NMR Studies of Quantum Gases Confined in Mesoporous Materials

MISSING IMAGE

Material Information

Title:
NMR Studies of Quantum Gases Confined in Mesoporous Materials
Physical Description:
1 online resource (81 p.)
Language:
english
Creator:
Ji, Yu
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Sullivan, Neil S
Committee Members:
Takano, Yasumasa
Cheng, Hai Ping
Kumar, Pradeep P
Angerhofer, Alexander

Subjects

Subjects / Keywords:
gases -- mesoporous -- quantum
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:
Low mass molecules such as H2, D2, HD and CH4 have interesting new properties when localized in constrained geometries such as mesoporous structures. The new propertis result from the highly quantum mechanical character of the translational and rotational degrees of freedom. We report the results of the use of nuclear relaxation spectroscopy to determine the interaction energies, the quantized translational modes are the dynamics fo these gases when trapped in the cages of the mesoporous structures. The sample system was probed using coherent pulse NMR techniques at 5.6 MHz. A special duplexer was made for this low frequency experiment. A series of experiments with different light gases and adsobent were carried out: i) CH4 in zeolite 13X for 1, 0.5 and 0.1 molecules per a-cage and 4 < T < 95 K. ii) HD in zeolite 13X for 1 and 0.5 molecules per a-cage and 1.6 < T < 20 K. iii) CH4 in metal organic frameworks for 1 and 0.5 molecules per a-cage and 4 < T < 95 K. iv) HD in metal organic frameworks for 1 and 0.5 molecules per a-cage and 4 < T < 95 K. The experiments are designed so that the nuclear spin-lattice relaxation times depend on the heat capacities and local excitations of the trapped molecules according to the bath model. The results indicate well defined excitation energies for low densities of CH4 and HD in the mesoporous structures. The values obtained for CH4 are consistent with Monte Carlo calculations. The results also confirm the high mobility and diffusivity of HD at low temperatures as observed by neutron scattering.
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 Yu Ji.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Sullivan, Neil S.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2012
System ID:
UFE0043980:00001


This item is only available as the following downloads:


Full Text

PAGE 1

NMRSTUDIESOFQUANTUMGASESCONFINEDINMESOPOROUSMATERIALSByYUJIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

PAGE 2

c2012YuJi 2

PAGE 3

Idedicatethistoeveryonethathelpedrevampthistemplate 3

PAGE 4

ACKNOWLEDGMENTS ItismyspecialpleasuretothankmyadvisorProf.N.S.Sullivanfortheopportunitytododoctoralresearchwithhim.WithouthissupportandencouragementIwouldnothaveundertakenthisprojectintheUnitedStates.Hiscomprehensiveunderstandingofphysicsaswellashispersonalinteractionwithmemadethisaveryproductivetime.BeinginvolvedinmanyprojectssuchasMicrokelvinfacilityandtheNationalHighMagneticFieldLaboratory,hestillputalotoftimeonmyprojectandhelpmeoutwhenevertherewasneedforit.Hisattitudetowardseducationhassetanexampleforme.IalsothankthepostdoctoralassociateJ.A.HamidawhowasanexpertonNMRexperiment.ShehelpedmetogetfamiliarwitheverythingwhenIwasfreshintothelab.PostdocC.HuanalsohelpedmealotonthebasicNMRconcepts.Theco-operationandfriendshipofmyfellowgraduatestudents,S.S.KimandY.Tangalwaysdiscusswithmeaboutmydifcultiesintheexperimentsandgavemeusefulcomments.MythanksgotoProfs.H.Cheng,P.Kumar,Y.TakanoandA.Angerhoferfortheirinterestandsupportwhileservingonmysupervisorycommittee.Specialthanksgotoallthesupportgroupsthatwereimportantforthesuccessofmywork.InthiscontexIwanttonamethemachineshopwhichproducedhighqualityparts,theelectronicshopwiththeirexpertiseandcryogenicgroupforsupplyingheliumandengineeringadvice.Finally,Ithankmyhusbandforhisskilledassistanceinthedraftingoftheguresandtheconstantencouragementhehasgivenmethroughoutmygraduatecareer. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1Quantumgasesinconstrainedgeometries ................. 13 1.1.1ZeroPointMotion ............................ 14 1.1.2DeBoerParameter ........................... 17 1.2PropertiesofH2andHD ............................ 17 1.2.1Ortho-andParahydrogen ....................... 17 1.2.2Ortho-paraConversion ......................... 19 1.2.3Quadrupolarinteraction ........................ 20 1.3PropertiesofCH4 ................................ 22 1.4PropertiesofZeolite .............................. 24 1.4.1StructureofZeolite ........................... 24 1.4.2MonteCarloSimulation-MethaneadsorptioninZeolite ...... 27 1.5PropertiesofZ-MOF .............................. 28 2THEORY-NUCLEARSPINRELAXATION .................... 30 2.1GeneralBathModel .............................. 30 2.2BathModelforMethaneandHDinZeolite .................. 40 2.2.1CouplingBetweenZeemanandTspecies .............. 41 2.2.2CouplingBetweenTspeciesandPhononsystemofCH4 ...... 41 2.2.3CouplingBetweenthePhononsystemsofCH4andZeolite .... 41 2.2.4ModelofHDinZeolite ......................... 45 3NMRMETHODS ................................... 47 3.1PulsedNMR .................................. 47 3.2PulseNMRApparatus ............................. 51 3.3CellDesign ................................... 53 4EXPERIMENTALDATAANDANALYSIS ...................... 59 4.1CH4inZeolite .................................. 59 4.2HDinZeolite .................................. 64 4.3CH4inZMOF .................................. 68 4.4HDinZMOF .................................. 70 5

PAGE 6

5CONCLUSION .................................... 75 REFERENCES ....................................... 78 BIOGRAPHICALSKETCH ................................ 81 6

PAGE 7

LISTOFTABLES Table page 2-1EnergyeigenvaluesEn,lHDinawellofwidthd=13A.ndesignatesthen,throotoftheBesselfunctionjl(kr). .......................... 46 4-1EnergyeigenvaluesEn,lHDinawellofwidthd=13A.ndesignatesthen,throotoftheBesselfunctionjl(kr). .......................... 74 7

PAGE 8

LISTOFFIGURES Figure page 1-1VariationofRMSdeviationhu2i1=2withdeBoer'squantumparameter. ..... 18 1-2EnergylevelsofCH4. ................................ 23 1-3EnergylevelsofCH4incubiceld. ......................... 24 1-4ZeolitestructuresfortypeAandX. ......................... 25 2-1SchematicrepresentationthreeenergybathsA,BandCwithaweallink(bottleneck)betweenBandCwithenergyowformCtothemainthermalbath. ...... 31 2-2Schematicrepresentationofthetwodecayratesforthethreebathsystemfor=3andRAB=RBL=)]TJ /F3 7.97 Tf 6.58 0 Td[(1XintheEq. 2 andEq. 2 .Theshortandlongtimedecaysaregivenby1.33Xand4.0X,respectively. .......... 39 2-3ObservedrelaxationforHDmoleculesinzeolite13Xfor1moleculepersodalitecageat5.0K.Twodistinctrelaxationtimesareobserved,correspondingtothetwo-bathrelaxationscenario. .......................... 40 2-4BathmodelfornuclearspinrelaxationofCH4connedinzeolite ........ 44 3-190x-180xpulseNMR ................................. 47 3-2IllustratingformationofanNMRspinechofora90-180pulsesequence. .. 48 3-3Pulseapparatusforquantumgasconnedinmesoporousmaterials. ...... 51 3-4DuplexerCircuits ................................... 52 3-5TransmitterCircuitsduringthepulses. ....................... 52 3-6Theactivecircuitofthetransmitter. ......................... 53 3-7PulsesGenerator. .................................. 54 3-8Pulsesandadjustabledacays.Pulsecanddincludebothasmalltimedelaytdandaslowdecays. ................................ 55 3-9ReceiverCircuit. ................................... 55 3-10SchematicillustrationofsamplecellshowingNMRcoilandthermallinkagetosamplefromcoldcap(green)andcoppershroud(red). ............. 56 3-11Schematicrepresentationofthesupportstructureandcoolingpathforthesamplecell.Theexteriorcoppercansitsinaliquidheliumbaththatcanbepumpedto1.4K. ................................... 57 3-12FIDofCH4inzeoliteat70Kafteraveraging50pulsesequences. ........ 57 8

PAGE 9

3-1390x-180xechoofCH4inzeoliteinourexperimentat70Kafteraveragingsignalsfor1moleculepersodelitecage. ...................... 58 4-1Adsorptionisothermformethaneonzeolite-13Xat77KforNstepswitheachstepcorrespondingtotheadditionof10)]TJ /F3 7.97 Tf 6.58 0 Td[(3moles.Thearrowindicatesthestepintheisothermwhichwetakeasthesignatureforcompletingllingat1moleculepersodalitecage.FigurereproducedwithpermissionfromJi.etal. ....... 60 4-2EchoamplitudeasafunctionofNMRpulsetimeintervalforHDinzeoliteat19Kwith1moleculepercage.Thesolidandbrokenlinesshowthettotheshorttimeandlongtimedecaysasexpectedinthe2-bathmodel. ....... 61 4-3EchoamplitudeasafunctionofNMRpulsetimeintervalforCH4inZMOFat35Kwith1moleculepercage.Thesolidandbrokenlinesrepresentthetsfortheshorttimeandtiemdecaysasexpectedinthe2-bathmodel. ...... 62 4-4Temperaturedependenceofthenuclearspin-latticerelaxationtimesobservedforCH4inthe-cagesofzeolite:diamonds,1moleculepercage;triangles,0.5moleculespercage;andsquares,0.05moleculespercage. ........ 63 4-5Comparisonoftheobservedtemperaturedependenceofthenuclearspin-latticerelaxationtime,T1,withthecalculatedvalue(solidredline),assumingindependentmolecularrotorstates.Thebroadpeaksat25Kand55K,areattributedtotwodistinctT-levels.Thesolidgreenlineisthecontributionfromthermalactivationofinter-cagediffusion.FigurereproducedwithpermissionfromJi.etal. .... 64 4-6Observedtemperaturedependenceofthenuclearspin-spinrelaxationtimeT2for1.00.2moleculespersodalitecageofzeolite13X.FigurereproducedwithpermissionfromJietal. ............................ 65 4-7Temperaturedependenceofnuclearspin-latticerelaxationtimesforHDadsorbedinzeolite13x;withcoveragesof1.0moleculespercage(squares)and0.5moleculespercage(triangles).ThesolidlineisabesttassumingSchottkylevelspecicheatsforthreediscreteenergylevels.Eachcontributionisshownbythebrokenlines. ................................. 66 4-8SchematicrepresentationofenergylevelsforthetranslationalmotionofHDmoleculesconstrainedtoasphericalcageof15Aindiameter. ......... 68 4-9Observedtemperaturedependanceofthenuclearspin-latticerelaxationtimeofHDinzeolite13Xathightemperatures.Thesolidline(red)isatforathermalactivationof733Kandanintrinsicturnelingrateof1.51011s)]TJ /F3 7.97 Tf 6.59 0 Td[(1. ....... 69 4-10Temperaturedependenceofnuclearspin-spinrelaxationtimesforHDadsorbedinzeolite13x;withcoveragesof1.0moleculespercage(squares)and0.5moleculespercage(triangles). ........................... 70 9

PAGE 10

4-11Temperaturedependenceofnuclearspin-latticerelaxationtimesforCH4adsorbedinZMOF. ....................................... 71 4-12Temperaturedependenceofnuclearspin-spinrelaxationtimesforCH4adsorbedinZMOF. ....................................... 72 4-13Temperaturedependenceofthenuclearspin-latticerelaxationtimeforHDinZ-MOF.foracoverage1.0moleculepercage.Thebrokenlineisthecalculateddependencedescribedinthetext. ......................... 73 4-14Temperaturedependenceofthenuclearspin-spinrelaxationtimeforHDinZ-MOF.Purple,x=0.1,blue,x=1.0moleculespercage.ThejumpatT=17Kmarkstheonsetofintercagediffusion. .................... 74 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNMRSTUDIESOFQUANTUMGASESCONFINEDINMESOPOROUSMATERIALSByYuJiMay2012Chair:NeilS.SullivanMajor:Physics LowmassmoleculessuchasH2,D2,HDandCH4haveinterestingnewpropertieswhenlocalizedinconstrainedgeometriessuchasmesoporousstructures.Thenewpropertisresultfromthehighlyquantummechanicalcharacterofthetranslationalandrotationaldegreesoffreedom.Wereporttheresultsfotheuseofnuclearrelaxationspectroscopytodeterminetheinteractionenergies,thequantizedtranslationalmodesarethedynamicsfothesegaseswhentrappedinthecagesofthemesoporousstructures. ThesamplesystemwasprobedusingcoherentpulseNMRtechniquesat5.6MHz.Aspecialduplexerwasmadeforthislowfrequencyexperiment. Aseriesofexperimentswithdifferentlightgasesandadsobentwerecarriedout: i)CH4inzeolite13Xfor1,0.5and0.1moleculesper-cageand4
PAGE 12

consistentwithMonteCarlocalculations.TheresultsalsoconrmthehighmobilityanddiffusivityofHDatlowtemperaturesasobservedbyneutronscattering. 12

PAGE 13

CHAPTER1INTRODUCTION 1.1Quantumgasesinconstrainedgeometries Thediscoveryofanewgenerationofcrystallineporousmaterialsintheformofnanoporousmetalorganicframeworks(MOFs)andcloselyrelatedmaterialsisofgreatinterestbecauseof(i)theabilitytostudythefundamentalpropertiesofhighsymmetrynanoclustersofquantumuidsandsolids(H2,HDand3He),and(ii)thepotentialforstoringlargevolumesofhydrogenintheinternalspaces.TheMOFstructureshaveextralargehomogeneousopencavitiesthatcanbetailoredwiththeuseofselectedmetalionsandfunctionalitiestomeetspecicinterests,rangingfrommolecularstoragetocatalysisandsensors.Examplesofthesematerialsincludemobilecrystallinematerials(MCMs),microporousmetalorganicmaterials(MMOMs),molecularbuildingblocks(MBBs),polyoxometallates(POMs),andcobaltnitratenetworks.Ofhighestinterestforthestudiesaretherecentlydevelopedmetal-liganddirectedassembliesthathavebeenusedtoorderrigiddirectionallargecavitiessuitableforforminglargemolecularclustersaswellashighstoragevolumes. LowmassmoleculessuchasH2,D2,HDandCH4haveanumberofpropertiesofapurelyquantummechanicalnaturethathavegeneratedconsiderableinterest,boththeoreticalandexperimental,becauseoftheirpronouncedquantumbehavior.Boththetranslationalandtherotationaldegreesoffreedommustbetreatedquantummechanically.Thetotalwavefunctiontotalistheproductofspaceandnuclearspinwavefunctions,total=space'spin,butbecauseofFermistatisticsand'arenotindependent[ 2 3 ].Asaresult,forH2therearetwodistinctmolecularspecies,orthoandparawithtotalnuclearspinI=1andI=0,andangularmomentumJoddandeven,respectively.ForCH4therearethreemolecularspecies,meta,paraandortho,correspondingtotheA,EandTrepresentationsofthetetrahedralgroup[ 4 ]withtotalnuclearspinI=2,1and0,respectively. 13

