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NMR Studies of 3He in Solid 4He

Permanent Link: http://ufdc.ufl.edu/UFE0043640/00001

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Title: NMR Studies of 3He in Solid 4He
Physical Description: 1 online resource (125 p.)
Language: english
Creator: Kim, Sungsu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: nmr -- quantumsolid -- solidhelium -- supersolid
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: Nuclear Magnetic Resonance (NMR) studies are performed on dilute concentrations of $^3$He in hcp solid $^4$He mixtures for a wide range of $^3$He concentrations grown under constant volume method. The nuclear spin-lattice relaxation times, $T_1$ and the nuclear spin-spin relaxation times, $T_2$ have been measured for $^3$He concentrations $16\leq x_3 \leq 2000~\text{ppm}$ using pulsed NMR techniques for a Larmor frequency of $\omega_{_L} = 2$~MHz in the temperature region where significant non-classical rotational inertia fractions (NCRIFs) have been reported. Dramatic changes are observed in the $T_1$ and $T_2$ at both the proposed supersolid onset temperatures and the phase separation temperatures. The spin relaxation anomalies are explained by a quantum tunneling model and a phenomenological fit to the temperature dependence of the relaxation is provided in terms of a thermally activated process related to the anomalies in shear modulus measurements. The $^3$He concentration dependence of $T_1$ and $T_2$ is discussed in terms of the impuriton model in the range of $10^{-5}\leq x_{_3}\leq10^{-2}$. The crossover concentration between the coherent motion region and the incoherent motion region is estimated from the experimental data as occurring for $^3$He concentrations $x_{_3} \sim10^{-4}$. The temperature dependence of nuclear spin relaxation times are studied for $^3$He concentrations, $500 \leq x_{_3} \leq2000$~ppm for a molar volume $V_M = 20.7$ cm$^3$. The formation of Fermi-liquid droplets of pure $^3$He are observed after phase separation. The temperature dependence suggests that the interface $^3$He atoms responsible for the nuclear spin relaxations are degenerate due to the spherical symmetry and exchange narrowing, not solid-like. The temperature independent plateaus attributed to the quantum exchange motional narrowing are observed at the temperature of $0.25\leq T\leq1.3$~K. The best fit for concentration dependence of the relaxation times shows deviations from the Landesman theory. The vacancy activation energy is determined to be $13.5 \pm 0.3$~K for a sample with $x_{_3} = 5\times10^{-4}$ and molar volume $V_{_M} = 20.9 $~cm$^3$.
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 Sungsu Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sullivan, Neil S.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043640:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043640/00001

Material Information

Title: NMR Studies of 3He in Solid 4He
Physical Description: 1 online resource (125 p.)
Language: english
Creator: Kim, Sungsu
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: nmr -- quantumsolid -- solidhelium -- supersolid
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: Nuclear Magnetic Resonance (NMR) studies are performed on dilute concentrations of $^3$He in hcp solid $^4$He mixtures for a wide range of $^3$He concentrations grown under constant volume method. The nuclear spin-lattice relaxation times, $T_1$ and the nuclear spin-spin relaxation times, $T_2$ have been measured for $^3$He concentrations $16\leq x_3 \leq 2000~\text{ppm}$ using pulsed NMR techniques for a Larmor frequency of $\omega_{_L} = 2$~MHz in the temperature region where significant non-classical rotational inertia fractions (NCRIFs) have been reported. Dramatic changes are observed in the $T_1$ and $T_2$ at both the proposed supersolid onset temperatures and the phase separation temperatures. The spin relaxation anomalies are explained by a quantum tunneling model and a phenomenological fit to the temperature dependence of the relaxation is provided in terms of a thermally activated process related to the anomalies in shear modulus measurements. The $^3$He concentration dependence of $T_1$ and $T_2$ is discussed in terms of the impuriton model in the range of $10^{-5}\leq x_{_3}\leq10^{-2}$. The crossover concentration between the coherent motion region and the incoherent motion region is estimated from the experimental data as occurring for $^3$He concentrations $x_{_3} \sim10^{-4}$. The temperature dependence of nuclear spin relaxation times are studied for $^3$He concentrations, $500 \leq x_{_3} \leq2000$~ppm for a molar volume $V_M = 20.7$ cm$^3$. The formation of Fermi-liquid droplets of pure $^3$He are observed after phase separation. The temperature dependence suggests that the interface $^3$He atoms responsible for the nuclear spin relaxations are degenerate due to the spherical symmetry and exchange narrowing, not solid-like. The temperature independent plateaus attributed to the quantum exchange motional narrowing are observed at the temperature of $0.25\leq T\leq1.3$~K. The best fit for concentration dependence of the relaxation times shows deviations from the Landesman theory. The vacancy activation energy is determined to be $13.5 \pm 0.3$~K for a sample with $x_{_3} = 5\times10^{-4}$ and molar volume $V_{_M} = 20.9 $~cm$^3$.
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 Sungsu Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sullivan, Neil S.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043640:00001


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NMRSTUDIESOF3HEINSOLID4HEBySUNGSUKIMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011 1

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c2011SungSuKim 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwishtothankallthosewhohelpedme.WithouttheirhelpIcouldnothavecompletedthisproject.Firstandforemost,Iwouldliketothankmyadvisor,Dr.Neil.S.Sullivanforhisextremelypatientguidance,adviceandsupportthroughoutallthiswork.IcannotthankenoughhimforgivingmeachancetoworkinhislabandagreatexperienceIhavehadduringmyPh.D.ManythanksshouldgotoLarry,whohadmountedtransistor,resistorsandcapacitorsonthelowtemperaturepreamplierandtoMarkwhohadmadeagreatNMRcell.IcannotthankmoretoGregandJohnforsupplyingliquidheliumduringallmeasurementsevenonChristmasday.ThisworkwouldnotbecompletedwithoutaspecialthankstoDr.Candela,Dr.Xia,Dr.HuanandDr.Yin.EspeciallyIwouldliketothanktoDr.CandelaforhelpingmewithdevelopingtheMathematicaprogramthatisusedtoanalyzethesmallNMRechosignaldatafromverydilutesamples.IlearnedsomanyvaluablelowtemperaturetechniquefromDr.Xia.TakingNMRdatamightbealittlepainful,notonlybecausethemeasurementsneedverylongmeasuringtimebutalsobecauseofthatreasonwehadtoworkliterallydayandnightduringmeasuringtosavetime.Iwouldliketoexpressmydeepestgratitudetomycommitteemembers,YoonseokLee,PradeepKumar,YasumasaTakano,andCliordR.Bowerswhoareverysupportive.Iwouldalsoliketothankalllabmembers,YuJi,YibingTangandDr.JahaHamida.DuringmyPh.D,everythinghasbeengratefulandI'vebeenblessedtohavemanyfriendshereandinKorea.Iamtrulyfortunatetohavebeenabletoenjoyandbenetfromsuchafriendshipwiththem.HwangandTJ,thankyouforcomingherefromKoreatomeetmeandforallgoodtimewespenttogether.IthanktoallKoreanstudentsindepartmentandIreallyenjoyedourregularbutmoreirregularmeeting.Jinmyung,Byunghee,Koo,Jeen,Sohyun,MinjunandInhea,thanksforhavinggoodtimewithme.Withoutfriendshipwiththem,mytimewouldhavebeenharsherthanitwouldbe. 4

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Specialthanksshouldgotoallmyfamilymembersfortheirconstantsupportsandbeliefinmydecision.Itrulythanktomyparentsandmygrandmother.WithouttheirloveandsupportIevenwouldn'thavestartedthislongjourneyandtheyhavebeenalwaysinmyside.Iwanttothankmysisters,SookyungandMikyungandmybrotherChoiwhan.Iwouldnotforgettothanktoallmybrother-in-lawsandsister-in-law.Iamtrulyanddeeplyindebtedtosomanypeoplethatthereisnowaytoacknowledgethemall,orevenanyofthemproperly.Ioermysincerestapologiestoanyoneungratefullyomitted. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1OverviewofNuclearSpinRelaxationsin3HeinSolid4He .......... 15 1.2TheoreticalBackgroundonDilute3He-4HeMixture ............. 16 1.2.1Pure3HeSystem ............................ 16 1.2.1.1Phonons ............................ 16 1.2.1.2Vacancywaves ........................ 17 1.2.1.3Tunnelingexcitations ..................... 18 1.2.2MassFluctuationWaves(MFW)forDilute3He-4HeMixtures ... 19 1.2.3NuclearSpinRelaxationTimesinDilute3He-4HeMixture ..... 20 1.2.3.1Region1-A .......................... 21 1.2.3.2Region1-B .......................... 24 1.2.3.3Region2 ............................ 25 1.2.3.4Region3 ............................ 27 1.2.4LandesmanModel:StrainField .................... 27 1.2.5HuangModel:MFW-MFWInteractionPotential ........... 30 2EXPERIMENTALDETAILS ............................ 33 2.1Overview .................................... 33 2.2DilutionRefrigerator .............................. 33 2.3NMRCoils ................................... 36 2.4LowTemperaturePreamplier ......................... 41 2.4.1SimpleSourceFollowerCircuit ..................... 42 2.4.2Performance ............................... 45 2.4.3SampleProbeandtheSignaltoNoiseRatio ............. 45 2.4.4Summary ................................. 48 2.5PulsedNMRMethod .............................. 48 2.5.1T1Measurements ............................ 49 2.5.2T2Measurements ............................ 51 2.6SampleGrowth ................................. 53 2.7Thermometry .................................. 56 2.7.13HeMeltingPressureThermometry .................. 56 2.7.2ResistanceThermometry ........................ 57 6

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2.8PressureMeasurements ............................. 58 3MICROSCOPICDYNAMICSOF3HEIMPURITIESINSOLID4HEINTHEPROPOSEDSUPERSOLIDPHASE ........................ 60 3.1Overview .................................... 60 3.2Concepts ..................................... 60 3.3ExperimentalDetail .............................. 61 3.4ResultandDiscussion ............................. 64 3.4.1AnomaliesinT1andT2 ......................... 64 3.4.2AnnealingEect ............................. 67 3.4.3ApplyingtheNMRTheorytothePeaksinT1andT2 ........ 68 3.4.4PossibleExplanationforT1andT2Anomalies ............ 69 3.5Summary .................................... 74 4CONCENTRATIONDEPENDENCEOFT1ANDT2 .............. 76 4.1Overview .................................... 76 4.2GeneralConcept ................................ 76 4.3ConcentrationDependenceofT1 ........................ 78 4.4ConcentrationDependenceofT2 ........................ 81 4.5Re-examinationofLandesman'sModel .................... 85 4.6Summary .................................... 85 5PHASESEPARATIONOFVERYDILUTE3HE-4HEMIXTURE ....... 88 5.1Overview .................................... 88 5.2Background ................................... 88 5.3ExperimentalDetail .............................. 89 5.4ExperimentalResult .............................. 91 5.5Discussion .................................... 93 5.6Summary .................................... 98 6NUCLEARSPIN-SPINRELAXATIONTIMESINHIGHTEMPERATUREREGION ....................................... 100 6.1Overview .................................... 100 6.2ResultandDiscussion ............................. 100 6.3Summary .................................... 102 7CONCLUSION .................................... 103 7.1Summary .................................... 103 7.1.1FirstMicroscopicDataintheProposedSupersolidPhase ...... 103 7.1.2CoherentandIncoherentQuantumMotioninDilute3HeinSolid4HeMixture ............................... 103 7.1.3NuclearSpinRelaxationMechanisminthePhaseSeparatedRegion 104 7

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7.1.4NewConsiderationofQuantumTunnelingMotionofthe3HeAtomsintheNuclearSpinRelaxations .................... 104 7.2FutureWork ................................... 105 7.2.1FrequencyDependentNMRMeasurements .............. 105 7.2.2FurtherStudiesfortheNuclearSpinRelaxationDependenceontheCrystalQuality ............................. 105 7.2.3MoreDatafromDierent3HeConcentrationSamples ........ 106 APPENDIX ACALCULATIONOFNMRSIGNALANDNOISE ................ 107 BDATAANALYSIS .................................. 109 CRELAXATIONTOPOLOGIES ........................... 116 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 125 8

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LISTOFTABLES Table page 4-1TableofT1andT2fromOtherGroupsandCalculatedJ34,M2andJeff ..... 81 5-1ComparisonofT1 ................................... 95 A-1SymbolsandUnits .................................. 108 C-1TopologicalRelaxationTime ............................. 117 9

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LISTOFFIGURES Figure page 1-1SchematicDrawingoftheVacancyWaveExcitation ................ 17 1-2SimpleSchematicDrawingoftheTunnelingExcitation .............. 18 1-3SchematicRepresentationoftheTemperatureDependenceofNuclearSpinRelaxationTimes ......................................... 21 1-4ScatteringPotentialintheHCPLattice ...................... 28 1-5SimpleSchematicDrawingofMassFluctuationWave(MFW)Motion ...... 31 2-1HighB/TFacility:Cryostat ............................. 34 2-2PhaseDiagramof3He-4HeMixture ........................ 35 2-3DilutionRefrigerator ................................. 36 2-4PhotoofReceivingCoil ............................... 37 2-5PhotoofTransmittingCoil ............................. 38 2-6SchematicofNMRCrossedCoils .......................... 38 2-7GeometryofaSimpleSaddleCoil .......................... 39 2-8OptimalAngleandField-UniformityParameterforaSaddleCoil ....... 40 2-9NMRMeasurementSet-Up:NMRcoils,PreamplierandTuningCapacitor .. 41 2-10LowTemperaturePreamplierCircuitDiagram .................. 42 2-11PhotoofPreamplier ................................. 43 2-12PhotoofaLowNoisePseudomorphicHighElectronMobilityTransistor(pHEMT) ............................................. 44 2-13Two-PulseNMRHahnEchoSignal ......................... 46 2-14Free-InductionDecay(FID)from500ppmSampleafter90oPulse ........ 50 2-15SchematicDrawingforSingleEchoPulseandCorrespondingSpinDescription 51 2-16FitofDataObtainedbyAveragedSingleEchoMethodforT2 .......... 52 2-17SchematicDrawingforCarr-Purcell-Meiboom-Gill(CPMG):Multi-EchoPulseSequence ........................................ 53 2-18SchematicRepresentationoftheBlockedCapillaryMethodforHeliumCrystalGrowth ........................................ 54 10

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2-19SolidicationintheNMRCell ............................ 55 2-20CalibrationDataofSiliconDiodeThermometer .................. 57 2-21CalibrationDataofCarbonResistanceThermometer ............... 58 2-22SchematicDrawingofthePressureStrainGauge ................. 59 3-1NMRCellDesign ................................... 62 3-2Determinationof3HeConcentration ........................ 63 3-3Temperaturevs.T1 .................................. 64 3-4Temperaturevs.T2 .................................. 66 3-5NormalizedT1Peaksof16ppmand24ppmSamples ............... 67 3-6PhotooftheTanglesofIndividualDislocationsintheCrystalofHelium .... 70 3-7SimpleSchematicforImpurityPinningtotheDislocationNetwork ....... 71 3-8ProbabilityandBindingEnergyof3He ....................... 73 4-1ObservedConcentrationDependenceoftheNuclearSpin-LatticeRelaxationTimesforDilute3HeinSolid4He .......................... 79 4-2ComparisonoftheConcentrationDependenceoftheNuclearSpin-SpinRelaxationTimesReportedintheLiteratureforDilute3Heinsolid4He ........... 82 4-3VariationoftheFunctionF(x3)=(4 3)T1T2x23(M2=!L)2asaFunctionof3HeConcentration ..................................... 86 5-1SchematicofPhaseSeparatedLiquid3HeInsideSolid4He ............ 90 5-2T1T2vs.x3 ...................................... 92 5-3Temperaturevs.T1 .................................. 93 5-4Temperaturevs.T2 .................................. 96 5-5ComparisonofPhaseSeparationTemperature ................... 97 5-6KineticsoftheGrowthof3HeNano-dropletsforx3=1000ppm ......... 98 6-1T2intheHighTemperatureRegionfor250ppm,500ppmand1000ppm .... 101 B-1Spin-EchoDatafromaSamplewith16ppm3He ................. 110 B-2Spin-EchoDatafromaSamplewith24ppm3He ................. 111 B-3Spin-EchoDatafromaSamplewith500ppm3He ................. 112 11

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B-4Spin-EchoDatafromaSamplewith900ppm3He ................. 113 B-5Spin-EchoDatafromaSamplewith1870ppm3He ................ 114 B-6SpinEchoDatafromaSamplewith2000ppm3He ................ 115 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNMRSTUDIESOF3HEINSOLID4HEBySungSuKimDecember2011Chair:N.S.SullivanMajor:PhysicsNuclearMagneticResonance(NMR)studiesareperformedondiluteconcentrationsof3Heinhcpsolid4Hemixturesforawiderangeof3Heconcentrationsgrownunderconstantvolumemethod.Thenuclearspin-latticerelaxationtimes,T1andthenuclearspin-spinrelaxationtimes,T2havebeenmeasuredfor3Heconcentrations16x32000ppmusingpulsedNMRtechniquesforaLarmorfrequencyof!L=2MHzinthetemperatureregionwheresignicantnon-classicalrotationalinertiafractions(NCRIFs)havebeenreported.DramaticchangesareobservedintheT1andT2atboththeproposedsupersolidonsettemperaturesandthephaseseparationtemperatures.Thespinrelaxationanomaliesareexplainedbyaquantumtunnelingmodelandaphenomenologicalttothetemperaturedependenceoftherelaxationisprovidedintermsofathermallyactivatedprocessrelatedtotheanomaliesinshearmodulusmeasurements.The3HeconcentrationdependenceofT1andT2isdiscussedintermsoftheimpuritonmodelintherangeof10)]TJ /F2 7.97 Tf 6.59 0 Td[(5x310)]TJ /F2 7.97 Tf 6.58 0 Td[(2.Thecrossoverconcentrationbetweenthecoherentmotionregionandtheincoherentmotionregionisestimatedfromtheexperimentaldataasoccurringfor3Heconcentrationsx310)]TJ /F2 7.97 Tf 6.58 0 Td[(4.Thetemperaturedependenceofnuclearspinrelaxationtimesarestudiedfor3Heconcentrations,500x32000ppmforamolarvolumeVM=20:7cm3.TheformationofFermi-liquiddropletsofpure3Heareobservedafterphaseseparation.Thetemperaturedependencesuggeststhattheinterface3Heatomsresponsibleforthenuclear 13

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spinrelaxationsaredegenerateduetothesphericalsymmetryandexchangenarrowing,notsolid-like.Thetemperatureindependentplateausattributedtothequantumexchangemotionalnarrowingareobservedatthetemperatureof0:25T1:3K.ThebesttforconcentrationdependenceoftherelaxationtimesshowsdeviationsfromtheLandesmantheory.Thevacancyactivationenergyisdeterminedtobe13:50:3Kforasamplewithx3=510)]TJ /F2 7.97 Tf 6.58 0 Td[(4andmolarvolumeVM=20:9cm3. 14

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CHAPTER1INTRODUCTION 1.1OverviewofNuclearSpinRelaxationsin3HeinSolid4HeNMRmeasurementsofthenuclearspin-latticerelaxationtimes(T1)andnuclearspin-spinrelaxationtimes(T2),giveusimportantinformationaboutthepropertiesofmaterialsonthemicroscopicscale.Forthecaseofdilutemixturesof3Heinsolid4He,whichisawellcharacterizedquantumsystem,largezero-pointenergyof3Heand4Heatomresultsinoverlapoftheirwavefunctionsandtheycanexchangetheirpositionsevenatabsolutezerotemperature.Duetothislargezero-pointmotion,3Heatoms,whichhaveoneunpairednuclearspin,movethroughthelatticebyquantumtunnelingandthismotionof3Hecanbedetectedbythenuclearmagneticresonance(NMR)measurements.Inthiscase,themeasurementsofnuclearspinrelaxationtimesof3Heatomsgiveusquantitativeinformationaboutthemechanismof3Hemotioninsidethe4Helattice,andfurthermorecangivecrucialinformationaboutthedynamicsofthe4Helattice.Nuclearspinrelaxationtimes(T1andT2)forverydilute3Heinsolid4Heweremeasuredinthisthesisworkandtherearetworeasonswhythisworkisimportant.1.TherehavebeenalargenumberofNMRexperimentsonpure3He 1 2 andnon-dilute3He-4Hemixtures 3 withtheoriesthatgivegooddescriptionsofthedatabuttherehavebeennoexperimentsonthissystemwithverydilute3Heinsolid4Hei.e.,x3100ppm.Totestandconrmtheexistingtheoryfornuclearspinrelaxationinthisquantumsystem,itisnecessarytocarryoutNMRmeasurementsonverydilutesamplesintheultralowtemperatureregion.2.SinceKimetal. 4 5 haveobservedtheanomaliesforsolid4Heintheresponsesoftorsionaloscillators,i.e.,thenon-classicalrotationalinertia(NCRI)whichcouldbethesignatureofthesupersolidphase,theoriginofthemicroscopicmechanismoftheNCRIhasbecomethefocusofalargenumberofexperimentsdesignedtounderstandthisunusualphase.NMRisanimportanttooltomeetthisdemandbecauseNMRisvery 15

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sensitivetothemotionofthe3Heattheatomiclevelsobytrackingthemotionofthe3Heinasolid4Helattice,onecanprobethemicroscopicdynamicsofthesolid4He.Thesepropertiesareverydiculttostudybecauseverydilute3Heinsolid4HecontainsanextremelysmallnumberofnuclearspinsandthisresultsinverysmallNMRsignalandlongspinlatticerelaxationtimes(T1). 1.2TheoreticalBackgroundonDilute3He-4HeMixture 1.2.1Pure3HeSystemTounderstandthepropertiesofdilute3Heinsolid4He,weneedtorstunderstandpure3Heexcitations.Itisbelievedthatthe3Hemotionin3He-4Hemixtureisverysimilarexceptfortheaddedeectofthe4Helattice.TheNMRrelaxationtimesaredeterminedbytheexcitationsofthesystem,sotoanalyzetheNMRexperiments,weneedtodeterminetheexcitedstatesofthesystem.Therearethreebasicexcitationsinpure3Hesystem:thephonons,vacancywavesandtunnelingexcitations. 1.2.1.1PhononsThephononexcitationsresultfromthedisplacementsoftheatomswithrespecttotheirequilibriumlatticesitesandusuallytheseexcitationsaretoosmallatlowtemperaturestogenerateconsiderableparticlemotioninthesystem.ThereforetheNMRrelaxation,whichmostlydependsonthemotionofthe3Heatoms,isbarelyaectedbythisexcitation.Inotherwords,thecontributionofthisexcitationtothenuclearspinrelaxationtimesisnotnormallydetectableinNMRmeasurements.Itisthereforegenerallyacceptedthatonemayignorethisexcitationwhenconsideringthenuclearspin-latticerelaxationtimes(T1)andthenuclearspin-spinrelaxationtimes(T2).Ifweconsideronlytwoexcitations(vacancywavesandtunnelingexcitations)inthesystem,theHamiltonianforpure3Hecanbeexpressedas 6 7 HPM=XR;"(R)b+RbR+XRR0;t(RR0)b+RbR0+1 2XR;00(R)b+Rb+R0bR0bR(1{1) 16

