesuperow[ 40 ],whereL0istheLIMlengthandD(10m)isthedipolelength.Recently,itwasnotedthattheregularanisotropyintroducedbycontrolleddeformationofaerogelcanalsorestorethesuperuidityoftheA-likephase.Forexample,Kunimatsuetal.[ 41 ]observedalargenegativeNMRfrequencyshiftintheA-likephaseandSatoetal.[ 42 ]alsofoundthestabilizedcoherentprecessionofmagnetizationintheA-likephase.Bothexperimentswereperformedincompressedaerogelsandtheirresultsareconsistentwiththecongurationoflparalleltomagneticeld,indicatingthatthelongrangeorderof~lisrestoredinaerogel. 32

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43 ]suggestedanotherphaseasacandidatefortheA-likephase,calledtherobustphase,whichhastheformofanESPstate,Aj= 3[^d(mj+inj)+^e(lj+ipj)]; where^dand^earethemutuallyorthogonalunitvectors,andthisorderparametershouldsatisfythecondition,AlAj+AjAl=jlconst:; whereAjisthemeanorderparameteroverthelargelengthscalecomparedtothedistancebetweenaerogelstrands.Underthiscondition,theinteractionwitharandomeld(byaerogel)vanishesandlong-rangeorderispreserved.AjfortheA-likephaseneedstobequasi-isotropicwithequalspinpairing(ESP).Therearesometheoretical[ 44 ]andexperimental[ 45 ]resultsthatmightsupporttheideaoftherobustphase,butitappearsthatthisphaseisnotconsideredasthethermodynamicallyfavoredstate[ 46 48 ]. 26 30 49 52 ]evenatlowpressuresbelowthebulkpolycriticalpoint(PCP).Gervaisetal.[ 26 ]haveperformedsystematicmeasurementsusingatransverseultrasoundtechniqueatseveralpressuresandmagneticelds.Basedontheirtrackingexperimentat33.4bar,theyconcludedthattheA-likephaseregionshouldlieinaverynarrowtemperaturewindow(20K)justbelowthesuperuidtransition,Tca.Furthermore,fromtheelddependentsuppressionoftheABtransition,theydeterminedastrongcouplingparameter,ga()(seeEq. 4 )forvepressures,andconcludedthatthepolycriticalpoint(PCP)didnotexistinaerogelsincenodivergenceinga()asafunctionofpressurewasobserved.Later,Vicenteetal.[ 29 ]performedtrackingexperiments 33

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TherelativesizeofthestepsforthesupercooledaerogelABtransitionasafunctionoftheturn-aroundtemperature.Thedashedverticallinesindicatetheaerogelsuperuidtransitiontemperature.TheA-andtheB-likephasescoexistintheshadedregions(seeRef.[ 29 ]).[FigurereproducedwithpermissionfromC.L.Vicenteetal.,Phys.Rev.B72,094519(2005).Copyright(2005)bytheAmericanPhsicalSociety.] at28.4and33.5barandidentiedthewarmingABtransitionsinzeromagneticeld(seeFig. 3-1 ).Acoupleofinterestingfeatureshavebeenaddressedbythoseauthors:thecoexistenceoftheA-likeandtheB-likephasesinthenarrowtemperatureregionbelowTca(seeFig. 3-4 )andthepositiveslopeoftheABtransitionline.TheobservationofapositiveslopewasalsomadebyBaumgardnerandOsheroff[ 50 ],andKadoetal.[ 52 ]inalowmagneticeld,28.4mT.AccordingtotheClasius-Claperonequation,dP dTAB=sBsA 34

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FigureofillustratingtheshiftupofPCPduetotheanisotropicscatteringandthedecreaseofstrongcouplingeffects. sincesB
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46 ]haveshowntheoreticallythatlocalanisotropytendstolowerthePCPinsteadofopeningtheA-likephaseallthewaydowntothepressuresuchasmagneticeldorglobalanisotropydo.Experimentalconrmationaboutthisissuehasnotbeenachievedclearlyyet. 53 ]consideredmagneticimpurityscatteringeffectsins-wavesuperconductorsintheBornlimit(phaseshift01).Theyshowedthattheboundstatesduetopair-breakingbyimpurityscatteringareformedinsidethegapattheexpenseofsmoothingoutthesquare-rootsingularitiesatthegapedge.Thenumberofboundstatesincreaseswithdisorder,whicheventuallyleadstogaplesssuperconductivity.Inthisweakscatteringlimit,thenon-magneticimpurityscatteringinanisotropicp-wavesuperconductor(orsuperuid)hasthesimilareffectstothatofthemagneticimpurityscatteringinans-wavesuperconductor.Intheunitarylimit,BuchholtzandZwicknagl[ 54 ]calculatedthedensityofstatesfortheisotropicp-wavesuperconductor.TheyfoundthatevenasmalldensityofimpuritiesgenerateanislandofimpurityboundstatescenteredattheFermienergyintheabsenceofamagneticeld.Inclosetobutnotexactlytheunitarylimit,theimpurityboundstatesareformedatapositioncenteredataniteenergy.Thesetheoreticalresultsareexactlyapplicabletothesuperuid3HeB-likephaseinaerogel,andqualitativelythesamefeaturesofdensityofstates(DOS)wereobtainedbySharmaandSauls[ 55 ]andHigashitanietal.[ 56 ].SharmaandSaulsshowedthatintheunitarylimitabandofexcitationsformed,centeredattheFermilevel,withenergies,0:67p 3-6 .Becauseoftheimpuritystates,thephysicalquantitiessuchasthermalconductivity,specicheat,andsoundattenuationareexpectedtofollowpowerlawsratherthantheexponential 36

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Upperpanel:ThetemperaturedependenceofshearviscositycoefcientnormalizedatT=TcforthevariousimpurityscatteringparameterTc.Lowerpanel:ThedensityofstatesatT=0forTc.NotethattheviscosityisnonzeroatT=0whendensityofstateisnonzero.[FigurereproducedwithpermissionfromS.Higashitanietal.,Phys.Rev.B71,134508(2005).Copyright(2005)bytheAmericanPhsicalSociety.] temperaturedependenceinthedeepsuperuidregion.ThisdeviationfromtheBCSpredictionbecomesmoresignicantatlowerpressureswherethepair-breakingeffectismoresevere.However,sincetheproleoftheimpuritystatesarelesssensitivetothespecicorderingsymmetry,thedifferencebetweenthedifferentphasesbecomesmoreevidentathigherpressures.Thethermalconductivitymeasurementsinthesuperuid 37

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57 58 ].Theyobservedthatthenormalizedthermalconductivityreachedanitevalueinthezerotemperaturelimit.Thisbehaviorwasattributedtothegaplesssuperuidity,specically,non-zeroDOSattheFermienergy.AnotherevidenceforgaplessbehaviorwasclaimedbyChoietal.[ 35 ]whomeasuredheatcapacity(Ca)andshowedanon-zerointerceptofCa=TasT!0.ArecentultrasoundattenuationmeasurementfromourgroupbyChoietal.[ 59 ]providedanotherevidenceforgaplesssuperuidityof3Heinaerogel.Theyobtainednitesoundattenuationinthezerotemperaturelimitandshowedthatitincreasedaspressuredecreased,whichisconsistentwithotherresults. Despitetheseexperimentalevidenceofgaplesssuperuidityin3He/aerogel,nomethodsequivalenttothetunnelingspectroscopyinsuperconductorsareavailableforthissystem. 60 ].Theymodiedtheconventionaltwouidhydrodynamicequationsallowingthemotionoftheaerogelmatrix,andobtainedthefollowingsecularequationforthesoundvelocity:(c2xc21)(c2xc22)+a 38

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60 ]. Golovetal.[ 61 62 ]appliedthismodeltothecaseofsuperuid3Heinaerogel.FromEq. 3 ,theyobtainedsimpliedrelationsforthefastandtheslowsoundvelocities.c2f=c211+a Thesoundvelocity,cf,isabout80%ofthebulk3Herstsoundvelocityandincreasesslightlyinthesuperuidstate.Theinequalitiesintheequationsarejustiedbycf350m/s,cs13m/s,c2<0:1m/sat29barand1mK,ca50m/sfor98%aerogel[ 61 ].Notethattheslowsoundismuchfasterthanthesecondsoundsuggestingthattherestoringforcemainlycomesfromtheaerogel.Fromtheslowsoundvelocitymeasurements,theydeterminedsuperuidfraction,s=,whichiscomparabletothosereportedbyPortoandParpia[ 2 ]. Unlikesoundvelocity,thedampingmechanisminliquid3Heinaerogelisdiverseandcomplex.Todate,onlyonetheoreticalmodelwasproposedtodescribesoundattenuationinthissystem.Higashitanietal.[ 56 ]andMiuraetal.[ 63 ]introducedaphenomenologicalexpressionforthecollisiondragforcedensity(~F)asanadditionaldampingsourceduetothetherelativemotionofnormaluidandaerogel,~F=1 wherefisthefrictionalrelaxationtimethatneedstobedeterminedthroughamicroscopiccalculationand~vn(a)isthevelocityofthenormaluid(aerogelstrand). 39