PAGE 14

1.1.1ZeroPointMotion Althoughallsolidsshowquantumeffectstoacertaindegree,solidsmadeofthelightestelementshavespecialpropertiesofpurelyquantummechanicaloriginunobservedforothersolids.Forthisreason,theyareknownasquantumcrystals.Themainpropertyofthequantumcrystalsisthattheconstituentatomsareonlyweaklylocalizedwithrespecttoquantumzeropointmotion(ZPM).Theparticleshaveonlysmallmassandthecohesiveforce(typicallyVanderWaals)areveryweak.Therefore,aparticlecanbevisualizedasoscillatingabouttheminimumofashallowpotentialwell.Thewell,beingweak,hasonlyasmallcurvatureandtheamplitudeoftheexcursionsaboutthemeanpositionisverylargecomparedtothatforaheavyclassicalsolidsuchasKr.Thefrequencyofthemotionsdeterminestheaveragequantumkineticenergy.Therefore,thereisalargeamplitudeofZPManddeviationsabouttheequilibriumpositionareasizablefractionofthenearestneighborseparation.TheimportantconsequencesofZPMarecharacterizedbycorrelations,exchange,andanharmonicityofthephononexcitations. InordertogaininsightintotheZPM,aharmonicapproximationispresented.ThestandardinteractionpotentialusedistheLennard-Jonespotential V(r)=4[( r)12)]TJ /F6 11.955 Tf 11.95 0 Td[(( r)6](1) whereisthehardcoreradiusandthepotentialminimum.TherstpartisamathematicallyconvenientformofarepulsiveinteractionandisaconsequenceofthePauliexclusionprinciplewhenelectroniccloudsstarttooverlap.ThesecondtermistheattractiveVanderWaalspotentialduetoinduceddipole-dipoleinteraction.Nopermanentdipolesareinvolvedbutonedipoleiscreatedbythermaluctuationeldsandanotherdipoleisinducedinaneighboringmolecule.AmoresophisticatedpotentialwasempiricallyfoundbySilveraandColdman[ 5 ] 14

PAGE 15

V(r)=exp()]TJ /F9 11.955 Tf 11.96 0 Td[(r)]TJ /F9 11.955 Tf 11.96 0 Td[(r2)+f(r)Xi=6,8,10Ci ri+C9 r9(1) with f(r)=exp)]TJ /F6 11.955 Tf 9.3 0 Td[([1.28Rm r)]TJ /F4 11.955 Tf 11.95 0 Td[(i]2forr<1.28Rm (1) =1forr>1.28Rm (1) wheretheconstantsare=1.713,=1.5671,=0.00993andC6=)]TJ /F6 11.955 Tf 9.3 0 Td[(12.14,C8=)]TJ /F6 11.955 Tf 9.3 0 Td[(215.2,C10=)]TJ /F6 11.955 Tf 9.3 0 Td[(4813.9andC9=143.1allinatomicunitsandRm=3.41Aatthewellminimumforwhichtheninthordertermisexcluded. ThesimpleLennard-JonespotentialisexpandeduptosecondorderinthedisplacementurelativetotheradiusofminimumpotentialenergywhichleadstoaharmonicoscillatorwithHamiltonian H=p2 2m+1 2ku2withk=ZV"(1) whereZisthenumberofnearestneighbors.Thegroundstateenergyofaharmonicoscillatoriswell-knowntobe E=3 2~!=3 2~r ZV" m(1) ThecorrespondingwavefunctionisofGaussianform (u)=1 (2W2)3=4exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2(u W)2(1) withwidthW.Thewidthdescribesthespreadofthespreadofthewavefunctioninspaceandthelocalizationofparticlesonthelattice.Theexpectationvaluesforthekineticandpotentialenergiesoverthisgroundstatewavefunctionare 15

PAGE 16

hjTji=3 4~2 mW2(1) hjVji=3 4ZV"W2(1) Theresultsimplytwoconsequences:rstly,theshallowpotentialwithsmallcurvaturereducesthepotentialenergyandsecondly,thesmallmassincreasesthekineticenergyraisingitsrelativeimportanceevenmore. Acalculationoftheminimumenergywithrespecttothewidthleadsto W=(~2 ZmV")1=4(1) intermsoftherelevantparameters.ItisespeciallylargeforsmallmandsmallV,whichisthecaseforquantumcrystals,suchashydrogen. Hydrogenhasalargekineticenergyrelativetoitssmallpotentialenergyandwouldbetoolocalizedforaboundstatesothatitreducesitskineticenergybyspreadingouttoahigherpotentialenergy.Thisprocessofspreadingthewavefunctionleadstoanewarrangementofthelatticeparameter.Aguidelineisr0)]TJ /F6 11.955 Tf 11.49 0 Td[(2hu2i1=2wherer0isthedistanceofminimalpotentialanduisthedisplacementaboutalatticesite.Ifthisquantityissmallerthanthehard-coreradiustheparticlesspendappreciabletimeineachothers'hardcoreswhichisanunphysicalsituation.Thestrongrepulsiveinteractionscostenergyandanewminimumenergymustbefound.Thesolutiontothedilemmaisanexpansionofthelatticefromr0to.TheprocedurereduceshTiandincreaseshPEi,whichreestablishesthesituationofclassicalsolidsandisacompromiseforacertainwidthW.Empiricallyitisfoundthat)]TJ /F3 7.97 Tf 6.58 0 Td[(21=2 arrangesitselfaroundunity. 16

PAGE 17

1.1.2DeBoerParameter Itisinterestingtodeneameasureforthequantumnatureofamolecule.ThiscanbeapproachedbyrewritingtheSchroedingerequationintermsofreducedquantitiesbyscalingthedistanceandthepotentialwiththeLennard-Jonesparameters r=r andV=V (1) whichresultsin [)]TJ /F13 11.955 Tf 21.29 8.09 Td[(~2 2m2(r)2+V]=E(1) Theprefactorofthereducedkineticenergytermexpressestherelativedominanceofthekineticenergycomparedtothepotentialenergy.ThisistherelevantquantityforquantumbehaviorandisdenedasthedeBoerparameter =~ p 2m2(1) whichislargeforsmallenergyandmass.Thisisconsistentwiththeearlierstatementsandcalculationsaboutthenatureofquantumcrystals.Theparameteris0.196forhydrogen[ 6 ]whichismuchlargerthanforclassicalcrystalswithvaluesbelow0.05. 1.2PropertiesofH2andHD 1.2.1Ortho-andParahydrogen Fortheunderstandingofmanyeffectsitisnecessarytobefamiliarwiththespecialpropertiesofhydrogen.Hydrogenisthesimplestexistingdiatomicmoleculeandcanthereforebeusedasamodelmoleculetounderstandfundamentalinteractionsandphenomena.Whenitiscooledbelow13.8Kitisalsothesimplestmolecularsolid.Thebeautyofhydrogenisthattheinvolvedinteractionsareunderstoodintermsofrstprinciples. 17

PAGE 18

Figure1-1. VariationofRMSdeviationhu2i1=2withdeBoer'squantumparameter. Hydrogenisspecialinitsproperties.Itisaveryinterestingsubstanceasitisexplicitlyquantum-mechanicalinitsnature.Therearethreerealizationsofthisfact:theexistenceoftwospecies(orthoandpara),thelargezero-pointmotionandthequantumrotorproperty.Hydrogenoccursnaturallyasadiatomicmolecule,Itsnuclearconstituentsarespin-1/2protonswiththeelectronicspincontributionspairedinasymmetric1P+gmoleculargroundstate.Thewavefunctioncanbewrittenasaproductofnuclear,rotationalandvibrationalwavefunctions.Thevibrationalgroundstateissymmetricanddoesnotinuenceconsiderationsofthelowtemperatureproperties.Thequantum-mechanicalrequirementofatotallyantisymmetricwavefunctionfortwoindistinguishablefermionscanthereforeberealizedintwodistinctways,leadingtotwodistinguishablemolecularspecies:ortho-andparahydrogen.Orthohydrogenhasasymmetricnuclearspinwavefunction(I=1)andanantisymmetricorbitalwavefunction(Jodd).Thenomenclatureissuchthatthespecieswiththehighestspinquantumnumberiscalledortho.Parahydrogenhasanantisymmetricnuclearwavefunction( 18

PAGE 19

I=0)andasymmetricorbitalwavefunction(Jeven).Itisimportanttonoticethatbothspeciesofhydrogenarebosons. Becauseofthesmallmoment-of-inertiaIforhydrogen,theseparationoftherotationalstatesisverylarge:EJ=BJJ(J+1)withBJ=~2 2I=85.4K.ThequantumnumberJisthereforeagoodquantumnumberatlowdensities.Theintermolecularinteractionswhichcanmixdifferentrotationalstatesareveryweak(typically1K)andthereforeanymixingofhighJstatesinthegroundstateisnegligible.Theground-stateistheJ=0parahydrogenstate.Thisstillholdswhenthesurface-moleculeinteractionoftherestrictedgeometryisincludedunlessthatinteractionismagnetic. 1.2.2Ortho-paraConversion Atstandardtemperatureandpressure,hydrogengascontainsabout25%oftheparaformand75%oftheorthoform,alsoknownasthenormalform.[ 7 ]Theequilibriumratiooforthohydrogentoparahydrogendependsontemperature,butbecausetheorthoformisanexcitedstateandhasahigherenergythantheparaform,itisunstableandcannotbepuried.Conversionishoweververyslowandsamplesofnon-equilibriumorthohydrogencanbepreparedforexperimentalstudies.Atverylowtemperatures,theequilibriumstateiscomposedalmostexclusivelyoftheparaform.Theliquidandgasphasethermalpropertiesofpureparahydrogendiffersignicantlyfromthoseofthenormalformbecauseofdifferencesinrotationalheatcapacities,asdiscussedmorefullyinSpinisomersofhydrogen.[ 8 ]Theorthoparadistinctionalsooccursinotherhydrogen-containingmoleculesorfunctionalgroups,suchaswaterandmethylene,butisoflittlesignicancefortheirthermalproperties.[ 9 ] TheuncatalyzedinterconversionbetweenparaandorthoH2increaseswithincreasingtemperature;thusrapidlycondensedH2containslargequantitiesofthehigh-energyorthoformthatconvertstotheparaformveryslowly.Theortho-paraconversionisveryslowbecausethetransitionbreaksbothnuclearspinsymmetryandorbitalspinsymmetryandisthusdoublyforbidden.Themagneticnuclearspin-spin 19

PAGE 20

interactionsaretheonlyinteractionthatleadtoconversioninthepuresystem.TheorthopararatioincondensedH2isanimportantconsiderationinthepreparationandstorageofliquidhydrogen:theconversionfromorthotoparaisexothermicandproducesenoughheattoevaporatesomeofthehydrogenliquid,leadingtolossofliqueedmaterial.Catalystsfortheortho-parainterconversion,suchasferricoxide,activatedcarbon,platinizedasbestos,rareearthmetals,uraniumcompounds,chromicoxide,orsomenickelcompounds,areusedduringhydrogencooling. Thereasonforstudyinghydrogen-deuteride(HD)isthatitisverysimilartoparahydrogenanddoesnothavetheorthoparaconversionasinH2.InHD,onehydrogenatomissubstitutedbyoneheavyhydrogenatom,thedeuteron.HDhasslightlymoremassthanhydrogenandthereforeslightlylesszero-pointmotion.Itotherwisebehavesverysimilartoparahydrogen.ThisisalsothereasonforanexceptionalpropertyofHD:itscenterofmassandcenterofrotationdonotcoincide.ThereplacementofhydrogenbyHDhastheadvantagethatparahydrogencanbesimulatedandNMRcanbeusedtostudyHDwhichisnotpossiblewithpara-H2.Thisisofspecialinterestinthecasewhereitisnecessarytodistinguishbetweenorientationalandtranslationalproperties.HDdoesnothaveorientationaldegreesoffreedombecauseitissphericallysymmetric.Orientationalorderingcanthereforenottakeplace.Thisisanothervaluabletesttodistinguishbetweenorientationalorderingandsupercooling. 1.2.3Quadrupolarinteraction Parahydrogenissphericallysymmetricwithnoelectricormagneticmoments.Thelackofnuclearspindegreesoffreedomforparahydrogenmakesitalsoundetectablebynuclearmagneticresonance.Thisisincontrasttoorthohydrogenwithspin1whichalsohasanelectricquadrupolemoment.Themoleculestendtoorientatlowtemperatureinordertominimizetheanisotropicinteraction. 20

PAGE 21

VQQ=( R))]TJ /F4 11.955 Tf 11.66 0 Td[(F(1,2)(1) whichoccurswhenthemolecularaxesarealignedata90degreeangle,aTcongurationbetweenquadrupoles.Hereisthelatticeparameter,Rtheintermoleculardistance,Fanangularfunctionoftheanglesmeasuredrelativetotheconnectinglinebetweenmolecules,and)]TJ /F1 11.955 Tf 10.09 0 Td[(=0.8K. Theorderparametersspecifyingthedegreesoffreedomforthisinteractionarecomponentsofthesecondrankquadrupolartensorwithrespecttothelocalreferenceaxes(x,y,z) Q=1 2J2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(3 4(JJ+JJ)(1) NotallthecomponentsQareindependent,andusingsymmetryargumentsonendsthattherearetwoadditionalparametersbesidesthethreelocalreferenceaxes.Thereasonforonlytwoadditionalparametersisthevanishingorbitalangularmomentum,whichmeansthathJii=0isquenched.Thisisthecasewhentime-reversalsymmetryisnotbrokenwhichcanbeassumedhere,eventhoughamagneticeldisapplied.Theratioofthemagneticinteractionenergy(mK)totheinvolvedrotationalenergies(BJ=~2 2I=85.4K)isthereasonforthisapproximation:themagneticenergyisonlyofnegligibleinuence.Theargumentforquenchingisrelatedtothenon-degeneratenatureoftherotationalgroundstate.Anon-degenerategroundstatemustberealotherwiserealandimaginarypartsofthewavefunctionwouldseparatelybesolutions,whichcontradictsnon-degeneracy.TheangularmomentumoperatorsareimaginaryandtheexpectationvaluehJiiwithi=x,y,zisthereforepurelyimaginary.Ontheotherhanditmustalsoberealduetohermiticityandtheonlyexpectationvaluesatisfyingbothconditionsiszero.AthighertemperaturesthisrequiresmorecarefulconsiderationbecauseJZ=1arealsooccupiedandcouldcontributetotherotationalZeemanenergy.Thisisnotthecase,however,becausethetransition 21