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wheretheoperatorb+RcreatesaparticleatRofspininthegroundstateofacompletesetofWannierstates.ThersttermistheHartreeenergyofthesystemandthesecondtermisthetunnelingterm,whichenablesparticlestomovefromlatticesiteRtoR0andthethirdtermrepresentsthehardcorerepulsionpart.Eqn. 1{1 describeswellthesystemthathastwosimpleexcitations(vacancywavesandtunnelingexcitations). 1.2.1.2Vacancywaves Figure1-1. Simpleschematicdrawingofthevacancywaveexcitations.(a)Thegroundstateofthelatticehasoneparticleateachlatticesite.(b)AvacancystateiscreatedbymovingaparticlefromR0toR.(c)ThevacancycanmovearoundinsidethelatticeduetothetunnelingtermintheparticlemotionHamiltonian(HPM)inEqn. 1{1 .(d)Thedoublyoccupiedstatealsocanpropagatebetweenthelatticesites. Inthegroundstateofthequantumsolid3He,normallyoneatomoccupiesonelatticesiteandthissolidusuallyiscalledacommensuratesolid.ThevacancyiscreatedbyapplyingthevacancycreationoperatorC+V(RR0)givenby, C+V(RR0)=b+RbR0(1)]TJ /F4 11.955 Tf 11.96 0 Td[(RR0)(1{2)Actuallythevacancycreationprocesscancreatetwolatticestateswhenitisappliedtothegroundstateofthesystem.OneisanemptylatticesiteandtheotherisdoublyoccupiedlatticesiteasshownschematicallyinFigure 1-1 .Inthisgure,forsimplicitythe 17

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spinstatesarenotshown.Thesetwostatespropagatecontinuouslythroughthecrystallattice.Theseexcitationsarecalledvacancywaves. 1.2.1.3TunnelingexcitationsThetunnelingexcitationwasdiscussedbyGuyerandZane 7 8 ,whondoutthatthetunnelingexcitationisaprocessassociatedwiththevacancystateresultingfromtheasymmetricwavefunctionofafermion.ThetunnelingmotionassociatedwiththevirtualvacancystateisshowninFigure 1-2 Figure1-2. Simpleschematicdrawingofthetunnelingexcitation.(a)Thesystemisinitiallyinthegroundstate.(b)TheparticleinthelatticeRtunnelstoR0.(c)OneoftwoparticlesinR0returnstothelatticesiteR.Thesystemgoesbacktothegroundstate. Therststepofthetunnelingexcitationprocessstartsfromthecommensurategroundstatewhereonelatticeisoccupiedbyoneatomasmentionedabove.ThenextstepstartswhentheatominthelatticesiteRmovestothelatticesiteR0.NowthelatticesiteRisempty.Asaresultthislatticesitebecomesa\vacancy"whilethelatticesiteR0becomesa\doublyoccupiedsite".Unlikethevacancywavesdiscussedintheprevioussectionwherethevacancyandthedoublyoccupiedsitepropagateawayfromeachother,inthiscaseoneoftwoparticlesinlatticesiteR0returnstothelatticesiteR.Asaresultofthisreturnprocess,thesystemgoesbacktothegroundstate(thecommensuratestate).Thevacancystatewhichexistsduringtheshorttimeiscalleda\virtualvacancystate". 18

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Manytheoreticalandexperimentalworks 9 10 showthatthevacancywavesarestronglycoupledwiththephononsandthetunnelingexcitations.However,thereareveryfewexperimentalstudiesoftheinteractionbetweenthetunnelingexcitationsandthephononswhilesometheoreticalworkshavebeenpublished. 11 12 1.2.2MassFluctuationWaves(MFW)forDilute3He-4HeMixturesIndilute3He-4Hemixtures,therearethreepurecrystalexcitationsasinpure3He:phonons,vacancywavesandthetunnelingexcitations.Howeverthereisalsoanadditionalexcitationthatisrelatedtothemotionsofatoms.Thesemotionshavetwoaspects,oneisthemotionofthe4Heatominthe3Hemediuminthecaseofdilute4Heinside3He,andtheotheristhemotionofthe3Heatominthe4Hemediuminthecaseofdilute3Heinside4He.Theseexcitationsresultingfromthemotionoftheatomsisrelatedtothemassuctuationwaves(MFWs).IfweassumethatthereareNlatticesitesinthegroundstateofthissystemthenthewavefunctionforthegroundstateisthesumofallcommensuratelatticestateswhichisgivenby 0=NYR=1R(xR)(1{3)whereR(xR)isthewavefunctionthatdescribesaparticleatxRandtheexcitedstateisgivenby RR0=b+RbR00(1{4)ThetunnelingtermintheparticleHamiltonianfordilute3He-4Hesystemisgivenbytheelementsofthematrix,hRR0jHj0iandthehard-coretermisgivenfromtheelementsofmatrix,hRR0jHjRR0i.Withthepropertheoreticalmodeforthedilute3He-4Hemixture,GuyercalculatedtheHamiltonianwhichgivesthetunnelingexcitationterminthedilute3He-4Hesystem, 13 HT=XRR00[t(RR0)t(RR0)=()]TJ /F4 11.955 Tf 9.3 0 Td[(0)]b+RbR0b+R00bR0(1{5) 19

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whereistheindexforthedierentstates.Ifweonlyconsiderthecase6=0(because=0correspondstothetunnelingofparticletotheneighboringsiteandthenoneparticlereturnstotheoriginalsitesleavingthesystemincommensuratestate:thetunnelingexcitation)thenEqn. 1{5 canbereducedbelow, HT=)]TJ /F1 11.955 Tf 9.29 0 Td[(2XRR0[t3(RR0)t4(RR0)=0]b+R4bR04b+R03bR3(1{6)Guyerdenedtheoperatorsas b+R4bR04b+R03bR3=a+Ra)]TJ /F5 7.97 Tf 0 -8.33 Td[(R0(1{7)wherea+RistheoperatorthatcreatesamassuctuationatRanda)]TJ /F5 7.97 Tf 0 -8.34 Td[(R0istheannihilationoperatorformassuctuation.Usingsimpliednotation M(RR0)34=XRR0t3(RR0)t4(RR0)=0(1{8)herewrotethetunnelingHamiltonianas HT=)]TJ /F1 11.955 Tf 9.3 0 Td[(2M(RR0)34a+Ra)]TJ /F5 7.97 Tf 0 -8.33 Td[(R0(1{9)Theoperatorthatmakesasingle3Heatomtunnelthroughthe4Helatticeisgivenby a+k=XRexp(ikR)a+R(1{10)Themassuctuationwave(MFW)isdenedbytheexcitationcreatedbytheoperatora+k.Themassuctuationwavesindilute3He-4Hemixtureinteractstronglywithphonons.Ontheotherhand,themassuctuationwavesdonotinteractstronglywithtunnelingexcitationsandthevacancywavesindilute3He-4Hemixturesbecausetheinteractionratesareproportionaltothe3Heconcentrationx3. 1.2.3NuclearSpinRelaxationTimesinDilute3He-4HeMixtureThewell-knowntemperaturedependenceofT1isshowninFigure 1-3 .Thenuclearspinrelaxationtimesofbothpure3Heanddilute3He-4Hemixturearesimilarinthehigh 20

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temperatureregion(Region1-AandRegion1-B).Inthelowtemperatureregion,thedilute3He-4Hemixtureundergoesawell-knownphaseseparationanditsnuclearspinrelaxationdeviatesfromthatofpure3He. Figure1-3. SchematicrepresentationofthetemperaturedependenceofT1.ThreequalitativemechanismsforT1indierenttemperatureregionscharacterizetherelaxationtopologies.Inregion1,T1isdeterminedbythedirectcouplingoftheZeemanenergytotheparticlemotionexcitation(thevacancywavesfor1-Aandthetunnelingexcitationfor1-B).Inregion2,T1isdeterminedbythecouplingbetweenexcitationsinthesystem,e.g.,tunneling-vacancy,andinthisregionanisotopicphaseseparationisobservedfordilute3He-4Hemixturelabeledby\PS".Inregion3,T1isdeterminedbythespatialdiusionoftheexcitations. 1.2.3.1Region1-AInregion1wherethesystemistightlycoupledtothethermalreservoir,therelaxationtimemeasurementgivesinformationaboutthevacanciesinthesystem.Inregion1-A,theenergyfromthenuclearspinsystemistransferredtothevacancywaveexcitations.Thefrequencyofthelocalelductuationassociatedwiththevacancy 21

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excitationisgivenby )]TJ /F2 7.97 Tf 6.59 0 Td[(1v=xvz!3(V;3)(1{11)wherexvisthethermallyactivatedvacancyconcentrationinthesystem,zisthenumberofnearneighbors,and!3(V;3)isthetunnelingfrequencyof3Heatomsintotheneighboringvacancysites.Inthehightemperatureregion,thefrequencyofthelocalelductuation,thatdeterminesthenuclearspinrelaxation,istheLarmorfrequency(!0=H0).!0ismuchlessthanthelocalelductuationfrequency()]TJ /F2 7.97 Tf 6.58 0 Td[(1v)becausethenumberofthermallyactivatedvacanciesincreaseasthetemperatureincreases.Asaresultinthehightemperatureregion,!0v1,andthespin-latticerelaxationtimesaregivenby 1 T1=(10 3)M2v!2dv(!0v1)(1{12)whereM2istheVanVlecksecondmoment,!disthelocaleldfrequencywithwhichthespinprecessesindipolareldofzneighborsatoms.Asthetemperatureislowered,thespin-latticerelaxationtime(T1)whichisindependentoftheLarmorfrequency(!0)becomesshorterandshorter.Becausethenumberofathermallyactivatedvacancydecreasesasthetemperatureisloweredandasaresultthevacancycorrelationtime(v)becomeslongerandlonger.FromEqn. 1{12 T1becomesshorterwithdecreasingtemperatureinthisregion.Attheresonancetemperatureatwhichthelocalelductuationfrequency()]TJ /F2 7.97 Tf 6.58 0 Td[(1v)isthesameasthefrequencyofnuclearprecession(Larmorfrequency!0),i.e.,!0v1,thespincanbeippedmosteectivelyresultingintheshortestT1shownasaminimuminFigure 1-3 T1=minimum(!0v1)(1{13)Asonelowersthetemperaturesfurther,i.e.,!0v1,thespin-latticerelaxationtime(T1)isgivenby 1 T1!2d !20v(!0v1)(1{14) 22

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Asthetemperaturedecreasesthenumberofvacancydecreasesresultinginweakerrelaxationrate.Thismeansthatthecorrelationtimevbecomeslonger.AsaresultthenuclearspinrelaxationtimebecomeslongerasitisclearfromEqn. 1{14 .Thespin-spinrelaxationtime(T2)hasmuchlessbizarretemperaturedependencethanthespin-latticerelaxationtime(T1),becausespin-spinrelaxationdoesnotinvolveenergytransferfromtheZeemansystemtoothersystem(itinvolvesenergytransferwithintheZeemansystemonly)butinsteadphasechangesforthenuclearmagnetization.Thespin-spinrelaxationoccursinthetransverseplaneduetotheuctuationofthelocaleldbythemotionofthevacancywaves.Thespin-spinrelaxationtimesinregion1-Aaregivenby 1 T2=2 3(M2=!0)f3 2+5 2[=(1+2)]+[=(1+42)]g(1{15)where=!0v,M2isthesecondmomentofthespectraldensityand!0andvaredenedabove.ForT2,wehavetoconsiderbothtransverselocaleld(Hd(!0)?)andz-directionlocaleld(Hd(!=0)z)unlikethecaseofT1forwhichweonlyneedtoconsiderthetransverselocaleld(Hd(!0)?)uctuationresponsibleforthespinip.TherstterminEqn. 1{15 isduetothezcomponentofthedipolareld.Inhightemperatureregion,!0v!0andtheanalysisissimilartothecaseforT1.Thespin-spinrelaxationtime,T2inthisregionisgivenby 1 T2(10 3)M2v(10 3)!2dv(1{16)ThetemperaturedependenceofT2inthisregionisduetothetemperaturedependenceofthevacancycorrelationtime,v.Inthisregion,theenergyowchaincanbeexpressedbythevacancywaverouteof\Zeeman!vacancywave!phonon!reservoir".TheenergyowtopologyforthisregionisexpressedbyZ-VPforbothpure3Hesystemanddilute3He-4HemixtureindicatingthattheweaklinkintheenergyowchainistheZeeman-vacancycoupling.ThedetaileddiscussionabouttherelaxationtopologiesisaddedintheAppendix C .TheenergytransferredfromtheRFpulsestoZeemansystemisdeliveredtovacancywaves 23

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whicharestronglyconnectedtothephononsandthethermalreservoirwhilethecouplingbetweenZeemansystemandthevacancywavesisrelativelyweak.Asthetemperaturedecreasesfurtherthespin-spinrelaxationtimes(T2)becomeshorteruntiltheparticlemotionsareonlyduetothetunnelingexcitationandthisisthestartingpointforregion1-Bthatwillbediscussedbelow. 1.2.3.2Region1-BInregion1-B,theenergytransferredfromtheRFeldtoZeemansystemisdeliveredtothetunnelingexcitations.InthiscasetheweaklinkintheenergyowchainisZeeman-tunnelingcoupling.ThetopologyoftheenergyowforthisregioncanbewrittenbyZ-TVPforpure3Hewhiletheenergyowtopologyfordilute3He-4HemixtureisZ-MFPbecauseindilute3Heinsolid4Hemixture,themassuctuationwavescompletelydominatethevacancywavesinregion1-B(attemperatureaslowas600mK).Sotheenergyowrouteof\Zeeman!massuctuationwave!phonon!reservoir"willbeaverygoodalternativetothevacancyrouteof\Zeeman!tunnelingwave!vacancywave!reservoir",ifthemassuctuationwavescangetridoftheenergytheyreceivedfromZeemansystem.Region1-Bappearswhentheparticlemotionduetothevacancywavesisnegligible.Becausethenumberofvacanciesistoosmalltobeeectivebelowacertaintemperature.InthiscasethecouplingoftheZeemansystemtothetunnelingexcitationdominatesthenuclearspinrelaxationandT1isgivenby T1=1 [J1(!0=!T)+4J1(2!0=!T)](1{17)whereJ1andJ2aregivenbyGaussianandLorentzianapproximationsrespectively. J1(!0=!T)=[1 M2=3!T]1=2exp()]TJ /F4 11.955 Tf 9.29 0 Td[(!20=2!2T)Gaussian(1{18) J1(!0=!T)=(M2=6!T)exp()]TJ /F4 11.955 Tf 9.3 0 Td[(!0=!T)Lorentzian(1{19)where!TisproportionaltoJand!0istheLarmorfrequency.WhenthenuclearprecessionfrequencyisslowerthanJ,whereJ=!T=bandbisaconstantthatdependson 24

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thechoiceofthecorrelationfunctionsofEqn. 1{18 andEqn. 1{19 .For!0=!T0,T1isgivenby T1!T !2d(1{20)whichisitssmallestvalue.IftheLarmorfrequencyislargerthanJand!0=!T!1,thenT1isgivenby T1!T !2d!T !2dexp[)]TJ /F2 7.97 Tf 10.5 4.71 Td[(1 2(!0=!T)2](1{21)Inthisregiontheuctuationofthelocaleldduetothetunnelingexcitationistemperatureindependent,i.e.,!TistemperatureindependentsoT1isalsotemperatureindependent.However!Tstronglydependsonthemolarvolume.ForT2,theparticlemotionsarealsorelatedtothetunnelingexcitationandfollowthesamediscussionaboveforT1andisgivenby 1 T2=2 3( 2)1=2(M2=!T)f3 2+5 2exp[)]TJ /F1 11.955 Tf 10.5 8.09 Td[(1 2(!0=!T)2]+exp[)]TJ /F1 11.955 Tf 9.29 0 Td[(2(!0=!2T)]g(1{22)wheretheGaussianapproximationisused.InthisregiontheVanVlecksecondmomentM2isrelatedtothe3Heconcentration,x3andthe!Tistemperatureindependent.ConsequentlyT2showsatemperatureindependentvalueandinthisregionT2onlydependsonthe3Heconcentrations.TheconcentrationdependenceofnuclearspinrelaxationtimesisdiscussedinSection 1.2.4 andSection 1.2.5 basedonthetheoreticalformalismofLandesmanandHuang. 1.2.3.3Region2Inthelowtemperatureregion,theZeemansystemandmassuctuationwavesarecoupled,andT1andT2aregivenby T11=fM2x33[1=(1+!223)]+[4=(1+4!2023)]g(1{23) T21=hM2x33f1+5 3[1=(1+!223)]+2 3[4=(1+4!2023)]gi(1{24) 25

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where)]TJ /F2 7.97 Tf 6.58 0 Td[(13isthesumoftwotunnelingrates:a3Heatomtunnelingratethrough4Heandthetunnelingratefora3Heatomintoavacantneighboringlatticesite.Inregion2,theZeemansystemisnottightlycoupledtothethermalreservoirorlattice,andtheenergyowaftertheRFpulseexcitationsisbottleneckedbytheparticlemotionexcitations.Inregion1,theparticlemotionexcitationsarestronglycoupledtothephononsandtheyarealwaysatthesametemperaturewiththelatticeorthermalreservoir.Whentheparticlemotionexcitationbecomesuncoupledfromthelatticeorthermalreservoir,region2occurs.Inthiscasetherelaxationprocessinvolvesthreeormoresystemsotherthanjutthetwosystemsinthecaseofregion1.Theintrinsicrelaxationtimesaredenedforthetwosystemrelaxationasinregion1,andthetopologicaltimesaredenedforthethreeormoresystemsasinregion2.Thetopologicaltimesaredenedby 1 Ttopological=(topologicalfactor)1 Tintrinsic(1{25)Thetopologicalfactorisdenedastheenergyconstantofthesystemswhichareinvolvedintherelaxationprocess.Inregion2,thesmallconcentrationsof3Hearemobilethroughthe4Helatticeandconstitutemassuctuationwaveswhichcoupletothephononsinthesystem.Asmentionedbefore,themassuctuationwavesof3Heatomsdonotinteractwiththetunnelingexcitations.TheenergyowtopologyisgivenbyZMF-Pandtheenergyowchaincanbeordered:Zeeman!massuctuationwave!phonon!reservoir.Theweaklinkintheenergyowchainismassuctuation-phononcoupling.Thetopologicalrelaxationtimesaregivenby 13 1 Ttopological=[kMF=(kz+kMF)]1 TMFP(1{26)whereTMFPistheintrinsicrelaxationtimewhenweonlyconsidertheweakcoupling(longrelaxationtime)intheenergyowchain,i.e.,relaxationfrommassuctuationwavestophonons.kMFandkzaretopologicalfactorsformassuctuationwavesystemandZeeman 26

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systemrespectivelyanddenedby kMF=dEMF d(1{27) kz=dEz d(1{28)whereEMFandEzaretheenergyformassuctuationandZeemansystemrespectivelyandisinversetemperature. 1.2.3.4Region3Inregion3,theenergyacquiredfromtheRFpulsesistransferredbythediusionoftheparticlemotionexcitationwhileitistransferredbythecouplingamongtheexcitationsinregion1and2.Asthetemperaturedecreasestheweakconnectionintheenergyowchainisthelinkofthemassuctuationwavestothephonons.Itthereforetakesalongtimefortheenergytoreachthephonons.OncephononsreceivetheenergytheytransferittothereservoirandtherelaxationisdeterminedbythecouplingeciencyofthetwophononsystemswhichistheKapitzaresistance 14 1.2.4LandesmanModel:StrainFieldLandesmancalculatedthenuclearspinrelaxationtimesofdilute3Heimpuritiesinsolid4Heinregion1-B.Heassumedthattherelaxationisbasedonthemodulationofthedipolarinteractionresultingfromthetunnelingofthe3Heatoms.Heconsideredtheconstraintsonthetunnelingfordilute3Heinsolid4Heduetothelargeelasticlatticeinteractions.Forthehcplatticethereisanelasticdeformationaroundeach3Heatom.Theelasticinteractionduetothelatticedistortionarounda3Heatomisinducedbythevirtualphononexchangeofthe3Heatsiteiandjandgivenby Vij=K(a Rij)3(1)]TJ /F1 11.955 Tf 11.96 0 Td[(3cos2)(1{29)aisthenearestneighbordistanceandistheanglebetweenRijandthetrigonalaxis.Landesmanassumed~=1andKJ34.AschematicrepresentationofthecongurationofthestraineldisshowninFigure 1-4 .Landesmancalculatedthediusionconstant 27

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Figure1-4. Theschematicoftheelasticinteraction.Thedistancebetweentwo3HeatomsthataresittinginthelatticesiteiandjisRijandtheistheanglebetweenRijandthetrigonalaxisinthehcplattice. usingsimplescatteringduetotheelasticinteractionsandfoundadiusionconstantofDx)]TJ /F2 7.97 Tf 6.58 0 Td[(4=33insteadofDx)]TJ /F2 7.97 Tf 6.59 0 Td[(13aspredictedbyothertheories. 15 16 HettedtheSussexdata 17 usinghisdiusionformulaandderivedthevalueforthetunnelingrateoftheisolated3Heimpuritiesfromsitetosite,J34=1:0MHzandtheelasticinteractionbetweentwo3Heatomssittingontheneighboringlatticesites,K=2=1200MHz.UptonumericalprefactorsLandesmancalculatesthe3HediusionconstantDLtobe DL=16 3\(4 3)a2J234 K(x3)4=3=0:36a2J234 Kx4=33(1{30) 28

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where=8:77.Landesmanexpressesthediusionintermsofaneectiverandom-walkjumprate)]TJ /F2 7.97 Tf 6.59 0 Td[(1ras DL=a2 6r(1{31)BycomparingEqn. 1{30 andEqn. 1{31 ,onecandeducetheexpressionfortheeectiverandomwalkjumpingrate)]TJ /F2 7.97 Tf 6.58 0 Td[(1ras )]TJ /F2 7.97 Tf 6.58 0 Td[(1r=2:2J234 Kx4=33(1{32)Whena3Hejumpsonelatticesite,thetypicalchangeinelasticenergyis ad(Ka3 r3) dr=3Ka4 r43Kx4=33(1{33)usingthemean3He-3Heseparationofra=x1=33.Intheabsenceoftheelasticenergya3HeatomwouldjumpattheratezJ34inabandofstatesofwidthJ34.Butwiththeelasticenergyduetoother3HeatomsthebandwidthisincreasedtoKx4=33.HencethedensityofstatesandjumpratearereducedbyafactorofJ34=Kx4=33,givingEqn. 1{32 uptotheprefactor.Landesmanderivedthespin-latticerelaxationtimeT1andspin-spinrelaxationtimeT2usingasimplescatteringmodelbyconsideringthescatteringofthe3Heatomsbytheelasticdistortioneldandfound T1=3 4!20 BM2Jeffx2=33=0:03!20K M2J234x2=33(1{34) T2=BJeff M2x4=33=23:3J234 M2Kx4=33(1{35)whereB=23:3.TheeectivetunnelingrateJeff=J234=K,andM2isthedipolarsecondmomentforpure3Hewiththesamelatticestructure.HecalculatedthecorrelationtimecthroughtheKuboformalismandfound 18 )]TJ /F2 7.97 Tf 6.59 0 Td[(1c=BJeff x1=33=23:3J234 Kx1=33(1{36) 29

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AsLandesmanpointedoutinhispaper,thereisalargedierencebetweentheNMRcorrelationfrequency(thelocaleldcorrelationfrequency),)]TJ /F2 7.97 Tf 6.59 0 Td[(1cinEqn. 1{36 andthejumpingfrequency,)]TJ /F2 7.97 Tf 6.58 0 Td[(1rinEqn. 1{32 (i.e.,)]TJ /F2 7.97 Tf 6.59 0 Td[(1c6=)]TJ /F2 7.97 Tf 6.59 0 Td[(1r).Asshowninthetwoequations,thejumpingfrequencyproportionaltox)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33ismuchlargerthantheactualNMRcorrelationfrequencywhichisproportionaltox)]TJ /F2 7.97 Tf 6.59 0 Td[(4=33. 1.2.5HuangModel:MFW-MFWInteractionPotentialHuangdiscussedtheeectofthelong-rangeinteractioninducedbythemassuctuationsonthemotionofthe3Heparticlethroughthe4Helattice.HederivedtheHamiltonianforthedilute3He-4Hemixtureintheformalismofsecondquantizationandaveragedasingleparticlecrystalstates.ThetotalHamiltonianforthedilute3Hesystemwith3Heconcentrationofx3isgivenby H(x3)=E0(x3)+HD+Hw+HI(1{37) HI=H(1)I+H(2)I(1{38)whereE0(x3)isthegroundstateenergyforthesystemwith3Heconcentrationofx3.HDgeneratesthedisplacementuctuationandHwisresponsibletothewidthuctuation.Themass{uctuation{mass{uctuation(MF-MF)interaction,HIisthesumofdirectuctuation,H(1)Iandindirectuctuation,H(2)I.HDandHwchangethegroundstateenergyslightlyanddescribethethermalexcitations,andatlowtemperatures,theMF-MFinteraction(HI)canaectthemotionofthe3Hein4He.Huangcalculatedtheeectofthisinteractionespeciallyonthedilute3Hein4He.Todiscusstheintermediateconcentrationregionherstdiscussedtheverydiluteregion,i.e.,MFWregion,illustratedinFigure 1-5 IntheMFWregion,3Heatomsaresorarethatthemeanseparationisgreaterthantherangeofthedistortionelds.OnemayassumethattheyformBlochwavesandpropagatewithinthelatticebutonce3Heatomsapproachoneanotherlessthanthedistancercthen3Heatomscannotpropagatefreelyanymore.Asaresult3Heatomsare 30