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3-7 [ 56 ].Afterintroducingthisdragforceandviscosity()intothehydrodynamicequationsofMcKennaelal.[ 60 ],Miuraetal.[ 63 ]obtainedtheextendeddispersionrelation,(z2xc21)(z2xc22)+i4! !fz2 !fn wherez=!=q;a=!=(!2!2q)=z2=(z2c2a);!q=caq.ThisequationismoregeneralthanEq. 3 sinceitdealswiththecaseofvn6=va.Fromthisdispersionrelation,theattenuationforfastsoundwasderived,f=!2=2cf Thisequationwillbeusedextensivelyinourlaterdiscussions. 40

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Thereducedtemperature(T=Tc)dependenceofthefrictionalrelaxationtimefnormalizedatT=TcfortheB-likephase.Fs2istakentobe10.01correspondingtothepressureof16bar.NotethatfvanishesatzerotemperaturebutwhenthesoundpropagatestothenodedirectionintheA-phase,itdependsontheparameterA,whereisthemeanfreetimeinthenormalstateandAisthemaximumvalueoftheorderparameterintheABMstate.i)IntheBornlimit,f=0forA=4,andf6=0forA=4.ii)Intheunitarylimit,f6=0foranyvalueofA.InthelimitofA1,ittakesthenormalstatevalueforbothBornandunitarylimits.ThisnoteisfromtheprivatecommunicationwithSeijiHigashitani.[FigurereproducedwithpermissionfromS.Higashitanietal.,Phys.Rev.B71,134508(2005).Copyright(2005)bytheAmericanPhsicalSociety.] 41

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60 ].Asaresult,twolongitudinalsoundmodesemergeinthiscompositemedium:onewiththespeedofsoundcloseto,butslightlylowerthan,thatoftheliquid(fastmode)andtheotherwithasignicantlylowerspeedofsound(slowmode)[ 61 ].Inthisexperiment,thelongitudinalfastsoundattenuationinsuperuid3Hein98%aerogelwasmeasuredatfrequenciesbetween3.69and11.3MHz.Theemploymentofthemultiplefrequencyexcitationsturnedouttobeextremelyvaluableinthiswork.TheexperimentwasperformedattheHighB/TFacilityoftheNationalHighMagneticFieldLaboratorylocatedinUniversityofFlorida. Figures 4-1 and 4-2 showaschematicdiagramandpicturesofthecell.Thebottompartofthecellismadeoutofpuresilverandcontainsabout14m2ofsilverpowderheatexchanger.ThetoppartofthecellmadeoutofcoinsilverwasgluedtothebottompartusingStycast2850FT,EmersonandCuming.Thetoppartofthesamplecellformsadiaphragmsothepressureofthecellcanbemeasuredcapacitively.Thevariationinthecellpressureduringthemeasurementwasaround0.1bar.Twobest-matchedLiNbO3transducers(9.6mmdiameter)withfundamentalresonancesof1.1MHzwereselectedfromsixtransducerstestedusingabroadbandcommercialspectrumanalyzerandahome-madeCWspectrometer.Figure 4-3 showsthelock-inoutputoftheCWspectrometerforthefrequencysweepsaroundthe11thharmonicsofthetransducers.Wechosethetransducers1(T1)and2(T2)asatransmitterandareceiver.ThetransducersweresupportedbyaMACORspacerforminga3.02mmsizeacousticcavity.Aerogelwith98%porositywasgrowninandaroundthiscavitytoensureoptimalacousticcouplingbetweentheaerogelandthetransducers.Theaerogelgrown 42

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Schematicdiagramoftheexperimentalcell. Figure4-2. Picturesoftheacousticcavityandtheassembledcell. 43

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Resonancetestforselectingbest-matchedpair. outsideofthecavitywascarefullyremoved,andcopperwireswereattachedtotheoutersurfaces(electrodes)ofthetransducersusingsilverepoxy.Inordertoreducetheringingofthetransducers,athinlayerofsilverepoxywasappliedtotheelectrode. Asmallpieceofacigarettepaperwithnumerousneedleholeswasplacedbetweeneachtransducerandthecellwalltointerruptbackreectionsfromthewallthroughthebulkliquid.ThesamplecellhousingthecavitywasplacedonthetopgoldplatedCu-angewhichisthermallyconnectedtotheCu-demagstagethroughthethreeCu-rodsweldedtoit.Ahomemadesuperconductingsolenoidmagnetlocatedintheinnervacuumspaceenclosedthecell.Themagnetwasthermallyanchoredtothemixingchamber(seeFig. 4-4 ).Themagneticeld,~B,waschosentobeperpendiculartothesoundwavevector~q,~B?~q,expecting~lk~qintheA-likephase. Twodifferentspectrometerswereusedinthiswork.AMATECbroadbandpulsedspectrometerwasusedforthemeasurementat29and19.5bar,andacommercial 44

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Pictureoftheexperimentalregionofthecryostat. spectrometer,LIBRA/NMRKITII(TecmagInc.,Houston,TX)wasusedfortherestofthepressures,33,25,and14bar.Bothspectrometerstransmitted3spulsesanddetectedthetransmittedsignals.AMATEC310broadbandgatedampliermixesgatingpulseswithacontinuoussinusoidalwave(12Vrmsrequiredtoproperlytriggertheinternalsynchronizationcircuits)toproduceanRFpulseofadesiredfrequency.ThisRFpulsewasfedtothetransmittertransducerthroughavariable(0-34dB)attenuator.ThesignalobtainedbythereceivertransducerwasampliedbyaMITEQAU-1534preamplieranddeliveredtotherststageinputofMATEC625broadbandreceiver.TheoverallschemeisshowninFig. 4-5 45

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Aschematicdiagramofthemeasurementscheme(MATECpulsedspectrometer). WithLIBRA/NMRKITIIsepctrometer,eachmeasurementwasobtainedbyaveragingeighttransmittersignalsproducedinaphasealternatingpulsesequence.Thelevelofexcitationusedinthisexperimentwassetintherangewhereneitherself-heatingnornonlinearitywasobserved.Atypicalsettingofthisspectrometerandtheoriginscriptsforhandlingdatacanbefoundelsewhere[ 64 ].Inonetemperaturesweep,themeasurementsatfourpre-determinedfrequencieswereperformedinacyclicmanner.Thetemperaturewasmonitoredbyameltingcurvethermometer(MCT)forT1mKandaPt-NMRthermometerforT1mK. Inspiteoftheefforttospoilthequalityfactorofthetransducers,sustainedringingswereobservedandwewereunabletoresolveechoesfollowingtheinitialreceived 46

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Atypicalreceivedsignalandanintegrationscheme. signal.Consequently,byintegratingaportionofthereceivedsignal,onlytherelativeattenuationcouldbedetermined.Figure 4-6 showsatypicalreceiversignalandanintegrationscheme.Theregionofintegrationwascarefullychosennottoincludeanyechoes.Ourmethodproducedconsistentrelativeattenuationforvariouschoicesoftheintegrationrangewithinthesafewindowdescribedabove.Therelativeattenuationinreferencetothevalueattheaerogelsuperuidtransitiontemperature(Tca)wasdeterminedby wheredisthesoundpathlengthandA(T)istheintegratedareaofthetransmittersignalattemperatureT. 47

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4.2.1Overview 65 ],3He-4Hemixture[ 66 67 ],3He[ 2 30 ],andliquidcrystals[ 68 69 ].Theeffectofaerogelonsuperuid3Heisexceptionallyinterestingbecauseitisap-wavetripletanisotropicsuperuidpossessingcontinuoussymmetry.Sincethediscoveryofsuperuiditiyof3Heinhighporosityaerogel[ 2 30 ],morethanadecadeoftheoreticalandexperimentaleffortshavebeeninvestedtounderstandthissystemandhaverevealedmanyinterestingphenomena.Thefragilenatureofp-wavepairingagainstimpurityscatteringwasimmediatelyrecognizedbythesignicantdepressionofsuperuidtransition[ 2 27 30 ],andthetheoreticaldescriptionsbasedonvariousisotropicimpurityscatteringmodelshaveprovidedasuccessfulaccountfortheobservedbehavior[ 22 25 28 ].Awidevarietyofexperimentalevidencereectingtheroleofaerogelasaneffectivepair-breakingagentarenowwelldocumented[ 34 ]. Forthepastfewyears,attentionhasbeenshiftedtounderstandingphenomenarelatedtoanenergyscalesmallerthanthecondensationenergy.Forexample,therelativestabilityamongpossiblesuperuidphases,specicallythetransitionbetweentwosuperuidphasesobservedinthissystem,theA-likeandtheB-likephases,hasbeeninvestigated.Intheabsenceofamagneticeld,thesupercooledA-likephaseappearsatallpressuresstudied,evenbelowthebulkpolycriticalpoint(PCP)[ 26 51 64 ],whileonlyaverynarrowregionwherethetwophasescoexistwasidentiedonwarming[ 29 ].Inthepresenceoflowmagneticelds,theB-liketoA-liketransitionwasobserved,onwarming,tofollowaquadraticelddependence[ 26 49 50 ],whichisreminiscentofthebulkABtransition,1TAB=Tc=g()(B=Bc)2,whereTABandTcaretheABtransitionandthesuperuidtransitiontemperatures,respectivelyandseeEq. 4 forBc.However,thesystematiceldandpressuredependencestudybyGervaisetal.[ 26 ]foundamonotonicincreaseing()withpressurewithoutshowing 48