PAGE 22

frequencyamongstthesestatesismuchhigherthanthedipolarfrequencyandisthereforeaveragedout.Alloff-diagonalmatrixelementsarealsozerobythepropertiesofangularmomentum. Oneadditionalorderparameterfororthohydrogenisthealignment =QZZ=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(1 23J2Z)]TJ /F6 11.955 Tf 11.95 0 Td[(2(1) ForJZ=0thealignmentis=1,whichdepictsaprolateellipsoidprobabilityforthewavefunction.ForthecaseofoblateellipsoidwhereJZ=1thealignmentis=)]TJ /F6 11.955 Tf 9.3 0 Td[(1=2. Theotheradditionalorderparameteristheeccentricitydescribedby =J2x)]TJ /F4 11.955 Tf 11.96 0 Td[(J2y(1) andiszeroforanyrotationallysymmetricbody. 1.3PropertiesofCH4 ThereisconsiderablefundamentalinterestintherotationaldegreesoffreedomofCH4innanoscalegeometriesbecauseoftheirspecialquantumproperties.Asbasisfunctionsforthetetrahedralrotorsweconsideronlythespatialfunctionsfortherotationaldegreesoffreedomgivenby(A)rot,(T)rot,and(E)rot,correspondingtotheA,EandTirreduciblerepresentationsofthetetrahedralgroupTd.(Therearebotheven(A1,T1)andodd(A2,T2)representations.)Similarly,weconsider'(A)spin,'(T)spin,and'(E)spinforthenuclearspindegreesoffreedom.Thetotalwavefunctiontotal=rot'spin,istheproductofthespaceandnuclearspinwavefunctions,butbecauseofFermistatisticsrotand'spinarenotindependent.totalmustbetotallyantisymmetricandthereforetransformaccordingtotheA2representationofthepermutationgroupS4.Asaresulttheonlyallowedstatesaregivenbytheconstructions:(A2)rot'(A1)spin,(E)rot'(E)spin,(T2)rot'(T1)spin,correspondingtothemeta-,para-andortho-moleculespinspecies,respectively. 22

PAGE 23

Figure1-2. EnergylevelsofCH4. ThethermodynamicandNMRpropertiesaredeterminedbythequantumnatureofthesestatesandtheirinteractionsintheconnedgeometries.Thegroundstateisthe5-folddegenerateA-statewithnuclearspinI=2.TheintermediateenergyT-state(I=1)is9-folddegenerate,andthehighenergyE-state(I=0)isnotobservedinNMRexperiments.Inbulksamples,CH4undergoescollectiveorder-disordertransitionsatT20Kinducedbythefrustratedanisotropicoctopole-octopoleintermolecularinteractions.AsimilartransitionatT=17KisobservedinmonolayerlmsongraphitebutstudiesoftherotationaldegreeoffreedomofCH4innanocageshas,tothebestofourknowledge,notbeencarriedout. Figure 1-2 showtheenergylevelsofthefreerotorstateandorderedmoleculesinbulkCH4andFigure 1-3 givestheenergylevelsindifferentcrystalelds. 23

PAGE 24

Figure1-3. EnergylevelsofCH4incubiceld. 1.4PropertiesofZeolite 1.4.1StructureofZeolite Zeoliteisanaturallyoccurringmineralgroupconsistingofover50differentminerals.ThenameZeoliteisGreekandmeansboilingstone.Aconsequenceofitsstructureandchemistryisthatitboilsandmeltstoaglassasitisheatedinaame.Beforereachinghightemperatures,above500C,thezeoltereleaseswaterbutdoesnotdisintegrate.Thisisincontrasttomostotherwater-bearingcrystalsandresultsfromtheporousstructureofzeolite.Thereareothernamesassociatedwithzeolitealludingtotheiruniqueproperties:molecularsieve,solidsolventorionexchanger.Itiswidelyusedbecauseof1)Selectiveadsorptionduetotheuniformporesizeofthezeolitestructure;and2)Highadsorptioncapacityforpolarsubstancesatlowconcentrations. Thereexistsagreatvarietyofnaturallyoccurringaswellassyntheticzeolites.Allofthemaredenedasaluminosilicateswithaframeworkstructureenclosingcavitiesoccupiedbylargeionsandwatermolecules,bothofwhichhaveconsiderablefreedomofmovement,permittingionexchangeandreversibledehydration.[ 10 ]They 24

PAGE 25

Figure1-4. ZeolitestructuresfortypeAandX. aregroupedintosodalites,Chabazites,Philipsites,AnalcimesandMordenites.[ 11 ]Themostcommonlyusedzeolitesarethesyntheticallypreparedandcommerciallyavailable(UnionCarbideCorporation)zeoliteXandAofthesodalite-faujasitegroup.Hereweonlydiscussthesetwostructures. Thestructureofzeoliteiscomplicatedbutcompletelyregularandresemblesajungle-gym.Figure 1-4 providesasimpliedviewofthestructureofzeoliteAandX.[ 12 ]Thefundamentalbuildingblockofanyzeoliteisatetrahedronoffouroxygenionswithacenteredaluminumorsiliconion.Eachoxygencarriestwooxygenionswithacenteredaluminumorsiliconion.Eachoxygencarriestwonegativecharges,oneofthembeingcompensatedbySi.TheremainingsinglechargeontheoxygenenablesittocombinewithotherSiionsandtoextendthecrystal.IfSiissubstitutedbyAlonelesschargeisneutralizedandanadditionalcation,suchassodiumorcalcium,isneeded.Theexchangeablecationsattachthemselveslooselytotheoxygensandactasbarriersinthechannelsconnectingthepores.Theseadditional,looselyboundcationsarethereasonforthefacilitatedexchangeofions,suchasNabyCa,inzeolites. Thestructureformedbythetetrahedraissimilartothestructuralunitinsodalite,thesodaliteunit.Itcontains24(Si,Al)and36oxygensandformsatruncatedoctahedron 25

PAGE 26

withonetetrahedronateachcorner.Tounderstandthisoneneedstoknowthatatruncatedoctahedronconsistsofeighthexagonalface,sixsquarefaces,24verticesand36edges. AtthispointthedifferencebetweenzeolitetypeAandXcomesintoplay.ForzeoliteAtheoctahedronassemblealongthesquareface(Figure 1-4 )whileforZeoliteXthehexagonalfacesconnect.Thisproducesacentral,truncatedcubicoctahedronwithaninternalcavitydiameterof9Afortype5A,consistingofeightsodaliteunitsonasimplecubiclattice.FortypeXatetrahedralarrangementof10sodaliteunits,asindiamond,isformedwithaninternalcavitydiameterof13Aandaquasi-sphericalshapewithfourcapscutoff.Thesecagesaretermedcagesorsuper-cages,incontrasttothecageswithintheinitialoctahedronof6.6Adiameter.Thecagesaretoosmalltobeocuppiedbecausetheirconnectingchannelsareonly2.2A.Theconnectingchannelsbetweenthecagesintype5Aareabout5Awide,formedbyaringoftwelveoxygenions.Bothtypesofsuper-cagesallowforoccupationbysmallmolecules. ThechemicalformulasforthealuminosilicateszeoliteAandXareNa56[(AlO2)56(SiO2)136]264H2OandNa86[(AlO2)86(SiO2)106]264H2Orespectively.TheformulaexpressesthesimilaritybetweenthetwotypeswiththedifferenceofanalteredAl-Siratio,whichleadstothedifferencesinthetwolatticestructures.ThedensityofzeoliteXdecreasesfrom1.97g/ccto1.31g/ccupondehydration.ThisisevenmoreprominentfortypeAwithareductionfrom1.99g/ccto1.27g/cc.Thevoidvolumeorporosityis50%forXand47%forA.Therelativesurfaceareaislargewith800m2/g.Thesymmetryiscubicinbothcases. Zeolitecanbesynthesized.Itistherebypossibletocreatevariationsthatdonotexistinnature.Theprocedureistoprepareahighlyreactivealuminosilicategelwhichisanaqueoussolutionofaluminate,sodiumsilicateandsodiumhydroxide.Thegelcrystallizesattemperaturesbetweenroomtemperatureand150centigradesunder 26

PAGE 27

atmosphericorelevatedpressures.Largenumbersofcrystallitenucleiareformedwhichgrowfromthesupersaturatedgel. Inadditiontothefascinatingmicroscopicstructurewhichisworthwhileinvestigatinginitself,zeoliteisvariablyusableinevery-dayapplications.ItisusedasawatersoftenerexploitingitscapabilityofexchangingNaforCa.Thepossibilityofcustom-designingthewidthofconnectingchannelsbyvaryingtheadsorbedneutralizingcationsopensawidevarietyofselectiveltering.Agoodexampleistheprocessofupgradinggasolinebyseparatingchainfrombranchinghydrocarbons.Thechainsslipthroughthechannelswhichisnotthecaseforthelessignitablebranchingstructure. Theioniccharacterofthealuminosilicatesalsoallowsforselectiveseparationofdifferentlypolarizablemolecules.Polarmoleculesandsaturatedhydrocarbonsaremorereadilyadsorbed. Zeolitesmayevenbeusedascarriersofvolatilecatalysts.Thecatalystsaretrappedintheporesandfacilitatethechemicalreactionbutarenotlostintheprocess. Itisimportanttokeepinmindthatzeolitesareveryregularporousmaterialswithawidespectrum[ 13 ]ofmonodiperseporesizesconnectedbychannelsofdiametersbetween3.8and10Aaccordingtothechosentypeofzeolite.Thisallowstheexperimentertoselectanappropriateporedimensionwiththeadvantageofmonodispersity.Theoutcomeandresultsofanexperimentarethereforemoreeasilyrelatabletoaspecicporeradius. 1.4.2MonteCarloSimulation-MethaneadsorptioninZeolite AMonteCarlocomputersimulationofCH4inzeoliteYcarriedoutbyS.Yashonath[ 14 ]showsthelowestenergysitetobenearoneofthehexagonalfaces,andCH4moleculesarefreetolibratewithrespecttolocalsymmetryaxes,dependingonthestrengthofthelocaleld.Atlowtemperaturesthemoleculesarelocatedinapotentialminimumnearoneofthefacesandrollorlibratewithrespecttotheaxisofthe 27

PAGE 28

localcrystaleld.Eventuallyatthehighesttemperaturesthebindingtothesurfaceisovercomeandmoleculescantravelbetweencages. 1.5PropertiesofZ-MOF Metal-OrganicFrameworksarecrystallinecompoundsconsistingofmetalionsorclusterscoordinatedtorigidorganicmoleculestoformone-,two-,orthree-dimensionalstructuresthatcanbeporous.Insomecases,theporesarestabletoeliminationoftheguestmolecules(oftensolvents)andcanbeusedforthestorageofgasessuchashydrogenandcarbondioxide. ThestudyofMOFsdevelopedfromthestudyofzeoliteswithverylittlechangeinsynthetictechnique.MOFsandzeolitesalikeareproducedalmostexclusivelybyhydrothermalorsolvothermaltechniques,wherecrystalsareslowlygrownfromahotsolutionofmetalprecursor,suchasmetalnitrates,andbridgingligands.Zeolitesynthesisoftenmakesuseofavarietyoftemplates,orstructure-directingcompounds,andafewexamplesoftemplating,particularlybyorganicanions,areseenintheMOFliteratureaswell.AparticulartemplatingapproachthatisusefulforMOFsintendedforgasstorageistheuseofmetal-bindingsolventssuchasN,N-diethylformamideandwater.Inthesecases,metalsitesareexposedwhenthesolventisfullyevacuated,allowinghydrogentobindatthesesites. MetalOrganicFrameworksattractattentionasmaterialsforadsorptivehydrogenstoragebecauseoftheirexceptionallyhighspecicsurfaceareasandchemically-tunablestructures.MOFscanbethoughtofasathree-dimensionalgridinwhichtheverticesaremetalionsorclustersofmetalionsthatareconnectedtoeachotherbyorganicmoleculescalledlinkers.HydrogenmoleculesarestoredinaMOFbyadsorptiononitssurface.Comparedtoanemptygascylinder,aMOF-lledgascylindercanstoremoregasbecauseofadsorptionthattakesplaceonthesurfaceofMOFs.(Notethatmolecularhydrogenadsorbstothesurface,notatomichydrogen.)Furthermore,MOFsarefreeofdead-volume,sothereisalmostnolossofstoragecapacityasa 28

PAGE 29

resultofspace-blockingbynon-accessiblevolume.Also,MOFshaveafullyreversibleuptake-and-releasebehavior:sincethestoragemechanismisbasedprimarilyonphysisorption,therearenolargeactivationbarrierstobeovercomewhenliberatingtheadsorbedhydrogen.ThestoragecapacityofaMOFislimitedbytheliquid-phasedensityofhydrogenbecausethebenetsprovidedbyMOFscanberealizedonlyifthehydrogenisinitsgaseousstate. 29

PAGE 30

CHAPTER2THEORY-NUCLEARSPINRELAXATION 2.1GeneralBathModel Inordertoevaluatemesoporousmaterialsfortheseapplicationsdescribedaboveitisimportanttodeterminethestrengthofinteractionsofthemoleculesbetweenthemselvesandwiththeconningwalls,andtodeterminethekineticsofthegasescontainedintheporousmaterials.Oneoftheidealtoolsforstudyingthefundamentalpropertiesofquantumgasestrappedinmesoporousstructuresisnuclearmagneticresonance(NMR).NMRisverysensitivetothemotionofthemoleculesasthemotionmodulatesthespin-spininteractionsandthereforedeterminesthenuclearspinrelaxationtimesthatcanbemeasuredoverawidetemperaturerange.Furthermoretherelaxationproceedsviathecouplingofthenuclearspindegreesoffreedomtothemolecularrotationalandtranslationaldegreesoffreedomwhiletrappedinthemesoporouscagesandtothephononsofthewallsofthemesoporousstructures.Thenuclearspindegreesoffreedomgenerallyreachinternalequilibriumveryrapidly(withthemselvesandthemolecularrotationalandtranslationalmotions)ontimesof10msec.determinedbythenuclearspin-spininteractionandthenuclearspin-rotationalinteraction,respectively.Thereis,however,abottleneckintheenergyowfromthemoleculardegreesoffreedomtothewallsbecauseofthelargeacousticmismatch(knownwidelyasaKapitzaresistance[ 15 ]).Thisbottleneckenablesonetostudythemoleculardegreesoffreedombymeasuringthenuclearspinrelaxationtimes.Thisprocessforreachingequilibriumiscalledthegeneralbathmodel.Ofparticularinterestisthethermalactivationofdiffusionfromonemesoporetoaneighboringpore. Wenowdevelopindetailthetheoryofnuclearspin-latticerelaxationprocessesthatarerelevantforthebathmodel,thatiswhentheexcitationofthemolecularsystemscanbedescribedintermsofcoupledthermalreservoirsthathavewelldenedtopologicalconnectionsandweaknuclearspincouplingstothelatticeorbetweendifferentdegrees 30