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Figure1-5. Thesimpleschematicofthemassuctuationwaves(MFW)motion.Ifa3Heatomisclosetoanother3Heatomatdistancelessthanrc,then3Heatomisscatteredduetothelargeinteractionpotentialbetweenthetwo3Heatoms. scatteredbyeachotherbecausetheenergydierenceduetothemotionofonerelativetotheotherismuchlargerthanthetunnelingrateJ34in3He-4He.TheMFW-MFWscatteringinteractionisgivenbyHuang'scalculation SI=V0( j)778(!R)]TJ 11.95 8.77 Td[()777(!R0j)(1{39)whereV0istheinteractionconstantbetweentwoMFWsandV010)]TJ /F2 7.97 Tf 6.59 0 Td[(2K.isthenearestneighbordistance.Thecross-sectionoftheMFW-MFWscatteringis c=r2=2(2V0 J34)1=2(1{40) 31

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Thedistanceofnearestapproachrisgivenbythefollowingrelation, J34=j()777(!)777(!rV0)( )777(!r)jr=)826(!r(1{41)Huangcalculatedthediusionconstantusingtheserelationsandfound DI=J34=1 2J34 x3(J34 3V0)1=2(1{42)whereistheMFW-MFWmeanfreepathwhichisgivenby =1 x3(J34 3V0)1=2(1{43)Byrequiringthattheaveragedistancebetweenatomsbegreaterthanthedistanceofthemaximumapproachrc,onecandeterminetheupperlimitfor3HeconcentrationforMFWregion.Abovethisconcentration3Hewillinteractstronglywithother3Heatomsandtheymoveincoherentlythroughthelattice.Heestimatedtheupper3Heconcentration,x310)]TJ /F2 7.97 Tf 6.59 0 Td[(3.Huangalsocalculatedthenuclearspin-spinrelaxationtimes(T2)basedonthismodelinthestronglyinteractingregionandthisisdiscussedindetailinChapter 4 byanalyzingtheconcentrationdependenceofourNMRdata. 32

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CHAPTER2EXPERIMENTALDETAILS 2.1OverviewThischapterwilldiscusstheexperimentaldetailsthatareusedforthemostoftheNMRmeasurementscarriedoutforthisthesiswork.AllNMRmeasurementsarecarriedoutinanultralowtemperatureenvironmentusingadilutionrefrigerator.Section 2.2 givesageneralbackgroundfortheuseofadilutionrefrigerator.AllNMRmeasurementsaredoneonverydilute3Heinsolid4Heandespeciallyforx3100ppm,thenumberofnuclearspinsinthesamplevolumeof0:15cm3isoftheoderof1017andthisisaverysmallnumber.ThemagnitudeoftheNMRechosignalisthereforeverysmall.AlowtemperaturehomemadepreamplierisconsequentlyneededtoimprovethesignaltonoiseratiobeforethesignalistransferedfromtheNMRcoiltotheroomtemperatureamplier.InSection 2.4 ,thelowtemperaturepreamplierisdiscussedindetail.ThepulsedNMRtechniqueisdiscussedinSection 2.5 .TheNMRsamplecellisconstructedasacrossedcoilarrangementasdiscussedinSection 2.3 ,andthegeneralmethodforgrowingtheheliumcrystalsisdiscussedinSection 2.6 .Section 2.7 andSection 2.8 describethemethodofthermometryusedatlowtemperaturesandgiveabriefsummaryofpressuremeasurementsrespectively. 2.2DilutionRefrigeratorMostNMRmeasurementsforthisthesisworkwerecarriedoutinthebay3highB/TfacilityinMicro-kelvinLaboratory,showninFigure 2-1 .Thechoiceofrefrigeratorforspecicmeasurementsdependsonthetemperaturerangewheretheactualexperimentswillbecarriedout.ForT0:25K,a3Heevaporationcryostatisusuallyused.Thetemperaturerangeofthedilutionrefrigeratoris0:003T1:0KandforT0:003K,adiabaticdemagnetizationisusuallyused.Ourmeasurementsaredoneintherangeof0:01T0:6Ksoadilutionrefrigeratorwithhighrefrigeratingcapabilityisused. 33

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Figure2-1. ThehighB/TfacilityinMicro-kelvinlaboratoryinUniversityofFlorida.PhotocourtesyofMicro-kelvinlaboratory. Thesimpleprocessofthedilutionrefrigeratorreliesoncertainthermodynamiccharacteristicsof3Heand4He.Thephasediagramof3He-4HemixtureisshowninFigure 2-2 .Attemperaturesbelowthetriplepoint,the3He-4Hemixturewillseparateintotwoliquidphases,a3Herichphaseanda4Herichphase,dividedbyaphaseseparationboundary.The3Herichphaseismostly3Heandthe4Herichphaseisamixtureof4Heand3Hewhichisalwayscomposedofatleast6%3He,downtothelowesttemperatures.Thisphaseisshowninthediagrambelowandtotheleftofthetriplepoint,alongtheequilibriumline.Thetwophasesaremaintainedinliquidform.Sincethereisaboundarybetweenbothphases,extraenergyisrequiredforparticlestogofromonephasetoanother.Ifwepumponthevaporabovethe4Herichliquidphasethenweremovemostly3Hebecauseofthelargervaporpressureof3He.Tomove3Heacrosstheboundarythedilute 34

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Figure2-2. Phasediagramof3He-4He.Thelambdalineisbetweenthesuperuidandnormalliquidphaseandthephase-separationlineisbetweenthesuperuidandunstablestatewherethe3Heand4Heareseparated. liquidphaseisconnectedtoastill(Figure 2-3 )whichcontainsbothliquidandvapor.Asaresult,3Hewillhavetoowacrossthephaseboundaryfromthe3Herichsidetothe4Herichsidetorestoreequilibrium.Crossingthisphaseboundarylineproducesthecoolingduetothelatentheatofmixing.Inotherwords,the3Herichphasesendsthe3Hetothedilutephaseandabsorbsenergyintheformofheatfromthephononsoftheliquidinthemixingchamberandasaresultthemixingchamberiscooleddown.Thesamplesareinthermalcontactwiththewallofthemixingchamberandtheyarecooleddownalso.The3Hewillcrossthephaseboundaryandjointhe4Herichphasebyrestoringequilibrium,andthe3Heatomslost 35

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Figure2-3. Theschematicofdilutionrefrigerator. duringthiscyclearereplenishedbyaconstantlycirculatingowof3He.Asetofheatexchangers(Figure 2-3 )isusedtocooltheincoming3Hetothetemperatureoftheliquidmixtureinthedilutephase.Inthiswaythedilutionrefrigeratorcoolsthemixingchamberandthusthesamplecontinuously.AsimpleschematicdrawingofadilutionrefrigeratorisshowninFigure 2-3 2.3NMRCoilsTheNMRcellisdesignedtohaveahorizontalcylindricalshapewithtwocoils:(i)asolenoidalNMRsignalreceivingcoilwoundaroundthecylindricalcell,and(ii)athermallyisolatedsaddleshapedradiofrequency(RF)excitationcoilfortheRFpulses. 36

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ThereceivingcoilandthetransmittingcoilwiththesamplellinglineattachedareshowninFigure 2-4 andFigure 2-5 respectively.Thesolenoidalreceivingcoilhas80100turnsofhighconductivitycopperwiretoachievethedesirablequalityfactorQandNMRcoilinductanceLs. Figure2-4. Photoofreceivingcoil.ThesamplellingcapillarylineisattachedwithStycast2850GT.OneofNMRsignaloutputlinegoestothepreamplierinputandtheotherisconnectedtotheground.PhotocourtesyofS.S.Kim. Figure 2-6 showsaschematicviewofthesetwocrossedcoils.Thiscrossedcoildesignoerstwofeatures.Thisdesigncanreducetheunwantedpick-upofRFexcitationinthereceivingcoilandprovidethesamplecellwithathermalisolationfromtheheatgeneratedintheRFexcitationcoil.Thecellwasmadefrompolycarbonateandoneendofthecellisopentoapressuregaugeandtheotherendissealedbyaninsulatingepoxy(Stycast2850GT)capthroughwhichacapillarylinepassestoprovideasamplellingline.Thetransmissioncoilisasaddle-shapedcoilanditcangenerateauniformRFmagneticeldperpendiculartothecylindricalaxisoftheNMRsamplecell. 19 { 21 Inourmeasurements,thesignalisextremelysmall(typically<1nV)andobtainingauniform 37

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Figure2-5. Thephotooftransmittingcoil.Thereceivingcoilisplacedinsidethetransmissioncoilandthellinglineisfromthereceivingcoil.PhotocourtesyofS.S.Kim. Figure2-6. TheschematicoftheNMRcrossedcoils. ACmagneticeldB1asillustratedinFigure 2-6 isveryimportant.IfB1isnotuniformthenitreducesthemagnitudeoftheNMRsignal. 38

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Figure2-7. Thegeometryofasimplesaddlecoil. FromthegeometryshowninFigure 2-7 ,theelduniformityparameterQandtheoptimalangle0isgivenasbelow Q=(15)]TJ /F1 11.955 Tf 11.95 0 Td[(3s) (2s2+2s3)(2{1) sin2(0=2)=(15+3s+6s2+12s3) 2(5+3s+4s2+8s3)(2{2)wheres=1+(h=D)2withhandDaretheheightanddiameterofthesaddlecoil.Coincidentally,intherange1h=D2,thecurveshowingQasafunctionofh=Disalmostidenticalinshapetothatof0asafunctionofh=D,andthereforesimplyrescalingtheverticalaxisasshowninFigure 2-8 sucestoindicatehowQdependsonh=D.For 39

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ourcase,h=D1:2andtheoptimalangle0isselectedabout121.5o.Figure 2-9 Figure2-8. Thegraphoftheangle0andtheeld-uniformityparameterQasafunctionofsaddlecoilphysicaldimension.Researchshowsthattheangle0andelduniformityfactorQhassamevalesintherangeof1h=D2. isaphotooftheNMRset-upshowingthecongurationofpreamplierdiscussedinsection 2.4 andNMRcoils.Withthisset-upwereacheddownto250mKbecauseofheatingfromthepreamplier.Wemovedthepreamplierawayfromthesamplecoiltoobtainlowertemperaturesdownto10mK.ThecircuitdiagramofthepreamplierisshowninFigure 3-1 40

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Figure2-9. NMRmeasurementset-up:NMRcoils,preamplierandtuningcapacitor.preamplierandthetuningcapacitorarelocatedclosetotheNMRcoils.SuperconductingwiresareusedtoconnectthereceivingcoiltothepreampliertoprovidehighelectricalconductivitybutlowthermalconductivityfromthepreampliercircuittotheNMRsample.PhotocourtesyoftheMicro-kelvinlaboratory. 2.4LowTemperaturePreamplierAlltheNMRmeasurementswerecarriedoutforverydilute3Heinsolid4Hesampleswherethenumberofnuclearspinsislessthan1017.TheNMRsignalisthereforetoosmalltobedetectedbytheusualNMRspectrometer.Forthiskindofmeasurementalowtemperaturepreamplierisnecessarytominimizethesignallossduetothemismatchofimpedancebetweenthesignalandthecoaxialcablewhichextendsfromlowtemperaturetotheroomtemperature. 41

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2.4.1SimpleSourceFollowerCircuit Figure2-10. Thecircuitdiagramofthelowtemperaturepreamplier.Thepseudomorphichighelectronmobilitytransistor(pHEMT),AgilenttypeATF-35143,isusedforasimplesourcefollowercircuit.TheoperatingpointofthetransistoriscontrolledbythebiasvoltageVB)]TJ /F1 11.955 Tf 10.99 1.79 Td[(tomatchthehighimpedanceoftheNMRresonancecoil(Ls)tothat(50)oftheRFcoaxialcablewhichtransfersthesignaltothelownoiseamplieratroomtemperature. Thecircuitdesignfortheamplierisasimplesourcefollowerusingapseudomorphichighelectronmobilitytransistor(pHEMT)asshowninFigure 2-10 .AlthoughapHEMTgeneratesappreciableheat(0:2)]TJ /F1 11.955 Tf 12.01 0 Td[(0:5mW),ithasahighgainatlowtemperaturesuptotheUHFrangeandtheplanargeometryallowsthepHEMTtobealignedparalleltothemagneticeld,eliminatingtheHalleects 22 .Thecircuitisoptimizedtooperateat2MHz 42

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atwhichthenuclearspin-latticerelaxationtimeis103satlowtemperature(10mK). Figure2-11. Thephotoofthelowtemperaturepreamplier.ThepreampliertakessignalfromNMRcoilwith1:6104.Theout-putimpedanceofthepreamplier(50)ismatchedwiththeroomtemperatureampliertoimprovethesignaltonoiseratio.PhotocourtesyoftheMicro-kelvinlaboratory. TheimpedancemismatchisthemainreasonfortheNMRsignallossatlowtemperatures.TheimpedanceoftheNMRsignalisgivenbyR=Q!L.Qisthequalityfactorofthecoiland!istheLarmorfrequencyandListheinductanceoftheNMRcoil.Forourcase,Q82,!2MHzandL100HandwiththesenumberstheimpedanceoftheNMRcoilis1:6104.AsshowninFigure 2-10 ,theimpedanceoftheNMRsignalis300timeslargerthanthatofcommercialcoaxialcables,sowithoutthepreamplieronlyasmallfractionofthesignalcanbetransferredtothecable. 43

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Figure2-12. Photoofalownoisepseudomorphichighelectronmobilitytransistor(pHEMT).ThemodelofATF-35143 23 issuitableforapplicationsincellularandpersonalcommunicationsservice(PCS)basestations,alowearthorbit(LEO)satellitesystems,multi-channelmulti-pointdistributionservice(MMDS),andothersystemsrequiringverylownoisegurewithgoodinterceptfrom450MHzto10GHzfrequencyrange.PhotocourtesyoftheAgilentTechnologiesInc. Forthesourcefollowercircuitthegainandimpedancearegivenby VoltageGain:AV=VOUT VIN=gmRs gmRs+11;forgmRs1(2{3) CurrentGain:AI=1(2{4) InputImpedance:ZI=1(2{5) OutputImpedance:ZOUT=Rs gmRs+11 gm(2{6)TherequiredtransferadmittancegmisachievedbytuningthebiasvoltageVB)]TJ /F1 11.955 Tf 10.99 1.79 Td[(tomatchtheoutputimpedanceoftheampliertothe50.Thisdesignhastheadvantageofminimizingthetendencyofthecircuittooscillatebecauseofthehighgainofthedevicesused.Theresistorswereminiaturemetal-lmresistorsandthecapacitorsminiatureceramicchipcapacitorstestedatliquidnitrogentemperaturespriortoassemblyforintegrityonthermalcycling.ThevalueforthesecomponentsareshownintheFigure 2-10 44

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Figure 2-11 isaphotoofthepreamplierandtheincorporatedAgilenttypeATF-35143pseudomorphichighelectronmobilityeldeecttransistor(pHEMT).ThelatterisshownseparatelyinFigure 2-12 .Thisdevicehaslowpowerdissipationcomparedtoothersinthesameclass.Thegatebiasissuppliedbyanexternalsupplytolimitthetotalpowerdissipationto0.5mWandwiththesourceresistancedeterminedtheoperatingpointandthusthetransconductanceofthepHEMT.Thecircuitwasoperatedat2MHzbecauseatthisfrequencythenuclearspin-latticerelaxationtime(T1)isabout100s.andthisisthemaximumtimelengthforwhichwecanobtainthetolerablesignal-to-noiseratiowithreasonableaveragingtimeatlowtemperatures. 2.4.2PerformanceAtypicalNMRspinechosignalforasamplecontaining500ppmof3Heinsolid4Heat400mKisshowninFigure 2-13 .Theobservedsignal/noiseis35aftersignalaveraging(10pulsesequences)forabandwidthof17kHz.Fromthecalibratedgainthiscorrespondstoanoisetemperatureofapproximately1.1K.Asmallcross-couplingfromthetransmissioncoilleadstoasmalluncertaintyinthegaincalibration.Thequotednoisetemperatureisanupperlimit. 2.4.3SampleProbeandtheSignaltoNoiseRatioInNMRsignaldetection,thesampleprobeandthecouplingbetweenthesampleprobeandthepreamplierarethemostcriticalpartsofanNMRapparatus.Formoresophisticatedexperiments,optimizingsamplesizeandsampleshapearealsoimportantbuttheNMRcoilscouplingtothepreampliercanbeessential.InthepulsedNMRexperiments,thesamplematerialreceivespulsesfromthetransmitterthatgeneratesaradiofrequencyB1eldperpendiculartothestaticDCB0eld.Theresponsetothesepulseshastobeconvertedintoasignalvoltageandfedtothepreamplier.Theelectricalnoiseisunavoidableanditoriginatesintheelectricalcomponentsthemselves.Usuallythecurrent-carryingelectronshavearandomelementsintheirmotionsduetotheirthermalenergy.Statisticalthermodynamicsshowsthatfora 45

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Figure2-13. NMRHahnechosignalobtainedafteratwopulsesequenceof90o)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F1 11.955 Tf 11.95 0 Td[(180o.Theredlineandblacklinearesignalsfromrealandimaginarycomponentsofthespinecho.Thesessignalsaretakenforasolidsamplewithx3=500ppmat400mK. systemintheequilibriumattemperatureT,theaverageenergyassociatedwiththermalexcitationsiskBT=2,wherekBisBoltzmann'sconstant.Forexample,inaresistortherecanbesomeelectronsthatslightlydeviatefromotherelectronswhichremainintheequilibriumstate.Thesedeviationsgenerateapotentialdierenceasauctuatingvoltage.ThisistheoriginofJohnsonnoiseandthemeansquaremagnitudeofthevoltageina 46

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frequencybandfisgivenby hv2i=4kBTRf(2{7)TheJohnsonnoiseisproportionaltothetemperatureandtheresistance.Anothercontributiontothenoiseis\shotnoise"andthisisduetothenitechargecarriedbyelectronsanditappearsasauctuationinthecurrentowingandthefrequencybandf. hi2i=2eIf(2{8)whereIisthecurrentandeistheelectroncharge.Thesetwocontributionstothenoiseareknownaswhitenoise.Thesecontributionsarefundamentallyduetorandomprocesses,sotheaverageofthenoisevoltageiszero.Themeansquarevoltagehasnitevaluesthuswhenwementionthenoisevoltageofcertainmagnitudewerefertotherootmeansquare(RMS)value.Thispropertyisimportantindataaveragingtoimprovethesignalnoiseratio.ThecommonwaytoreducethenoiseintheNMRmeasurementistorepeatthemeasurementsandthenaveragethem.Forexample,ifwedothemeasurementsNtimesandaddthemtocalculatetheaveragevalueofanoisevoltagethenthenoiseincreaseswiththesquarerootofN,i.e., i=NXi=1vi=p Nvi(2{9)SoifwedivideitbyNtheaveragednoisevalueasbelow, Pi=Ni=1vi N=1 p Nvi(2{10)thentheaveragenoiseactuallydecreasesbythefactorof1=p N.InthiswaybyaveragingtheNMRsignalwecanimprovethesignaltonoiseratio.Typicallyforfrequenciesbelow100MHz,thenoiseintroducedbytheamplierisignoredandthedominantnoiseistheJohnsonnoisefromthetunedNMRcoilwhichisgivenbyEqn. 2{7 IntheparallelresonantcongurationthecoilresistanceisR=Q!L.Thesizeofthesignalafter=2 47

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pulseistobe vs=NA!0B0(2{11)whereNisthenumberofturns,Aisthecrosssectionofthecoiland0isthemagneticsusceptibility.Thesignaltonoiseratioisgivenbytheratioofthesignalvoltagetothenoisevoltage. (S=N)=NA!0B0 (4kBTRf)1=2(2{12)whereisthellingfactorthatisusuallyunitywhenthevolumeofthesamplecellisequaltothevolumeoftheNMRcoil.NMRcellwasdesignedtohave1.ThedetailsofthecalculationsforthesignalandnoisevoltageforourNMRmeasurementsaregiveinAppendixA. 2.4.4SummaryAlowpowerpreamplieremployingapseudomorphictransistorwasusedtorealizelownoisetemperatures(TN),TN1:1K,forpulsedNMRexperimentsatmilli-kelvintemperaturesbyisolatingtheamplierandNMRcoilsfromthesamplecellitself.ThecircuitiseasytoconstructandinprinciplecanoperateuptoUHFfrequencies.Thedesignallowstheuseofhighpowerpulsesforstraightforwardpulsesequences.Theimprovementoversimplepassivematchingusingacapacitivetransformertomatcharesonantcoiltoa50transmissionlineisbetterthanafactorof260. 2.5PulsedNMRMethodThepulsedNMRtechniquewasusedtomeasurespin-latticerelaxationtimes(T1)andspin-spinrelaxationtimes(T2).InordertounderstandthepulsedNMRmethodsweneedtoconsidertheeectofashortradiofrequencypulseontheequilibriummagnetizationIz.TheRFpulsegeneratesanoscillatingRFmagneticeldB1cos!atwhere!aisveryclosetotheLarmorfrequency!0.TheinteractionwiththenuclearspinisgivenbytheHamiltonian, H=~B1cos!atIx+~BoIz(2{13) 48

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where!0=Bointheappliedstaticmagneticeldandisthegyromagneticratiofor3He.Inareferenceframerotatingatfrequency!aaboutthez-axis,wemaywrite H=~B1Ix+~(!0)]TJ /F4 11.955 Tf 11.95 0 Td[(!a)Iz(2{14)ThelastformisnegligiblesotheeectofB1appliedfortimetwisgivenbytheunitaryoperator U=eiHtw=ei(~B1tw)Ix(2{15)whichisequivalenttoarotationbyangle=~B1twaboutthex-axis.Bychoosingtwand/orB1onecanthereforerotatethespinbyanydesiredangle.Typicalvalueofare90oand180oasdescribedbelow. 2.5.1T1MeasurementsInordertodetermineT1,wemeasuredeither(i)thefree-inductiondecay(FID)afterasingle90opulse,or(ii)asinglespinechomethodfora90o)]TJ /F1 11.955 Tf 12.28 0 Td[(180opulsesequence.WhenweusedtheFIDtomeasureT1,wechangedthedelayingtimetbetweenpulsesandapplied90opulsesforeachmeasurement.ThetypicalFIDdatafroma500ppmsampleisshowninFigure 2-14 .ItisimportanttonotethatimmediatelyaftertheRFpulsethereisadeadtimeduetotherecoveryofthereceivingamplierafterthepulse.Thisdeadtimetd'10Q=!0'100s.WeincreasedthedelaytimegraduallyforeachFIDmeasurement.AsweincreasedthedelaytimetheFIDsignalsizealsoincreasedexponentially.ThetofFIDsignalsizewithrespecttothedelaytimesgivestheT1values.TheotherpulsesequenceweusedforT1measurementsisthesingleechopulsedmethod.TheequilibriummagnetizationIzisdestroyedandthenoneappliesa(90o)]TJ /F4 11.955 Tf 12.14 0 Td[()]TJ /F1 11.955 Tf -452.25 -23.91 Td[(180o)pulsesequencetoobserveanechoat2.Onethenwaitsatimebeforerepeatingtoobservethegrowthoftheechoastbecomescomparabletotherelaxationtime.Thetimeintervalbetween90oand180opulsesisxedas=4ms.Theechoheightsaregivenby h(t)=h(1)[1)]TJ /F1 11.955 Tf 11.95 0 Td[(exp()]TJ /F4 11.955 Tf 9.3 0 Td[(t T1)]exp()]TJ /F1 11.955 Tf 9.3 0 Td[(2 T2)(2{16) 49