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In1996,Volovik[ 39 ]discussedthesignicanceofthequenchedrandomanisotropicdisorderpresentedbythestrand-likeaerogelstructureanditsinteractionwiththeanisotropicorderparameter.ThiscouplingisthoughttobeparticularlyimportantintheA-phase,wheretheorderparameterisdoublyanisotropicinthesensethattherotationalsymmetriesinspinandorbitalspacearebrokenseparately.Vicenteetal.[ 29 ]arguedthattheaerogelstrandsgeneratedorbitaleldsemulatingtheroleofamagneticeld,therebygivingrisetosimilarprofoundeffectsontheA-liketoB-liketransition.Theyfurthersuggestedtheuseofuniaxiallydeformedaerogeltoamplifyandtosystematicallyinvestigatetheeffectoftheanisotropicdisorder[ 29 ].AseriesofcalculationsbyAoyamaandIkeda[ 46 70 ]areconsonantwiththeseideasandpredictawidenedA-likephaseregioninauniaxiallydeformedaerogel,theappearanceofanovelsuperuidphaseinuniaxiallystretchedaerogel,andachangeofthePCPlocationinthephasediagram. UnliketheB-likephase,theclearidenticationoftheA-likephaseinaerogelhasnotbeenmade.However,someoftherecentNMRmeasurementsusinguniaxiallydeformedaerogels[ 41 71 ]providecompellingevidencethattheA-likephasepossessestheABMpairingsymmetry,albeitwithunusualtexturalcongurations.ThefreeenergycalculationbyIkedaandAoyama[ 72 ]alsofoundthedisorderedABMphaseasthemoststableamongthevariousplausiblepairingstates,suchasthe 49

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73 ],theplanar,andtherobust[ 43 ]phases.Furthermore,thethirdsuperuidphaseobservedin98%aerogelinthepresenceofhighmagneticelds[ 74 ]fortiesthisidentication.Therefore,wewillcontinueourdiscussionwiththeassumptionthattheA-likephaseobservedatleastin98%aerogelhasthesamepairingsymmetryasthebulkA-phase. Withthisnotion,weconductedlongitudinalultrasoundattenuationmeasurementsinthesuperuidphasesof3Hein98%porositysilicaaerogel.Ourmeasurementswereperformedinthepresenceofmagneticelds,0to4.44kG,andatvarioussamplepressuresrangingfrom14to33bar.Atthehighesteld,theexistenceofthemeta-stableA-likephasepersistedtothelowesttemperatures,therebyallowingthesoundattenuationintheA-likephasetobemeasuredovertheentirerangeofthetemperaturesstudied.Inlowermagneticelds,wewereabletoidentifythetransitionsbetweenthetwophasesoncoolingandwarming,andherein,aP-B-Tphasediagramofthissystemispresented. 4.2.2.1LongitudinalSoundAttenuationandtheA-BTransitioninAerogel 4-7 4-9 4-11 ,and 4-13 showtherelativeultrasoundattenuationsobtainedat29,19.5,33,and25barinthepresenceofmagneticeldsrangingfromzeroto4.44kG,respectively.AllthedatashownweretakenonwarmingaftercoolingthoughthesupercooledA-liketoB-liketransitionataxedexternalmagneticeld,exceptforB=4.44kG,wherenosupercooledtransitionwasobserveddownto200K. Therefore,thewarmingtraceatthehighesteldshouldbeintheA-likephasefortheentiretemperaturerange,probablyinthemeta-stableA-likephaseinthelowtemperatureregion.Thesuperuidtransitionismarkedbyaslightdecreaseinattenuationaround1.95mKfor29bar(Fig. 4-7 ),1.61mKfor19.5bar(Fig. 4-9 ),2.1mKfor33bar(Fig. 4-11 ),and1.85mKfor25bar(Fig. 4-13 ).Thezeroeldattenuation,whichessentiallyrepresentstheB-likephaseattenuationexceptfora 50

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Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat29barinthepresenceofvariousmagneticelds.AllthedataweretakenonwarmingaftercoolingthroughtheA-liketoB-liketransitionexceptforB=4.44kG,wherenosupercooledtransitionwasobserved.ThearrowspointthepositionswheretheB-liketoA-likephasetransitionsoccur.Inset:Magniedviewofzeroeldattenuationnearthesuperuidtransitionindicatedbytheverticalline. verynarrowregion(100K)rightbelowTca,canbedirectlycomparedwiththeabsoluteattenuationmeasurementsbyChoietal.[ 75 ]performedunderalmostidenticalexperimentalconditions.Thefeaturesobservedinthecurrentexperiment,namelythebroadshoulderstructureappearingintherange1:0
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TheABtransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat29bar.Thered(black)tracerepresentstheattenuationintheA-like(B-like)phase.Theswitchingbehaviorbetweenthetwotracesisdemonstratedforeacheldasmarkedbyanarrow. betweenthetwophasesatanyintermediateeldwhereaswitchingfromonetracetoanotheroccurs.ItisexpectedthattheattenuationintheA-likephaseishigherthanintheB-likephaseundertheassumptionthatitistheABMstate,sincethesoundpresumablypropagatesalongthenodedirectioninourexperimentalconguration.However,unlikeinthebulk,thedifferenceinattenuationbetweentheA-likeandtheB-likephasesismuchsmallerandsubtlebecauseoftheabsenceoftheorderparametercollectivemodes,whicharethengerprintsofspecicpairingsymmetry,andthepresenceoftheimpuritystatesresidinginthegap.OnecanseethesubtledifferenceintheattenuationbetweentwophasesinFigs. 4-7 4-9 4-11 ,and 4-13 .Atalltemperatures,theattenuationintheA-likephaseisslightlylargerthanintheB-likephase,whilethelargestdifferenceisobservedinthezerotemperaturelimit.Forthis 52

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Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat19.5barinthepresenceofvariousmagneticelds.AllthedataweretakenonwarmingaftercoolingthroughtheA-liketoB-liketransitionexceptforB=4.44kG,wherenosupercooledtransitionwasobserved.ThearrowspointthepositionswheretheB-liketoA-likephasetransitionsoccur.Inset:Magniedviewofzeroeldattenuationnearthesuperuidtransitionindicatedbytheverticalline. reason,theacousticsignatureoftheABtransitioninaerogelisnotasclearasinthebulk.Despitethissmalldifferenceinattenuation,theB-liketoA-liketransitionfeaturesarenoticeableinmostofthecases(indicatedbythearrowsinFigs. 4-7 ).However,inthetemperatureregionwheretwophasesshowalmostidenticalattenuation,asin0:7
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TheABtransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat19.5bar.Thered(black)tracerepresentstheattenuationintheA-like(B-like)phase.Theswitchingbehaviorbetweenthetwotracesisdemonstratedforeacheldasmarkedbyanarrow. non-trivialfrequencydependenciesoftheattenuationobservedinaerogelandwillbediscussedinnextsectionofthischapter. Thelowestnitemagneticeldusedinthisexperimentwas1.11kG,andtwoattenuationmeasurementsperformedinthiseldat33barareshowninFig. 4-15 .Thesedatawerecollectedwithtwodifferentwarmingratesof1.4K/min(invertedtriangles)and1.7K/min(regulartriangles).Bothmeasurementsproducedthesametransitiontemperaturedespitethedifferenceinthewarmingratebyabout20%. InFig. 4-16 ,thewidthoftheA-likephase,T=TcaTABa,asafunctionofB2,alongwiththeresultsobtainedintheloweldregionbyGervaisetal.,isplotted.WithintheGinzburg-Landau(G-L)limit,wecanperformanalysisthatissimilartoworkusedtodescribethebulkliquid[ 76 ].Specically,thesuppressionoftheB-likephaseinnite 54

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Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat33barinthepresenceofvariousmagneticelds.AllthedataweretakenonwarmingaftercoolingthroughtheA-liketoB-liketransitionexceptforB=4.44kG,wherenosupercooledtransitionwasobserved.ThearrowspointthepositionswheretheB-liketoA-likephasetransitionsoccur.Inset:Magniedviewofzeroeldattenuationnearthesuperuidtransitionindicatedbytheverticalline. magneticeldscanbewrittenas Here,Bcrepresentsacharacteristiceldscaledirectlyrelatedtothetransitiontemperature,namely wherekB,;(x),andFa0aretheBoltzmannconstant,thegyromagneticratiofora3Henuclei,theRiemannzetafunction,andaFermiliquidparameter,respectively.Inaddition,thestrongcouplingparameterg()isafunctionofthepressure-dependent 55