PAGE 31

offreedom(e.g.betweentunnelingexcitationsandlatticephonons).Thismultiplebathmodelhasbeenusedtodescribethenuclearspinrelaxationofsolid3Heatlowtemperatures[ 16 ],soliddeuteriumatintermediatetemperatures[ 17 ]andthediffusionofHDimpuritiesinsolidhydrogen[ 18 19 ].FollowingGuyer,RichardsonandZane[ 16 ]weconsiderthreeenergybathsA,BandCwhosenuclearspinscanbeconsideredasquasi-independent(e.g.ortho-H2,para-D2andHDimpuritiesinapara-H2matrix)thatareweaklycoupledtogetherandforwhichonlyoneoftheenergybathsisstronglycoupledtothelattice(Figure 2-1 ).BathAconsistsofthenuclearZeemanenergyandcanincludethegroundstateoftheatomicormolecularstatesandBandCareassociatedwithotherdegreesoffreedom,quantumtunneling,phononsetc.thatareinvolvedinthetransferofenergytothethermalbath.IfthereisabottleneckintheowofenergybetweenbathsBandCtheobservedrelaxationdependsontheheatcapacitiesofthebaths. Figure2-1. SchematicrepresentationthreeenergybathsA,BandCwithaweallink(bottleneck)betweenBandCwithenergyowformCtothemainthermalbath. Forthecaseofsolid3He,AisthenuclearspinsystemandBrepresentstheexcitationsduetoatomicmotion(quantumtunnelingmotionsand/orvacancymotion)andCisthephononsystemforsolid3Hewhichisincontractwiththecontainingwallwhichislinkedtothethermalbath.InthiscasetheheatcapacityforbathAis 31

PAGE 32

CA=1 2NkB(~!L=kBT)2wherekBisBoltzmann'sconstant,!ListhenuclearLarmorfrequencyandTistheabsolutetemperature.ForbathBtheheatcapacityisCB=3 8zNkB(~J=kBT)2whereJistheexchangefrequencyduetoquantumtunneling,andforC,CC=234NkB(T=D)3whereDistheDebyetemperatureforsolid3He.(Weusethelowtemperaturelimitforthecurrentexperiments.) ForthecaseofmethanethebathAincludesthenuclearZeemanenergyandthegroundstatefortherotationaldegreesoffreedom.Therearethreemolecularstatesforamethanemolecule,A,TandE,correspondingtothedifferentspin-symmetrycongurationsforatetrahedralgroupoffermions.TheEstatehasnototalnuclearspinandplaysnoroleinthenuclearrelaxationprocesses.TheexcitedstatesarethereforegivenbytheTspeciesofwhichthereareatleasttwo,T1andT2,correspondingtothedifferentsymmetrysitesinazeolitecage.ThenuclearZeemanenergyandtherotationalgroundstateoftheAspeciesareconsideredasonesinglebathbecausetheyaretightlycoupled.TheheatcapacityforbathBisthesumofthecontributionsfortheT1andT2excitations,withC=Pi=1,2giNikB(i=T)2exp()]TJ /F6 11.955 Tf 9.29 0 Td[(i=T)=(1+giexp()]TJ /F6 11.955 Tf 9.29 0 Td[(i=T))2whereiaretheexcitationenergiesfortheT1andT2statesandgiaretherelativedegeneracies. ForHDmoleculestrappedinmesoporousstructures,AisthecombinedbathfortheZeemanandthegroundstateforthetranslationaldegreesoffreedom,andBisthebathcorrespondingtotheexcitedstatesfortheHDmolecules(wavenumbernon-zero).InthiscaseCAisexpectedtobethelowtemperatureDebyeheatcapacity(forthegroundstate)andCBisthesumoftheSchottkyheatcapacitiesforeachexcitedenergylevel. Forthethreebaths,A,BandCofFigure 2-1 ,thetotalHamiltonianforthecompletesystemiswrittenas H=HA+HB+HC+HAB+HBC(2) 32

PAGE 33

whereHA,HBandHCaretheHamiltoniansforsystemsA,B,andCrespectively,andHABandHBCrepresentstheinteractionbetweenAandB,andBandC,respectively.ItisassumedthatthethreebathsareinthermalequilibriumwiththemselvesandbehaveindependentlyexceptfortheweakcouplingsHABandHBC,i.e.[HA,HB]=0while[HA,HAB]6=0and[HB,HBC]6=0,andsimilarlyforHBandHC.TheevolutionofAfollowingaperturbationfromequilibrium(e.g.bytheapplicationofanRFpulse)isdeterminedby d dt=d dtTr(HA)=Tr[d dtHA](2) whereisthedensitymatrixforthecompletesystem.ThecalculationcanbedevelopedfollowingthedescriptionofAbragam[ 20 ]usingtheinteractionrepresentation(toremovethetrivialtime-dependenceoftheLamorfrequency)withd dt=[HAB,]. d dt=)]TJ /F13 11.955 Tf 9.3 0 Td[(~2Z10dTr[HA<[HAB(),[HAB(0),()]TJ /F9 11.955 Tf 11.96 0 Td[(0)]]>](2) where0istheequilibriumdensitymatrix.Thenextstepistoassumethateachbathcanbedescribedbyaspintemperature,orinversespintemperature)]TJ /F3 7.97 Tf 6.58 0 Td[(1A=kBTA(t):i.e. =exp[)]TJ /F9 11.955 Tf 9.3 0 Td[(AHA]exp[)]TJ /F9 11.955 Tf 9.3 0 Td[(BHB](2) AstheB-systemapproachesthetemperatureoftheA-system(afteraperturbationfromequilibrium),wecanwrite =[1)]TJ /F9 11.955 Tf 11.95 0 Td[(A(t))]TJ /F9 11.955 Tf 11.96 0 Td[(B(t)]0(A)(2) where0(A)istheequilibriumdensitymatrixforHAforinversetemperatureA.UsingEq. 2 andEq. 2 wend 33

PAGE 34

d dt=[A(t))]TJ /F9 11.955 Tf 11.95 0 Td[(B]~2Z10d(2) wheretheaverage =Tr[exp()]TJ /F9 11.955 Tf 9.3 0 Td[(H0)HA(t)[HAB(t),[HAB(),HA(t)]]](2) FromEq. 2 wecanalsowrite d dt=)]TJ /F9 11.955 Tf 9.3 0 Td[(A(t)=)2(2) NotingthattheheatcapacityisgivenbyCA=d dt=d dTd d=kAd dT,wecandeneaheatcapacityconstantkA=dEA dandsimilarlyfork.Thecross-relaxationrateT)]TJ /F3 7.97 Tf 6.58 0 Td[(1ABbetweentheAandtheBreservoirisdenedby@A @t=)]TJ /F4 11.955 Tf 9.3 0 Td[(T)]TJ /F3 7.97 Tf 6.59 0 Td[(1AB(A(t))]TJ /F9 11.955 Tf 12.04 0 Td[(B)andusingEq. 2 wend T)]TJ /F3 7.97 Tf 6.58 0 Td[(1AB=~2kAZ10d(2) Fromenergyconservationd dt=0andusingd dt=kA@A @tandd dt=kB@B @twecanemploy@B @t=)]TJ /F4 11.955 Tf 9.3 0 Td[(T)]TJ /F3 7.97 Tf 6.59 0 Td[(1BA(B)]TJ /F9 11.955 Tf 11.95 0 Td[(A)toshowthat T)]TJ /F3 7.97 Tf 6.59 0 Td[(1BA=T)]TJ /F3 7.97 Tf 6.58 0 Td[(1ABkA kBorRBA=RABkA kB(2) whereRAB=T)]TJ /F3 7.97 Tf 6.58 0 Td[(1ABandsimilarlyforRAB.Eq.A.10isaveryimportantresultfortheanalysisoftheexperimentalresultsdiscussedbelowbecausethetotalheatcapacitykA+kBanditscomponentscanbeverydifferentdependingontherelevantdegreesoffreedomforthedifferentenergybathsandtheirtemperaturedependence.Thisdependenceontheheatcapacitieswillleadtosignicantchangesintheobservedrelaxationenergiesassociatedwiththedifferentmodesthatdeterminetheheat 34

PAGE 35

capacitiesoftheenergybaths;hencetheterm,nuclearrelaxationspectroscopy.Examplesofthisdependencearetherelaxationratesfordifferentmolecularspeciessuchastheorthoandparaspeciesinhydrogen,deuterium[ 17 ][ 18 ]andinmethane[ 21 ] Forthethree-bathsystemwehavetwocoupledlineardifferentialequations: @A @t=)]TJ /F4 11.955 Tf 9.3 0 Td[(RAB(A)]TJ /F9 11.955 Tf 11.96 0 Td[(B)(2) @B @t=)]TJ /F4 11.955 Tf 9.3 0 Td[(RBA(B)]TJ /F9 11.955 Tf 11.95 0 Td[(A))]TJ /F4 11.955 Tf 11.95 0 Td[(RBC(B)]TJ /F9 11.955 Tf 11.95 0 Td[(C)(2) @C @t=)]TJ /F4 11.955 Tf 9.3 0 Td[(RCB(C)]TJ /F9 11.955 Tf 11.96 0 Td[(B))]TJ /F4 11.955 Tf 11.96 0 Td[(RCL(C)]TJ /F9 11.955 Tf 11.96 0 Td[(L)(2) UsingRBA=RABkA kBwecanwriteEq.A.12as @B @t=)]TJ /F4 11.955 Tf 10.86 8.09 Td[(kA kBRAB(B)]TJ /F9 11.955 Tf 11.95 0 Td[(A))]TJ /F4 11.955 Tf 11.95 0 Td[(RBC(B)]TJ /F9 11.955 Tf 11.95 0 Td[(C)(2) andifCreachesequilibriumrapidlycomparedtothebottleneckbetweenBandC,Equation 2 becomes@C @t=0. WenowconsiderthecaseforwhichbathsAandBcomerapidlytoacommontemperature,allowingonetoexpressEquation 2 asB)]TJ /F9 11.955 Tf 12.53 0 Td[(A=1 RAB@A @t1 RAB@B @t.InsertingthisrelationinEquation 2 wehave @B @t=)]TJ /F4 11.955 Tf 10.81 8.09 Td[(RBA RAB@B @t)]TJ /F4 11.955 Tf 11.96 0 Td[(RBC(B)]TJ /F9 11.955 Tf 11.96 0 Td[(C)(2) NowfollowingthepremisethatthelinkBCistheweakest,thatisthatAandBreachequilibriummuchfasterthanBandC,Eq. 2 canberewrittenas @B @t[1+RBA RAB]=)]TJ /F4 11.955 Tf 9.3 0 Td[(RBC(B)]TJ /F9 11.955 Tf 11.95 0 Td[(C)(2) 35

PAGE 36

TherelaxationrateforB,thelongesttimeconstantinthesystem,andthustheeffectiveoverallnuclearspin-latticerelaxationratethatisobserved,isgivenby Re=RBC=[1+kA=kB](2) wherewehaveusedkBRBA=kARAB.Equivalently,theeffectivenuclearspinrelaxationtimeis e=[(kA+kB)=kB]X(2) whereX=R)]TJ /F3 7.97 Tf 6.58 0 Td[(1BCisthecross-correlationtimebetweenbathsBandC. AsnotedbyGuyer,RichardsonandZane[ 16 ],ifthecouplingofthebathAandBisslowerthanthecouplingbetweenBandC,theninsteadofEq. 2 onobservese=R)]TJ /F3 7.97 Tf 6.59 0 Td[(1AB.TheanalysisalsoapplieswhenbathCandthethermalreservoiraremergedintoonebath,exceptnowEq. 2 becomes e=[(kA+kB)=kB]BL(2) whereBListhecouplingofBtothelattice.Theimportantpointisthatinbothcaseswherethereisabottleneckbetweentwobathstheeffectiverelaxationtimedependsontheheatcapacities(Eq.A.18andEq. 2 ,andthusmeasurementsofthetemperaturedependenceoverarangethatcoversthecorrespondingenergiescanbeusedtodeterminekAandkBandthustheexcitationspectrum. InmostcasesoneobservesinadditiontotheprincipalrelaxationgivenbyEq. 2 andEq. 2 ,asmallershorter(orsometimesslower)componenttotherelaxation.InordertoanalyzethisfeatureweneedtogobeyondthetreatmentofGuyer,RichardsonandZaneandconsidermoregeneraltimedependencesoftheform eA)]TJ /F9 11.955 Tf 11.96 0 Td[(A(t)=Ae)]TJ /F5 7.97 Tf 6.58 0 Td[(R+t+Ae)]TJ /F5 7.97 Tf 6.58 0 Td[(R)]TJ /F5 7.97 Tf 6.25 1.08 Td[(t(2) 36

PAGE 37

and eB)]TJ /F9 11.955 Tf 11.95 0 Td[(B(t)=Be)]TJ /F5 7.97 Tf 6.58 0 Td[(R+t+Be)]TJ /F5 7.97 Tf 6.59 0 Td[(R)]TJ /F5 7.97 Tf 6.25 1.08 Td[(t(2) whereeAandeBaretheequilibriuminversetemperatures.Theinitialconditionsare 0A=eA)]TJ /F6 11.955 Tf 11.96 0 Td[((A+A)and0B=eB)]TJ /F6 11.955 Tf 11.96 0 Td[((B+B)(2) WeareprincipallyinterestedinthecasewheresystemBisweaklycoupledtoeitheranotherbathortothelattice.SubstitutingEq. 2 andEq. 2 intherateequationswendfortherateconstants R=1 2(RAB+RBA+RAL+RBL)p [(RAB+RBA)2+(RAL)]TJ /F4 11.955 Tf 11.96 0 Td[(RBL)2+2(RAB)]TJ /F4 11.955 Tf 11.96 0 Td[(RBA)(RAL)]TJ /F4 11.955 Tf 11.95 0 Td[(RBL)] andfortheamplitudeswehave A=RAL+RAB)]TJ /F4 11.955 Tf 11.95 0 Td[(R)]TJ ET q .478 w 123.88 -383.53 m 208.01 -383.53 l S Q BT /F4 11.955 Tf 143.53 -394.72 Td[(R+)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 27.93 9.99 Td[((eA)]TJ /F9 11.955 Tf 11.95 0 Td[(0A))]TJ /F4 11.955 Tf 25.2 8.09 Td[(RAB R+)]TJ /F4 11.955 Tf 11.95 0 Td[(R)]TJ /F6 11.955 Tf 8.28 9.99 Td[((eB)]TJ /F9 11.955 Tf 11.96 0 Td[(0B)(2) A=R+)]TJ /F4 11.955 Tf 11.96 0 Td[(RAL)]TJ /F4 11.955 Tf 11.96 0 Td[(RAB R+)]TJ /F4 11.955 Tf 11.95 0 Td[(R)]TJ /F6 11.955 Tf 27.93 9.99 Td[((eA)]TJ /F9 11.955 Tf 11.95 0 Td[(0A))]TJ /F4 11.955 Tf 25.2 8.09 Td[(RAB R+)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 8.29 9.99 Td[((eB)]TJ /F9 11.955 Tf 11.96 0 Td[(0B)(2) B=RAL+RAB)]TJ /F4 11.955 Tf 11.95 0 Td[(R)]TJ ET q .478 w 104.59 -492.11 m 188.72 -492.11 l S Q BT /F4 11.955 Tf 124.24 -503.3 Td[(R+)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 27.93 9.99 Td[((eA)]TJ /F9 11.955 Tf 11.95 0 Td[(0A))]TJ /F4 11.955 Tf 13.15 8.09 Td[(RAB+RAL)]TJ /F4 11.955 Tf 11.96 0 Td[(R+ R+)]TJ /F4 11.955 Tf 11.95 0 Td[(R)]TJ /F6 11.955 Tf 27.94 9.99 Td[((eB)]TJ /F9 11.955 Tf 11.96 0 Td[(0B)(2) and B=R+)]TJ /F4 11.955 Tf 11.96 0 Td[(RAL)]TJ /F4 11.955 Tf 11.96 0 Td[(RAB R+)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 27.93 9.99 Td[((eA)]TJ /F9 11.955 Tf 11.95 0 Td[(0A))]TJ /F4 11.955 Tf 13.15 8.09 Td[(RAB+RAL)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ ET q .478 w 256.12 -570.31 m 340.24 -570.31 l S Q BT /F4 11.955 Tf 275.77 -581.5 Td[(R+)]TJ /F4 11.955 Tf 11.96 0 Td[(R)]TJ /F6 11.955 Tf 27.93 9.99 Td[((eB)]TJ /F9 11.955 Tf 11.96 0 Td[(0B)(2) 37