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Figure2-14. Thefree-inductiondecay(FID)from500ppmsampleafter90opulse.Theblacktraceisthein-phasecomponentoftheFIDandtheredtraceistheoutofphasecomponent. whereh(1)istheechoheightoftheequilibriumstate.Wedeterminedtheh(1)sizefromthevaluesoftheechoheightthataremeasuredwitha45hourwaitingtime.T1issensitivetothevariationinh(1),somorethanonemeasurementatt=5wasmadeintheexperiments.Withxedwhichismostlychosentobe4ms,asweincreasethewaitingtimettheechoheightincreasesexponentiallywithtimetandthebesttusingEqn. 2{16 givestheT1values. 50

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Figure2-15. Schematicdrawingfortheformationofasingleechoandcorrespondingspinevolution. AschematicdrawingforthesingleechopulsesequenceandcorrespondingevolutionofthespinareshowninFigure 2-15 .WeusedtheFIDsequencetomeasureT1forrelativelyhigh3Heconcentrationsamples.Thesingleechosequencewasusedforverydilutesamples(16ppmand24ppmsamples).Thedierencebetweenthevaluesofspin-latticerelaxationtimesobtainedfromFIDandsingle-echomethodwasnegligible. 2.5.2T2MeasurementsThespin-spinlatticerelaxationtimes(T2)aremeasuredbyusingamultispinechomethod,theCarr-Purcell-Meiboom-Gill(CPMG)andalsotheaveragedsinglespinechomethod.Theaveragedsinglespinechomethodneedslongermeasuringtimesbecauseafterapplying90o)]TJ /F4 11.955 Tf 12.12 0 Td[()]TJ /F1 11.955 Tf 12.12 0 Td[(180opulsewehavetowaitaverylongtime(>T1)untilwestartthenextmeasurementwithadierentvalueof.WeusedthismethodasanalternativetochecktheaccuracyoftheT2valuesacquiredbyCPMGmethod.Figure 2-16 showsoneofthetsoftheechoheightsobtainedbytheaveragedsingleechomethodtoobtain 51

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T2values.Inaveragedsingleechomethod,asweincreasethetimeintervalwithxedwaitingtimenuclearspinechoheightsdecreasesexponentially. Figure2-16. FitofdataobtainedbytheaveragedsingleechomethodforT2.Theredlineisattosingleexponentialdecay. Themulti-spinechomethodcanreducetheamountofdatatakingtimeandalsominimizethediusioneectsontheechoheightswhichisincurredinthesingle90o)]TJ /F4 11.955 Tf 12.09 0 Td[()]TJ /F1 11.955 Tf -454 -23.9 Td[(180osequencemethod.TheCarr-Purcell-Meiboom-Gillsequencewasusedformulti-echosequences,[90o)]TJ /F4 11.955 Tf 11.95 0 Td[()]TJ /F1 11.955 Tf 11.96 0 Td[((180o)]TJ /F1 11.955 Tf 11.96 0 Td[(2)n]andtheechoheightattimeisgivenby, h()=h(0)exp()]TJ /F1 11.955 Tf 9.29 0 Td[(2=T2)]TJ /F1 11.955 Tf 11.96 0 Td[(22G2D23=3n2)(2{17) 52

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whereGisthemagneticeldgradient,Disthediusionconstant,andnisthenumberof180opulses.Figure 2-17 showstheschematicdrawingforCPMGsequence. Figure2-17. TheschematicdrawingforCarr-Purcell-Meiboom-Gill(CPMG)withfour180opulses.Asbecomeslongerthenthesizeofthenuclearspin-echodecreasesfollowingEqn. 2{17 .(Weusedeight180opulsesbutforsimplicityfourpulsesareshowninthisgure.) Shorteningthetimeintervalbetweenspinechoesreducesthediusion-inducedshorteningofT2andimprovestheresolutionofshortT2components.Increasingthenumberofechoesincreasessignal-to-noise(SNR)andimprovestheresolutionoflongT2components.Weneedtooptimizethenumberofechoesandthetimeintervalbetweenechoestooptimizedatatakingtimeandobtaindatawithgoodresolution.Allpulsesareappliedalongthepositivex-axisandtheinitialpreparation90opulsetipsthenetmagnetizationinthezdirectionM0tobealongthepositivey-axis.Forourmeasurementswemostlyusedeight180opulsesandaveragedfor5daysforverydilutesamples. 2.6SampleGrowthTheblockedcapillarymethodwasusedtogrowasolidcrystalofthe3He-4Hemixtureanditsdensityiskeptconstant 24 { 27 .Thecapillaryisweaklythermallyanchoredinsidethecryostatatthe1Kpot,thestill,theheatexchanger,andthemixingchamber.During 53

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Figure2-18. Schematicrepresentationoftheblockedcapillarymethodforheliumcrystalgrowth.Atroomtemperature,3He-4HegasmixtureispressurizedtolltheNMRsamplecellandasmallexternalreservoirandthegasmixtureiscondensedinthecellat4.2K.Aftersubsequentcoolingthesystem,solidicationstartsatthecoldestpartofthellinglineandcrystallitesgrowfromthellinglineintothecell.Duringsolidicationthepressuredropsbecausetheheliumsamplesolidieswithconstantvolumei.e.constantdensity. llingfromagasmixtureof3He-4Heatroomtemperature,thecapillarylineiskeptatahighertemperature(around4.2K)topreventsolidicationofheliuminsidethecapillarylinebeforethepressureoftheNMRsamplecellreachesthedesirablepressure. 54

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Byincreasingthepressureofthegasmixtureof3He-4HewecancondensethemixtureintotheNMRcellandduringthiscondensationthegasmixtureexperiencesbothcoolingandpressurizinguntilitreachesthevaluesneededforcrystalgrowthinthecell.Afterthepressureofthecellreachesthenalpressurethesystemiscooleddownuntilithitsthemeltingtemperature.Thecapillarybecomesblockedandthepressurefollowsthemeltingcurveasobservedbythedecreaseofthesamplepressuremeasuredbytheinsitupressuregauge,asshowninFigure 2-18 .Themaximumpressureisaround46barforallsamplesbeforecondensingthegasmixture.Withthepressuremaintainedhigherthanthemeltingpressure,theNMRcelliscooleddowntothesolidregionofthephasediagram. Figure2-19. TheschematicofthesolidicationoftheheliumintheNMRcell.Whenthesystemiscooleddownthesolidicationofheliumstartsfromthecapillarylineandtheredarrowshowsthedirectionofthecrystallizationinthecell. Byrunningthedilutionrefrigeratoratfullcapacity,thecapillarylineiscooleddownandisblockedatsomepointwhichisthecoldestpartinthellingline.ThisblockingenablesustomaintainthesolidsampleintheNMRcellandthedirectionofthe 55

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solidicationisfromthecapillaryline(llingline)intothecell,asshownschematicallyinFigure 2-19 .However,the3Hehasamuchlargerzeropintmotionthan4He,andasaresultithasahigherbindinginteractionwithwalls.3Hecanthereforepreferentiallyplateoutonthecapillarywallsratherthaninthecell.Asaresultthedensityof3Heinsidethe4Heusuallychangesbytheorderof10%.BycoolingaliquidsampleintheNMRsamplecellfrom4K,theresultantsolidsamplepressureistypically2730barsandthisis40%lowerthanthestartingpressure. 2.7ThermometryInourexperimentinordertodeterminethecorrecttemperatureweusedseveralthermometers,selectingtheonesappropriatetothetemperaturerange.WhenwecooledthesystembyrunningthedilutionrefrigeratorfromroomtemperaturetocondensethegasmixtureintotheNMRcellattemperature4.2K,weusedaDT-670silicondiodethermometer.Figure 2-20 showsthecalibrateddataforhightemperatureandthephotoofthesilicondiodethermometer.Thecalibrationiscarriedoutusinga3Hemeltingpressurethermometer.Below1.2K,acarbonresistancethermometeristypicallyusedtoreadthetemperatureofthesample.ThecalibrationdataofcarbonresistancethermometerisshowninFigure 2-21 .Mostofourmeasurementsweredoneintherangeof0:01T0:4K,soalmostalltemperaturesarereadfromthecarbonresistancethermometer. 2.7.13HeMeltingPressureThermometryThemeltingcurveof3Heexhibitsapronouncedtemperaturedependenceasreportedinmanypapers 29 { 31 .Thistemperaturedependencecanbeusedasathermometerparticularlyinthetemperaturerangeof1T250mK. 29 { 33 Thetemperatureoftheminimumofthemeltingcurve,thesuperuidtransitionsofliquid3Heandthenuclear 56

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Figure2-20. Calibrationdataofsilicondiodethermometer.InsetphotoistheLakeshoresilicondiodeDT-670series. 28 anti-ferromagneticorderingtransitionofsolid3Heonthemeltingcurveprovidewelldenedxedpoints.Theadvantagesofusinga3Hemeltingcurvethermometerarethehighresolutionandreproducibility.Theweakpointofthisthermometeristhelargespecicheatofliquid3Heinthethermometerandalsotheneedforagashandlingsystemwithacapillaryllingline.Forourexperimentweuseda3Hemeltingthermometertocalibrateourcarbonresistancethermometer.Oursampletemperaturesaredecidedbythevalueofcarbonresistancethermometerintherangeofmeasurementtemperature. 2.7.2ResistanceThermometryResistancethermometryexploitsthetemperaturedependenceofmetalsorsemiconductorsanditisthemostwidelyusedmethodinlowtemperaturemeasurementseventhoughthethermalconductivity,thermalcontactandself-heatingofthedevicemightbeaweak 57

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Figure2-21. Calibrationdataofcarbonresistancethermometer. point.Thecarbonresistoriswidelyusedforlowtemperaturethermometerandthisisnotmanufacturedasspecicthermometer.ThereforeitisverycheapandhasverystableR-Tbehavior. 2.8PressureMeasurementsAStraty-Adamsstraingauge 34 wasusedtomeasurethepressureofthesampleandisplacedinoneendoftheNMRcell.Intheconstructionofthecapacitorthetwomovableplatesareelectricallyinsulatedbyaspacer,suchasKapton,mylarandaglassshimbutthosematerialsshowalargetemperatureandpressuredependenceatverylowtemperaturesandbecauseofthisproperty,thestraingaugecanalsobeusedasathermometer. 35 Forthisreasonthecapacitivestraingaugewithaplasticspacercannotworkasanaccuratepressuregauge. 58

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Inourexperiment,weusedanewself-containedcapacitivepressuretransducerwithoutanyspacerandthedesignofthecompactstraingaugeisshowninFigure 2-22 .Theeddycurrentheatingandthelargeheatcapacityofconventionaldesignsareavoided Figure2-22. SchematicdrawingofthepressuregaugewhichisplacedinsidetheNMRcelltomeasureanaccuratesamplepressure.ThisgureisreproducedwiththepermissionfromRef. 36 byusingcoinsilverforthediaphragmsandhigh-puritytitaniumforthecapacitiveplates.BeforecarryingouttheactualNMRmeasurementswetestedtheNMRcellwiththispressuregaugeinsidethecellandthissystemworkedverywelluptoveryhighpressures(P46bar).ThenalpressureofthesolidsamplesandthemolarvolumeVmofthesamplesweredeterminedbythePVTdataofGrillyandMills 37 andtheequationderivedbyMullin, 38 Vm(p;x3)=x3V3(p)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(x3)V4(p))]TJ /F1 11.955 Tf 11.96 0 Td[(0:4x3(1)]TJ /F4 11.955 Tf 11.96 0 Td[(x3)(2{18)usingthemeasuredpressureofthesamplecell.HereV3andV4arethemolarvolumesofpure3Heandthe4He,respectively. 59

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CHAPTER3MICROSCOPICDYNAMICSOF3HEIMPURITIESINSOLID4HEINTHEPROPOSEDSUPERSOLIDPHASE 3.1OverviewThischapterhasfocusedoninvestigatingthemicroscopicdynamicsof3Heatomsinsolid4HebymeasuringtheNMRrelaxationtimesT1andT2intheregionwheresignicantnon-classicalrotationalinertiafractions(NCRIFs)havebeenreportedforsolid4He.For3Heconcentrationsx3=16ppmand24ppm,changesareobservedforboththespin-latticerelaxationtimeT1andthespin-spinrelaxationtimeT2atthetemperaturescorrespondingtotheonsetofNCRIFsand,atlowertemperatures,tothe3He-4Hephaseseparation.ThemagnitudesofT1andT2attemperaturesabovethephaseseparationagreeroughlywithexistingtheorybasedonthetunnelingof3Heimpuritiesintheelasticstraineldduetoisotopicmismatch.However,adistinctpeakinT1andalesswell-resolvedfeatureinT2areobservednearthereportedNCRIFsonsettemperature,incontrasttothetemperature-independentrelaxationtimespredictedbythetunnelingtheory. 3.2ConceptsThediscoveryofanon-classicalrotationalinertiafraction(NCRIF)insolid4HebyKimandChan 4 5 hasgeneratedenormousinterestbecausetheobservedNCRIFcouldbethesignatureofasupersolidstate. 39 Severalindependentexperiments 40 41 haveshownthattheNCRIFmagnitudeandtemperaturedependencearestronglydependentondefectssuchas3Heimpuritiesandthequalityofthecrystalsintermsofthedensityofdislocations.Furthermorerecentstudiesoftheelasticpropertiesofsolid4HebyBeamishandcolleagues 42 haverevealedasignicantfrequencydependentchangeintheelasticshearmoduluswithanenhanceddissipationpeakhavingatemperaturedependencecomparabletothatobservedfortheNCRIF.Theseresultsfortheshearmodulussuggestthatthedynamicsofthe4Helatticeplaysanimportantroleinthelowtemperaturebulkpropertiesofsolid4Heandratherthanobservingaphasetransitiontoasupersolidstate 60

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onemaybeobservingathermallyexciteddynamicalresponse.Itisthereforeimportanttostudythemicroscopicdynamicsof3Heimpuritiestoprobethedynamicsofthe4HelatticeandunderstandtheirrelationtotheNCRIFandshear-modulusphenomena.ThemeasurementoftheNMRrelaxationtimesofdilute3Heimpuritiesinsolid4Heatlowtemperatureisoneofthemosteectiveexperimentstomeetthisneed.TheNMRrelaxationratesaredeterminedbyquantumtunneling(via3He-4Heatomexchange)andthescatteringofthediusingatomsbythecrystaldeformationeldaroundthe3Heimpuritiesandotherlatticedefects.TheNMRrelaxationratesarethereforeverysensitivetotheelasticpropertiesofthesolid4Heandonanychangesinthecrystalgroundstatethatwouldmodifythetunnelingrate.Althoughlow-temperatureNMRdatafor3Heimpuritiesinsolid4HehavebeenreportedinRef. 43 44 forhigher3Heconcentrationsx3100ppm,weobtainedtherstdirectNMRdataonisolated3Heimpuritiesintheregionofx3andtemperaturesinwhichNCRIFhasclearlybeenobserved.RecentlyTodaetal. 43 44 reportedsimultaneousNCRIFandNMRdataforasamplewithx3=10ppm,buttheNMRsignalwasonlyobservableintheirstudyfrom3Heatomsinphase-separatedclustersandnot3Heinsolutioninthe4Helattice. 3.3ExperimentalDetailAsdiscussedinChapter 2 ,thesampleswerepreparedbymixinghighpuritygasesandcondensingthemixtureathighpressure(46.2bar)intoapolycarbonatecellthatcontainedapressuregauge(Figure 3-1 ),andthensolidifyingthesamplesusingtheblockedcapillarymethod.Thermalcontacttothesamplewasprovidedbyasolidsilvercoldngerextendingfromthedilutionrefrigerator.Toobtainthelowesttemperature10mKthepreamplierthatwasatrstmountedclosetotheNMRcell(thisrstcongurationisshowninFigure 2-9 )wasmovedtothe4Kplatewhichis1.8mfromthesamplecellasshowninFigure 3-1 .ThismovereducedtheheatinputgeneratedbythepreamplierintotheNMRprobe. 61

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Figure3-1. SchematicrepresentationofthelowtemperatureNMRcell.Thepreamplierandtuningcapacitorarelocatedona4Kcoldplatelocatedadistanceof1.8mfromthesamplecell.TheRFtransmittingandreceivingcoilsaresimpliedinthisgure.[FigurereproducedwithpermissionfromKimetal.,Phys.Rev.Lett.106,185303(2011).Copyright(2011)bytheAmericanPhysicalSociety.] Arstsamplewithx3=16ppmwasannealedfor24hoursjustbelowthemeltingpoint,whileasecondsamplewithx3=24ppmwasannealedforonly30minutes.Forbothsamplesthenalpressuremeasuredinsituatlowtemperaturewas27:750:05bar,correspondingtoamolarvolumeVm20:80:1cm3.AsdescribedinSection 2.5 ,standardpulsedNMRtechniqueswereusedtomeasurethenuclearspinrelaxationtimes:magnetizationrecoveryfollowingaspinechotomeasureT1,andaCPMG(Carr-Purcell-Meiboom-Gill 45 )multiple-echosequencetomeasureT2. 62

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ThestudieswerecarriedoutforaLarmorfrequencyof!L=2=2MHzastheexpectedrelaxationtimesextrapolatedfrompreviousstudies 46 { 49 wouldbeprohibitivelylongathigherfrequencies. Figure3-2. Nuclearspinechoamplitudesfordeterminationofthesampleconcentration.Todeterminetheconcentrationofeachsampleweusedacalibrationmethod.A1isthespinechoamplitudeofthecalibrated1000ppm,andA2andA3isthespinechoamplitudesforeachsample.TodeterminetheconcentrationofsampleswecalculatedtheratioA1 A3forthe16ppmsampleandA2 A3forthe24ppmsample. Inordertodeterminethe3HeconcentrationsofsampleswemeasuredtheamplitudeoftheNMRechoesathightemperatures(150T350mK)whichfollowstheCurielawandcomparedthesignaltoastandardreferencesamplewithx3=1000ppmasshowninFigure 3-2 63

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3.4ResultandDiscussion 3.4.1AnomaliesinT1andT2Theobservedtemperaturedependencesofthenuclearspin-latticerelaxationtimeT1forsampleswithx3=16ppmand24ppmarecomparedwiththedependenceforamuchhigherconcentrationsample,x3=500ppminFigure 3-3 .Apronouncedpeak Figure3-3. Temperaturedependenceofthenuclearspin-latticerelaxationtimeT1forsampleswithx3=16ppmand24ppmcomparedtoahighconcentrationsample(x3=500ppm,fromRef. 50 ).ApeakinT1isobservedforbothsamplesatT175mK.Inaddition,forthex3=16ppmsampleT1dropsbyafactorof100below85mKdueto3He-4Hephaseseparation.Solid(open)circlesshowdatatakenonwarming(cooling).ThedashedgreenlinesrepresentT1valuescalculatedfortheimpuritonmodelofLandesman.[FigurereproducedwithpermissionfromKimetal.,Phys.Rev.Lett.106,185303(2011).Copyright(2011)bytheAmericanPhysicalSociety.] atT175mKisobservedforboththe16ppmand24ppmsamplesincontrastwith 64

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theweaktemperaturedependenceobservedforthex3=500ppmsample.Sampleswithhigherconcentrations(500ppmx32000ppm)werealsostudied 50 51 andhadtemperaturedependencessimilartothatofthe500ppmsampleexceptfortheexpectedshiftinthephaseseparationtemperature.ThepositionofthepeakinT1correspondscloselytothesaturationtemperatureT90observedforNCRIFatthisx3 40 whereT90isthetemperatureatwhichthenormalizedNCRIFis90%ofthelowtemperaturelimitingvalue.Byallowingforexpectedshiftswithpressureandx3,theT1peakweobservealsocorrelateswellwiththesheardissipationpeakobservedbySyschenkoetal. 42 Finally,theT1peakoccursatapproximatelythesametemperatureforthisx3valueasthesharpultrasonicabsorptionanomalyreportedbyHoetal. 52 InparallelwiththeobservationsforT1,alesswellresolvedpeak,whichisfollowedbyaminimum,inT2isobservedforx3=16ppmatthesametemperatureforwhichtheT1peakisseen(Figure 3-4 ).AtlowertemperatureslargechangesinT1andT2areobserveddueto3He-4HephaseseparationatTps85mK.ThisvalueofTpsagreeswellwiththepredictionsofEdwardsandBalibar. 53 Thesamefeaturesareobservedforthesamplewith24ppm:astrongpeakinT1atT175mK(Figure 3-3 )andaweakerpeakinT2(Figure 3-4 ).TheT1andT2featuresatT175mKappearunrelatedtothelargechangesinT1andT2associatedwiththephaseseparationatlowertemperatures.Toverifythisfact,the24ppmsamplewaspurposelynevercooledtothephase-separationtemperature.Thethermalhysteresisaround85mKinT1andT2reinforcestheviewthatthechangesatthistemperaturearedueto3He-4Hephaseseparation. 54 TheNMRechoamplitudes(Figure 3-2 )becometemperatureindependentbelowthephaseseparationasexpectedfortheformationofdegenerateliquid3Hedroplets. 51 TheresultsforthephaseseparationtemperaturesareingoodagreementwiththosereportedbyTodaetal.forx3100ppm. 43 Thesampleswithx3=16ppmandx3=24ppmhavesignicantlydierenttemperaturedependencesforT2atT200mK.AsT2(unlikeT1)issensitive 65

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Figure3-4. Temperaturedependenceofthenuclearspin-spinrelaxationtimeT2forsampleswithx3=16ppmand24ppm.Inadditiontoastrong,hystereticdropinT2forx3=16ppmbelow100mKduetophaseseparation,bothsamplesshowpoorly-resolvedfeaturesnearthetemperatureatwhichtheT1peaksareobserved,T175mK.AsT2reectsthespectraldensityof3Hemotionnearzerofrequencyitissensitivetoslowandstatic3HeredistributionsthatdonotaectT1.ThismayalsocontributetothetemperaturedependenceoftheT2valuesobservedatT200mKforthex3=24ppmsample,whichwascooledafterlessannealingthanthex3=16ppmsample.[FigurereproducedwithpermissionfromKimetal.,Phys.Rev.Lett.106,185303(2011).Copyright(2011)bytheAmericanPhysicalSociety.] 66

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toveryslowandstaticchangesinthepositionsofthe3Heimpurities,itseemsthattheT2datacannotclearlyresolvetheeectsleadingtotheT1peakfromothereects.Inparticular,thehigher-temperaturedatainFigure 3-4 suggestthatT2issensitivetothecrystalquality. 3.4.2AnnealingEect Figure3-5. ThenormalizedT1peaksof16ppmand24ppmsample.Thesolidbluelinerepresentsatforthe24ppmsampleandthesolidredlineisthetfor16theppmsample.TheabsolutevalueforT1forthe16ppmsampleismuchlargerthanthatof24ppmsamplebecausethespin-latticerelaxationtimeisproportionaltothex3concentrationasT1x)]TJ /F2 7.97 Tf 6.59 0 Td[(4=33orT1x)]TJ /F2 7.97 Tf 6.59 0 Td[(4=33indilute3He-4Hemixture.Ifthepeaksarenormalizedremovingtheconcentrationdependencethenonlytheannealingeectisseeninthepeaks. 67