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TheABtransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat33bar.Thered(black)tracerepresentstheattenuationintheA-like(B-like)phase.Thetop(bottom)panelsshowthetracestakenusing6.22MHz(8.73MHz)excitations.Theswitchingbehaviorbetweenthetwotracesisdemonstratedforeacheldasmarkedbyanarrow. 77 ],andcanbewrittenasg()=245 whereijk=i+j+k.Intheweakcouplinglimit,g()!1,andthestrongcouplingeffectscauseittoincrease. Inordertoilluminatetheoverallelddependence,thedatapresentedinFigure 4-16 arerecastedasT=B2inFigure 4-17 .AsnotedbyTangetal.[ 76 ],oneoftheadvantagesofthisplotisthattheintersectionofthecurvewiththeB=0axisgivesthestrongcouplingparameter,g(),andtheslopeofthecurveisrelatedtothecoefcient 56

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Temperaturedependenceofrelativelongitudinalsoundattenuationsusinga6.22MHzexcitationat25barinthepresenceofvariousmagneticelds.AllthedataweretakenonwarmingaftercoolingthroughtheA-liketoB-liketransitionexceptforB=4.44kG,wherenosupercooledtransitionwasobserved.ThearrowspointthepositionswheretheB-liketoA-likephasetransitionsoccur.Inset:Magniedviewofzeroeldattenuationnearthesuperuidtransitionindicatedbytheverticalline. ofthehigherordercorrection,ascanbeseeninEq. 4 .Ourg()valuesextractedbyextrapolatingtozeroeldareshowninFigure 4-18 .TheparametersusedaresummarizedinTable 4-1 Table4-1. Parametersfordeterminingg(). P(bar)Tca(mK)Fa0Bc(kG)g() Inthesamegure,g()ofthebulkbyTangetal.(opencircles)andof98%aerogelbyGervaisetal.(solidcricles)areincludedforcomparison.Additionally,we 57

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TheABtransitionfeaturesinsoundattenuationusinga6.22MHzexcitationat25bar.Thered(black)tracerepresentstheattenuationintheA-like(B-like)phase.Thetop(bottom)panelsshowthetracestakenusing6.22MHz(8.73MHz)excitations.Theswitchingbehaviorbetweenthetwotracesisdemonstratedforeacheldasmarkedbyanarrow. reproducedthetheoreticalcalculation[ 26 ]basedonthehomogeneousscatteringmodel(HSM)[ 22 ]withtherescaledstrongcouplingcorrectionsbythefactorofTca=Tcfortwodifferentmeanfreepathvaluesof`=150(dot-dashedline)and200nm(dashedline).Althoughourg()valueat19.5barisingoodagreementwiththatofGervaisetal.,thediscrepancybetweenthetwosetsofthedatabecomeslargerathigherpressures.However,g()inaerogelfrombothmeasurementsissubstantiallysmallerthanthatofthebulkvalueatthecorrespondingpressure.Forthebulk,g()growsquicklyandapproachesthePCPaspredictedbytheG-Ltheory.However,nosuchbehaviorisseeninaerogel.Althoughtheerrorbarsinourdataareratherlarge,ourresultsliebetweenthetwotheoreticalcurves.Itisalsointerestingtoobservethatthesignofthequartic 58

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Temperaturedependenceofattenuationat33barusing6.22MHzexcitation.TheattenuationintheB-like(B=0)andtheA-like(B=4.44kG)phasesarealreadyshowninFigures 4-7 .ForB=1.11kG,theattenuationwasmeasuredwithtwowarmingratesof1.4K(invertedtriangles)and1.7K(triangles).Inset:MagniedviewoftheregionoftheABtransitioninaerogel. correctionisnegativeathigherpressuresandseemstochangeitssignatP19.5bar(seeFigure 4-17 ),whichneedstobecomparedwiththebulkcasewherethesigncrossoveroccursatP6.7bar[ 76 ].Basedontheseobservations,onecouldarguethatthepresenceofaerogelreducesthestrongcouplingeffectsand,ineffect,thephasediagramofthissystemisshiftedupinpressure. 59

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MagneticelddependenceofthewidthoftheA-likephase,T=TcaTABa.Forcomparison,ourresultsareplottedalongwiththosefromGervaisetal.(solidcircles)[ 26 ].ThedatapointsfromGervaisetal.weretakenattheslightlydifferentpressuresof33.4,28,25,and20bar,respectively. silvercellbody.Toalleviatethisproblem,weslowlydemagnetizedthemainmagnetofthenucleardemagnetizationstageduringaeldsweep(typically0.14G/min).ThispassiveprocedurelimitedthetemperaturevariationduringanIFSto50K. InFig. 4-19 ,themagnitudesoftheintegratedacousticsignalstakenatfourdifferentfrequenciesduringanisothermaleldsweepat25barand0.3mKaredisplayed.Thetemperaturevariationduringthisprocessisalsoshowninthesamegure.Thesamplewascooledfromthenormaluidinthepresenceofamagneticeldof4.44kGto0.3mK.Afterestablishingequilibrium,themagneticeldwasslowlyreducedattherateof4G/min[ 78 ]togothroughtheA-liketoB-liketransition.Therefore,theB-likephasewassupposedtobeinducedthroughaprimarynucleation,andthiscaseistheonlyinstanceofaprimarynucleationtransitionobservedbyIFSinourwork.Forthe 60

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MagneticelddependenceofthewidthoftheA-likephasescaledbyB2.Thequadraticcoefcient,g()isdeterminedbytheintersectionoftheeachcurvewiththeB=0axis,Eq. 4 entiresweepprocess,thetemperatureremainedwithin30Karound0.27mK.Thesmoothchangeinmagnitudesatallfrequenciescanbeobservedfrom4.3to4.0kG,indicatingthetransitionfromtheA-liketoB-likephase.Thedifferenceinthemagnitudeoftheacousticsignalbetweentwophasesmatcheswellwiththeattenuationdifferencedeterminedfromthetemperaturesweepmeasurements. ForB.4.0kG(intheB-likephase),theattenuationexhibitsaweakelddependence,mostnotablyat11.3MHz.Thisbehaviorcannotbesimplyattributedtothetemperaturevariationsduringtheeldsweepbecausetheattenuationshowsaveryweaktemperaturedependencearound0.3mK(seeFigs. 4-7 and 4-14 ).Onecanspeculatethatthisvariationinattenuationmightberelatedtotheprogressivedistortionofthegapinducedbymagneticeld,astheisotropicBWstateevolvesthroughthedistortedBWstatetotheplanarstateandeventuallytotheABMphasewiththenode 61

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Pressuredependenceofg().Thepresentdata(solidsquares)aregivenwiththedatabyGervaisetal.(solidcircles)[ 26 ]foraerogelandbyTangetal.(opencircles)[ 76 ]forthebulkliquid.Thedashedanddot-dashedlinesarefromhomogeneousscatteringmodel(HSM)withthetransportmeanfreepath,`=200and150nm,respectively(seeRef.[ 26 ]fordetails). alongthesoundpropagationdirection[ 19 ].Theincrease(decrease)inthemagnitude(attenuation)intheloweldregioncouldbeduetotheenhancement(reduction)inthecomponentofthegapperpendicular(parallel)tothemagneticeld.IntheA-likephaseatthehighesteld,thesoundpropagatesinthenodedirection,resultinginahigherattenuation. SeveraladditionalIFSstudieswereconductedatvariouscombinationsofpressureandtemperature,wherethesamplewascooledfromthenormalstateataxedeldtoatemperatureintheB-likephaseviathesuperuidandthesupercooledA-liketoB-liketransitions.Then,themagneticeldwasrampedupthroughtheB-liketoA-liketransitionanddecreasedagainbackthroughthetransition,ifnecessary.Figure 4-20 showstheIFSresultsat14barandT0.27mK.Thephasetransitionoccurs 62

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Resultsoftheisothermaleldsweep(IFS)at0.3mKandP=25bar.Themagnitudesoftheintegratedacousticsignals,A(T),measuredusing4differentexcitationfrequenciesaredisplayedasafunctionofmagneticeld.ThetemperaturevariationduringtheIFSisalsoshowninthebottompanel. overaratherbroadrangeofeld(B0.5kG),butnoappreciablehysteresiswasobserved.TheresultsoftwootherIFSstudiesat29bar(T0.86and1.38mK)areshowninFigs. 4-21 and 4-22 .ForT0.86mK(Fig. 4-21 ),thetransitioncanonlybeidentiedinthe3.69MHzmeasurements(B0.2kG).Brussaardetal.[ 49 ]observedhystereticbehaviorintheelddrivenABtransitionintheirmeasurementsatT0.335mKandP=7.4barusinganoscillatingaerogelsampleattachedtoavibratingwire.ThemagneticeldsweepwasperformedinthepresenceofaeldgradientinwhichasingleABphaseboundarywasmovingthroughthesampleduringtheprocess.TheyproposedthepinningoftheABphaseboundarybytheaerogelstrandsasamechanismfortheobservedhysteresis.Furthermore,based 63

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Resultsoftheisothermaleldsweepat14bar. Figure4-21. Resultsoftheisothermaleldsweepat29barandT0:86mK. 64