PAGE 38

NotethatA+A=eA)]TJ /F9 11.955 Tf 10.55 0 Td[(0AandB+B=eB)]TJ /F9 11.955 Tf 10.55 0 Td[(0Basrequired.Dening=RBA RAB=kA kB,theratiooftheheatcapacities,weexaminetwolimitingcasesthatareparticularlyrelevanttotheexperimentsunderdiscussion: (i)BathBisolatedfromthelatticebutstronglycoupledbathA,i.e. RAB>RALRBL (ii)BathAmorestronglycoupledtothelatticethantobathB,i.e. RAL>RABRBLIncase(i)thereisaninitialrelaxationgivenbytherelaxationtime R+=1+ RAB[1+RBL RAB1 (1+)2+......](2) followedbyalongtimerelaxationgivenby R)]TJ /F6 11.955 Tf 10.41 1.79 Td[(=RBL 1+[1)]TJ /F4 11.955 Tf 13.93 8.09 Td[(RBL RAB1 (1+)2+......](2) wheretherelaxationtimesa=R)]TJ /F3 7.97 Tf 6.59 0 Td[(1aetc.areinagreementwithTableVofRef.[ 16 ].ThecorrespondingamplitudesdependontheinitialconditionsandwewillexaminethepracticalcaseforwhichtheAspinsaresaturated(0A=0)andtheBspinsareclosetothermalequilibrium0B=eB.InthiscasewendthattherecoveryoftheAspinamplitudesaftersaturationaregivenby (a)= 1+eA(1)]TJ /F3 7.97 Tf 13.48 4.71 Td[(1 )eA,forthefastrelaxingcomponent,and (b)= 1+eA1 eA,fortheslowrelaxingcomponent.Inthecasesofinterestinthisstudy,>1andA>A.Figure 2-2 showsaschematicillustrationofthisrecoveryfor>1. Boththeratesandtheamplitudesdependontheradiooftheheatcapacities(orthemagnetizationsifthenuclearspeciesaredifferent).Thisistheprincipalpointforthestudiesreportedherewherethecross-relaxationandlatticerelaxationtimesareconstantorwellknownslowlyvaryingfunctions.Theobservedrelaxationwilldependstronglyontheheatcapacitiesoftheenergyreservoirsandmeasurementsoftherelaxationtimesasafunctionoftemperaturecanthereforebeusedtodetermine 38

PAGE 39

Figure2-2. Schematicrepresentationofthetwodecayratesforthethreebathsystemfor=3andRAB=RBL=)]TJ /F3 7.97 Tf 6.59 0 Td[(1XintheEq. 2 andEq. 2 .Theshortandlongtimedecaysaregivenby1.33Xand4.0X,respectively. therelevantheatcapacities.(ForreferencethereadershouldnotethatwhileGuyer,RichardsonandZane[ 16 ]derivesimilarrateequations(AppendixA),theydonotcalculatetheamplitudesandonlygivetheratestorstorder.) Incase(ii)forwhichRBL>RBARAL,therelaxationtimesaregivenbyashorttermrelaxation: )]TJ /F2 11.955 Tf 10.4 5.35 Td[(=BL[1)]TJ /F9 11.955 Tf 13.61 8.09 Td[(BL BA+...](2) followedbyalongtermrelaxation; +=BA [1+BL BA+...](2) 39

PAGE 40

TheamplitudesofthefastandlongtimerelaxationsforrecoveryfollowingsaturationoftheBspinsforthecaseRAL>RABRBL(orAL
PAGE 41

weakcouplingsbetweentheseenergybathsanddeterminewhetherthebathmodelappliesinthiscase. 2.2.1CouplingBetweenZeemanandTspecies ThematrixelementsofthecompletespinrotationinteractionareobtainedbyNijman[ 22 ]. <1m0;1M0gjHSRj1m;1Mg>=)]TJ /F4 11.955 Tf 9.3 0 Td[(hCtm0M0;mM(2) whereHSRisthespin-rotationinterationC=4.20.3kHzisthespinrotationconstantandtm0M0;mMthenomalizedmatrixelements(oforderunity).OnethencalculatestherelaxationtimebetweenZeemanenergyandTspeciesZT10)]TJ /F3 7.97 Tf 6.59 0 Td[(4s. 2.2.2CouplingBetweenTspeciesandPhononsystemofCH4 Yamamoto[ 23 ]calculatedthecouplingparameterfortheelectrostaticoctupoleinteractionsbetweenCH4moleculeswithseveralassumptionsandapproximations:(1)Onlytherotationaldegreesoffreedomofeachmoleculearetakenintoaccount,andthecentersofmassbeingheldxedatasiteinthefcclattice;(2)Neighboringmoleculesinteractwitheachotherviatheirelectrostaticoctupoles,thevalueoftheoctupolemomentbeingtheonlyparameterattheirdisposal;(3)Themolecular-eld-approximationmethodisemployed;(4)ThecalculationsarecarriedoutinthesubspacewithJ4,Jbeingtherotationalquantumnumber;and(5)Moleculesofnuclear-spinspeciesAandTareassumedtodistributerandomlyinthelattice,speciesEbeingomittedbecauseofitssmallconcentration. Theoctupolecouplingparameter=I23=R7,whereI3istheoctupolemomentandRisthedistantbetweentheneighboringmolecules.=3.5KatheliumtemperaturewithR=4.65A.Forourcase10)]TJ /F3 7.97 Tf 6.59 0 Td[(3K,whichmeansthatTP10)]TJ /F3 7.97 Tf 6.58 0 Td[(2s. 2.2.3CouplingBetweenthePhononsystemsofCH4andZeolite TheDebyemodelisamethoddevelopedbyPeterDebyein1912[ 24 ]forestimatingthephononcontributiontothespecicheatinasolid.ConsideracubeofsideL,from 41

PAGE 42

theparticleinaboxmodel,theresonatingmodesofthesonicdisturbancesinsidetheboxhavewavelengthsgivenbyn=2L=n,wherenisaninteger.TheenergyofaphononisEn=hn,wherehisplank'sconstantandnisthefrequencyofthephonon.Makingtheapproximationthatthefrequencyisinverselyproportionaltothewavelength,wehave: En=hn=hcs n=hcsn 2L(2) inwhichcsisthespeedofsoundinsidethesolid.Inthreedimensionswehave: E2n=(hcsn 2L)2(n2x+n2y+n2z)(2) Theapproximationthatthefrequencyisinverselyproportionaltothewavelength(givingaconstantspeedofsound)isgoodforlow-energyphononsbutnotforhigh-energyphonons.ThisisoneofthelimitationsoftheDebyemodel,andcorrespondstoinaccuraciesoftheresultsatintermediatetemperatures,whereasbothatlowtemperaturesandalsoathightemperaturestheyareexact. Thetotalenergyintheboxis: U=XnxXnyXnzEnN(En)(2) whichcanbeapproximatedas U=Z3p N0Z3p N0Z3p N0E(n)3 eE(n)=kT)]TJ /F6 11.955 Tf 11.96 0 Td[(1dnxdnydnz(2) Theenergyintegralbecomes: U=3 2ZR0hcsn 2Ln2 ehcsn=kT)]TJ /F6 11.955 Tf 11.95 0 Td[(1dn(2) Changingtheintegrationvariabletox=hcsn 2LkT, 42

PAGE 43

U=3 2(2LkT hcS)ZhcsR=2LkT0x3 ex)]TJ /F6 11.955 Tf 11.96 0 Td[(1dx(2) whichleadstothespecicinternalenergy: U Nk=9T(T TD)3ZTD=T0x3 ex)]TJ /F6 11.955 Tf 11.96 0 Td[(1dx=3TD3(TD T)(2) whereD3(x)isthethirdDebyefunctionandTDistheDebyetemperature TD=hcsR 2LkT=hcs 2Lk3r 6N =hcs 2k3r 6N V(2) Inourcase,TD300KandT10K.TheaveragefrequencyoftheCH4moleculesistherefore =E h=U Nh108Hz(2) FromFrankPobell[ 15 ],thetransferofenergy,however,fromthetranslational(phonon)degreesoffreedomoftheCH4moleculestothephononsofthesurroundingwallisveryineffectivebecauseofthemismatchinenergyandfrequencyofthetwosetsofphonons.ThisisthefamousKapitzaresistanceproblem[ 25 ].IfthevelocityofphononsinCH4isvCandthatinthesolidzeoliteisvS,wehavetheSnell'slaw sinC=sinS=vC=vS(2) fortheanglesatwhichthephononscrosstheboundary.BecausevC200m/swhereasvS5000m/sforzeolite,thecriticalangleofincidenceatwhichphononsfromCH4mayenterthesolidisverysmall, crit=arcsin(vc=vz)3(2) Thefractionofphononshittingtheinterfacethatfallintothecriticalconeis 43

PAGE 44

f=1 2(vc vz)210)]TJ /F3 7.97 Tf 6.59 0 Td[(3(2) However,becauseofthedifferenceinacousticimpedanceZ=v,notevenallofthesephononsaretransmitted.Theenergytransmissioncoefcientisgivenby t=4ZcZz Zc+Zz4cvc zVz210)]TJ /F3 7.97 Tf 6.59 0 Td[(3(2) Thereforeonlyafraction ft=2Cv3C Zv3Z10)]TJ /F3 7.97 Tf 6.59 0 Td[(5(2) ofthephononswillenterthesolid.Thespin-latticerelaxationtimeistherefore )]TJ /F3 7.97 Tf 6.58 0 Td[(1PP=f1 10010Hz(2) Whereisthedensity,visthesoundvelocityinsolid,andthesubscriptsCandZstandsforCH4andzeoliterespectively.TherelaxationratebetweenthephononsystemofCH4andzeoliteisthereforePP0.1s. Accordingtothecalculatedrelaxationratesbetweenthesefourbaths,theweaklinkintheowofenergyfromtheZeemansystemtotheultimatethermalbathshowninFigure 2-4 isthephononbottleneckPPbetweenthelocalizedmethanemoleculesandthesurroundingzeolitelattice.TherelaxationdiagramisshowninFigure 2-4 Figure2-4. BathmodelfornuclearspinrelaxationofCH4connedinzeolite Oncewesetthisbathmodel,wecancalculatethespin-latticerelaxation[ 16 ]: 44

PAGE 45

T)]TJ /F3 7.97 Tf 6.58 0 Td[(11=PP(kP kZ+kT+kP)(2) wherekxistheenergyconstantforeachenergybathX,kx=dEx dwhichisproportionaltotheheatcapacitiesC=dE dT.Eq. 4 canthenbewrittenas T)]TJ /F3 7.97 Tf 6.58 0 Td[(11=PP(CP CZ+CT+CP)(2) Accordingly,therelaxationtimeisgivenby T1=PP(1+CZ+CT CP)(2) where CZ=dEZ dT=d dT()]TJ /F4 11.955 Tf 10.49 8.09 Td[(N 4~2!20 kBT)=NkB 4(~!0 kBT)2(2) CT=dET dT=d dT(e)]TJ /F3 7.97 Tf 6.59 0 Td[(=T 1+e)]TJ /F3 7.97 Tf 6.59 0 Td[(=TNkB)=NkB( 2T)21 cosh2( 2T)(2) CP=dEP dT=234(T D)3,atlowtemperature,D100K (2) =(E T)2eE=T (eE=T)]TJ /F6 11.955 Tf 11.95 0 Td[(1)2athightemperature,E80K (2) 2.2.4ModelofHDinZeolite InthecaseofHDconnedinzeolite,anewinterestinthelighteratomsormoleculesistotestforthepredictionofthequantizationofthetranslationalmotionofatomsconstrainedtothecenterofasphericalcage.Thisquantizationisoftengivenasanmodeltextbookexamplebuthasnotbeenrealizedexperimentallyforthesesimplesystems.ForasphericalpotentialofradiusatheenergylevelsaresimplygivenbyEn,l=(~k)2=2mwhere~kisthenthrootoftheBesselfunctionjl(ka).ForHDina 45

PAGE 46

13Acage,thelowestenergylevelsareoftheorderofafewK(Table 4-1 )andeasilyobservablebymeasuringthenuclearspinrelaxationratesinthesituationwherethebathmodelofSec. 2.1 isvalid,thisprocessiscalledrelaxationspectroscopy. Table2-1. EnergyeigenvaluesEn,lHDinawellofwidthd=13A.ndesignatesthen,throotoftheBesselfunctionjl(kr). l=1l=2l=3l=4l=5 n=101.964.417.4110.8n=25.629.4813.910.9n=314.99 Fromthebathmodel,theseenergylevelsshouldbeobservableinthemeasurementsofthespin-latticerelaxationtimes. 46

PAGE 47

CHAPTER3NMRMETHODS 3.1PulsedNMR RFpulsetechnologiesareusedtoobservenuclearspindynamicsandtherebythemotionsofatomsandmoleculesindifferentmaterials.EachRFpulserotatesthenuclearmagnetizationbyanangle=H1twwhereH1istheamplitudeoftheRFmagneticeldgeneratedbythepulseandtwisthewidthofthepulse.Thephaseofthepulsewithrespecttoareferencesignaldeterminestheaxisaboutwhichtherotationismade:x-axisfor=0180,andyaxisfor=90270.Theexperimentissetuptoobservenuclearspinechoes;usinga90x-180xpulsesequenceforliquidsanda90x-90ysequenceforsolids.TheformercasecanbedemonstratedbyFigure 3-1 .ForaliquidtheprincipalperturbationontheprecessingnuclearspinisH=~!IZwhere!representsthelocaleldinhomogeneities.Theothertermsofspin-spininteractionaveragetozeroduetothemotionintheliquid.Thedephasingofthespinintime(Figure 3-2 ac)canbethereforereversedbya180degreepulse,leadingtoanechoattime2(Figure 3-2 e). Figure3-1. 90x-180xpulseNMR TheHamiltonianforthesystemintherotatingframe,duringtheapplicationofapulsealongthex-direction,isgivenby 47