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Annealingisawellknownmethodtoimprovethequalityofcrystals.Annealingisperformedbywarmingthesampleclosetothemeltingpoint(1.2K)forsometime,andcoolingdownslowlyafterwardsat27bar.The16ppmsamplewasannealedfor24hourswhilethe24ppmsamplewasannealedfor30min.AsshowninFigure 3-5 thenormalizedT1peakheightof16ppmsampleis10%smallerthanthe24ppmsample.ThebesttusingLorentzianfunctionshowsthatthepeakpositionis173.6mKfor16ppmsampleand178.5mKfor24ppmsample.AnnealingdecreasesthepeakinT1andthisagreeswellwithotherobservationsrelatedtoNCRIandshearmodulusmeasurements.RittnerandReppy 41 havefoundthatannealingtheheliumsamplereducestheperiodshiftoftheirtorsionaloscillatorintheirexperiment.AlthoughPenzevetal.reportedtheobservationofa10%increaseoftheNCRIafterannealing, 55 andKondoetal.foundnoeectofannealing, 56 itisnowgenerallyacceptedthatannealingreducesthedisorderinthecrystalandconsequentlyitalsodecreasesthemagnitudeoftheNCRI.TheespeciallycarefulanalysisofannealingbyRittnerandReppy 57 andbyClark,WestandChan 58 havefurtherconrmedthisviewpoint.AsshownclearlyintheFigure 3-5 ,thisfeaturesagreewiththepeaksinthenuclearspinrelaxationtimes. 3.4.3ApplyingtheNMRTheorytothePeaksinT1andT2Thenuclearspindynamicsof3Heimpuritiesinsolid4Hehavebeenstudiedextensivelyforrelativelyhighconcentrations(x390ppm)andhightemperatures(T350mK). 46 { 49 Theresultshavebeendescribedintermsofmobile3Heimpuritiestunnelingthroughthe4Hematrixby3He-4Heexchange(J34)withamutualscatteringduetotheelasticdeformationeldsurroundingeachimpurity(K(r)=K0r)]TJ /F2 7.97 Tf 6.59 0 Td[(3). 13 59 AreasonablettotheT1andT2valuesobservedforverylowconcentrations(20ppm)athightemperature(T200mK)isobtainedusingtheLandesmanmodel 59 asshownbydashedgreenlinesinFigure 3-3 andFigure 3-4 68

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Landesman 59 showsthatforhighconcentrations(x3(J34=K0)2,where10)]TJ /F2 7.97 Tf 6.58 0 Td[(6(J34=K0)210)]TJ /F2 7.97 Tf 6.59 0 Td[(5),thecorrelationtimeisgivenbyaneectiveexchangefrequencyJeff=J234=K0whichisdeterminedfromtheelasticstraineldKduetolatticedeformationinducedbythedierenceofoccupiedvolumebetween3Heand4He.Landesman'smodelisdiscussedindetailinChapter 4 .ApplyingLandesman'smodelthevalueofT1forrelativelyhightemperatures(T200mK)iscalculatedusingtheparameterscitedbyLandesmaninhispaper.ThisimpuritonmodelleadstoT1=5700sandT2=1:34sforx3=16ppmincloseagreementwiththeobservedvaluesshownbythedashedgreenlinesinFigure 3-3 .ItisimportanttonotethatinthissimplemodelT1_J)]TJ /F2 7.97 Tf 6.59 0 Td[(1effandT2_Jeff.OnewouldthereforeexpectapeakinT1tobeaccompaniedbyadipinT2ifthespectraldensityoftherelaxationtime(G(!))isspeciedbyasinglecharacteristictime.Theexperimentalresultsshowthatthisisnotnecessarilythecase.Thisisnotsurprisingastherearetwocharacteristictimes:thediusiontimeforone3Heatomtodiusetothesiteofanother3Heatom(tDx)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33=J34)andthescatteringtimefortheinteractioneldK(tCj<)777(!rr)778(!K>2j1=2=J234). 59 TheresultsclearlyshowthatanadditionaldynamicaleectcontributestotheNMRrelaxationratesforT175mK.The3Heatomscanalsotunnelasweaklyboundpairs. 60 Thepair-tunnelingmodelexplainstheresonantdipsinT1observed 48 at!L=1:3and2.6MHzsinceT1directlymeasuresuctuationsin3He-4Heseparationsatfrequenciesof!Land2!L.However,noneofthesemodelspredicttemperaturedependentrelaxationtimessotheycannotexplaintheobservedT1peakseventhoughtheycanexplaintheT1andT2valueroughlyinthehighertemperatureregioni.e.beforethepeaksappear. 3.4.4PossibleExplanationforT1andT2AnomaliesTheT1peakoccursatroughlythesametemperatureatwhichtorsionaloscillatorandshearmodulusanomaliesareobserved, 40 42 anditistemptingtoinferaconnectionbetweenallthreephenomena.Onepossibilityisthatthesharpre-entrantpeakinuctuationsnear175mKsignalsaphasetransition,possiblyassociatedwithsupersolidity. 69

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However,otherexplanationsnotinvolvingaphasetransitionmustbeconsidered. Figure3-6. Tanglesofindividualdislocationsinthecrystalofhelium.ThiscrystalpictureisfromatalkgivenbyBeamishatthe25thInternationalConferenceonLowTemperaturePhysics.PhotocourtesyofJohnBeamish. Typicallysolidheliumcrystalscontaindefectssuchasisotopicimpurities(3Heatoms),dislocationsandgrainboundaries.SeveralresearcherssuggestthatatlowtemperatureeachdislocationlinehasnodeswithotherdislocationlinesandthisleadstoathreedimensionalnetworkasshowninFigure 3-6 .Thetypicaldislocationdensityof4Hecrystaldependsonthecrystalgrowthmethod.ThedislocationdensityNdinpoorcrystalstypicallygrownbytheconstantvolumemethodisaboutNd109cm)]TJ /F2 7.97 Tf 6.59 0 Td[(2. 61 62 Thedislocationdensityingoodcrystalsgrownbytheconstantvolumemethod 63 orbytheconstanttemperaturemethod 64 above0:5K 70

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isNd105or107cm)]TJ /F2 7.97 Tf 6.58 0 Td[(2.Thedislocationdensityofthebestcrystalsthataregrownbytheconstanttemperaturemethodbelow0:2KisNd10cm)]TJ /F2 7.97 Tf 6.58 0 Td[(2or102cm)]TJ /F2 7.97 Tf 6.59 0 Td[(2. 65 Thisdislocationnetworkcouldbepinnedbyintersectionnodesofthenetworkand Figure3-7. Simpleschematicforimpuritypinningtothedislocationnetwork. iftheisotopic3Heatomsexistinthecrystalthentheyalsocanpinthedislocationnetwork. 66 { 69 AsshowninFigure 3-7 ,thedistancebetweentwonodesisLN,thelengthofthedislocationbetweentwo3HeimpuritiesisLIPandaisdistancebetweennearestneighbors.Ifthetemperatureisloweredandtheconcentrationof3HeisincreasedthenLIPislessthanLN.Inthiscasetheimpuritypinningisdominant. 70 AsshowninRef. 40 thex3-dependentonsettemperatureforthetorsional-oscillatoranomaliesagreeswellwiththetemperature,TIP(x3),belowwhich3Hepinningofdislocationlinesisexpectedtodominateoverpinningbydislocation-networknodes 71

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TIP=B ln(a x3LN)(3{1)whereB0:5Kisthebindingenergyofa3Heimpuritytoadislocationline,LNistheaveragelengthofdislocationsegmentsbetweennodes,anda=3:710)]TJ /F2 7.97 Tf 6.59 0 Td[(10misthenearest-neighbordistance.InthismodelthereareroughlyLN=abindingsitesfora3Heimpurityonaninter-nodesegmentofadislocationline,andeachbindingsiteisoccupiedwithprobability Poc=x3 x3+e)]TJ /F5 7.97 Tf 6.59 0 Td[(B=T(3{2)Foratotallengthofdislocationlinesperunitvolume0:2=L2N1011m)]TJ /F2 7.97 Tf 6.59 0 Td[(2for4Hecrystalsgrownbytheblocked-capillarymethod, 40 theconcentrationof3Hebindingsitesondislocationsrelativetothenumberof4Heatomsinthesampleisxd10)]TJ /F2 7.97 Tf 6.59 0 Td[(8,muchsmallerthantheconcentrationsof3Heatomsx3presentinourNMRexperiments.ItwouldbesurprisingifsuchasmallconcentrationofbindingsitesxdcouldhaveameasurableeectonT1.However,thereareotherindicationsthatthedensityofdislocationsorotherdefectsmustbemuchlargerthantheestimatequotedabove,iftheNCRIFobservationsareexplainedeitherbysuperowalongdislocationscores 58 orasadirectmechanicaleectunconnectedwithsupersolidity. 71 Figure 3-8 showsthetemperaturedependenceofT1alongwiththeoccupationprobabilityfor3Hebindingsitesandanothermodeldiscussedbelow.Withtheexpected3He-dislocationbindingenergyB0:5KitcanbeseenthatthesignicanttransitioninPoc(T)occursatmuchlowertemperaturesthanthepeakinT1asshowninFigure 3-8 (b).IflargerbindingenergiesB=1:8)]TJ /F1 11.955 Tf 12.21 0 Td[(1:9Kareassumed,thedropinPoc(T)occursattemperaturesclosetothoseoftheNMRanomaly.However,itisnotclearhowthepartialoccupationof3Hebindingsiteswouldleadtoreduceductuationsof3HeinteratomicvectorsasinferredfromthepeakinT1.AnotherphenomenologicalmodelweconsideristoassociatetheT1anomalieswithathermallyactivatedrelaxationpeakashasbeenusedsuccessfullytodescribethe 72

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Figure3-8. (a)Measuredspin-latticerelaxationtimesfortwolow3Heconcentrationsamples(datapoints)alongwithtstoaphenomenologicalmodelbasedonthermally-activatedrelaxationofunknowndegreesoffreedom(smoothcurves).(b)Temperature-dependentprobabilityforadefectsuchasadislocationtobeoccupiedbya3Heimpurity.HereBisthe3Hethermalactivationenergy(a)ordefectbindingenergy(b).[FigurereproducedwithpermissionfromKimetal.,Phys.Rev.Lett.106,185303(2011).Copyright(2011)bytheAmericanPhysicalSociety.] 73

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shear-modulusshifts. 42 ThesmoothcurvesinFigure 3-8 (a)aretstotheform T1=[R0)]TJ /F1 11.955 Tf 11.95 0 Td[(2R1! 1+(!)2])]TJ /F2 7.97 Tf 6.59 0 Td[(1;!=!0eB=T(3{3)HereR0andR1arettingparametersgivingthebackgroundrelaxationrateandheightofthepeakinT1,Bistheactivationenergy,0isanattemptfrequency,and!isthefrequencyatwhichrelaxationisbeingprobed(e.g.!L=1:26107s)]TJ /F2 7.97 Tf 6.59 0 Td[(1).ForthedottedanddashedcurvestheactivationenergywasxedatB=0:5K(theapproximatebindingenergyofa3HeimpuritytoadislocationinferredinRef. 40 whilefortheothercurvesBwasallowedtoincreasetoimprovethettothedata.Notethatthesetsdetermineonlythecombination!0(not!or0separately);forthebestttothex3=16ppmdata!0=2:910)]TJ /F2 7.97 Tf 6.58 0 Td[(5andB=1:81K.ItmustbeemphasizedthatthettingfunctionsusedinFigure 3-8 (a)arephenomenologicalandnotderivedfromasimplemicroscopicmodeloftheNMRrelaxation.Forexample,asimpleresonancebetweenthethermally-activatedmotionof3HeimpuritiesandtheLarmorfrequency!LwouldordinarilyleadtoaminimuminT1(increaseductuationsattheLarmorfrequency),notapeakasweobserve.Ontheotherhand,relaxationof3Heimpuritiesinsolid4Heisknowntobecontrolledbyanon-monotonicspectraldensityofuctuationswithasharpfeatureatthefrequency(3MHz)atwhichnearest-neighbor3Hepairs\walk"throughthelatticeviaquantumtunneling. 60 Thereforeamorecomplexmechanismsuchasthedisruptionofthequantumwalkingmotionof3HepairsbyresonantuctuationsofdislocationlinesmightbeneededtoexplaintheNMRdata. 3.5SummaryTheNMRmeasurementscanonlybeunderstoodintermsofasharpchangeintheuctuationspectrumandthesamechangeswouldalsoplayadominantroleintheanomaliesobservedforthesoundattenuationandtheshearmodulus.Specically,iftheuctuationspectrumisassociatedwithcriticalbehaviorataphasetransition,theattenuationandNMRrelaxationrateswouldvaryasT1jT)]TJ /F4 11.955 Tf 12.21 0 Td[(T0jwith 74

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=1=2inagreementwiththeNMRandsoundattenuationresults.Alternatively,ifacollectivebutnon-criticalchangeoccurredinthelatticedynamics,theassociatedchangeintheelasticpropertiesofthesolidwouldresultinchangesofboththeobservedshearmodulusandtheNMRrelaxationrates.Thelatterhasbeenshowntodependontheelasticstrainsurrounding3Heimpurities. 59 Furtherstudiesforawidefrequencyrangeandforsamplesgrownatconstantpressuretoproducehigherqualitycrystalsareneededtodistinguishbetweentheseinterpretations.However,itisclearthatasuccessfulmicroscopicmodelofthelatticedynamicsofsolid4HemustexplainnotonlytheNCRIFobservations,butalsothecoincidentanomaliesthathavebeenobservedinultrasound, 52 shearmodulus, 42 andNMRrelaxationasreportedhere. 75

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CHAPTER4CONCENTRATIONDEPENDENCEOFT1ANDT2 4.1OverviewTheconcentration(x3)dependenceofthespin-latticerelaxationtime,T1,andspin-spinrelaxationtime,T2,inthelowtemperatureregionwherethetemperatureindependentquantumtunnelingdeterminesthespinrelaxationisdiscussed.Intheintermediateconcentrationregime,impuritiesinteractcontinuouslywiththestraineldofhighconcentrations 59 ormassuctuation-massuctuation(MF-MF)potential 15 andtheseinteractionsstronglyattenuatethetunnelingmotionsofthe3Heimpurities.Thisdampedtunnelingmotionof3Heatomsmodulatesthenuclearspinrelaxationtimes.Intheverydiluteregimethe3Heimpuritiesbehavesasindividualparticlesorindividualmassuctuationwaves(MFWs)thatmovethroughthelatticebytunnelingandarescatteredbythecollisionsbetweeneachother.Thereisacontroversialcrossoverconcentrationabovewhichthemotionof3Heatomsasdeterminedbycollisionsnolongerdominatesthemotionof3Heatomsandinstead3Heatomsmoveundertheinuenceofcontinuousinteractions.Landesmanpredicts 59 thiscrossoverconcentrationismuchhigherthan(J34=K0)210)]TJ /F2 7.97 Tf 6.59 0 Td[(5or(J34=K0)210)]TJ /F2 7.97 Tf 6.58 0 Td[(6whileHuangestimatesitas(J34=3V0)3=410)]TJ /F2 7.97 Tf 6.59 0 Td[(3.HowevertheanalysisofoverawiderangeofconcentrationsdependenceofT1andT2measurementsusingthenewdataprovidedbythisthesisworkconrmthatthecrossoverconcentrationisabout10)]TJ /F2 7.97 Tf 6.59 0 Td[(4andthecorrelationtimesforT1andT2aredierent. 4.2GeneralConceptInquantumcrystalsthezero-pointmotion,resultingfromsmallatomicmassandtheweakattractiveinteractionbetweentheatoms,islargeenoughtoleadtoasignicantoverlapofthewavefunctionsofeachatomandasaresultofthisoverlapatomscanexchangetheirpositionsinthelattice 13 76

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Dilute3Heinsolid4Hemixtureisanidealsystemtotestthisquantummobilityofatoms.Ifasmallamountof3Heimpuritiesareaddedtotheperfect4Hecrystal,the3Hemaytunnelthroughthelatticebyexchangingthepositionwitha4HeneighboratarateofJ34oritmaypairwithanother3Hetoform(3He)2molecules 60 andtunnelwhilebeingassistedbythepairmotionof3He 72 .AndreevandLifshitzrstpointedoutthatitispossibletospeakofabranchofdefectonexcitationswhoseenergy"cantakeonallpossiblevaluesinabandofwidth"whichisproportionaltothedefecttunnelingfrequency,"=z}J,wherezisthenumberofnearestneighbors.Whenthetunnelingfrequency(rate)islargecomparedtothecollisionratebetweenimpuritiesandthephonons,thecoherentquantumtunnelingisdominant.Viceversa,whenthetunnelingrateissmallcomparedtotheimpurityscatteringratebythephonons,thecoherentquantumtunnelinglosesthecoherencyandtheimpuritymotionbecomesarandomwalkthroughthelattice 46 .Coherenttunnelingoccursatlowtemperaturesandincoherenttunnelingdominatesathighertemperaturesbecauseoftheincreasingnumberofthermalphonons.Howeverataxedtemperaturewherethetemperatureindependentquantumtunnelingmotionisdominant,therearetwo3Heconcentrationregionsintheimpuritonmodel.Theintermediateconcentrationregioniswherethe3Heimpuritiescancontinuouslyinteractwitheachotherthroughthestraineldresultingfromthedistortionofthelatticeduetothedierenceofvolumeoccupiedby3He,4He,andvacancies.Thestraineldaectsthemotionofimpuritiesandcontrolsthetunnelingrateof3Heimpurities.Analternativeexplanationforthemotionof3Heinthisregionconsiderstheinteractionbetweenthewelldenedexcitations:massuctuationwave(MFW).InthismodeloneformulatestheinteractionbetweentheMFWsintheaveragedcrystalinsteadoftherealcrystal.Anotherregionistheverydiluteregionwherethemeanseparationof3Heimpuritiesareoutsidetheeectiverangeofanystraineldormassuctuationinteractionpotential 77

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formostofthetime.Thereforetheymaybeconsideredasadilutegas,each3Hetravelingfreely(exchangingcoherently)overseverallatticespacingsuntilencounteringanother3Heimpurityorotherdefectssuchasvacancies,whereitisscattered.Severaltheorists 13 15 calculatedtheconcentrationlimitas10)]TJ /F2 7.97 Tf 6.59 0 Td[(3abovewhichtheimpuritonmodelisinvalid.Asthe3Heimpurityconcentrationx3increases,the3Heatom'smotionmustchangefrompropagatingasindividualballisticparticlesbetweencollisionstooneofcontinuousinteractioninthestrongeldenvironment.Theprecisevalueofthecrossoverconcentrationisveryimportantforunderstandingthepropertyofthequantummotionof3Heimpurities.Howeverthecrossoverconcentrationof3Heforthetransitionfromtheverydilutegas-likeregiontothestronginteractionregionhasnotbeenclearlydemonstratedbytheexperimentaldatapriortothestudiesreportedhere.Inthissection,theconcentrationdependenceofT1andT2willbediscussedbasedontheimpuritonmodelandthecrossover3Heconcentrationwillbeestimatedfromtheexperimentaldata. 4.3ConcentrationDependenceofT1Fig. 4-1 showsallT1dataandthedatareportedbyothergroup 49 54 73 { 76 withtheconcentrationrangingfrom10)]TJ /F2 7.97 Tf 6.59 0 Td[(5to10)]TJ /F2 7.97 Tf 6.59 0 Td[(2,andspecicvaluesareshowninTable 4-1 .Forintermediateconcentrationregion,Eqn. 1{34 isusedwiththevaluesofM2andJeff,thatLandesmanobtainedforhiselasticinteractionmodelgivenbelow M2=5:2108s)]TJ /F2 7.97 Tf 6.58 0 Td[(1(4{1) Jeff=2=1:8kHz(4{2)Eqn. 1{36 givesthecorrelationvaluesasc=3:810)]TJ /F2 7.97 Tf 6.59 0 Td[(6x)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33sandfromthecalculationusingEqn. 1{34 andallnumbersintroducedbyLandesman,thespin-latticerelaxationtime(T1)iscalculatedasT1=0:8610)]TJ /F2 7.97 Tf 6.58 0 Td[(3x)]TJ /F2 7.97 Tf 6.58 0 Td[(2=33s.ThegreenlineinFigure 4-1 showsthisvaluescalculatedbyusingLandesman'smodel. 78

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Figure4-1. Theobservedconcentrationdependenceofthenuclearspin-latticerelaxationtimesfordilute3Heinsolid4He.Theredlinerepresentsthebesttusingthecollisionmodelinthediluteregion.ThegreenlineisthetobtainedbyapplyingtheLandesman'smodelintheintermediateregion.ThebesttusingLandesman'snumbersgivesthecorrelationtimefortheintermediateregionc=3:810)]TJ /F2 7.97 Tf 6.58 0 Td[(6x)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33s.Thecorrelationtimeforthediluteregionisobtainedasch=1:410)]TJ /F2 7.97 Tf 6.58 0 Td[(8x)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33s.(Experimentaldata:Kimetal., 73 Schratteretal., 49 }Schusteretal., 74 4 Allenetal., 75 5 Hirayoshietal., 76 Greenbergetal. 54 ) Whentheimpurityconcentrationx3decreasesthiscontinualinteractingsystembecomesinvalidbecausethe3He-3Hecollisionsarenoteective,andtheonlycharacteristic 79

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timeisgivenby3He-4Hehoppingbetweensites.Thecharacteristictimeisgivenby chsep g=1 x1=3zJ(4{3)wherethegroupvelocityisg=azJ34andtheseparationdistancebetweentwo3Heatomsissep=a=x1=3.zisthenumberofthenearestneighborsandinhcpstructurez=12.Thenearest-neighbordistanceisa=3:710)]TJ /F2 7.97 Tf 6.59 0 Td[(10m.If3HeatomsoccupyabandofwidthJ34 ~thenthemaximum(mid-band)impuritonvelocityisgazJ34=2:410)]TJ /F2 7.97 Tf 6.59 0 Td[(3(m/s).Landesman'svaluesyield ch=1:410)]TJ /F2 7.97 Tf 6.59 0 Td[(8x)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33s.(4{4)FromEqn. 1{34 ,Eqn. 1{36 andEqn. 4{4 theT1canbegivenindiluteregion, T1=3 4!2L M2)]TJ /F2 7.97 Tf 6.58 0 Td[(1c=3 4!2L M2)]TJ /F2 7.97 Tf 6.59 0 Td[(1ch=2:810)]TJ /F2 7.97 Tf 6.59 0 Td[(3x)]TJ /F2 7.97 Tf 6.59 0 Td[(4=3s.(4{5)ThesevaluesareshowninFigure 4-1 asasolidredlinewhichwellmatcheswiththedata.Thesimplestmodelforscatteringduetotheelasticinteractionwouldbethat3HeatomspropagateballisticallyuntiltheyarewithinadistanceRJKofeachother,whereRJKisthedistancerbetweentwo3Heatomsatwhichtheelasticinteractionenergy~K(a=r)3equalsthetunnelingenergy~J34.RJK=(K=J34)1=3a=11a=4:010)]TJ /F2 7.97 Tf 6.58 0 Td[(9(m)=4(nm).Thisgivesascatteringcross-sectionR2ij,ameanfreepatha3x3andadiusioncoecientDdilg, Ddil=ga3 R2ijx3=J34a2(J34=K)2=3 x3=710)]TJ /F2 7.97 Tf 6.59 0 Td[(15 x3(m2=s)(4{6)ComparingEqn. 1{30 totheEqn. 4{6 andassumingwhicheverdiusioncoecientislowestdominates,therewillbeaverymildcrossoverfromDdilx)]TJ /F2 7.97 Tf 6.59 0 Td[(13forx3J34=K.Landesmanhadestimatedthatthecrossoverconcentrationabovewhichthecontinuouslystrongelasticeldinteractionisdominantisabout10)]TJ /F2 7.97 Tf 6.59 0 Td[(5or10)]TJ /F2 7.97 Tf 6.58 0 Td[(6.However,theadditionallowerconcentrationdatalabeledas\Kimetal."intheFigure 4-1 ,shows 80