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Resultsoftheisothermaleldsweep(rampuponly)at29barandT1:38mK. onthisscenario,theymadeanargumentthattheABtransitionsdeterminedbyaconventionaltemperaturesweepmethod,specicallythosebyGervaisetal.,mightnotprovidereliablethermodynamictransitionpointsduetosupecoolingandsuperwarmingcausedbythepinning,suggestingthenitewidthofthetransitionisanevidenceoftheexistenceofarangeofpinningpotentialstrengths[ 79 ].WewouldliketopointoutthattheexperimentsbyGervaisetal.andbyuswereperformedwithoutadesignedeldgradient.Inthiscase,itisalsoplausiblethattherandomdisorderpresentedbyaerogel,morespecicallyanisotropicdisorder,couldcausethebroadeningofthetransition[ 29 80 ].Theeffectofroundingbydisorderisalsoapparentinthesuperuidtransition,whichisasecondordertransitionanddoesnotinvolveaninterfacialboundary.ImryandWortis[ 80 ]havemadeaheuristicargumentabouttheinuenceofrandomimpuritiesonarstordertransition.Theypredictedvariousdegreesofroundinginthetransitionduetouctuations(inhomogeneities)oftherandommicroscopicimpuritiesthroughthe 65

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81 ]validforsecondordertransition.ItisworthnotingthattheLancastergroupalsoreportedasimilardegreeofhysteresisineld(mT)inthebulkABtransitioninducedbyasimilarmethod[ 33 ].Theeldsweepperformedat29bararound0.86mKinFig. 4-21 seemstoshowaglimpseofhysteresisinthe3.69MHzdata.However,weacknowledgethathysteresisatthelevelofmTcannotberesolvedfromourmeasurements,andthewidthofthetransitioniscertainlylargerthananyhysteresisthatmightexist. 4-23 .Forbothmethods,themid-pointofthetransitioninTorBwaschosenasthetransitionpointandtheactualwidthofthetransitionisrepresentedbytheerrorbar.ThewidthinBistranslatedintothetemperaturewidthusingthemeasuredelddependenceoftheABtransitioninaerogel(seeFigs. 4-7 ).Thetransitionpointsdeterminedbythetwodifferentmethodsexhibitself-consistencywithintheresolutionofourmeasurements.Forexample,theIFStransitionpointat14barwasobservedat3.33kGandliesontheextensionoftheTSCFmeasurementsat3.33kG,andthe3.7kGIFSpointisrightonthelinefor3.85kGfromtheTSCF.WecouldnothaveobtainedtheIFSpointat4.21kGat25barbytheconventionalTSCGatthiseld. Theemergingphasediagram,Fig. 4-23 ,fromourmeasurementsunambiguouslyrevealsthattheABphaseboundaryin98%aerogelrecedestowardthemeltingpressureandzerotemperaturecornerinresponsetotheincreasingeld.Thistendencyisrobustevenwhenallowingforthepossibilityofsuperwarming,whichmightshiftthetransitiontemperaturedown.Thisphasediagramisindrasticcontrasttothatofthebulk[ 82 ].Firstly,theslopeoftheconstanteldphaseboundaryispositiveinaerogelbutnegativeinbulkformostofthecorrespondingpressurerange.Secondly,thephaseboundaryinthebulkrecedestowardP19bar,whichisincloseproximitytothebulk 66

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Phasediagramofsuperuid3Hein98%aerogel.Thesolidtrianglesrepresenttheaerogelsuperuidtransition.TheABtransitionsinaerogelobtainedbytheTSCFareinsolidcirclesandbytheIFSinsolidstars.Thesolidlinesgoingthroughthedatapointsareguidesforeyesbutconformstotheconstanteldphaseboundariesfor1.11,2.22,2.75,3.33,and3.85kG,respectivelyfromrighttoleft.Forcomparison,theconstanteldABphaseboundariesforthebulkliquidareshownbythedottedlines[ 82 ]for1,3,5,5.5,and5.8kG,respectively.Thenumbersrightnexttothestarsymbolsindicatethemid-eldstrengthofthetransition. PCP,ratherthantowardthemetingpressure.ItisnoteworthythattheslopeofthebulkABphasetransitionlineactuallychangesitssignaroundthePCP,withapositiveslopeforP
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Here,Airepresentstheorderparameterofasuperuidstatewithspin()andorbital(i)indices[ 20 ].Themagneticeldcouplesthroughthespinchanneloftheorderparameter.WithtwodistinctsymmetriesintheAandBphaseorderparameters,thisquadraticcontributionliftsthedegeneracyinthesuperuidtransitiontemperature,therebypushingtheA-phaseTcslightlyabovethatoftheBphase.Asaresult,anarrowregionoftheA-phasemustbewedgedbetweenthenormalandtheBphaseforP
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22 ],fa=gaaiAiAjaj: where^aisaunitvectorpointinginthedirectionoftheaerogelstrand.ThesimilaritybetweenEqs. 4 and 4 isapparent.Theeffectoftheorbitaleldproducedbytheaerogelstrandswasestimatedtobecomparabletotheeffectproducedbyamagneticeld1kGinthecaseofcompletealignment[ 29 ].Ithasbeenexperimentallydemonstratedthatuniaxialcompressionindeedinducesopticalbirefringenceproportionaltothestrainand,consequently,globalanisotropyintothesystem[ 84 85 ]. Inagloballyisotropicaerogel,however,thelocalanisotropycomesintoplayonlywheno.a,wherearepresentsthecorrelationlengthoftheaerogelandoisthepaircorrelationlength[ 29 ].Intheotherlimit,thelocalanisotropyissimplyaveragedouttoproducenoeffect.AsdiscussedbyVicenteetal.,thisnetlocalanisotropyshouldemulatetheeffectofmagneticeldevenintheabsenceofmagneticeldinagloballyisotropicaerogel.FurthermoreaninhomogeneityinthelocalanisotropywouldcauseabroadeningoftheABtransitioninaerogelinwhichthemixtureoftheAandBphasescoexists[ 80 ].Consideringa40-50nmin98%aerogel,thislocalanisotropyeffectinagloballyisotropicaerogelshouldbemorepronouncedathigherpressuresbutisexpectedtotailoffasthepressuredecreasestothepointwhereoa,whichoccursaround10bar.TheimpressiveagreementinTcabetweentheexperimentsandthetheoryofSaulsandSharma[ 28 ]wasachievedbyincorporatingtheaerogelcorrelationlengthintothedepairingparameterofthehomogeneousisotropicscatteringmodel[ 22 ]. Althoughtheaerogelsampleusedinthisworkissupposedtobeisotropic,wecannotruleoutthepossibilityofhavingaweakglobalanisotropybuiltintothissamplefromthesamplepreparationortheshrinkageoccurringduringcondensationof3He. 69

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Relativeattenuationat6.22MHzforzeromagneticeldoncoolingandwarmingand4.44kGonwarming.Inset:Theaerogelsuperuidtransition(Tca)andtheABliketransitionsonwarming(TABw)andcooling(TABc)areindicated. Ineithercase,theobservedbehaviorinthisworkaswellasotherscanbeexplainedcoherently[ 84 86 ]. 4-24 ,weplottherelativeattenuationoncoolingandwarminginzeromagneticeldalongwiththewarmingtraceat4.44kG.ThemetastableA-likephaseregionextendsto1.6mKinzeroeld.ItisnoteworthythattheattenuationinthezeroeldmetastableA-likephaseispracticallyidenticaltothatoftheA-likephaseatthehighesteld.Consideringourexperimentalconguration,weexpecttohave~H?^lk~qthroughoutthesampleat4.44kGandauniform^l-textureinadipole-lockedstatesincethesurfaceorientationeffectisinharmonywiththeeldcoupling.Consequently,thesoundattenuationwouldexhibitanisotropicbehavior[ 87 ].Incontrast,theImry-Ma 70

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39 88 ]shouldshowisotropicattenuationinzeroeldsincethe^l-textureinthisstateisdisordered.Therefore,ifthezeroeldA-likephasewastheImry-Mastate,itisreasonabletoexpectdifferentattenuationinthehigheldA-likephaseunlessitiscompletelydipoleunlockedstate.Ontheotherhand,ithasbeendemonstratedthatthesoundattenuationinaerogelisstronglymodiedbythepresenceofimpuritystates,especiallyatlowpressureswherethepair-breakingeffectisstronger[ 59 ].Theanisotropyintheattenuationinhigheldsmightbeweakenedbythepresenceofimpuritystates.Itwillbeinterestingtodirectlymeasuretheanisotropyintheattenuationinzeroandnitemagneticeldsaswellasinuniaxiallycompressedaerogelwhereinterestingtexturalcongurationshavebeenobserved[ 41 71 89 ]. 4.3.1Overview 90 ].Soundpropagationinthesuperlfuidphasesisalsoexpectedtoremaininthehydrodynamiclimit[ 91 92 ]butisfurthercomplicatedbythepresenceofimpuritystatesinducedbypair-breakingscattering[ 59 92 ].Thedetailsofimpuritystatesdependonthetypeofpairingmechanismandscatteringstrength.Therefore,thefullunderstandingofsoundpropagationinthissystemisnottrivialandshould 71