PAGE 48

Figure3-2. IllustratingformationofanNMRspinechofora90-180pulsesequence. H=IZ(H0)]TJ /F9 11.955 Tf 13.15 8.09 Td[(!L )+IxH1(3) wheretheH1eldismuchlargerthantheeffectiveeldH0)]TJ /F8 7.97 Tf 13.9 5.11 Td[(!L inthez-direction.isthenucleargyro-magneticratiothatdenestheLamorfrequency!L=H0ThisjustiestheapproximationH=)]TJ /F9 11.955 Tf 9.3 0 Td[(H1Ixfortheshorttimeofthepulse.ThusapplyingH1meansthatthenuclearspinwavefunctionacquireatimedependencegiventheunitaryoperator R()=eiHt=eiH1Ixt=eiIx(3) whichinequivalenttoarotationaboutthex-axisbyanangle=H1t.BychoosingthetimetforthedurationoftheRFpulsecanbe90or180degreesoranothervalue.DuetothestrengthoftheappliedRFeld,H1ideallyturnsthespinsinanegligibletime.Thepulsesequencedescribedinthefollowingistheoriginal90x--180x-sequenceproposedbyHahnandisgraphicallyshowninFigure 3-2 48

PAGE 49

ThesteadyeldH0leadstoanexcessorientationofspinsinz-direction(Figure 3-2 A).Afterthesteadystatehasbeenreached,theH1-eldisswitchedonforashorttimeinordertotiltthemagnetizationby90degrees(Figure 3-2 B)fromMzinto-My: Mz(t= 2H1)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(i 2IxMz(0)e+i 2Ix=)]TJ /F4 11.955 Tf 9.3 0 Td[(My(3) ThesamecoilproducingH1picksupthevoltageinducedbythemagnetizationrotatinginthex-yplane.Withoutrelaxationeffects,asexpressedbyT1andT2,thealternatinginducedvoltagewouldpersistforever.InrealityT2playsarolebuttheT1effectisnegligible,consideringT1>>T2forasolid.Thisgivesrisetodephasingwhichdestroystheideallypersistingvoltagebyphasecancellation(Figure 3-2 C).Afreeinductiondecay(FID)resultsfromthesuperpositionofallindividualspinsignalsatslightlydifferentelds.ThesignaldecreasesexponentiallyinthecaseofaLorentzianlinewithatimeconstantT2.ThisT2isasumofalleffects,especiallytheinhomogeneityofH0.Itcausessomespinstoadvance,otherstolagbehind.Butthisisnotwhatreallydeterminesthephysics.Thegoalistoeliminatethisunphysicalinuencebyapplyinga180degreepulse (+)=e)]TJ /F5 7.97 Tf 6.58 0 Td[(iIx()]TJ /F9 11.955 Tf 9.3 0 Td[(y)e+iIx=y(3) The180pulseproducesamirrorimage:theslowestspinsareaheadofthefasterspins(Figure 3-2 D).The-pulseactslikeatimereversaloperator.Afterthesametimebetweenthe=2andthe-pulsethefastestspinshavecaughtupwiththeslowest.Sincetheinhomogeneityofthesteadymagneticeldhasnotchangedduringthetime2thealmostsameFIDoccursafterthistime(Figure 3-2 E).Thedifferenceisthatthevoltagehastheoppositesignoftherotatedmagnetizationandadecreasedamplitude.Forthe90x--180x-sequencetheecho,calledordinaryecho,hasthesamesignastheoriginalFID.Thereductioninsignalamplitudeiscausedbyeffectsthatgiveinsight 49

PAGE 50

intothemicroscopicbehavior.Duetocollisionsduringthetwotimeintervalsonlyadiminishedspinechooccurs,e.g.asaresultofdiffusioneffects. The90x-90yprocesscannotbedemonstratedsimplybysuchapicturebecausetheperturbingHamiltoniancontainstensorialoperators,butiseasytocalculate. H12pert=X(3i1zi2z)]TJ /F4 11.955 Tf 11.96 0 Td[(i1i2)d=(3I2z)]TJ /F4 11.955 Tf 11.95 0 Td[(I2)d(3) whereizisthezcomponentofthespin.Aftera90degreepulsearoundxaxis,=Iy,aftertimetbecomes =e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)tIye+d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)t(3) Followedbya90degreespulsearoundyaxis,themagneticmomentisnow =Re)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)tIye+d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)tR+(3) whereRdemonstratesthevectorofthe90degreerotationaroundyaxis.Aftertime, =e)]TJ /F5 7.97 Tf 6.58 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)Re)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)tIye+d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)tR+e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2) (3) =e)]TJ /F5 7.97 Tf 6.58 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)(Re)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)tR+)(RIyR+)(Re+d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)tR+)e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2) (3) =e)]TJ /F5 7.97 Tf 6.58 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2x)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)t(RIyR+)e+d(3I2x)]TJ /F5 7.97 Tf 6.58 0 Td[(I2)te)]TJ /F5 7.97 Tf 6.58 0 Td[(d(3I2z)]TJ /F5 7.97 Tf 6.59 0 Td[(I2) (3) (3) Ift=, =e+d(3I2y)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)t(RIyR+)e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2y)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)(3) andtheNMRsignalisgivenby S=Trne+d(3I2y)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)t(RIyR+)e)]TJ /F5 7.97 Tf 6.59 0 Td[(d(3I2y)]TJ /F5 7.97 Tf 6.59 0 Td[(I2)Iyo=1(3) 50

PAGE 51

Thisproducesanechoatt=2calledasolidechoandisoppositesigntotheordinaryecho. 3.2PulseNMRApparatus Figure3-3. Pulseapparatusforquantumgasconnedinmesoporousmaterials. Figure 3-3 showsthecompletepulseapparatus.Theradiofrequency(RF)generatorsuppliesacontinuousfrequencyof5.6MHz,with0.6Vamplitude.Thisfrequencycorrespondstoamagneticeld1315gaussaccordingtotheLamorrelationship.TheRFissplitintotwochannels,oneinphasewiththeRFgenerator,andtheother90degreeoutofphase.Thesechannelsarefedtogates(aandb)thatareopenedbylogicpulsesfromthecomputer.Thepulsesequencecanbeadjustedintermsofthepulselengthandtimeintervalasneeded.Afterpassingthroughtwoampliers,thesignalgoestotheduplexer,whichisaveryimportantpartoftheapparatus.Theduplexerisusedtoconnectthetransmitter,thesampleandthereceiver.Therequirementbecomesclearwhenconsideringthepowerofthetransmitter,300W,whichismuchlargerthanthe10pWpoweroftheNMRsignalthatonewantstoobserve 51

PAGE 52

immediatelyafterthepulse.Detailsoftheduplexerwillbediscussedlater.Aftertheduplexer,signalsfromthesamplearefedtoanamplierandmixedwiththereferenceRFandmonitoredbytheoscilloscope. Figure3-4. DuplexerCircuits TheduplexerisshowninFigure 3-4 .Itconsistsoftwoparts:transmitterandreceiver.Transmitter:Whenthepulseison,thelargeRFpulsegoesthroughthediodepairD0andismatchedtothehighimpedanceofthesamplecoilcircuit(C1,C2,L1,C3andLS).ThisisapowerampliertotheNMRcoil.ToprotectthereceiverduringtheRFpowerpulse,twodiodeshunts(DS)[ 26 ]areconnectedafterthesamplecoilsothatonlyaverysmallsignalfromthetransmitterpassestothereceiverduringtheRFpulse.Figure 3-5 showstheeffectivecircuitofthetransmitterduringthepulseandtheactivecircuitisshowninFigure 3-6 Figure3-5. TransmitterCircuitsduringthepulses. 52

PAGE 53

Figure3-6. Theactivecircuitofthetransmitter. Thispartresonateswhen!0C02L0s=1,R r=(C1 C02)2whereR=Q!Ls,Qistherequalityfactor50inourmeasurement. Thediodeshuntsareturnedonbythetwopulsescandd(equalandopposite)synchronouslywiththelogicpulsesaandb.Pulsescanddaregeneratedbyalong-tailtransistorpairthataddsbothadelayandanadjustabledecaytimeshowninFigure 3-7 .Theadjustabledecayisusedtopreventanadditionalringupthatcanoccurwhenthediodeshuntsaretrunedoff.Iftheresistanceofthediodeshuntswhenconductingisr,DS3reducestheRFpulseamplitudebyafactorofr R310)]TJ /F3 7.97 Tf 6.58 0 Td[(2ofthelevelatpointD. Receiver:Whenthepulseisoff,thesignalfromthesampleissosmallthatitislessthanthenoiseinthebandwidthofLS.TheleftdiodepairD0stopsanyleakagefromthesignalgeneratorandstopsthesamplesignalfromgoingleft.Thediodeshuntsarealsoclosed,sonowLSismatchedtoR2,R3andC4andthesignalistransmittedtoamplier1.TheeffectivecircuitnowisshowninFigure 3-9 Similarly,whenthereceiverresonates,!20LSC3=1,!20L1C4=1and!20L2C5=1.C5andL2aremadetomatchtotheinputoftheampliertoincreasetheS/N.Inourexperiment,S/Nisnormallyabout20afterfurtheramplierandphasedetector. 3.3CellDesign AsuccessfulversionoftheNMRsamplecelldesignisshowninFigure 3-10 andFigure 3-11 .Arelativelargevolumewasdesiredduetothelowsensitivityofanadsorbedsamplewith1020spins(onlyapproximately10)]TJ /F3 7.97 Tf 6.58 0 Td[(3moles).Inaddition,itisveryimportantthatthedesignallowforeaseofadmissionandextractionofmolecular 53

PAGE 54

Figure3-7. PulsesGenerator. H2orotheradsorbents.Thecellprovidesgoodthermalcontacttothesampleatlowtemperatureviaabrushofcopperhairspenetratingthesamplecell. WewereabletomeettheserequirementsinthedesignoftheNMRsamplecellshowninFigure 3-10 .TheoveralldimensionsoftheKel-Fcellare1inchlong,0.5inchdiameter.Amongthematerialsavailable,Kel-FwaschosenfortheNMRsamplechamberduetoitslowprotoncontent(estimated100ppm),thusminimizingthebackground1HNMRsignal.Inaddition,itiswellknownthatKel-Fiseasilymachinable,hasgoodthermalpropertiesandisdurableatlowtemperatures.TheRFcoilconsistsof6turnsofcopperwire.VacuumsealsareprovidedbyIndiumO-ringsattheange.16non-magneticstainlesssteelscrewsfastentheangeviatwobrasshalf-noonringsattheend.Asmallamountofprotonlessuorosilicatedlubricantwasappliedtotheindiumtofacilitatetheassembly.Abrushofenamel-freecopperwires(0.025mmdiameter)wassoft-solderedtothecoldmetalinordertoassurethermalcontacttothesample.The 54

PAGE 55

Figure3-8. Pulsesandadjustabledacays.Pulsecanddincludebothasmalltimedelaytdandaslowdecays. Figure3-9. ReceiverCircuit. resistorusedforheatingtheNMRcellwerefoundtobebothnon-magneticandstableonthermalcyclingtolowtemperatures.Thetemperatureofthecellismeasuredusingacalibratedcarbonglassthermometerandthetemperaturewasregulatedbycomparingtheresistancevalue(measuredbyresistancebridge)withapredeterminedvalueandusingthedifferencetoheatthesamplecell.Theheaterwasmadefromwirearoundtheexteriorofthecell(Figure 3-10 ).Inthismannerthetemperaturecouldberegulatedtowithin0.5Kat100Kand0.1Kat1K. 55

PAGE 56

Figure3-10. SchematicillustrationofsamplecellshowingNMRcoilandthermallinkagetosamplefromcoldcap(green)andcoppershroud(red). TheNMRsamplecellwasinitiallyleaktestedinaheliumatmospherewhilebeingpumpedonbyleakdetectoratroomtemperatureand77K.Thecellperformedsatisfactorallyevenafterseveralthermalcyclesandcontinuedtoremainleak-tightwellbelow4K. ExamplesoftheechoandFIDforCH4inzeoliteareshowninFigure 3-12 andFigure 3-13 56

PAGE 57

Figure3-11. Schematicrepresentationofthesupportstructureandcoolingpathforthesamplecell.Theexteriorcoppercansitsinaliquidheliumbaththatcanbepumpedto1.4K. Figure3-12. FIDofCH4inzeoliteat70Kafteraveraging50pulsesequences. 57

PAGE 58

Figure3-13. 90x-180xechoofCH4inzeoliteinourexperimentat70Kafteraveragingsignalsfor1moleculepersodelitecage. 58

PAGE 59

CHAPTER4EXPERIMENTALDATAANDANALYSIS TherelaxationtimesweremeasuredforbothCH4andHDinaconventionalzeolitestructure(13X)andanadvancedmetalorganicframworks(Aldrich).WeexpecttherelaxationtimesforCH4tobedetectedbytherotationalenergylevel,andthoseforHDbythetranslationalenergylevelsofHDmoleculesintheconningcageofthezeoliteorMOFstructure. 4.1CH4inZeolite Thesampleswerepreparedbyadsorbingmethaneoncrushedzeolite-13Xthatwassubsequentlybakedat200Cinvacuum.ThevolumeofCH4requiredforobtaining1moleculepercagewasdeterminedbymeasuringtheadsorptionisothermsat77K(Figure 4-1 ).Thesharpriseintheequilibriumpressure(correspondingto0.0110.0005molesfora1g.sample)wastakenasthesignaturefortheadsorptionof1moleculeperavailablecage. Apreliminarysetofmeasurementswascarriedouttodeterminetherateofconversionofthemolecularspecies.Followinginitialcondensationofasampleandcoolingto4K(forwhichtheequilibriumspecieswouldbe99%A-species),free-inductiondecays(FIDs)weremeasuredrstly,ataxedtemperatureof4Koverseveraldays,andsecondly,asafunctionoftemperatureto100KtodeterminethechangeinspeciesconcentrationthatcouldbefollowedbychangesintheFIDamplitudes.Thepurityofthesamplegaswas99.99%ofwhichthemainimpuritywasO2whichisknownfrombulkstudiestobetheprincipalgeneratorofconversionduetoitsparamagneticmoment.Forbulksamplesofsolidmethanetheconversionrateistypically1h.at4Kandlessthan200s.forT>30K[ 27 ].Weexpectfasterratesforthezeolitesbecauseoftherelativelyhigherconcentrationsofparamagneticimpurities.OurobservationsshowedthattheFIDintensitieswereconsistentwiththethermalequilibriumvaluesofthesampleswherethewaitingtimeforequilibriumafter 59

PAGE 60

Figure4-1. Adsorptionisothermformethaneonzeolite-13Xat77KforNstepswitheachstepcorrespondingtotheadditionof10)]TJ /F3 7.97 Tf 6.58 0 Td[(3moles.Thearrowindicatesthestepintheisothermwhichwetakeasthesignatureforcompletingllingat1moleculepersodalitecage.FigurereproducedwithpermissionfromJi.etal. atemperaturechangewas15min.Theresultsimplyconversionratesfasterthan15minandallexperimentswerethereforecarriedoutafterwaitingforatleast1h.aftereachtemperaturechangetoensurethatequilibiriumspeciesconcentrationswererealized. TypicalexamplesofthedecayorrecoveryofNMRspinechosareshowninFigure 4-2 andFigure 4-3 .Insomecasesoneobservesashort-timerecoveryaswellasalong-timerecovery.TherelativeweightsofthetworecoverytimesdependsontherelativeweightsoftheheatcapacitiesasshowninSec. 2.1 .Theseobservationsstronglysupporttheuseofthebathmodeltoanalizetheresults.ForthedecaysshowninFigure 4-2 andFigure 4-3 theerrorbarsindeterminingtheslowrelaxationistypically10%. ThevaluesoftherelaxationtimesdeterminedfromtheanalysisaboveforthelongrecoverytimesfordifferentquantitiesofCH4adsorbedonzeolite-13Xareshownin 60