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thatthecrossoverconcentrationisabout100ppm(10)]TJ /F2 7.97 Tf 6.59 0 Td[(4)whichisalittlehigherthanthatofLandesman'sprediction. Table4-1. T1andT2fromothergroups.J34,M2andJeffarecalculatedusingtheLandesman'sformula.J34=p KJe,whereK=21200Hz 59 .M2=4 3T1T2x23 !2L,Je=r 4 3T1 T2(23:3)2 !2Lx2=33.~Allvalusesarenormalizedto2MHzandtheredvaluesareshowingthenormalizedvalues. x3(ppm)f=!L 2(MHz)T1(s)T2(ms)Vm(cm3/mol)J34(MHz)M2(sec)]TJ /F16 4.981 Tf 5.4 0 Td[(2)Jeff(sec)]TJ /F16 4.981 Tf 5.4 0 Td[(1) 16 73 25873124720.80.67.94E+0948.3624 73 23212229820.80.95.28E+09101.6100 49 2620340020.951.852.37E+09452.8100 49 2450190020.71.733.72E+09397.3242 49 2259123020.952.062.52E+09565.7242 49 2180200020.72.552.37E+09865.3499 49 211842020.952.173.10E+09623.4499 49 2120130020.72.861.75E+091087500 50 217052320.82.092.31E+09580500 75 2110150020.953.031.69E+091221790 75 275.621020.952.203.46E+09641.8790 49 29835020.74.167.43E+082301790 49 28020020.952.142.85E+09608.9900 50 291.819520.82.103.72E+09586.61190 49 257.510520.952.122.06E+09596.51870 50 255.514420.82.495.73E+09825.62000 50 2293120.82.022.35E+09542.72500 49 242.88020.952.411.52E+097735000 49 235.657.420.952.611.17E+09904.57500 49 23149.620.952.791.05E+09103110100 49 227.138.520.952.841.05E+09107320000 76 3( 2 )62( 27.6 )1620.892.308.19E+0870428000 75 -504020.8--4.4ConcentrationDependenceofT2Figure 4-2 showstheT2datawithrespecttothe3Heconcentration.AlldatapointsareselectedfromthesamegroupasT1shownintheFigure 4-1 .Thegreenlineresultsfromthettotheelasticstraineldmodel(Eqn. 1{35 )ofLandesmanusingthesamecorrelationtimeusedforT1,c=3:810)]TJ /F2 7.97 Tf 6.59 0 Td[(6x)]TJ /F2 7.97 Tf 6.59 0 Td[(1=33s.Thistisverypoor 81

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fortheT2dataandthisisnotsunrisingbecauseLandesmanalreadyhadmentionedinhispaper 59 thathismodelisnotconsistentwithT2data.Huangdidadetailed Figure4-2. Comparisonoftheconcentrationdependenceofthenuclearspin-spinrelaxationtimesreportedintheliteraturefordilute3Heinsolid4He.ThegreenlineshowsLandesman'selasticeldinteractionmodelusinghisvalues.ThetthatworkedforT1verywelldoesnotworkforT2.TheredlinesarefromHuang'smodel 15 intheintermediateconcentrationregion.Thedottedredlineshowsx3independentT2inducedfromtheindividualballisticcollisionmodel.(Experimentaldata:Kimetal., 73 Schratteretal., 49 }Schusteretal., 74 4 Allenetal., 75 5 Hirayoshietal. 76 ) calculationforthediusionconstantandthespin-spinrelaxationtimeT2applyingmass-uctuation-mass-uctuationinteractiontotheverylow3Heconcentrationmixture. 82

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Inhismassuctuationwave(MFW)scatteringmodel,theinteractionbetweentwoMFWsprohibitsthe3Hetunnelinginside4Heanddisturbsthecoherentmotionofthe3Heimpurities.TheinteractionbetweentwoMFWsissimilartotheelasticstraineldinLandesman'smodel.InhispaperHuangderivedthespin-spinrelaxationtimeT2atzeroLarmorfrequencyas T2(0)=W2(x3)RR0 M2(1)x)]TJ /F2 7.97 Tf 6.58 0 Td[(13(4{7)whereM2(1)isthesecondmomentforx3=1andW2(x3)RR0istherateoftransitionofa3HeatRtothesiteR00byinterchangewith4Hewhileaspectator3HeisatR0.HuangcalculatedM2(1)3:6!d(RR0)2,where!d(RR0)isthefrequencyoftheparticle(MWF)motionduetothedipolareld.Fortherigidlattice,Huangused!d(RR0)==jR)]TJ /F4 11.955 Tf 13.02 0 Td[(R0j35103rad/s,where=104rad/sG,=10)]TJ /F2 7.97 Tf 6.59 0 Td[(23erg/G,andjR)]TJ /F4 11.955 Tf 11.95 0 Td[(R0j=3:51010m.WithallthesevalueshesimpliedEqn. 4{7 asgivenbelow, T2(0)W2(x3)RR0 3:6!d(RR0)x3W2(x3)RR0 9107x)]TJ /F2 7.97 Tf 6.59 0 Td[(13(4{8)HuangobtainedJ34=V0=2:510)]TJ /F2 7.97 Tf 6.58 0 Td[(3byttingthediusiondataandcalculatedW2(x3)RR0=3:8104.InsertingthisvalueintoEqn. 4{8 resultsinT2=4:2210)]TJ /F2 7.97 Tf 6.59 0 Td[(4x)]TJ /F2 7.97 Tf 6.58 0 Td[(13andthisfunctionispresentedwithblacksolidlineinFigure 4-2 .TheredlineT2=1:6910)]TJ /F2 7.97 Tf 6.58 0 Td[(4x)]TJ /F2 7.97 Tf 6.58 0 Td[(13forJ34=V0=110)]TJ /F2 7.97 Tf 6.59 0 Td[(3whichisusedfortheT1datainLandesman'smodel.ThebluelinestandsforT2=1:910)]TJ /F2 7.97 Tf 6.59 0 Td[(4x)]TJ /F2 7.97 Tf 6.59 0 Td[(13resultedfromthealgebraicexpressionforW2(x3)RR0ofHuang'smodel.Huangsuggestedthattheconcentrationrangeforincoherentmotioninducedbythismass-uctuationmass-uctuationinteractionis10)]TJ /F2 7.97 Tf 6.59 0 Td[(3x310)]TJ /F2 7.97 Tf 6.59 0 Td[(2.Below10)]TJ /F2 7.97 Tf 6.58 0 Td[(3,theverydiluteregioncalledtheMFWregion,the3HeatomsareveryfarfromeachotherandHuangassumedthe3Heatomsthenpropagatewiththedispersionrelationandwhenthetwo3Heatomsareclosertoeachotherthanthecriticaldistancerctheycannotpropagateanymorebythequantumtunneling.Becausetheenergydierenceinducedby 83

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thisscatteringismuchlargerthanJ34.ThisscatteringdisturbsthetunnelingmotionandinthiswayMFW-MFWscatteringmostlydeterminestherelaxationtimesof3Heatoms.HuangcalculatedthemeanfreepathforMFW-MFWscatteringas mfp=1 x3(J34 3V0)1=2(4{9)whereisthedistancebetweenthe3HeatomsandV0istheMFW-MFWinteractionconstantandJ34istheexchangerateofthepairof3He-4Heinamixture.Thevelocityofthe3Heparticleisgivenby v=J34(4{10)Thecharacteristictimethatthendeterminesthespinrelaxationtimeof3HeintheMFWregionis =mfp v(4{11)InsertingEqn. 4{9 andEqn. 4{10 intoEqn. 4{11 givesthecharacteristictimeas =1 x3(3V0J34)1=2(4{12)IfoneusesHuang'svalueforJ34=3V010)]TJ /F2 7.97 Tf 6.59 0 Td[(3anduseJ34=6106,thenoneobtainsV0=2109andEqn. 4{12 becomes =1=x3 3:14(321096106)1=2=0:1610)]TJ /F2 7.97 Tf 6.58 0 Td[(81 x3(s)(4{13)Inourcase,theLarmorfrequencyis!L=2MHzand1=!L<.Inthiscasethespin-spinrelaxationtimeT2isgivenby T2=1 x3M2(4{14)IfoneputsEqn. 4{13 andEqn. 4{1 intoEqn. 4{14 thenthespinspinrelaxationtimeintheMFWregionis T2=1 x3M2=x3 x35:21080:1610)]TJ /F2 7.97 Tf 6.58 0 Td[(8=1:2(s)(4{15) 84

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InFigure 4-2 thedottedredlineshowsthisT2intheMFWregion(InFigure 4-2 itislabeledas\dil"tobeconsistentwiththeT1analysis.)ThevalueofT2fromtheHuang'sMFWmodelagreesquitewellwithourdataintheverydiluteregion.UnfortunatelyHuangdidnotworkontheT1becauseofthecomplexityofinteractionamongfrequencies:J34,J33andW2(x3)RR0.HuangsuggestedthatthecrossoverconcentrationfromMFWregion(inourcaseverydiluteregion)tomass-uctuationmass-uctuationinteractionregion(inourcaseitiscalled\intermediateregion")isaboutx310)]TJ /F2 7.97 Tf 6.58 0 Td[(3.Howeverfromouranalysisusingadditionalnewdataweobtainedintheverydiluteregionitisclearthatthecrossoverconcentrationisaboutx3<10)]TJ /F2 7.97 Tf 6.58 0 Td[(4andthisisinagoodagreementwiththeanalysisforT1insection 4.3 4.5Re-examinationofLandesman'sModelFig. 4-3 showstheproductofT1andT2foreachconcentrationofx3forseveralindependentexperiments.MultiplicationofEqn. 1{34 withEqn. 1{35 gives 4 3T1T2x23(M2 !L)2=1(4{16)T1andT2arefromFigure 4-1 andFigure 4-2 andinsertingEqn. 4{1 andusing2MHzfortheLarmorfrequencyonecancalculate4 3T1T2x23(M2 !L)2foreachconcentration,andfromLandesman'smodelthisshouldbeunityifT1andT2havethesamecorrelationtime.HoweverasshowninFigure 4-3 ,thisisnotunityfortheconcentrationx310)]TJ /F2 7.97 Tf 6.59 0 Td[(2. 4.6SummaryThenewdataobtainedfromthisthesisworkmakeitpossibletoinvestigatefurtherthetwotheoreticalmodels 15 59 thathavebeenusedtoexplaintheconcentrationdependenceofthenuclearspinrelaxationsinverydilute3Heinsolid4He.Amorereliabledeterminationofthecross-overconcentrationbetweencoherentandincoherentmotionof3Heimpuritiesinthe4Helatticewasmade.Eventhoughthesetwomodelsusesimilarconceptsthecross-overconcentrationsaredierent.Landesmanfoundtheupperlimit 85

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Figure4-3. VariationofthefunctionF(x3)=(4 3)T1T2x23(M2=!L)2asafunctionof3Heconcentration.ForauniquecorrelationtimeF(x3)=1:0.(Experimentaldata:Kimetal., 73 Schratteretal., 49 }Schusteretal., 74 4 Allenetal., 75 5 Hirayoshietal. 76 ) concentrationforthecoherent3Hemotiontobex3510)]TJ /F2 7.97 Tf 6.59 0 Td[(5whileHuangestimatesthecross-overoccurringatx310)]TJ /F2 7.97 Tf 6.59 0 Td[(3.Thedatasetenhancedbytheexperimentsreportedhereconrmsthatthecross-overconcentrationisx310)]TJ /F2 7.97 Tf 6.58 0 Td[(4andthisvalueisconsistentwiththeanalysisforbothT1andT2.HuangdeducedthisvaluefromthediusionconstantandT2intheverydiluteregion(MFWregion,borrowinghistermusedinhisliterature)buttherewerenoexperimental 86

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dataforhimtoapplyhismodelatverydilutevaluesofx3.Thenewdatafromthisthesisworkareabletocheckhismodelforincoherentmotionandhismodelagreeswellwiththedata. 87

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CHAPTER5PHASESEPARATIONOFVERYDILUTE3HE-4HEMIXTURE 5.1OverviewThetemperaturedependenceofthenuclearspin-latticerelaxationtimes(T1)andnuclearspin-spinrelaxationtimes(T2)aremeasuredforawiderangeof3Heconcentrationsindilutemixturesof3Heinsolid4He.DramaticchangesinT1andT2occurwhenphaseseparationoccursbecausetheseparatedphaseconsistsofessentiallypure3Heandpure4He.Theseisotopicphaseseparationsareobservedforallsampleswith3Heconcentrationsintherange,500x32000ppm,foramolarvolumeVm=20:7cm3.ThetemperaturedependenceoftheamplitudesoftheNMRsignalsafterphaseseparationconrmsthatthephase-separated3HeatomsformFermi-liquiddropletsbelowthephaseseparationtemperaturesandsuggeststhattheinterfacesbetween3Hedropletsand4Heatomsareresponsibleforthenuclearspinrelaxation.Thespin-spinrelaxationtimes(T2)afterphaseseparationincreasedabruptlyandbecametemperatureindependent.Thisfeatureissimilartothatobservedforthehigherx3concentrationsamplereportedbefore. 54 5.2BackgroundStudiesofsolidandliquid3Heclustersandlmshaverevealedaricharrayofnewphenomenaforboththemagneticpropertiesandthedynamicsatlowtemperatures,butacompleteunderstandingoftheinterfaceinteractionsandkineticshasbeenelusive.ThepropertiesofthesesystemsareverydierentfrombulkpropertiesatverylowtemperaturesbecausetheDeBrogliewavelengthbecomeslargerthantherangeofatomicinteractionsandthisresultsininterestingnewquantumproperties.AsnotedbyCollinsetal. 77 earlyNMRstudiesofthelowtemperaturepropertiesofseveralsystemsledtoawidebeliefthata2to3atomlayerofa2D-likesolidformsattheinterfacebetweentheliquid3Heandtheconningsurface. 78 79 Howeverrecentstudiesofphaseseparated3He-4HesolidsolutionsbyKingsleyetal. 80 didnotobservethetemperaturedependenceexpectedin 88

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thiscasebutinsteadatemperatureindependentrelaxationatthelowesttemperatures.ThisobservationisalsoconsistentwiththeresultsreportedbyHebraletal. 81 andbyMikhinetal. 82 Inaddition,theobserveddependenceoftherelaxationonmagneticeldvariesconsiderablywithexperimentalconditionsfromnegligibledependencetolineardependenceforconstrainedgeometriesincontrasttotheclassicalquadraticdependenceathighmagneticelds 83 forbulksystemsandthisvariationneedstobeunderstood.InordertoresolvethesequestionswehaveperformedNMRstudiesoftherelaxationofthenuclearspinsin3Heclustersordropletsformedbyphaseseparationforlow3Heconcentrationsfromcarefullyannealedbulksamplesof3He-4Hesolidsolutions.Thenuclearspin-lattice(T1)relaxationtimesweremeasuredforeachconcentrationstudiedoverawidetemperaturerange(15T300mK).ThevaluesexpectedforT1dependontherelativestrengthsoftheuctuationspectrumofthe3HemotionattheLarmorfrequency.TheresultsofthesestudiesarealsoimportantfordevelopinganunderstandingoftheNMRpropertiesofverydilute3He(x330ppm)intheregionwheresupersolidphenomenaareobserved. 5 5.3ExperimentalDetailForthesestudiesofthephaseseparationitismostimportanttohavereliablevaluesofthe3Heconcentrations.Areferencegasmixturewaspreparedwitha1000ppmof3Heconcentrationbymixingknownvolumesofhighpurity3Heand4Hegas.TheNMRechoamplitudesofastandardRFpulsesequenceforthisreferencemixtureweremeasuredatagiventemperaturewhichiswellabovethephaseseparationtemperature.ThismeasurementprovidedareferencepointforcomparisonwithlowerconcentrationsamplesusingthesamemethodasdescribedinChapter 3 andshowninFigure 3-2 tocalculatethecorrectconcentrationforeachsample.Lowerconcentrationsampleswerepreparedbydilutingthereferencegassamplewithultrapure4He.Becausesome3Heatomscanbelostbypreferentialcondensationonthewallsofthelongcapillarylines,NMRamplitudes 89

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ataxedtemperatureof300mKwereusedtodeterminethetrueconcentrationofthesamples,asdiscussedinSection 3.3 .Thesolidsampleswereformedusingtheblockedcapillarymethod,andallhadpressuresatlowtemperatureof29:00:03barasmeasuredbyanin-situpressuregauge.Atthispressureatlowtemperaturesbulk4Heissolidandbulk3Heisliquid.Thusthesimplestmodelofthephase-separatedsystemwouldbedropletsofliquid3Hedispersedinamatrixofsolid4HeasshowninFigure 5-1 .Asdescribedearlierthecapillarywas Figure5-1. Schematicofphaseseparated3He-4Hemixture.Underthepressureandtemperaturewhere3Heisliquidand4Heissolid,themixturephaseseparatesintothe3Herichphase(liquiddropletof3Heatoms)and4Herichphase(4Helattice). heatsunkatthe1Kpotofadilutionrefrigeratorbutotherwisethermallyisolated.Thesampleswereannealedfor24hoursjustbelowthemeltingcurvebeforecoolingtolowtemperaturesfordatataking.Thedetailsoftheexperimentalcellhavebeenreportedelsewhere. 50 StandardNMRpulsetechniqueswereusedtomeasurethenuclearspin-lattice(T1)relaxationtimesusing90x-180xRFpulsesat2MHz.AsdiscussedinChapter 2 the 90

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relativelylowfrequencywasusedtokeeptherelaxationtimesmanageableatthelowesttemperaturesandstillprovideadequatesignal-to-noiseratios.Inordertoenhancethesignal-to-noiseratios,weemployedalowtemperaturepreamplierwithapseudomorphiceldeecttransistor(refertoSection 2.4 )thatcouldbeoperatedinamagneticeldanddownto250mK. 84 5.4ExperimentalResultThevariationoftheamplitudeoftheNMRechoesforaxedRFpulsesequenceareshowninFigure 5-2 .Aclearbreakintheparamagnetic(T)]TJ /F2 7.97 Tf 6.58 0 Td[(1)dependenceisobservedforeachconcentration.ForhightemperaturestheechoamplitudesfollowCurie'slawc=C=Tandforlowtemperaturetheyfollow,CF=(3=2)C=TF,whichisconstantbelowtheFermitemperatureTFasshowninFigure 5-2 .AtemperatureindependentbehaviorisobservedbelowthisabruptchangeconsistentwiththeformationofdegenerateFermidroplets.Thisbreakintemperaturedependenceistakenasthephaseseparationtemperature.Anabruptchangeisalsoobservedforthetemperaturedependenceofthespin-latticerelaxationtimesasshowninFigure 5-3 .Itisimportanttonotethatwhile3Heatomsareexpectedtobecomeboundtodislocationsandothercrystaldefectsinthesolid4Hematrix, 40 58 69 85 86 thetotalnumberofdislocationsissuchthatlessthan1%ofthe3Heatomsatthelowestconcentrationsstudiedareexpectedtobeintheseboundstates. 87 TheNMRamplitudesthereforearenotexpectedtobeaectedbyanychangesinthenumberofbound3Heatoms.Theeectonrelaxationtimes,however,cannotbeimmediatelydiscountediftheboundsitesareeectiverelaxationcenters.Forthedislocationstobeeectiveforrelaxationtheboundatomswouldneedtohavehighmobilitiescomparedtothemotionatthewallofthedroplets,whichprovidesacompetingrelaxationchannel.ThemostsignicantfeatureofthetemperaturedependenceshowninFigure 5-3 isthatforallsamplestheT1valuesbecometemperatureindependentatthelowesttemperatures.Theorderofmagnitudeoftheratesarecomparabletothosereportedby 91

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Figure5-2. ObservedtemperaturedependenceoftheNMRechoamplitudefor3Heconcentrations500x32000ppminsolid4He.TheparamagneticbehaviorshownbythebrokenlinechangesabruptlytoatemperatureindependentbehavioratthephaseseparationtemperaturewiththeformationofFermi-liquiddroplets.[FigurereproducedwithpermissionfromJ.LowTemp.Phys.162,167,(2010).Copyright(2010)bytheJournalofLowTemperaturePhysics.] Kingsleyetal. 80 Thisbehaviorsupportstheviewthattheatomsatthewallofthedropletaredegenerate.TheechoamplitudesthenbecometemperatureindependentbecauseoftheconstantFermi-liquidsusceptibility.Forthelimitinglowtemperaturevaluesthereisasmallbutconsistentconcentrationdependence. 92

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Figure5-3. Temperaturedependenceofthenuclearspin-latticerelaxationtimeatlowtemperaturesfor3Heconcentration500x32000ppm,showingacharacteristictemperatureindependencefortherelaxationfortheFermidropletsatlowtemperatures.[FigurereproducedwithpermissionfromJ.LowTemp.Phys.162,167,(2010).Copyright(2010)bytheJournalofLowTemperaturePhysics.] 5.5DiscussionTherelaxationratescanbeinterpretedintermsofathree-bathmodel 77 88 inwhichtherelaxationconsistsofathreestageprocess:afastdiusionofmagnetizationthroughthecoreofthe3Henano-clustertotheboundarywiththe4Helatticewhereaslowerspincouplingoccursbetweenthebulkliquidatomsandtheatomsattheboundaryfollowed 93

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byastrongercoupingoftheboundaryatomstothephononsinthesolid4Hematrix.InthisscenarioGuyer,RichardsonandZaneshowedthattheoverallrelaxationtimeT1=(1+Mcore=Mwall)XwhereXrepresentstheintrinsic3Hecore-3Hewallrelaxationtimeatthewall.McoreandMwalldesignatethetotalmagnetizationsofthecoreandwallatoms,respectively.NotethatthisexpressiondiersfromtheformalexpressionsofGuyer,RichardsonandZane 13 whogiveT1=(1+kcore=kwall)Xwherekcoreandkwallaretherespectiveheatcapacitiesoftheatomsinthecoreandthebulk,respectively.TheworkofChapellierandcoworkers, 74 studyingrelaxationof3Heatomsincontactwithuorocarbonspheres,showedconvincinglythatitisthemagnetizationsthatmustbeconsiderediftheinteractionbetweenbathsisduetonuclearspin-spincouplings.(Theuseofheatcapacitiesisappropriateforcouplingstootherdegreesoffreedomsuchasthemotionofvacancies.)Ifthe3Heatomsatthedropletwallwerenon-degenerateasinbulksolid3He,T1wouldincreasewithdecreasingtemperaturecontrarytowhatisobserved.Conversely,ifthewallatomsaredegeneratetherelaxationtimesaretemperatureindependentandgivenby T1=(ncore=nwall)X=1 3NR0 a0X(5{1)wherencoreandnwallarethenumberofatomsinthecoreandbulk,respectively.Nisthenumberofatomiclayerinthequasi-2Dlocalizedwalllayerthatplayanactiveroleintherelaxation,R0istheradiusofthedropletanda0istheatomicseparationinthebulk.NMRstudiesofthespindiusionineldgradients 88 giveR03m.Forrapid3HeexchangemotionatthewalllayerJ3>!L(theLarmorfrequency),X=J3=M2whereM2istheeectivesecondmoment5.0108s)]TJ /F2 7.97 Tf 6.59 0 Td[(2.ForN1andJ33.0107s)]TJ /F2 7.97 Tf 6.58 0 Td[(1(thebulk3Heexchangetime),weestimateT117020sinreasonableagreementwiththelowtemperaturelimitofFigure 5-3 .Intable 5-1 wecomparethisestimatedlimitforT1withthosereportedintheliteratureforsimilarexperimentsbutathigherconcentrations.ThedataisconsistentexceptforthehighconcentrationresultsofGreenbergetal. 54 Thedata 94