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Absoluteattenuationsforpressuresfrom8to34barasafunctionoftemperatureat9.5MHz(onwarmingexcept8bar). befollowedbyextensivetheoreticalandexperimentalstudies.ArecentultrasoundattenuationmeasurementperformedintheB-likephaseat9.5MHzexposedmanyinterestingfeaturessuchastheabsenceoftheorderparametercollectivemodesandthenitezerotemperatureattenuationevincingtheexistenceofimpuritystatesandgaplesssuperuidity[ 59 ](seeFig. 4-25 ).Theseobservationsareconsistentwiththetheoreticalpredictionsbasedonahydrodynamictwo-uidmodelasmentionedearlier[ 92 ].Figure 4-26 showstheattenuationratio0=cobtainedfromtheresultsshowninFig. 4-25 ,where0istheattenuationinthezerotemperaturelimitandcattenuationatthesuperuidtransition.ThisattenuationratioprovidesthelowerboundofthedensityofstatesattheFermilevelaccordingtothecalculationbyHigashitanietal.[ 56 ],anditshowsthatthedensityofstatesstartsincreasingsignicantlybelow15bar. 72

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Aratioofthezerotemperatureattenuation0totheoneatsuperuidtransitiontemperaturec.Redcirclesaretheattenuationatthesuperuidtemperatureasafunctionofpressure. Inthissection,wepresentultrasoundattenuationoftheB-likephaseofsuperuid3Hein98%porosityusing4differentfrequenciesbetween3.6to11.3MHz.Ourresultsrevealnon-trivialfrequencydependencesinattenuation,graduallydepartingfromthe!2-dependenceexpectedinthissystem. 93 ].Thismodel,basedontheLandau-Boltzmanntransporttheory,considerstheimpurityscatteringoffaerogelaswellastherelativemotionbetweenaerogeland3Heliquid.A3Hequasiparticleimpingedonaerogelstrandtransfersmomentumandcausesadraggedmotionofaerogel,thecollisionaldrageffect.Whenthisprocessgeneratesrelativemotionbetweenthesetwocomponents,itgivesrisetoanadditionaldampingmechanism.Theirtheoryprovidedasatisfactoryaccountforthelongitudinalsoundattenuationmeasurementsconductedinthe 73

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90 ].Thecollisionaldrageffectisonlypronouncedinthehydrodynamiclimitandthesoundattenuationisexpectedtofollowaquadraticfrequencydependence(seeEq. 3 ).Althoughthereisnodirectexperimentalconrmationforthe!2-dependenceofattenuation,thisclaimissupportedbytheobservationofstrongfrequencydependenceinattenuation[ 59 90 ].Inaddition,theelasticscatteringmeanfreepathhasbeenexperimentallydeterminedin98%aerogelbythermalconductivity(90nm)[ 58 ],spindiffusion(130nm)[ 94 ],andsoundattenuation(120nm)[ 59 ],whichmakes!<1for!<20MHzforthewholepressurerangeandatalltemperature. Withthisnotion,wededucedtheabsoluteattenuationfromtherelativeattenuationmeasuredinthisexperimentusingthenormaluidabsoluteattenuationat9.5MHzasaxedpointthrough whereristheattenuationat9.5MHzataerogelsuperuidtransitiontemperature,Tca,fromRef.[ 59 ]and!r=2=9:5MHz. SoundattenuationcalculatedfollowingthisrecipeisdisplayedinFig. 4-27 for33barandinFig. 4-28 for25and14bar.Thetracesshownweresmoothedthroughthe10pointsslidingaveragelter.Thesoundattenuationat9.5MHzforthecorrespondingpressureisalsoreproducedinthesamepanel.Theyallsharesimilarqualitativefeatures:theabsenceofcollectivemodesandpair-breakingedge,theshoulderstructureappearingaround0.6T=Tca,andnon-exponentialtemperaturedependenceleadingtothenitezerotemperatureattenuation.AllofthesefeaturesareinqualitativeagreementwiththetheoryofHigashitanietal.Intheirtheorysimilartothetwo-uidhydrodynamicmodeldescribedinRef.[ 60 ],thecollisionaldrageffectisincludedintheformofamutualfrictionbetweenthenormalcomponentandtheaerogel.Therefore, 74

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(a)Temperaturedependenceofrelativeattenuationat3.69,6.22,and11.30MHztakenat33bar.Inset:thetemperaturedependenceofabsoluteattenuationat9.5MHzatthesamesamplepressureisreproducedfromRef.[ 59 ].(b)Temperaturedependenceofabsoluteattenuationforallfrequenciesat33bar.Seetextforthedetails. 75

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Temperaturedependenceofabsoluteattenuationforthreefrequenciesalongwiththepreviousmeasurementat9.5MHzfor25bar(a)and14bar(b)ofsamplepressures. 76

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60 ]arisesfromthefrictionaswellastheusualshearviscosity()(seeEq. 3 ). Theircalculationcapturestheimportantfeaturesobservedinourexperimentsandthedirectcomparisonwiththeexperimentalresultsat9.5MHzcanbefoundinRef.[ 59 ].Forexample,thebumpinattenuationaround0.6T=Tcaisdirectlyrelatedtothefrictionaldampingfromthecollisionaldrageffect.Thefrictionalrelaxationtime,f,initiallyincreasesrightbelowTca(evenabovethenormaluidvalueforhighpressure)duetoratherrapidopeningofsuperuidgap.feventuallyapproacheszeroasT!0,displayingabroadpeak.Thiseffectmanifestsinthesizeofthepeakinfismorepronouncedatahigherpressurewherethepair-breakingeffectislesssignicant,andaccordingly,thebumpstructureintheattenuationgraduallyfadesoutasthesamplepressureisloweredasshowninFigs. 4-27 and 4-28 Weplotthesoundattenuationasafunctionoffrequencyataconstanttemperature(Fig. 4-29 ).ThesoundattenuationisnormalizedbytheoneatTca(effectivelybythesquareofthesoundfrequency,f2)andthedataforthesamesetofthereducedtemperaturesarechosenforallpressuresinthisplot.OurassumptionoftheclassichydrodynamicbehaviorinthenormaluidenforcesaatlineforT=Tca=1.Onecanclearlyseetheevolutionofthefrequencydependencedeviatingfromthe!2-dependenceastemperaturelowers.For33and25bar,theattenuationestablishesaquitestrongfrequencydependencebeyondthequadraticbehaviorinthezerotemperaturelimitaftergoingthroughanon-monotonicfrequencydependenceoncooling.For14bar,however,theattenuationshowsaquitedifferentbehaviorthantheoneobservedathigherpressures.Downtothelowesttemperature,theattenuationat14barseemstopossessastructureratherthanfollowingamonotonicfrequencydependence.At25and33bar,thenon-monotonicdependenceseemstobeassociatedtothebroadbumpstructureintheattenuation.However,thesimilarfrequencydependencepersists 77

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Soundattenuationasafunctionoffrequencyforselectreducedtemperaturesat33(a),25(b),and14(c)bar.ThesoundattenuationisnormalizedbytheoneatTca,effectivelybyf2.Thelinesgoingthroughthedatapointsareguidesforeyes. 78

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Importantparametersestimatedforthreepressuresusedinthiswork. P 2~ 2~ ~! (! (MHz) 0.30 96.17 62 0.180.200.2625 0.27 91.13 57 0.200.220.2914 0.23 77.37 31 0.360.390.53 downtothelowesttemperatureat14barwheretheanomalousbumpstructurealmostvanishes. Thisnon-monotonicbehaviorisnotwellunderstoodyetbutthispersistentfeaturedowntothelowesttemperatureseemstoberelatedtotheexistenceoftheimpurityboundstatesinsidethegapasseeninFig. 4-26 .Morespecically,webelievethattheprogressivedeviationfromthe!2-dependenceisdirectlyrelatedtothegapstructure.Table 4-2 listsseveralimportantquantitiespertinenttoourdiscussion.TheaveragegapinaerogelatzerotemperatureusedinthistableisfromHalperinetal.[ 34 ].Thesoundfrequencyusedinthistableis11.3MHz,whichisthehighestfrequencyemployedinthiswork.ThesuperuidgapsatnitetemperaturesareobtainedassumingthesametemperaturedependenceasinthebulkandusingthespecicheatjumpmeasurementsbyChoietal.[ 35 ].Thesuperuidgapissignicantlysuppressedatallpressuresandthedegreeofthesuppressionismuchsevereratlowerpressure,14bar.Althoughthesoundisinthehydrodynamiclimit,thesoundfrequencyisyetcomparabletothesizeofthesuperuidgapatalltemperatures.Thisuniquesconditioncannotberealizedinbulksuperuidwhereareasonablyhighfrequencysoundinevitablyentersintothezerosoundlimitatlowtemperature.Therefore,thesoundattenuationcannotbedescribedpurelyfromthehydrodynamicmechanismandshouldinvolvemechanismsofresonantquasiparticleand/orpairexcitations.Therefore,thedetailedspectrumoftheimpuritystatesshouldbeconsidered. Hirschfeldetal.[ 95 ]calculatedtheelectromagneticabsorptioninisotropicaswellasanisotropicp-wavesuperconductingstatesconsideringvariousabsorption 79