PAGE 61

Figure4-2. EchoamplitudeasafunctionofNMRpulsetimeintervalforHDinzeoliteat19Kwith1moleculepercage.Thesolidandbrokenlinesshowthettotheshorttimeandlongtimedecaysasexpectedinthe2-bathmodel. Figure 4-4 asafunctionoftemperature.Threefeaturesareobserved:twopeaksat27Kand46K,andadeepminimumat78K.ThetwopeaksareattributedtotwotunnelingstatesfortheTmolecularspeciesassociatedwithtwodifferentsitesforlocalizationoftheCH4molecules.Inthetwo-bathmodel,bathTconsistsofthethermalexcitationstothedistinctrotationalstatesTaandTb.ThegroundstateforthemolecularrotationsreferredtointheliteratureasthesymmetricAstateformsbathZ.TheheatcapacityforbathTthereforeconsistsoftwoSchottky-likeheatcapacitiesgivingrisetothetwopeaksintheobservedrelaxationtimes.Thevaluesoftheenergypeaks(27Kand49K)areclosetothosereportedforearlierheatcapacitymeasurementsthatwereseenasweakbumpsinthetotalheatcapacity[ 28 ].ThepeaksintheobservedrelaxationtimesaremuchsharperthanthoseforsimpleSchottkyheatcapacitiesandthisisexpectedbecausetheexcitedstatesarepocketstatescorrespondingtosmallamplitudemolecularlibrationalmotionsabouttheirequilibriumorientations. 61

PAGE 62

Figure4-3. EchoamplitudeasafunctionofNMRpulsetimeintervalforCH4inZMOFat35Kwith1moleculepercage.Thesolidandbrokenlinesrepresentthetsfortheshorttimeandtiemdecaysasexpectedinthe2-bathmodel. ThebroadminimumathightemperaturesisinterpretedasaclassicBloembergen-Purcell-Poundminimum[ 29 ]associatedwithathermallyactivateddiffusionforpassagefromone-cagetoadjacentcageswithanintercagetunnelingof=0exp()]TJ /F4 11.955 Tf 9.29 0 Td[(EA=T).ThetshowninFigure 4-5 yieldsanactivationenergyEA=2600Kandatunnelingrate0=1.21015s)]TJ /F6 11.955 Tf 9.3 0 Td[(1.ThisvalueiscomparabletothevalueestimatedfromMonteCarlostudies[ 14 ]. ClearlythettothedataforT1inthelowtemperatureregion(Figure 4-5 )doesnotdescribetherelativelysharppeaksat28Kand49K.ThesepeaksresemblethesharppeaksseenbyneutronscatteringforCH4absorbedongraphite[ 30 ]andweinferthattherotationaldegreesoffreedomcannotbetreatedassimpleindependentrotors.ThisviewisqualitativelyconsistentwithrecentcomputersimulationsofKumaretal.[ 31 ]whonotethattheanisotropicinteractionbetweenaCH4moleculeandthezeolitesurfaceattheentrancetothetunnelbetween-cagesfavorsoneparticularorientationoveranotherbyabout0.5kJ/mol(60K)inpassagefromonecagetoanother,whichiscomparabletothepeaksweobservedforT1inthesestudies.Theauthorsalsonoteanappreciableorientationaldependenceonthelocalizationenergyofamoleculeata 62

PAGE 63

Figure4-4. Temperaturedependenceofthenuclearspin-latticerelaxationtimesobservedforCH4inthe-cagesofzeolite:diamonds,1moleculepercage;triangles,0.5moleculespercage;andsquares,0.05moleculespercage. particularsiteinthecage,withtotalbindingenergiesofU=3.5-5.8kJ/mol(420-696K),considerablylowerthanourvalueofEact.Anorientationaldependancewasalsoseenincomputersimulations[ 32 ]ofthepassageofCH4moleculesbetweencagesinzeoliteandthisisconsistentwiththeexistanceofdistinctrotationalenergiesformoleculeswithinacage.ThetranslationalmotionsareexpectedtodetermineT2whichinfeaturelessintheregionoftheT1maxima. Thespin-spinrelaxationtimeT2showsnegligibletemperaturedependenceuntil78Kwherealargeincreaseoccurs.Thisjumpat78Kisinterpretedasthesignatureofthequasi-meltingseeninthesimulationsreportedbyYashonathetal.[ 14 ]whichcorrespondstothermallyactivatedmoleculardiffusionfromonecagetoanother.Theatlowtemperatureregionisattributedtoastrongcouplingbetweenthedominantlow 63

PAGE 64

Figure4-5. Comparisonoftheobservedtemperaturedependenceofthenuclearspin-latticerelaxationtime,T1,withthecalculatedvalue(solidredline),assumingindependentmolecularrotorstates.Thebroadpeaksat25Kand55K,areattributedtotwodistinctT-levels.Thesolidgreenlineisthecontributionfromthermalactivationofinter-cagediffusion.FigurereproducedwithpermissionfromJi.etal. temperaturespecies,theA-species,andparamagneticimpuritiesatthesurfaceofthezeolite. 4.2HDinZeolite InordertocarryoutamorefundamentalstudyofthemolecularrelaxationanddiffusioninmesoporousstructureswestudiedthebehaviorofHDtrappedinthecagesofzeolite.Whilemethanehasseveraldistinctmolecularspecies(ortho,metaandpara),correspondingtothedifferentcombinationsofrotationalsymmetryandnuclearspinsymmetry,HDmoleculesdonothavethesepropertiesandexistasasinglespecieswithaweakelectricdipolemoment.WechoseHDratherthanH2,becauselikeCH4,H2hastwomolecularspecies,ortho-H2(withtotalnuclearspinI=1andorbitalangularmomentumJ=1)andpara-H2(withI=0andJ=0).Onlyortho-H2canbeobservedbynuclearmagneticresonanceandthisspeciesconvertstopara-H2slowlyviamagnetic 64

PAGE 65

Figure4-6. Observedtemperaturedependenceofthenuclearspin-spinrelaxationtimeT2for1.00.2moleculespersodalitecageofzeolite13X.FigurereproducedwithpermissionfromJietal. interactions.Inaddition,ortho-H2hasanelectricquadrupolemomentthatresultsininterestingmolecularalignmentcongurationsbutthesecanbedifculttointerpret[ 33 ].TherelaxationofHDmoleculesontheotherhandisrelateddirectlytotheirtranslationaldegreesoffreedomandthespin-spininteractionsbetweenthesemoleculesandbetweenothermoleculesinthewallsofthecages.HDexperimentsarethereforeidealfordeterminingthemoleculardiffusioninmesoporousmaterialsandforexploringtheeffectsofconnementonthemolecules,i.e.arethetranslationaldegreesoffreedomquantizedasexpectedforaperfectsphericalcage. Theresultsofmeasurementsofthenuclearspin-latticerelaxationtime,T1,overawidetemperaturerangeareshowninFigure 4-7 fortwodifferentdensities,(i)1.0moleculepercageand(ii)0.5moleculespercage,asdeterminedfromtheadsorptionisotherms.Distinctpeaksareseenforeachlling,x,butatdifferenttemperaturesforthedifferentllings:at2.9K,4.9Kand7.3Kforx=1.0;andat2.1K,4.5Kand12.3Kforx=0.5.InFigure 4-7 weshowthetassumingdistinctSchottkyheatcapacitycontributions(illustratedbythebrokenlines)forthedatafor0.5moleculespercage.TheSchottky 65

PAGE 66

Figure4-7. Temperaturedependenceofnuclearspin-latticerelaxationtimesforHDadsorbedinzeolite13x;withcoveragesof1.0moleculespercage(squares)and0.5moleculespercage(triangles).ThesolidlineisabesttassumingSchottkylevelspecicheatsforthreediscreteenergylevels.Eachcontributionisshownbythebrokenlines. heatcapacityform(Ci=NkB(i T)2exp()]TJ /F3 7.97 Tf 10.49 5.11 Td[(i T)=[1+exp()]TJ /F3 7.97 Tf 10.5 5.11 Td[(i T)]2foreachexcitedstatei)canonlybeconsideredasanapproximationfortheweaklytrappedmoleculesandisvalidonlyiftheexcitedstatescanbetreatedtoagoodapproximationasdiscreteenergylevels.InthebathmodelofFigure 2-1 ,CB=PiCifortheiexcitedstates,andCAisthelowtemperatureDebyeheatcapacityforthemoleculargroundstate.InFigure 4-7 wehaveassumedarbitraryamplitudesforeachcontributiontotthedata.ThevaluesfortheexcitationlevelsshowninFigure 4-7 areverydifferentfromtheprincipaladsorptionenergiesof80Kand40K,respectively,fortheS1andS2bondingsitesforhydrogeninzeolite13Xasdeterminedfromneutronscatteringexperiments[ 34 35 ].Thetwopossibleoriginsforthesediscreteenergiesare:(i)thelowlyingenergybarriersbetweenthepotentialsattheS1andS2sitesformotionbetween 66

PAGE 67

theseneighboringsites,and(ii)theexistenceofdiscreteexcitedenergylevelsinsidethedeepbindingpotentials. Ifweconsiderasimplemoleculeofmassmtrappedinasphericalcageofradiusr,theenergylevelscorrespondingtothetranslationalmotionofthemoleculearequantized[ 36 37 ]withenergyEl,n=2l,n~2=2mr2wherel,nisnthrootofthelthordersphericalBesselfunction.AsillustratedinFigure 4-8 theenergylevelsfora15Acageform=3areoftheorderofafewKandshouldbeobservable. TheoriginorthestrickingpeaksintherelaxationtimeshasdifferentoriginsforCH4andHDinamesoporousstructure.ForCH4asmentionedpreviouslytherelaxationisattributedtoexcitationtotherotationallevelsTaandTb.ForHDwhichhasnorotationallevels,thedistrictpeaksareattributedtodiscretelevelsassociatedwiththequantizationofthetranslationalmotioninanapproximatlyspherecage.Thesphericalcageisaverypoorapproximationtothemesoporouscagesofzeolitewhichhaveseveralopenchannelsbutthemodeldoesprovideanorderofmagnitudeestimateforthetranslationalenergies.Thevaluesobservedexperimentallyareindeedcomparabletothoseexpectedinthismodel.Inadditiontheenergiesappeartodecreaseasthellingisreducedandthiscanbeunderstoodintermsofaneffectiveincreaseintheboundingdimensionforthelowerxvalueswithreductionoftheblockingofthetunnelstothecagesandthusleadingtoadecreaseintheexpectedenergies.Theseestimatesareveryapproximateandcomputersimulationsforthezeolitegeometriescouldprovideamuchbetterttotheobservedfeatures. ItisimportanttonotethatdespitethehighvaluesfortheabsorptionenergyoftheS1andS2sites,themoleculesarestillhighlymobile.Therelaxationtimesdecreaseexponentiallybelow12K.Figure 4-9 accordingtoathermallyactivatedprocesswithatunnelingrate)]TJ /F3 7.97 Tf 6.58 0 Td[(1T=)]TJ /F3 7.97 Tf 6.58 0 Td[(10exp()]TJ /F4 11.955 Tf 9.3 0 Td[(EA=kBT)wheretheactivationenergyEA=733Kand)]TJ /F3 7.97 Tf 6.59 0 Td[(10=1.51010)]TJ /F3 7.97 Tf 6.59 0 Td[(11s)]TJ /F3 7.97 Tf 6.59 0 Td[(1.Thesevaluesleadtoadiffusionrateof8.22.810)]TJ /F3 7.97 Tf 6.58 0 Td[(6cm2s)]TJ /F3 7.97 Tf 6.59 0 Td[(1atT=19.5Kwhichistobecomparedwiththevalueof15410)]TJ /F3 7.97 Tf 6.58 0 Td[(5cm2s)]TJ /F3 7.97 Tf 6.58 0 Td[(1obtainedfrom 67

PAGE 68

Figure4-8. SchematicrepresentationofenergylevelsforthetranslationalmotionofHDmoleculesconstrainedtoasphericalcageof15Aindiameter. recentinelasticneutronscatteringmeasurementsofCoulombetal.[ 38 ].Theimportantfeatureoftheseresultsisthattheyclearlyshowthatthereexistsahighmobilityforlowdensitiesofhydrogeninzeolitedowntoverylowtemperatures. InFigure 4-10 weshowthetemperaturedependenceofthenuclearspin-spinrelaxationtimes.AnabrupttransitionisobservedatT=14Kwhichisattributedtotheonsetofintercagediffusion.Thethermalactivationrepresentedbythesolidredlinebelow14Kcorrespondstoajumpfrequency)]TJ /F3 7.97 Tf 6.58 0 Td[(10exp()]TJ /F4 11.955 Tf 9.3 0 Td[(EAC=T)with)]TJ /F3 7.97 Tf 6.59 0 Td[(10=1010s)]TJ /F3 7.97 Tf 6.58 0 Td[(1andEAC=20020K.ThetemperaturedependenceofT2forT>17KfollowsthatexpectedforadilutegasforwhichT2/=Twheretheviscosity/T1=2.[ 20 ] 4.3CH4inZMOF Figure 4-11 showstheresultsofmeasurementsofthenulearspin-latticerelaxationtimeT1forCH4adsorbedonzeolitelikemetalorganicframeworks(ZIF-8).Threepeaksareobservedat16K,30Kand46KwhichareattributedtothreetunnelingstatesfortheTmolecularspeciesassociatedwiththreedifferentsitesinaZMOFcage.The30Kand46KpeaksareconsistantwithourmeasurementofCH4connedinZeolite.However, 68

PAGE 69

Figure4-9. Observedtemperaturedependanceofthenuclearspin-latticerelaxationtimeofHDinzeolite13Xathightemperatures.Thesolidline(red)isatforathermalactivationof733Kandanintrinsicturnelingrateof1.51011s)]TJ /F3 7.97 Tf 6.59 0 Td[(1. theZMOFcageislargerandmorecomplecatedthanthezeolite-cage,whichmakesmoretunnelingstateforTspecies.TheheatcapacityforbathTthereforeconsistsofthreeSchottky-likeheatcapacitiesgivingrisetothethreepeaksintheobservedrelaxationtimes.ThepeaksintheobservedrelaxationtimesaremuchsharperthanthoseforsimpleSchottkyheatcapacitiesandthisisexpectedbecausetheexcitedstatesarepocketstatescorrespondingtosmallamplitudemolecularlibrationalmotionsabouttheirequilibriumorientations. Thedeepminimumathightemperatureisassociatedwiththethermallyactivateddiffusionforpassagefromonecagetoadjacentcageswithandintercagetunneling)]TJ /F3 7.97 Tf 6.59 0 Td[(1V=)]TJ /F3 7.97 Tf 6.59 0 Td[(1V0exp()]TJ /F4 11.955 Tf 9.3 0 Td[(EA=T).ThettingshowninredlineyieldsanactivationenergyEA=2500KandatunnelingrateV0=1.21015s)]TJ /F3 7.97 Tf 6.59 0 Td[(1. Thespin-spinrelaxationtimeT2(Figure 4-12 )showsnegligibletemperaturedependenceuntil80Kwherealargeincreaseoccurs.Thisjumpat80Kisinterpreted 69