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inFigure 5-3 indicatesaweakconcentrationdependenceandthissuggestsadependenceofthedropletsizeoninitialconcentration. Table5-1. ComparisonofourlowtemperaturelimitforT1withvaluesreportedbyothergroups.TmaxisthemaximumtemperaturewheretemperatureindependenceofT1isobserved.(*Figure 5-3 ofRef. 79 reanalyzedintermsoftwodecayrates,**Fig.4ofRef. 80 ***Fig.10ofRef. 54 ) Groupsx3(%)Tmax(mK)T1(second) Thiswork 51 0.2100170Shvarts 79 11.08228Kingsley 80 133240Greensberg 54 218040 ForhighmagneticeldsaquadraticelddependencewouldbeexpectedasthespectraldensitiesforrelaxationaregivenbyG(!)=J)]TJ /F2 7.97 Tf 6.59 0 Td[(13=(1+(!L=J3)2)andleadtoX=!L2=(M2J3)for!L>J3.Thisclassicalbehaviorisobservedforbulksystems. 83 Forhighlyheterogeneoussystemswheretheinterfacelayersareessentiallyamorphousalinearelddependenceisobserved. 74 Thisisattributedtotheexistenceofabroadspectrumoflocalexchangefrequencies.AsaresulttherelaxationisdominatedbyfavoredsiteswheretheexchangefrequencymatchestheLarmorfrequency(correspondingtoaT1minimum)andthisleadstoalinearelddependence. 89 90 Figure 5-4 showsthetemperaturedependenceofthespin-spinrelaxationtimes(T2)foreach3Heconcentration.Thespin-spinrelaxationtimesshoweddramaticchangesasthetemperaturewasloweredthroughthephaseseparationtemperature.TheT2sinFigure 5-4 showthecoolingdatafrom300mKto15mKforsampleswith3Heconcentrationsintherange16x32000ppm.TheT2sforallsamplesincreasebyafactorof10ormore.Thevariationofthephaseseparationtemperatureswithinitial3HeconcentrationsisgiveninFigure 5-5 whereweshowthephaseseparationtemperaturesinferredfromtheamplitudechangesandthechangesinthetemperaturedependenceofthespinrelaxation 95

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Figure5-4. Temperaturedependenceofthenuclearspin-spinrelaxationtimes(T2)atlowtemperaturesfor3Heconcentration500x32000ppm.AfterphaseseparationT2increaseddramaticallyforallsamples. times.TheresultsareinreasonablygoodagreementwiththetheoryofEdwardsandBalibar 53 andwiththevaluesreportedintheliteraturebyothergroups. 91 92 Wehavealsofollowedthekineticsforthegrowthofthe3HedropletsbymeasuringthetimedependenceoftheamplitudesoftheNMRechoesfollowingasuddendropinthetemperaturefrom119mKto110mKforasamplewithx3=1000ppm.Althoughthereisasmallinitialdelay(probablyduetothenucleationtime)thesubsequentgrowthshownin 96

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Figure5-5. ComparisonoftheobservedphaseseparationtemperatureswiththeEdwards-Balibarpredictionfor3Heinsolid4Heforconcentrationsof16x32000ppm.Opensquaresandtriangles(red)representtheT1minimumandamplitudemaximumtemperature,respectivelyforthecurrentexperiment.Theblackstarisfromultrasoundmeasurements 91 andbluecrossesrefertothedatafrompressuremeasurements. 92 [FigurereproducedwithpermissionfromJ.LowTemp.Phys.162,167,(2010).Copyright(2010)bytheJournalofLowTemperaturePhysics.] Figure 5-6 isexponentialwithatimeconstantof2.40h.ThisvalueisingoodagreementwiththeresultsofSmithetal. 93 forx3=1%,afterscalingtherelevantparametersforthedierenceininitialconcentrations.Theinitialdelaymayberelatedtoalongnucleationperiodforverydilutesamples. 97

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Figure5-6. Kineticsofthegrowthof3Henano-dropletsforx3=1000ppmforatemperaturestepof9mKatthephaseseparationtemperature.Thesolid(blue)lineisattoalongtimeexponentialgrowthwithtimeconstantd=2:40h.[FigurereproducedwithpermissionfromJ.LowTemp.Phys.162,167,(2010).Copyright(2010)bytheJournalofLowTemperaturePhysics.] 5.6SummarySystematicmeasurementsofthe3Henuclearspin-latticerelaxationratesofphaseseparated3HeFermi-liquiddropletsinsolid4Hehavebeenshowntohaveacharacteristictemperature-independentvalueatverylowtemperatures.Thisbehaviorisconsistentwithotherstudiesandleadstotheconclusionthatforlowdensitysolid4Hematricesthe 98

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3Helayerofthewallresponsiblefortherelaxationisdegenerate.Theobservedphaseseparationtemperaturesfor16x32000ppmareingoodagreementwiththetheoryofEdwardsandBalibar. 99

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CHAPTER6NUCLEARSPIN-SPINRELAXATIONTIMESINHIGHTEMPERATUREREGION 6.1OverviewInadditiontomeasuringthenuclearspin-spinrelaxationtimes(T2)intheregionwherethetemperatureindependentquantumtunnelingisdominantwehavealsomeasuredT2athightemperaturewherethethermallyactivatedvacancymotiondeterminestherelaxation.Inthepreviouschaptersmeasurementsofthespin-spinrelaxationtime,T2,wereusedtomeasurethetunnelingrateanditsdensitydependence.AtemperatureindependentplateauforT2wasobservedandattributedtotheexchangemotionalnarrowingwasobserveddowntothelowesttemperaturestudied.Noevidenceofanexchange-phononbottleneckforthespin-latticerelaxationisseendownto40mK.Thevacancyactivationenergyisdeterminedtobe13.5Kforasamplewithx3=510)]TJ /F2 7.97 Tf 6.59 0 Td[(4andmolarvolumeof20.9cm3.Forthesemeasurements,welocatedthepreamplierclosetotheNMRcellasshowninFigure 2-9 andwiththissetupthelowesttemperaturewas250mK.Thesamplesweregrownbyablockedcapillarymethod.ThemoreexperimentaldetailswerediscussedinChapter 2 6.2ResultandDiscussionThetemperaturedependenceofthevaluesmeasuredforthespin-spinrelaxationtimes,T2,fordierent3HeconcentrationsisshowninFigure 6-1 .Athightemperaturesawell-knownstrongtemperaturedependenceisobservedandisattributedtothethermalactivationofhighlymobilevacancies. 94 Therelaxationinthisregionisdeterminedbythefrequencyofmodulationofthedipole-dipoleinteractionbythemotionofvacancies.Theindividualjumprate)]TJ /F2 7.97 Tf 6.58 0 Td[(1Iismuchfasterthanthe3He-4Hetunnelingratesothatsmallconcentrationofvacanciescanbeveryeectiveincausingnuclearspinrelaxation. 95 The3Heatomsseeanetmodulationfrequency)]TJ /F2 7.97 Tf 6.59 0 Td[(1R=xv(T))]TJ /F2 7.97 Tf 6.59 0 Td[(1Iwherexv(T)istheconcentrationofvacanciesattemperatureT.Inaclassicalmodelxv(T)=eW=TwhereW=+mwiththevacancyformationenergyandmthepotentialenergy 100

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Figure6-1. Observedtemperaturedependenceofthenuclearspin-spinrelaxationtime,T2,for3Heinsolid4HeatamolarvolumeVm=20:9cm3forthreedierentfractionalconcentrations:10)]TJ /F2 7.97 Tf 6.58 0 Td[(3,510)]TJ /F2 7.97 Tf 6.58 0 Td[(4and2:510)]TJ /F2 7.97 Tf 6.59 0 Td[(4.[FigurereproducedwithpermissionfromJ.LowTemp.Phys.158,584,(2010).Copyright(2010)bytheJournalofLowTemperaturePhysics.] barriertomoveavacancytotheneighboringsite.Thesolidheliumis,however,quantumsolid,andformsabandofexcitedstates 95 withwidthbecauseoftheirmotions.TheconcentrationofthevacancyattemperatureTisgivenby xv(T)=4(T )3=2e)]TJ /F2 7.97 Tf 6.59 0 Td[(()]TJ /F7 5.978 Tf 7.78 3.26 Td[( 2)=2(6{1)TheresultshowninFigure 6-1 clearlyshowtheexponentialdependence,butthepowerlawpre-factorisdiculttodistinguishfromtheclassicalbehavior. 101

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TheminimumatT=1:1KisthefamiliarBPPminimum 96 whentheLarmorfrequencyisclosetothevacancymodulationfrequency.FromtheslopeofthetemperaturedependenceandthevalueoftheminimumvalueofT2wecandeterminethevacancyformationenergyEvac=(m)]TJ /F2 7.97 Tf 13.62 4.71 Td[( 2)andthefrequency!vac=)]TJ /F2 7.97 Tf 6.59 0 Td[(1Iwithwhichavacancymovesthroughthelattice.ThisanalysisyieldedEvac=13:5Kand!vac=1:1109s)]TJ /F2 7.97 Tf 6.58 0 Td[(1fora500ppmsamplewithamolarvolumeVm=20:9cm3.Attemperaturesbelow0.7K,weobservetemperatureindependentplateausforT2forallsamplesstudied.Thisregion,previouslystudieddownto0.4K 75 ,isinterpretedasresultingfromthequantumtunnelingofthe3Heatomsthroughthe4Helattice. 6.3SummaryAstrongtemperaturedependenceisobservedfortemperatureT0:9K,thisdependenceallowustodeterminethethermalactivationenergyEvac=13:50:5Kforvacancyformationforamolarvolumeof20.9cm3.Avacancytunnelingrateof1:1109s)]TJ /F2 7.97 Tf 6.59 0 Td[(1wasalsodecreased.Thesevaluesareingoodagreementwithstudiesforhigherconcentrations. 102

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CHAPTER7CONCLUSION 7.1SummaryThenuclearspin-latticerelaxationtimes(T1)andnuclearspin-spinrelaxationtimes(T2)areinvestigatedforsolid3He-4Hemixtureswithawiderangeof3Heconcentrations(16x32000ppm)atlowtemperaturesdownto10mK. 7.1.1FirstMicroscopicDataintheProposedSupersolidPhaseThedynamicsof3Heatomsinsolid4Heintheregionwhereasignicantnon-classicalrotationalinertiafractions(NCRIFs)havebeenstudiedatthemicroscopicscale.Forthisstudyweusedx3=16ppmandx3=24ppmsamplesandthespin-latticerelaxationtimes(T1)showstronganomaliesintheproposedsupersolidphase.TheanomaliesappearedinbothT1andT2forbothsamples.FirsttheseanomaliesareinvestigatedwithexistingNMRtheorybyLandesmanandHuang.Howeverthosemodelscouldnotexplainthepeaksintherelaxationtimes.Anotherresultistheobservationofthephase-separationattemperatureslowerthantheproposedsupersolidphaseandwededucethattheanomaliesarenotrelatedtotheisotopicphaseseparationwhichiswellstudiedboththeoreticallyandexperimentally.AphenomenologicalthermallyactivationmodelcanreproducethepeaksinT1andthe3Hebindingmechanismprovidescloseagreementwithpeaktemperature.ThesetwophenomenologicalmodelsarenotfromtheNMRrelaxationmodel.Howeverthedatashowninthisthesisaretherstnon-invasivemicroscopicresultsreportedfortheproposedsupersolidphaseandtherstnuclearspinrelaxationtimesintheverydiluteandultralowtemperatureregion. 7.1.2CoherentandIncoherentQuantumMotioninDilute3HeinSolid4HeMixtureIndilute3He-4Hemixture,therearedistinctivetwoquantummotionsinthelowtemperatureregion.Oneisthecoherentquantumtunnelingmotionwhichoccursinverydilute3He-4Hemixturewherethe3Hemotionisoutsidetherangeoftheeectivestaineldormassuctuationpotential. 103

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Theotheristheincoherentmotionwhichresultsfromtheinteractionbetween3Heatomsandstraineldduetothedistortioninthe4Helatticesurroundingeach3Heimpurityatom,ortheinteractionbetweenmassuctuationwaves(MFWs)wherethecontinuousinteractionisdominant.Inthisregiontheinteractiondisturbsthecoherentmotionandataxedtemperaturethesemotionsstronglydependontheconcentrationof3He.However,althoughthereisalargeamountofdataandseveralmodelsexistthatdiscussthecrossover3Heconcentrationbetweenthesetworegions,thereisacontroversialdiluteregionwherethereisnodatapriortothisthesiswork.Ournewdatafortheverydiluteregionconrmsthecriticalconcentrationof3Hemoreclearly.TheanalysisoftheconcentrationdependenceofT1andT2forawiderangeofconcentrationfromourmeasurementsshowsthatthecrossoverconcentrationisabout10)]TJ /F2 7.97 Tf 6.58 0 Td[(4. 7.1.3NuclearSpinRelaxationMechanisminthePhaseSeparatedRegionThetemperaturedependenceofthenuclearspin-latticerelaxationtimes(T1)andnuclearspin-spinrelaxationtimes(T2)forawiderangeof3Heconcentrations,16x32000ppm,ondilutemixturesof3Heinsolid4Heshowsthatthephase-separated3HeatomsformaFermi-liquiddropletbelowphaseseparationtemperature.Theinterfacesbetween3Hedropletsand4Heatomsareresponsibleforthenuclearspinrelaxations,andthequantumstateof3Heattheinterfacesisdegenerate.T1becomestemperatureindependentafterformationofliquiddropletandT2increasesabruptlyandbecomestemperatureindependent. 7.1.4NewConsiderationofQuantumTunnelingMotionofthe3HeAtomsintheNuclearSpinRelaxationsAtsucientlylowtemperatures,thequantumtunnelingmotionof3Heisdominantinthenuclearspinrelaxation.Inthisregionthenuclearspinrelaxationistemperatureindependentanddependsonlyontheconcentrationof3Heandthedensity.The 104

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measurementsofthespin-spinrelaxationtimes(T2)showthetemperatureindependentplateauattributedtotheexchangemotionalnarrowingdownto250mK.ThebesttofthevariationofT2with3HeconcentrationshowsthatT2/x)]TJ /F2 7.97 Tf 6.59 0 Td[(1:890:13ratherthanx)]TJ /F2 7.97 Tf 6.59 0 Td[(13andnoexchangephononbottleneckisobservedatT=40mK.Thevacancyactivationenergyisabout13.5Kforasamplewithx3=510)]TJ /F2 7.97 Tf 6.59 0 Td[(4formolarvolumeVm=20:9cm3. 7.2FutureWorkWehavesuccessfullyperformedNMRmeasurementsonverydilute3He-4Hemixturesatultralowtemperatures.Almostalldataareforpreviouslyunexploredtemperatureandconcentrationregionsandgivevaluableinformationforunderstandingunrevealedquantumphenomena.Howevertherearestillseveralstudiesthatareneededtoproperlyunderstandthemicroscopicmechanism,whichisnowcrucialinordertodeterminetheunderlyingphysicsoftheproposedsupersolidregion,andtocomprehendthedilute3He-4Hemixture,auniqueandperfectsystemtostudyquantumeects. 7.2.1FrequencyDependentNMRMeasurementsFurtherstudiesforawidefrequencyrangeofNMRmeasurementsareneededtodeterminewhetherthepeaksin3Henuclearspinrelaxationarerelatedtoanonmonotonicspectraldensityuctuationswithasharpfeatureatthefrequency(2:8MHz)ornot.Nearthisfrequencythenearestneighbor3Hemoleculesmovetogetherthroughthe4Helatticeviaquantumtunnelingmotions. 48 60 ThereforethemeasurementsofthenuclearspinrelaxationtimeswithdierentLarmorfrequenciescanhelptoresolveamorecomplexrelaxationmechanism. 7.2.2FurtherStudiesfortheNuclearSpinRelaxationDependenceontheCrystalQualityTheearlymeasurementofKimetal.andotherresearchgroups 4 5 41 55 56 97 havefoundthatthepossiblesignatureofsupersolidphasestronglydependsonthequalityofthecrystalsandtheconditionofcrystalgrowth.OurNMRsamplesaregrownbytheblocked 105

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capillarytechniquewhichisalsocalledtheconstantvolumemethodbecausevolumesarekeptconstantduringsolidication.Usuallycrystalspreparedbythismethodhavemanydefectsandstudiesshowthatthesedefectsaecttheanomaliesinnon-classicalrotationalinertiafractions(NCRIFs).ThefractionoftheNCRIFsisdecreasedwhencrystaldefectsaredecreased.Itiswellknownthatannealingdecreasescrystaldefects 57 58 andourpresentexperimentsperformedafterdierentannealingonthe16ppmand24ppmsamples,observedtheeectonthepeaksinthenuclearspinrelaxationtimes.Howeverresearchelsewhereshowssamplesgownunderconstantpressurehavelessdefectsthansamplesgrownunderconstantvolume.Forfutureexperiments,NMRstudyondilute3He-4Hemixturegrownbyconstantpressureneedstobecarriedouttoelucidatetheeectofthedefectsontherelaxationtimes. 7.2.3MoreDatafromDierent3HeConcentrationSamplesInChapter 4 ,3Heconcentrationdependenceofthenuclearrelaxationsisdiscussedinthetemperatureindependentregion,wheretherelaxationismostlybasedonthequantumtunneling.Inthisregionthe3Hemotioncanbeeithercoherentorincoherentdependingonlyonthe3Heconcentration.Ourdataconrmthatthecrossover3Heconcentrationbetweenthecoherentandincoherentregionsisaboutx310)]TJ /F2 7.97 Tf 6.59 0 Td[(4.Howeveradditionalnuclearspinrelaxationmeasurementsintheregion10x3500ppmarestillneededtoobtainmoreaccuratevaluesforthecross-overconcentration. 106

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APPENDIXACALCULATIONOFNMRSIGNALANDNOISE TheJohnsonnoiseinducedbythetunedNMRcoilisgiveby vn=p 4kBTRf(A{1) =149:6(nV)(A{2)usingkB=1:410)]TJ /F2 7.97 Tf 6.59 0 Td[(23m)]TJ /F2 7.97 Tf 6.59 0 Td[(2kgs)]TJ /F2 7.97 Tf 6.58 0 Td[(2K)]TJ /F2 7.97 Tf 6.58 0 Td[(1,T=0:4K,R=5105andf=2104s)]TJ /F2 7.97 Tf 6.59 0 Td[(1.TheNMRsignalsizeforatunedNNRcoilforpure3Heisgivenby vs=NAoB2o(A{3) =NAo!2o (A{4) =6:19105(nV)(A{5)whereNisthenumberofturn,Aisthecrosssectionareaofthecoil,0isthemagneticsusceptibilityof3He,isthegyromagneticratioof3Heand!0istheLarmorfrequency.Thevaluesusedforcalculationaregivenbelow N=80 A=0:0210)]TJ /F2 7.97 Tf 6.59 0 Td[(3m2 o=510)]TJ /F2 7.97 Tf 6.59 0 Td[(7 =2:037108s)]TJ /F2 7.97 Tf 6.59 0 Td[(1T)]TJ /F2 7.97 Tf 6.58 0 Td[(1 !o=Bo=12:56106s)]TJ /F2 7.97 Tf 6.58 0 Td[(1Forthe16ppmsamplethesignalvoltageis vs(x3=16ppm)=1610)]TJ /F2 7.97 Tf 6.58 0 Td[(66:19105(nV) =9:9(nV)(A{6) 107

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Forthe100ppmsamplethesignalvoltageis vs(x3=100ppm)=10010)]TJ /F2 7.97 Tf 6.59 0 Td[(66:19105(nV) =61:9(nV)(A{7)Forthe16ppmsamplethetypicalsignaltonoiseratio(S/N)isgivenby (S=N)=vs vn=NAoB2o p 4kBTRf(A{8)whereisthellingfactorthatisdenedbytheratioofthevolumecoveredbyNMRcoiltothevolumeoccupiedbythesample.ForourNMRdesignthellingfactorisalmostunity(1).For16ppm, (S=N)=vs vn=9:9 149:6=0:066(A{9)For100ppm, (S=N)=vs vn=61:9 149:6=0:413(A{10)Below100ppmthesignaltonoiseratioisverypoorandsignalaveragingwasneededandcomplextwasalsoneededtoreadspinechoheightaccurately. TableA-1. Symbolsusedinequations. SymbolUnit KKelvinVVoltage;VoltsSecondmMeterICurrent;AmpareJJoulekgKilogramHHenryTTeslaTTemperatureNNumberofturnoftheNMRcoilACrosssectionareaoftheNMRcoil 108

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APPENDIXBDATAANALYSISAlldatawereanalyzedbyttingthewholeshapeofthespin-echosignal.Aphasecorrectionwascarriedouttoreadanaccuratespin-echoheight.WecombinedaGaussianfunctionshapewiththephasecorrectiontermtondthebestvaluesforthespin-echoheightandthespin-echowidth.InNMR,wemeasurethesignalsrelativetotherotatingframewhichisrotatingwithLarmorfrequency!.TheNMRsignalsevolveaccordingtoCexp(i!t)whichcanbewrittenintermsofrealandimaginarycomponents: Cexp(i!t)=Ccos(!t)+iCsin(i!t)(B{1)whereCisthedecayingtermforthespin-echorelatedtothediusionandT2relaxation.Forquadraturedetection,boththerealandimaginarycomponentsoftheNMRsignalaremeasuredbyusingtwodetectorchannelsseparatedby90degrees.Therstoneiscalledtherealchannelwhichproducesacosinewave.Thesecondquadraturechanneliscalledtheimaginarychannelwhichisshiftedby90degreesfromtherealchannelandproducesasinewavesignal.TheFouriertransformationofthesignalswithoutphasecorrectionresultsinaphase-distortedline-shapewhichshowsamixtureofabsorptiveanddispersiveline-shapes.Thisdistortionofline-shapemakesitdiculttoreadanaccuratespin-echoheightandaline-width,especiallywhentheNMRsignalsareverysmall.Inthiscase,asmalldistortioninducesalargeuncertaintyinreadingthespin-echoheight.Thedistortionoftheline-shapecanbecorrectedbyaddingphaseterminEqn. B{1 ineithertimeorfrequencydomains.Thisiscalledthephasingofspectrumorphasecorrection.Weusedcomplexcurve-tmethodofMathematicaforthetandthefunctionusedforphasecorrectionisgivenby Curfit=aexp[)]TJ /F1 11.955 Tf 9.3 0 Td[((j)]TJ /F4 11.955 Tf 11.96 0 Td[(jc) (2w2)]exp[i(f(j)]TJ /F4 11.955 Tf 11.95 0 Td[(jc)+)](B{2) 109

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wherejcisthecenterofthespin-echo,aistheparameterfortheamplitudeadjustment,wisthefullwidthathalfmaximum(FWHM)ofthepeakandfandarethettingparametersforthephasecorrection.Someoftherawdatafromeach3Heconcentration(16,24,500,900,1870,2000ppm)withtsareshowninthenextgures.AllgraphsshowthebesttsfortherealandimaginarycomponentsandthephaseoftheNMRsignals. FigureB-1. Spin-echofromasamplewith16ppm3Heat10mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged32timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((5ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 110

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FigureB-2. Spin-echofromasamplewith24ppm3Heat125mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged16timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((5ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 111

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FigureB-3. Spin-echofromasamplewith500ppm3Heat25mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged8timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((10ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 112

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FigureB-4. Spin-echofromasamplewith900ppm3Heat25mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged8timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((4ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 113

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FigureB-5. Spin-echofromasamplewith1870ppm3Heat25mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged4timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((4ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 114

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FigureB-6. Spin-echofromasamplewith2000ppm3Heat10mK.Thesolidlinesaretstothedata.(Re-tfortherealcomponents,Im-tfortheimaginarycomponents,Abs-tfortheabsorptivecomponentsandArg-tforthephasecomponents.)Thesignalwasaveraged2timesbyapplying90o)]TJ /F4 11.955 Tf 11.95 0 Td[((4ms))]TJ /F1 11.955 Tf 11.96 0 Td[(180opulseseachtime. 115