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Schematicdiagramofresonantscatterings. mechanismsinvolvingimpurityboundstates.AsketchofvariousabsorptionmechanismsareillustratedinFig. 4-30 .Thesoundattenuationshouldpossessthesimilarmechanismsasdescribedinthisdiagram.Alargeelectromagneticabsorptionoccurswhentheresonantscatteringtakesplace.Forthescatteringphaseshiftclosetounitarylimit(6==2),whichis,webelieve,thecaseofaerogel,therearethreedifferentresonantscatteringprocesseswhichcontributetothetotalabsorption:I.scatteringfromlledtoemptyboundstates,II.scatteringfromlledstatesatgapedgetounoccupiedboundstates,III.scatteringfromlledboundstatestotheemptygapedge.TheyshowedthatthecontributionfromtheprocessIwouldberelativelysmallerthanthosefromtheprocessesIIandIIIduetothelargedensityofstatesatthegapedges.Thereisanotherpossibleprocess,pair-breakingmechanismfor~!=2G,where2Gistheenergygapinthepresenceofimpurities,butthisdoesnotcontributetodeterminetheabsorption 80

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95 ].Inaerogel,theproleoftheimpuritystatesmightbeclosetoasingledomestructurecenteredattheFermienergyratherthanhavingtwoseparatedomesasshowninFig. 4-30 .Therelativecontributionbetweenthedifferentprocessescouldbequitedifferentatlowerpressureswheretheseverepair-breakingcausessignicantlysmearedgapedgesandconsiderableweightofimpuritystates.Furthermore,thepresenceofthermallyexcitedquaiparticlescannotbeignoredconsideringtherangeofsoundfrequencyusedinourwork. Thefactthateachstrandofaerogelhasanitesize(3-5nm)whichismuchlargerthanFhasacoupleofimportantimplications.Itshouldputthequasiparticlescatteringatleastclosetotheunitarylimitandalsointheintermediateregimebetweenthepoint-likeimpurityandsurfacescattering.Itisinterestingtoponderthelatteraspectfurther.Recently,Nagaietal.[ 96 ]consideredtheeffectsofsurfaceAndreevboundstatesontransverseacousticimpedance.Inthecaseofdiffusivescattering,weaksingularitiesareexpectedtoappearwhentheexcitationfrequencyisequaltothesizeofthegapbetweenthelowerboundofthebandofimpuritystatesandtheemptygapedge(+),andtheupperboundofimpuritystatesandtheemptygapedge().Inhightemperatures,theformercontributionissupposedtobethedominantprocess.ThiseffecthasbeenobservedasaresonantpeakinthetransverseacousticimpedancemeasurementbyAokietal.[ 97 ].Inthelowtemperaturelimitwherethegapisfullydeveloped,thecontributionfromthelatterprocessisgettingmoresignicantatagivenexcitationfrequency.Webelievethattheultrasoundattenuationinsuperuid3Hein98%aerogelshouldreectthesimilarprocessesdescribedaboveresultinginnon-trivialfrequencydependenceinattenuation.Wedenitelyneedmoredetailedexperimentalinvestigationstofullyunderstandthisphenomenasuchasthemeasurementsinthewiderandnerrangeinfrequency. Weobservedthatthepropertyofaerogel,asreectedinsoundattenuation,changeaftergoingthroughacool-downandwarm-upcycle.Figure 4-31 showstherelative(a) 81

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4-27 ).Nofurthersignicantchangesinattenuationhasbeenobservedaftermultiplethermalcycles.However,themainfeaturesinthefrequencydependenceremainqualitativelythesame(Figs. 4-29 and 4-32 ). 82

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(a)Temperaturedependenceofrelativeattenuationat3.69,6.22,8.73,9.50and11.30MHztakenat29bar.(b)Temperaturedependenceofabsoluteattenuationforallfrequenciesat29bar.Thetemperaturedependenceofabsoluteattenuationat9.5MHzatthesamesamplepressureisreproducedfromRef.[ 59 ]. 83

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Soundattenuationasafunctionoffrequencyforselectreducedtemperaturesat29bar.ThesoundattenuationisnormalizedbytheoneatTca,effectivelybyf2.Thelinesgoingthroughthedatapointsareguidesforeyes. 84

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98 ].Amongthem,the(imaginary)squashingandrealsquashingmodes,whichexistatp FortheA-phase,thegapisnotisotropic,sothatthepair-breakingoccursforanyfrequencyduetothegapnodes.Asaconsequence,theclassicationofOPCMisdifferentfromtheB-phase(see[ 20 ]fordetail).TherearethreeOPCMsthatcoupletozerosound,calledclapping,normalappingandsuperappingmodes.Theexcitationofthesemodesisstronglyanisotropic.Itdependsontherelativeorientationof^lwithrespectto^q,thedirectionofthesoundpropagation.Whentheyareparallel,theappingandtheclappingmodesareabsent,andwhentheyareperpendicular,theappingmodeisabsent. 85

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86

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(a)Quarterwavelengthforheavybackingmaterials(b)Halfwavelengthforpolymerbackingmaterials. tocheckthepolarityofthelmistobringamultimeterandconnecttheleadstotheelectrodesofthePVDFlm,andsendawarmbreathtothelm.Ifitshowsapositivevoltage,thesideconnectedtothepositivemultimeterleadisthepositivepolingside. TheresonancefrequencyandthebandwidthofaPVDFtransducerdependonthethickness(d)andthebackingmaterial.ItiswellknownthatPVDFtransducersareoperatedatthicknessmodesofd=(2n1)=4forheavybackingand(2n1)=2,forpolymerbacking(n=1;2;3;;;),whereisthewavelengthandvisthesoundvelocity(2200m/s)inPVDF(Fig. 5-1 ).Thesemodescorrespondtotheresonancefrequenciesoffr=(2n1)v=4dand(2n1)v=2d,respectively.Whenthereisaquarter-wavebackingplatewithalargeracousticimpedancethanPVDF,itexhibitsnewresonancepeaksnearthefrequenciesthatmeetthe=2condition.Ineffect,thebackingplateplaystheroleofacompletereector[ 99 100 ].APVDFtransducerwithapolymerbackinghasthewidestbandwidthbutthelowestsensitivity.Itisimportantthatthebondinglayershouldbesignicantlythinner(
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Drawingofcopperbacking. Sincetheyhavelargedielectricandmechanicalinternal(viscoelastic)losses,whichareseveraltensoftimeslargerthanthoseofceramictransducermaterials,PVDFtransducershaveasignicantamountofelectricpowerdissipationandinternaldamping.Ohigashiintroducedthelossfactors,taneandtanmtoaccountforthedielectricandviscoelasticlosses,respectively[ 101 ].ThesefactorsareincorporatedintothemodelssuchasMason'smodelandKLM(Krimholtz-Leedom-Matthaei)modelfortheoreticalconsideration[ 102 ].ToimprovetheefciencyofaPVDFtransducer,itissometimessynthesizedwithothermaterial.Forexample,P(VDF-TrFE)isacopolymerofPVDFandtriuoroethylene.ThismaterialissuperiortoPVDFinmanyaspectsandcomparabletoclassicalPZTtransducersexceptforitsmechanicalstrength[ 103 ]. 88

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Pictureofacousticcavitieswithaerogel(a)andwithoutaerogel(b). backingpiecewithsilverepoxy(EPO-TEK,H20E).Thebackingpiecewasmadeoutofcopper.Theoppositesideofthecoppercylinderwasmachinedoutinaconeshapetopreventthedirectbackreection[ 104 ](seeFig. 5-2 ).Thesideofthecoppercylinderwasalsothreadedtodiffusethewallreections. Figure 5-3 showsthesetwoacousticcavities,leftonewithaerogelandrightonewithoutaerogel.ThetopandbottomviewsofthisassemblyareshowninFig. 5-4 .The 89

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Pictureofcellwithtwoacousticcavitiesinit:(a)Bottomview(b)Topview. cellwasinstalledinThule,theultralowtemperaturecryostatinLee'sgroupequippedwithaCudemagnetizationstage(Fig. 5-5 ).Temperaturewasmonitoredbyameltingcurvethermometer(MCT).TheMatecbroadbandspectrometerusedinthepreviousworkwasalsousedforthisstudy.Unfortunately,wecouldnotdetectanyacousticresponsesfromtheaerogelcavitysuggestingthatthecouplingbetweentheaerogelandthetransducersestablishedbysimplecompressionwasnotgoodenough.Therefore,inthiswork,wepresentthedataobtainedonlyfromthebulkliquid3Hewithoutaerogel. 5-6 showstheresultsofthesensitivitystudyconductedat12.1mKusingthebulk3Hecavity.Themagnitudewascalculatedbyintegratingtheareaoftherstreceivedsignalatagivenexcitationfrequencywhilekeepingtheexcitationlevelofthetransmitterconstant.Sinceithasaheavybacking,PVDFshouldoperatein(2n1)=4modeandthereforetherstresonance(n=1)isexpectedtoappearat20MHz(=v=4d)withaverylowQasshowninFig. 5-6 .SimilarresponseshavebeenobservedbyFrankelandGranroth[ 105 106 ]using9mthickPVDFlms.Figure 5-7 showsthesensitivityofaPVDFtransducerinliquid4Heat1barand30mKobtainedbyGranrothetal..Thepositionoftheirmainpeak(resonance)around50MHzisconsistentwithoursconsideringthedifferenceinthelmthickness. 90

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Pictureofexperimentalcellset-up. Duetothevaryingsensitivity,thelinearresponseregimesarealsodifferentfordifferentfrequencies.Figures 5-8 and 5-9 showtheresultsofthelinearitytestconductedat0.7mKat6MHzand20MHz.Eachguredisplaysthereceiversignalsobtainedatdifferentlevelsoftransmitterexcitation.The28Vrmspulsewasfedtothetransmitterthroughvariouslevelsofattenuation.Eachreceiversignalwasnormalizedinreferencetoagivenreceiversignalbyproperlycompensatingtheattenuationdifferences.For6MHz,thereceiversignalat0dBexcitationwaschosenasthereferenceandthenormalizedsignalsaredisplayedintheinsetofFig. 5-8 .Basedonthesemeasurements,weconcludethatthelinearoutputregimeliesbetween1V 91

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Integratedmagnitude(inarbitraryunits)versusexcitationfrequencyforliquid3Heat28.1barand12.1mK. and5V.Inourstudy,wemaintainedthereceiversignallevelswithinthelinearresponseregimebyadjustingtheattenuationleveloftheexcitation. Figure 5-10 showsthesoundattenuationinnormalandsuperuid3Hefor6,11,17and24MHzat28.1bar.TherelativeattenuationwasobtainedbyintegratingtheareaoftherstreceivedsignalandwascalibratedusingthefactthattheattenuationisfrequencyindependentinthezerosoundregimeexceptnearTcanditbecomesalmostzerobelowthetemperature,0.5Tc[ 107 ].Basedonouranalyses,0.2cm1baselineshiftinattenuationaffectedthettingparametersdescribedbelowby1%.Thecrossoverbetweentherstandzerosoundregimesisclearlyseeninattenuationandmovestothehighertemperaturewiththefrequencyasexpected.BelowTc,theattenuationpeaksassociatedwiththepair-breakingandtheOPCMaremanifest. 92

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Integratedmagnitude(inarbitraryunits)versusfrequencyresponsemeasuredbyGranrothetal.inliquid4Heat1barand30mK.ThisplotappearedintheposterpresentationsofRefs.[ 105 ]and[ 106 ]butwerenotincludedintheproceedings. Table5-1. Fittingparameters Frequency(MHz)P1(cm1)P2(smK2) 242:479(1:581)1050:67(0:84)172:544(1:581)1050:77(0:99)112:478(1:581)1050:70(0:85)63:121(1:581)1050:78(0:91) Theresultsofourt(redcurve)areshowninFigs. 5-11 ,andsummarizedinTable 5-1 ThebehavioroflongitudinalsoundinthenormaluidiswelldescribedbytheviscoelasticmodelofRudnick[ 10 ](seeEq. 2 ).Since/1=T2,weusedtheequationshownbelowtotourdatawithtwottingparametersofP1andP2.P1 93

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Linearitytestfor6MHzpulse.Inset:normalizedoutputsignalsagainsttheonefor0dBattenuatedinput(black). Figure5-9. Linearitytestfor20MHzpulse.Inset:normalizedoutputsignalsagainsttheonefor24dBattenuatedinput(red). 94

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Soundattenuationinnormalandsuperuid3Heatfourdifferentsoundfrequency.Inset:MagniedviewnearTc: 6 ],(c0c1)=c21=P11:581105whichissubstantiallysmallerthanourvalues.WehavealsotriedattoourdatawithP1xedat1:581105(bluecurves).Thisprocedureproducedunsatisfactorytstoourdata(bluecurves).Thereareseveraladditionalmechanismsforattenuationthatweneedtoconsider.Therstisthecontributionfromthewallscatteringofthequasiparticles.Whenthesoundfrequencyisoftheorderoforlargerthanthequasiparticlerelaxationrate(1=),orthemeanfreepath(`=vF)iscomparabletotheviscouspenetrationdepth(=(2=!)1=2),theuidbeginstoslipatthewall.Thisslipeffectcausesanadditionaldamping,andthecorrespondingattenuation(w)isrepresentedinthehydrodynamic 95

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Datattingfor24MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. Figure5-12. Datattingfor17MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. 96

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Datattingfor11MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. Figure5-14. Datattingfor6MHz.Redcurveisatheoreticalttingwithtwofreeparameters,P1andP2andbluecurvewithP1xed. 97

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ValuesofT2byseveralgroups Carlessetal.[ 110 ]Wheatly[ 111 ]Rudnick[ 10 ] regimebyw=! whereRistheradiusofthecylindricalresonator.Thisadditionalattenuationisproportionalto1=Tandp 2 shouldbedetermined.Althoughwehavenotdonethisanalysis,wedonotbeleivethatthewallscatteringissignicantenoughtoexplainthediscrepancy.Otherpossiblecorrectionsarefromthediffractionofthewaveduetothenitesizeofthetransducers,andnon-parallelismofthetwotransducers.Theeffectsofnon-parallelismofthetransducersinsuperuid3HewerediscussedindetailbyWatson[ 108 ]andGranrothetal.[ 105 ].Theyshowedthatthiseffectcanbesignicantespeciallywhentheattenuationofthemediumissmall.Botheffectsdependonthespeedofsoundinthemedium[ 109 ].However,sincetherelativeattenuationwasobtainedfromtherstreceivedsignalonly,thoseeffectsshouldenterinourmeasurementsonlythroughthetemperaturedependentquantity.Thechangeinthesoundvelocityoccurringinthecrossoverregimeat28barisonly0.6%,andtherefore,thisvelocitychangehasanegligibleeffectontherstreceivedsignal.Usingtheangularerrorof=4104radiansestimatedbyGranrothetal.[ 105 ],weestimatedthepossiblechangeofsignalamplitudeduetothisvelocitychangetobelessthan0.1%. ThevaluesofT2obtainedbytheseveralgroupsaresummarizedinTable 5-2 .Thisvaluetendstoincreaseasthepressuredecreases.OurP2valuesareingood 98

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99

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Longitudinalultrasoundattenuationmeasurementswereconductedina98%uncompressedaerogelinthepresenceofmagneticelds.Utilizingthemeta-stableA-likephasethatextendeddowntothelowesttemperaturein4.44kG,wewereabletoestablishthetemperaturedependenceoftheattenuationintheA-likephaseovertheentiresuperuidregion.ThisarrangementallowedustodeterminetheABtransitionsinaerogelinvariousmagneticelds.Basedonthetransitionpointsonwarming,aPTBphasediagramofthissystemisconstructed.Thekeyfeaturesofthephasediagramcanbeunderstoodonthebasisoftwofundamentalpoints:rstly,thestrongcouplingeffectissignicantlyreducedinthissystembyimpurityscattering,andsecondly,theanisotropicdisorderpresentedintheformofaerogelstrandsplaysanimportantrolethatemulatestheeffectofamagneticeld. Intheabsenceofamagneticeld,theA-likephasehasnotbeenunderstoodatthesatisfactorylevelyet.Ourmeasurementssuggestthatthepracticallyidenticalbehavioroflongitudinalultrasoundattenuationtotheoneinthehighestmagneticeld(4.44kG)mayindicateadipoleunlockedLIMstatefortheA-likephase.Ifthisisthecase,itwouldbeinterestingtostudyaboutthemechanismsthattwoisotropicstatesshowthedifferentattenuation(seeFig. 4-24 ). Ourresultsoffrequencydependentultrasoundattenuationofferthedecisiveevidenceofthegaplesssuperuidityinsuperuid3Heinaerogel.AccordingtoFig. 4-29 ,theimpurityboundstatesaremuchmoreconspicuousatlowerpressure,turningthephaseintothecompletelygaplessregime,andtheresonantimpurityscatteringsmayplayaimportantrolefortheobservedbreakdownof!2dependenceofsoundattenuationinsuperuidstateinaerogel.Inordertounderstandthedetailofattenuationmechanism,themeasurementinawiderandnerfrequencyrangeisrequired. 100

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ByoungHeeMoonwasborninsmallruraldistrictinChunchengbuk-DoofSouthKorea.HewenttoSeoulcityforcollegeeducationandgraduatedfromYonseiUniversity.Hestudiedtheoreticalnuclearphysicsinmastercourseinthesameuniversity.Afterservingthearmyfor26months,hedecidedtogoabroadandcametoUniversityofFloridain2002andjoinedtheexperimentallowtemperaturegroupofDr.YoonLeein2003andreceivedhisPhDinMay2010. 108