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APPENDIXCRELAXATIONTOPOLOGIESTherelaxationoftheenergyinvolvestwosystemstherelaxationtimecanbecalculatedbyconsideringthecouplingoftwosystems.Therelaxationtimefortwosystemsistheintrinsicrelaxationtime.Let'sconsideratwobathsystemrst.Iftherelaxationofenergyinvolvetwosystems:system-1andsystem-2,andtheenergyowchainissystem-1!system-2!reservoir.System-2isstronglycoupledwiththethermalreservoirandthetemperatureofsystem-2issameasthereservoir.Theintrinsicrelaxationtimebetweensystem-1andsystem-2,T12isgivenby T12=)]TJ /F1 11.955 Tf 36.44 8.09 Td[(1 _1(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2)(C{1)where1istheinversetemperatureofthesystem-1and2istheinversetemperatureofthesystem-2.Ifthereisanothersystem-0whichcombinedwiththesystem-1thenenergyowinvolvesthreebaths:system-0,system-1andsystem-2.Inthiscasetherelaxationtimeforthreesystemscannotbeexpressedbythesimpleintrinsicrelaxationtime.Thesystem-2isstillstronglyconnectedtothethermalreservoirandtheweaklinkintheenergyowchainisthecouplingbetweensystem-1andsystem-2andthenotationis01-2.Theobservedspin-latticerelaxationtimeT1forthreesystemsisgivenby T1=[(k1+k0) (k1)]T12(C{2)where k0=dE0 d;k1=dE1 d(C{3)andE0andE1aretheenergiesforsystem-0andsystem-1respectively.WecallT1thetopologicalrelaxationtimecontrasttotheintrinsicrelaxationtimedenedinEqn. C{1 .TopologicaltimesdiscussedintheChapter 1 andtheChapter 5 areshownintheTable C-1 .Whentherelaxationisassociatedwithspin-spincouplingsratherthan 116

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themotionofvacanciesforwhichtheheatcapacitiesareused,onemustconsiderthemagnetizationinthetopologicalfactorasdiscussedinChapter 5 TableC-1. Thetopologicalrelaxationtime.(>Z-Zeeman,V-Vacancywaves,P-Phonon,T-Tunneling,MF-Massuctuationwaves,C-Core,W-Wall.) NotationIntrinsicrelax-ationtimeTopologicalrelaxationtimeRegion Z-VPTZVT1=TZVRegion1-Aforpure3Heanddilute3Heinsolid4HeZ-TVPTZTT1=TZTRegion1-Bforpure3HeZ-MFPTZMFT1=TZMFRegion1-Bfordilute3Heinsolid4HeZMF-PTMFPT1=[(kZ+kMF)=kMF]TMFPRegion2fordilute3Heinsolid4HeCW-PTWPT1=[(Mcore+Mwall)=Mwall]TWPPhaseseparatedregionforFermi-liquiddropletand2D-solidwall. 117

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REFERENCES 1. Goodkind,J.M.andFairbank,W.M.Nuclearspinrelaxationinsolid3He.Phys.Rev.Lett.4,458{460,May(1960). 2. Hunt,E.R.,Richardson,R.C.,Thompson,J.R.,Guyer,R.A.,andMeyer,H.Nuclearrelaxationinsolid3Heatlowtemperatures.Phys.Rev.163,181{185,Nov(1967). 3. Miyoshi,D.S.,Cotts,R.M.,Greenberg,A.S.,andRichardson,R.C.Nuclearmagneticresonanceinsolidhelium-3-helium-4mixturesbetween0.3and2.0K.Phys.Rev.A2,870{882,Sep(1970). 4. Kim,E.andChan,M.H.W.Observationofsuperowinsolidhelium.Science305(5692),1941(2004). 5. Kim,E.andChan,M.H.W.Probableobservationofasupersolidheliumphase.Nature427(6971),225(2004). 6. Mullin,W.J.Cellmodelofabose-condensedsolid.Phys.Rev.Lett.26,611{614,Mar(1971). 7. Guyer,R.A.andZane,L.I.Massuctuationwaves.Phys.Rev.Lett.24,660{663,Mar(1970). 8. Guyer,R.A.andZane,L.I.Tunnelingandexchangeinquantumsolids.Phys.Rev.188,445{468,Dec(1969). 9. Carruthers,P.Theoryofthermalconductivityofsolidsatlowtemperatures.Rev.Mod.Phys.33,92{138,Jan(1961). 10. Garwin,R.L.andReich,H.A.Heatcapacityoftheexchangebathinsolid3He.Phys.Rev.Lett.12,354{357,Mar(1964). 11. Nosanow,L.H.andVarma,C.M.Exchangeincrystalline3He.Phys.Rev.Lett.20,912{915,Apr(1968). 12. Nosanow,L.H.andWerthamer,N.R.Calculationsofsoundvelocitiesincrystallineheliumatzerotemperature.Phys.Rev.Lett.15,618{621,Oct(1965). 13. Guyer,R.A.,Richardson,R.C.,andZane,L.I.Excitationsinquantumcrystals(asurveyofnmrexperimentsinsolidhelium).Rev.Mod.Phys.43,532{600,Oct(1971). 14. Pollack,G.L.Kapitzaresistance.Rev.Mod.Phys.41,48{81,Jan(1969). 15. Huang,W.,Goldberg,H.A.,andGuyer,R.A.Quantum-crystalalloysi:Mass-uctuationwaves.Phys.Rev.B11(9),3374{3392,May(1975). 16. Widom,A.andRichards,M.G.Quantumtheoryofdiusionwithapplicationtosolidhelium.Phys.Rev.A6(3),1196{1199,Sep(1972). 118

PAGE 119

17. M.G.Richards,J.P.andWidom,A.Proc.13thInt.Conf.onLowtemperaturephysics2,67,Mar(1972). 18. Kubo,R.Generalizedcumulantexpansionmethod.JournalofthePhysicalSocietyofJapan17(7),1100{1120(1962). 19. Hoult,D.andRichards,R.Thesignal-to-noiseratioofthenuclearmagneticresonanceexperiment.JournalofMagneticResonance(1969)24(1),71{85(1976). 20. Ginsberg,D.M.andMelchner,M.J.Optimumgeometryofsaddleshapedcoilsforgeneratingauniformmagneticeld.Rev.Sci.Instrum.41(1),121{123(1970). 21. Firester,A.H.Designofsquarehelmholtzcoilsystems.Rev.Sci.Instrum.37(9),1264{1265(1966). 22. Bodart,J.R.,Garcia,B.M.,Phelps,L.,Sullivan,N.S.,Moulton,W.G.,andKuhns,P.Theeectofhighmagneticeldsonjunctioneldeecttransistordeviceperformance.ReviewofScienticInstruments69(1),319{320(1998). 23. AgilentTechnologiesInc.,SantaClara,CA95051.(2008). 24. Lie-zhao,C.,Brewer,D.F.,Girit,C.,Smith,E.N.,andReppy,J.D.Flowandtorsionaloscillatormeasurementsonliquidheliuminrestrictedgeometriesunderpressure.Phys.Rev.B33(1),106{117,Jan(1986). 25. Adams,E.D.,Uhlig,K.,Tang,Y.-H.,andHaas,G.E.Solidicationandsuperuidityof4Heinconnedgeometries.Phys.Rev.Lett.52(25),2249{2252,Jun(1984). 26. Beamish,J.R.,Hikata,A.,Tell,L.,andElbaum,C.Solidicationandsuperuidityof4Heinporousvycorglass.Phys.Rev.Lett.50(6),425{428,Feb(1983). 27. Csathy,G.A.,Kim,E.,andChan,M.H.W.Condensationof3Heandreentrantsuperuidityinsubmonolayer3He-4HemixturelmsonH2.Phys.Rev.Lett.88(4),045301,Jan(2002). 28. LakeShoreCryotronicsInc.,575McCorkleWesterville,OH43082.(2008). 29. Greywall,D.S.Melting-curvethermometryatmillikelvintemperatures3He.Phys.Rev.B31,2675{2683,Mar(1985). 30. Grilly,E.R.Pressure-volume-temperaturerelationsinliquidandsolid3He.JournalofLowTemperaturePhysics4,615{635(1971).10.1007/BF00628297. 31. Halperin,W.P.,Rasmussen,F.B.,Archie,C.N.,andRichardson,R.C.Propertiesofmelting3He:Specicheat,entropy,latentheat,andtemperature.JournalofLowTemperaturePhysics31,617{698(1978).10.1007/BF00116046. 32. Greywall,D.S.Specicheatofnormalliquid3He.Phys.Rev.B27,2747{2766,Mar(1983). 119

PAGE 120

33. Fukuyama,H.,Ishimoto,H.,Tazaki,T.,andOgawa,S.Meltingcurvebelow15mK3He.Phys.Rev.B36,8921{8924,Dec(1987). 34. Straty,G.C.andAdams,E.D.Highlysensitivecapacitivepressuregauge.40(11),1393{1397(1969). 35. Xia,J.,Choi,H.,Lee,Y.,andSullivan,N.Kaptoncapacitancethermometryatlowtemperaturesandinhighmagneticelds.JournalofLowTemperaturePhysics148,899{902(2007).10.1007/s10909-007-9433-7. 36. J.S.Xia,L.Y.E.D.A.andSullivan,N.S.Acompactcapacitivepressuretransducer.JournalofPhysics,Conf.Series150,012054{012057(2009). 37. Grilly,E.R.andMills,R.L.Pvtrelationsin4Henearthemeltingcurveandthe[lambda]-line.AnnalsofPhysics18(2),250{263(1962). 38. Mullin,W.J.Theoryofphaseseparationinsolid3He-4Hemixtures.Phys.Rev.Lett.20(6),254{257,Feb(1968). 39. Leggett,A.J.Canasolidbesuperuid?Phys.Rev.Lett.25(22),1543{1546,Nov(1970). 40. Kim,E.,Xia,J.S.,West,J.T.,Lin,X.,Clark,A.C.,andChan,M.H.W.Eectof3Heimpuritiesonthenonclassicalresponsetooscillationofsolid4He.Phys.Rev.Lett.100(6),065301,Feb(2008). 41. Rittner,A.S.C.andReppy,J.D.Observationofclassicalrotationalinertiaandnonclassicalsupersolidsignalsinsolid4Hebelow250mK.Phys.Rev.Lett.97,165301,Oct(2006). 42. Syshchenko,O.,Day,J.,andBeamish,J.Frequencydependenceanddissipationinthedynamicsofsolidhelium.Phys.Rev.Lett.104(19),195301,May(2010). 43. Ryo,Gumann,P.,Kosaka,K.,Kanemoto,M.,Onoe,W.,andSasaki,Y.Simultaneousmeasurementoftorsionaloscillationsandnmrofverydilute3Heinsolid4He.Phys.Rev.B81(21),214515,Jun(2010). 44. Toda,R.,Onoe,W.,Kanemoto,M.,Kakuda,T.,Tanaka,Y.,andSasaki,Y.Simultaneousmeasurementoftorsionaloscillatorandnmrinsolid4Hewith10ppmof3He.JournalofLowTemperaturePhysics162,476{482(2011).10.1007/s10909-010-0279-z. 45. Carr,H.Y.andPurcell,E.M.Eectsofdiusiononfreeprecessioninnuclearmagneticresonanceexperiments.Phys.Rev.94(3),630{638,May(1954). 46. Richards,M.G.,Pope,J.,andWidom,A.Evidenceforisotopicimpuritonsinsolidhelium.Phys.Rev.Lett.29(11),708{711,Sep(1972). 120

PAGE 121

47. Grigoriev,V.N.,Esel'son,B.N.,Mikheev,V.A.,Slusarev,V.A.,Strzhemechny,M.A.,andShulman,Y.E.3Heimpurityexcitationsinsolid4He.J.LowTemp.Phys.13,65{79,OCT(1973). 48. Richards,M.G.,Smith,J.H.,Tofts,P.S.,andMullin,W.J.Frequencydependenceoft1for3Heinsolid4He.Phys.Rev.Lett.34(25),1545{1547,Jun(1975). 49. Schratter,J.,Allen,A.R.,andRichards,M.G.Concentrationdependenceofd,t1,andt2fordilutesolutionsof3Heinsolid4He.J.LowTemp.Phys.57,179{187,OCT(1984). 50. Kim,S.S.,Huan,C.,Yin,L.,Xia,J.,Candela,D.,andSullivan,N.S.NMRStudiesof3HeImpuritiesin4HeintheProposedSupersolidPhase.J.LowTemp.Phys.158,584,February(2010). 51. Huan,C.,Kim,S.,Yin,L.,Xia,J.,Candela,D.,andSullivan,N.Nmrstudiesof3Hedropletsindilute3He-4Hesolidsolutions.JournalofLowTemperaturePhysics162,167{173(2011).10.1007/s10909-010-0260-x. 52. Ho,P.-C.,Bindloss,I.P.,andGoodkind,J.M.Anewanomalyinsolid4He.J.LowTemp.Phys109,409{421(1997). 53. Edwards,D.O.andBalibar,S.Calculationofthephasediagramof4He-3Hesolidandliquidmixtures.Phys.Rev.B39(7),4083{4097,Mar(1989). 54. Greenberg,A.S.,Thomlinson,W.C.,andRichardson,R.C.Nuclearmagneticresonancestudiesofisotopicimpuritytunnelinginsolidhcp4He.J.LowTemp.Phys.8,3{28,Jul(1972). 55. Penzev,A.,Yasuta,Y.,andKubota,M.Annealingeectforsupersolidfractionin4He.JournalofLowTemperaturePhysics148,677{681(2007).10.1007/s10909-007-9452-4. 56. Kondo,M.,Takada,S.,Shibayama,Y.,andShirahama,K.Observationofnon-classicalrotationalinertiainbulksolid4He.JournalofLowTemperaturePhysics148,695{699(2007).10.1007/s10909-007-9471-1. 57. Rittner,A.andReppy,J.Theannealingprocessinsolid4He.JournalofLowTemperaturePhysics148,671{675(2007).10.1007/s10909-007-9447-1. 58. Clark,A.C.,West,J.T.,andChan,M.H.W.Nonclassicalrotationalinertiainheliumcrystals.Phys.Rev.Lett.99(13),135302,Sep(2007). 59. Landesman,A.Theoryofmagneticrelaxationfordilute3Heatomstunnelinginhcpsolid4He.PhysicsLettersA54(2),137{139(1975). 60. Mullin,W.J.,Guyer,R.A.,andGoldberg,H.A.Moleculesinsolid3He-4He.Phys.Rev.Lett.35(15),1007{1010,Oct(1975). 121

PAGE 122

61. Castles,S.H.andAdams,E.D.Low-temperaturespecic-heatanomalyofbcc3He.Phys.Rev.Lett.30(22),1125{1128,May(1973). 62. Castles,S.H.andAdams,E.D.Specicheatofsolidhelium.JournalofLowTemperaturePhysics19,397{431(1975).10.1007/BF00114090. 63. Iwasa,I.,Araki,K.,andSuzuki,H.Temperatureandfrequencydependenceofthesoundvelocityinhcp4Hecrystals.JournalofthePhysicalSocietyofJapan46(4),1119{1126(1979). 64. Tsymbalenko,V.L.Anomalousreectionofsoundatafacetofa4Hequantumcrystal.LowTemperaturePhysics21(2),129{132(1995). 65. Ruutu,J.P.,Hakonen,P.J.,Babkin,A.V.,Parshin,A.Y.,andTvalashvili,G.Growthof4He-crystalsatmk-temperatures.JournalofLowTemperaturePhysics112,117{164(1998).10.1023/A:1022246029056. 66. Granato,A.andLucke,K.Theoryofmechanicaldampingduetodislocations.JournalofAppliedPhysics27(6),583{593(1956). 67. Iwasa,I.andSuzuki,H.Soundvelocityandattenuationinhcp4Hecrystalscontaining3Heimpurities.JournalofthePhysicalSocietyofJapan49(5),1722{1730(1980). 68. Paalanen,M.A.,Bishop,D.J.,andDail,H.W.Dislocationmotioninhcp4He.Phys.Rev.Lett.46(10),664{667,Mar(1981). 69. Day,J.andBeamish,J.Low-temperatureshearmoduluschangesinsolid4Heandconnectiontosupersolidity.Nature450,853,Dec(2007). 70. Wanner,R.,Iwasa,I.,andWales,S.Evidencefordislocationsinsolidh.c.p.helium-4fromsoundvelocityexperiments.SolidStateCommunications18(7),853{856(1976). 71. Reppy,J.D.Nonsuperuidoriginofthenonclassicalrotationalinertiainabulksampleofsolid4He.Phys.Rev.Lett.104(25),255301,Jun(2010). 72. Sacco,J.E.andWidom,A.Quantummobility3Heimpurityatomsinsolid4He.Phys.Rev.B17(1),204{214,Jan(1978). 73. Kim,S.S.,Huan,C.,Yin,L.,Xia,J.,Candela,D.,andSullivan,N.S.Nmrstudyofthedynamicsof3Heimpuritiesintheproposedsupersolidstateofsolid4He.Phys.Rev.Lett.106,185303,May(2011). 74. Godfrin,H.,Frossati,G.,Thoulouze,D.,Chapellier,M.,andClark,W.G.Nmrmeasurementsonliquid3Heconnedinaluminaandplatinumpowdersandgrafoil.J.Phys.Colloques39,C6{287{C6{289(1978). 75. Allen,A.,Richards,M.G.,andSchratter,J.Anomaloustemperaturedependenceofdandt2fordilutesolutionsof3Heinsolid4He.J.LowTemp.Phys.47,289{320,May(1982). 122

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76. Hirayoshi,Y.,Mizusaki,T.,Maekawa,S.,andHirai,A.Astudyof3He-4Hetunnelingmotioninsolid3He-4Hemixturesbymeansoft1measurements.PhysicsLettersA57(4),359{360(1976). 77. Collin,E.,Triqueneaux,S.,Bunkov,Y.M.,andGodfrin,H.Fast-exchangemodelvisualizedwith3Heconnedinaerogel:Afermiliquidincontactwithaferromagneticsolid.Phys.Rev.B80(9),094422,Sep(2009). 78. Schier,P.,O'Keefe,M.T.,Oshero,D.D.,andFukuyama,H.Lowtemperaturemagnetizationof3Helms.JournalofLowTemperaturePhysics94,489{513(1994). 79. Shvarts,K.K.N.M.E.A.J.X.E.S.V.Magneticorderingin3Henanoclusters.JournalofLowTemperaturePhysics126,169{178(2002). 80. S.C.J.Kingsley,I.K.L.R.V.M.J.S.andCowan,B.NMRstuyof3Hebubblesinphase-separatedsolid3He-4Hemixtures.JournalofLowTemperaturePhysics110,339{404(1998).10.1023/A:1022592918545. 81. Hebral,B.,Greenberg,A.S.,Beal-Monod,M.T.,Papoular,M.,Frossati,G.,Godfrin,H.,andThoulouze,D.Fermi-liquiddropletsinliquid-solidsolutionsoftheheliumisotopes.Phys.Rev.Lett.46(1),42{45,Jan(1981). 82. Mikhin,N.P.,Omelaenko,N.F.,Polev,A.V.,Rudavskii,E.Y.,andShvarts,V.A.Thermomagneticrelaxationintwo-phasesolid3He-4Hemixturesatultralowtemperatures.LowTemperaturePhysics24(12),845{851(1998). 83. vanSteenbergen,A.S.,Perenboom,J.A.A.J.,Wiegers,S.A.J.,andMaan,J.C.Classicalbehaviourinthequantumliquid3He-4He:Observationofab2dependenceofthesurfacerelaxation.PhysicaB:CondensedMatter246-247,164{170(1998). 84. Huan,C.,Kim,S.S.,Yin,L.,Xia,J.,Candela,D.,andSullivan,N.S.Anoveldesignofalowtemperaturepreamplierforpulsednmrstudiesofdilute3Heinsolid4He.J.LowTemp.Phys.158,692,February(2010). 85. Boninsegni,M.,Kuklov,A.B.,Pollet,L.,Prokof'ev,N.V.,Svistunov,B.V.,andTroyer,M.Luttingerliquidinthecoreofascrewdislocationinhelium-4.Phys.Rev.Lett.99(3),035301,Jul(2007). 86. Toner,J.Quencheddislocationenhancedsupersolidordering.Phys.Rev.Lett.100(3),035302,Jan(2008). 87. Fraass,B.A.andSimmons,R.O.X-raystudyofphasesandstructuresin3He-4Hesolidsolutions.Phys.Rev.B36(1),97,Jul(1987). 88. Poole,M.,Saunders,J.,andCowan,B.Stagesofhomogeneousnucleationinsolidisotopicheliummixtures.Phys.Rev.Lett.100(7),075301,Feb(2008). 123

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89. Genio,E.B.,Ihas,G.G.,andSullivan,N.S.Nuclearspinrelaxationofpowderedmetallicantimonyinliquid3He.JournalofLowTemperaturePhysics112,21{45(1998).10.1023/A:1022289811309. 90. Hammel,P.C.andRichardson,R.C.Relaxationofnuclearmagnetizationofliquid3Heinconnedgeometries.Phys.Rev.Lett.52(16),1441{1444,Apr(1984). 91. Goodkind,J.M.Aphasetransitioninsolid4He.AIPConferenceProceedings850(1),329{330(2006). 92. Gan'shin,A.N.,Grigor'ev,V.N.,Maidanov,V.A.,Omelaenko,N.F.,Penzev,A.A.,Rudavskii,E.Y.,andRybalko,A.S.Phaseseparationcurveofdilute3He-4Hehcpsolidsolutions.LowTemperaturePhysics26(12),869{873(2000). 93. Smith,A.,Maidanov,V.A.,Rudavskii,E.Y.,Grigor'ev,V.N.,Slezov,V.V.,Poole,M.,Saunders,J.,andCowan,B.Phaseseparationinsolid3He)]TJ /F2 7.97 Tf 9.3 4.34 Td[(4Hemixtures:Comparisonwiththeoryofhomogeneousnucleation.Phys.Rev.B67(24),245314,Jun(2003). 94. Reich,H.A.Nuclearmagneticresonanceinsolidhelium-3.Phys.Rev.129(2),630{643,Jan(1963). 95. Sullivan,N.,Deville,G.,andLandesman,A.Vacancy-inducednuclear-spin-latticerelaxationinsolid3He.Phys.Rev.B11,1858{1865,Mar(1975). 96. Bloembergen,N.,Purcell,E.M.,andPound,R.V.Relaxationeectsinnuclearmagneticresonanceabsorption.Phys.Rev.73(7),679{712,Apr(1948). 97. Aoki,Y.,Graves,J.C.,andKojima,H.Oscillationfrequencydependenceofnonclassicalrotationinertiaofsolid4He.Phys.Rev.Lett.99,015301,Jul(2007). 124

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BIOGRAPHICALSKETCH SungSuKimwasbornasthethirddaughteroffourchildrenofherparentsinSeoul,SouthKorea.Sheusedtolearnballetfor6yearsandshethoughtshewouldbeaballerinasomeday.Howeveronceshestartedtolearnmathematicsandphysics,herinterestmovedtophysics.Shegraduatedthehighschoolwithspecializingscienceandregistereduniversityformajoringphysics.Aftergraduatingwithabachelor'sdegreeinphysics,sheworkedforsemiconductorcompanyinKoreaasaQCengineer.Inthiscompanysheworkedinthesemiconductorsurfaceinspectionlabbyexaminingtheimpuritiesandimperfectionsonthesemiconductorchip.AcoupleofyearslatershedecidedtoseekahigherdegreeinphysicsandsheenrolledthegraduateschoolinSeoul,Korea.Shestudiedthecarbonnano-tubesinasingleelectrontransistorstructuresbyinvestigatingthecurrent-voltagecharacteristicsandtheCoulomb-blockadeoscillation.Shereceivedhermaster'sdegreeinphysicsinKoreaandshecametotheUniversityofFlorida,USAtogetaPh.Dinphysics.Shortlyaftersheinvolvedinahighfrequencyelectronparamagneticresonanceexperimentsontheorganicquantummagnetandsinglemoleculemagnet,shechangedhereldtothenuclearmagneticresonance(NMR)researchtoexplorethemicroscopicspindynamicsofquantumsolidandsupersolidunderProf.Sullivan.Afteragreatexperienceinthisresearch,shewillgraduatewithPh.DdegreeinphysicsinDecember2011.Shewillcontinuetheacademicresearchcareerasapostdoctoralresearcherforacoupleofyearsmore. 125