PAGE 70

Figure4-10. Temperaturedependenceofnuclearspin-spinrelaxationtimesforHDadsorbedinzeolite13x;withcoveragesof1.0moleculespercage(squares)and0.5moleculespercage(triangles). asthesignatureofthequasi-meltingseeninthesimulationsreportedbyYashonathetal.[ 14 ]whichcorespondstothermallyactivatedmoleculardiffusionfromonecagetoanother. 4.4HDinZMOF Theobservedtemperaturedependenceofthenuclearspin-latticerelaxationtimesforthetemperaturerange1.6
PAGE 71

Figure4-11. Temperaturedependenceofnuclearspin-latticerelaxationtimesforCH4adsorbedinZMOF. TheobservedrelaxationfollowinganRFpulsedependsonthecouplingofthenuclearspindegreesoffreedomtootherdegreesoffreedom(translationalmotion)andultimatelytothephononsoftheMOFstructure.AfteranRFpulsethenuclearspinsreachacommonspintemperatureveryrapidlyonatimescaleM)]TJ /F3 7.97 Tf 6.59 0 Td[(1=2210)]TJ /F3 7.97 Tf 6.59 0 Td[(3s.whereM2istherigidlatticeNMRsecondmoment. Thecouplingofthetranslationalmotionofthemoleculestothephononbathis,however,muchslowerbecauseoftheacousticmismatch[ 15 ]betweenthephononsoftheMOFandthetranslationalexcitations.Fromthismismatchweestimateanintrinsicbottleneckrelaxationtimeint10)]TJ /F6 11.955 Tf 12.21 0 Td[(50s.Inthiscasetheobservedrelaxationisdeterminedbytheheatcapacitiesofthedifferentthermalbathsintherelaxationprocess,andtheobservedrelaxationhasbeenshownbyGuyer,RichardsonandZane[ 16 ]tobegivenby 71

PAGE 72

Figure4-12. Temperaturedependenceofnuclearspin-spinrelaxationtimesforCH4adsorbedinZMOF. T1(obs)=(1+CT CP)int(4) whereCTandCPare,respectively,theheatcapacitiesofthetranslationalexcitationsEk,loftheHDmoleculesandthephononsofthethermalbath.WewouldexpectthatCTwouldbegivenbyasumofSchottkyheatcapacities[ 39 ]butasseeninFigure 4-13 thetemperaturedependenciesaremuchsharper.ToobtainamorerealistictweusedaGaussiancontributionforeachEn,lascalculatedbyStroudetal.[ 28 ]foracellmodelofCH4inzeolite.ThecorrespondingtisshownbythebrokenlineinFigure 4-13 .Wehavealsomeasuredtherelaxationtimesforacoverageofx=0.1.ThereismorescatterintheresultsbutthepeaksinT1areobservedatthesametemperatures. InFigure 4-14 weshowthetemperaturedependenceofthenuclearspin-spinrelaxationtimes.AnabrupttransitionisobservedatT=17Kwhichisattributedtotheonsetofintercagediffusion.Thethermalactivationrepresentedbythesolid 72

PAGE 73

Figure4-13. Temperaturedependenceofthenuclearspin-latticerelaxationtimeforHDinZ-MOF.foracoverage1.0moleculepercage.Thebrokenlineisthecalculateddependencedescribedinthetext. redlinebelow17Kcorrespondstoajumpfrequency)]TJ /F3 7.97 Tf 6.59 0 Td[(10exp()]TJ /F4 11.955 Tf 9.29 0 Td[(EAC=T)with)]TJ /F3 7.97 Tf 6.58 0 Td[(10=3.52.01010s)]TJ /F3 7.97 Tf 6.59 0 Td[(1andEAC=723K.ThesevaluesareingoodagreementwiththosereportedbyneutronscatteringstudiesforHDinAlPO4andMCM-41.[ 38 40 ] ThetemperaturedependenceofT2forT>17KfollowsthatexpectedforadilutegasforwhichT2/=Twheretheviscosity/T1=2.[ 20 ] Foraneffectivediameterd,theenergylevelsaregivenbyEn,l=(~kn,l)2=2md2wherekn,larethenthrootsofthel-thordersphericalBesselfunction.[ 36 ]Theseenergyvaluesarelistedintable 4-1 ford=15A,thelargestspherethatcanbeinscribedintheZ-MOFsupercage. 73

PAGE 74

Figure4-14. Temperaturedependenceofthenuclearspin-spinrelaxationtimeforHDinZ-MOF.Purple,x=0.1,blue,x=1.0moleculespercage.ThejumpatT=17Kmarkstheonsetofintercagediffusion. Table4-1. EnergyeigenvaluesEn,lHDinawellofwidthd=13A.ndesignatesthen,throotoftheBesselfunctionjl(kr). l=1l=2l=3l=4l=5 n=101.964.417.4110.8n=25.629.4813.910.9n=314.99 74

PAGE 75

CHAPTER5CONCLUSION Nuclearmagneticresonanceexperimentshavebeenusedtoexplorethefundamentalpropertiesofhighsymmetrynanoclustersofquantumuidsandsolids.BecauseofthelightmassandweakinteractionsofmoleculesandasH2andCH4,themoleculeshavealargequantumzeropointmotionandtheextentoftheirwavefunctionscanbecomecomparabletothedimensionsoftheconningcagesinmodernmesoporousstructuressuchasthezeolitesandmetalorganicframeworks.Whilealmostallexperimentsofthesegasesinmesoporousmeterialshavebeenlimitedtolargedensitieswithseveralmoleculespercage,thesestudiesarededicatedtoobservingthequantumpropertiesofsinglemoleculesinthecage.Thequantizationofthemotions,translationalandrotationalwereexpectedtoleadtonewphenomenasuchastheexistenceofdiscreteenergylevelsforhydrogenincageafewAindiameter.Theexperimentweredesignedtodeterminetheinteractionsanddynamicsofhydrogenandmethaneinporousmeterialstodeterminetheusefulnessofthesemeterialsforenergystroage.Thestudiesalsoprobetheunusualquantumfeaturesoflightatomsandmoleculesinconstrainedgeometries.Theexperimentsinselectedporousmeterialsto(i)testforexpectednewquantumpropertiesand(ii)determinetheusefulnessofthesemeterialsforenergystorage. InordertocarryouttheexperimentsuniqueinstrumentationwasdevelopedthatallowedmeasurementsoftheadsorptionisothermsofthegasesunderstudyandNMRrelaxationinthesameapparatus.Precisionmeasurementsoftheisothermswithaccuratethermalregulationoftheexperimentalcelloverthelongperiodsrequiredforadsorptionenabledthedeterminationforsinglemoleculeoccupancyofthecages.Thesingal/noiseoftheNMRmeasurementsrequiredthedevelopmentofacustom-builtduplexerintegratedwithacomputercontrolledsignalaverager.Themeasurementsofthetemperaturedependenceoftheprotonnuclearspin-latticerelaxationtimesrevealed 75

PAGE 76

theexistenceofdistinctivepeaksintherelaxationtimesforbothCH4andHDmoleculesconstrainedtotheinteriorofthemesoporouscagesofzeoliteforconcentrationsequaltoorlessthan1moleculepercage.Tothebestofourknowledgethesefeatureshavenotbeenpreviouslyobserved.Thistemperaturedependenceisinterpretedintermsofdiscretelowenergyexcitationsforthemolecules.TheoriginoftheexcitationsisdifferentforCH4andHD.Byusingawell-knowndescriptionofthecouplingbetweendifferentenergybathscorrespondingtotheseexcitationsthepeaksintherelaxationtimesarerelatedtothepeaksintheheatcapacitiesoftherelevantbathscontributedbytheseexcitations.Thisanalysisisonlyvalidwhenbottlenecksoccurintheenergyowbetweenthebathsduetotheweakcouplings. ForthecaseofCH4thelevelsareinterpretedintermsofexcitedstatesfortheTrotationalstatesofCH4.Thereareclearlytwosuchstatescorrespondingtothetwoclassesofsymmetriesfortheinteractionswiththefacesoftheconningcages(squareandhexagonal).Tothebestofourknowledgesthisfeaturehasnotbeenpreviouslyreported.ForHDtheenergylevelsareclosetothosecalculatedfortheexpectedquantizationofthetranslationallevelsofmoleculesinsphericalcagesof15Aindiameter.DetailedtheoreticalcalculationsforthemicroscopicmolecularmotionintheconnectedcagesofZeolitelikestructuresareneededtorelatetheresultstothepropertiesofthecage.Inadditiontoobservingtheenergystatesthemoleculardynamicswasdetermined.UnderstandingthedynamicsiscriticalfordeterminingtheusefulnessofthematerialsforH2andCH4storagebecausetheratesareimportantforefcienttransferinpracticalapplications.Thespin-spinrelaxationtimesenabledthecleardeterminationoftheenergybarriesforintercagediffusionthatcanonlybedeterminedindirectlyfromothermeasurements.Inadditiontheresultsprovidevaluesforthefundamentaldiffusionfrequency.ThedetailedresultsshowastrikinglyhighmobilityforhydrogenmoleculeintheMOFstructuresdowntoT10Kconsistentwith 76

PAGE 77

neutronscatteringstuduies.ThisresultimpliesthattheMOFstructureshavehighpromiseasstorageelementsforH2andCH4. 77

PAGE 78

REFERENCES [1] N.S.SullivanYuJi,J.A.Hamida.Nmrstudiesofquantumrotorsconnedinzeolite.J.LowTemp.Phys.,162(121),2010. [2] I.F.Silvera.Thesolidmolecularhydrogensinthecondensedphase:Fundamentalsandstaticproperties.Rev.Mod.Phys,52:393,1980. [3] L.Pauling.Therotationalmotionofmoleculesincrystals.Phys.Rev.,36:430,1930. [4] K.Tamita.Statesfosolidmethaneasinferredfromnuclearmagneticresonance.Phys.Rev.,89:429,1953. [5] I.F.SilveraandV.V.Goldman.Theisotropicintermolecularpotentialforh2andd2inthesolidandgasphases.Chem.Phys.,69(4):4209,1980. [6] M.Rall.Nmrstudiesofmolecularhydrogenconnedtotheporesofzeolite.PH.D.thesis,1991. [7] VladimirI.TikhonovandAlexanderA.Volkov.Separationofwaterintoitsorthoandparaisomers.Scinece,296:5577,2002. [8] JamesHritz.Nasaglennresearchcenterglennsafetymanual,documentgrc-mqsa.001.Nasa,296:5577,2006. [9] AvshalomC.Shinitzky,M;Elitzur.Ortho-paraspinisomersoftheprotonsinthemethylenegroup.Chirality,18:754,2006. [10] J.V.Smith.Denitionofazeolite.MineralogicalSocietyofAmerica,1,1963. [11] D.W.Breck.zeolite.J.Chem,41:678,1964. [12] Ltd.XinyuanTechnologyCo.Xinyuanmolecularsieve,2008.URL http://www.molecularsieve.org/Zeolite_Molecular_Sieve.htm [13] R.M.Barrer.ZeolitesandClayMineralsasSolventsandMolecularSieves.AcademicPress,London,NewYork,Sanfrancisco,1978. [14] S.Yashonath,J.M.Thomas,A.K.Nowak,andA.K.Cheetham.Thesiting,energeticsandmobilityofsaturatedhydrocarbonsinsidezeoliticcages:methaneinzeolitey.Nature,331(4):601,1988. [15] F.Pobell.MatterandMethodsatLowTemperatures.1.Springer-Weriag,Germany,1992. [16] R.A.Guyer,R.C.Richardson,andL.I.Zane.Excitationsinquantumcrystaks:Asurveyofnmrexperimentsinsolidhelium.Rev.Mod.Phys.,43:532,1971.doi:10.1103/RevModPhys.43.532. 78

PAGE 79

[17] F.Weinhaus,H.Meyer,andS.M.Myers.Studyofthenuclear-magnetic-resonancelineshapeinhcpd2.Phys.Rev.B,3:626,1971.doi:10.1106/0375-9601(71)90483-X. [18] D.Zhou,M.Rall,J.P.Brison,andN.S.Sullivan.Nmrstudiesofvacancymotioninsoplidhydrogen.Phys.Rev.B,42:1929,1990.doi:10.1103/PhysRevB.42.1929. [19] M.Rall,D.Zhou,ErikaG.Kisvarsanyi,andN.S.Sullivan.Nuclearspin-spinrelaxationofisotopicimpuritiesinsoldhydrogen.Phys.Rev.B,45:2800,1992.doi:10.1103/PhysRevB.45.2800. [20] A.Abragam.TheprinciplesofNuclearMagnetism.1961. [21] J.C.Jansen,M.Stoecker,H.G.Kange,andJ.Weitkamp(editors).Advancedzeolitescienceandapplicationsinstudiesinsurfacescienceandcatalysis.85,1994. [22] A.J.Nijman.InvestigationofSolidmethanebyNMRathighpressureandlowtemperature.Thesis,1977. [23] T.YamamotoandY.Kataoka.Quantumstatisticalmechanicalstudyonphasetransitionsinsolidmethane.Phys.Rev.Lett,20:3,1993. [24] Debye.Zurtheoriederspezischenwaerme.AnnalenderPhysik,39:789,1912. [25] P.L.Kapitza.Zh.eksp.teor.z.Zh.Eksp.Teor.Fiz,11-1,1941. [26] E.B.Genio,G.G.Ihas,andN.S.Sullivan.Nuclearspinrelaxationofpowderedmetallicantimonyinliquid3he.J.LowTemp.Phys,112:21,1993. [27] S.Buchman,D.Candala,W.T.Vetterling,andR.V.Pound.Spin-speciesconversionrateinsolidch4inthetemperaturerange4-23k.Phys.Rev.B,26(1459),1982. [28] H.J.F.Stroud,E.Richards,P.Limcharden,andN.F.Parsonage.JCSFaradayTrans.,172(942),1976. [29] N.Bloembergen,E.M.Purcell,andR.V.Pound.Relaxationeffectsinnuclearmagneticresonanceabsorption.Phys.Rev.,73(679),1948. [30] J.Z.LareseandR.J.Rollenfson.Rotationaltransitionsinmonolayermolecularsolids.Phys.Rev.B,31:3408,1985. [31] A.V.A.Kumar,S.Yashonath,M.sluiter,andY.Kawazoe.Rotationalmotionofmethanewithintheconnesofzeolitenacaa:Moleculardynamicsandabinitiocalculations.Phys.Rev.E,65:011203,2001. 79

PAGE 80

[32] R.Chitra,A.V.A.Kumar,andS.Yashonath.Translational-orientationalcouplingduringthepassageofmethanethroughthebottleneckinzeolitea.J.Chem.Phys.,114-1:11,2001. [33] N.S.Sullivan.Lowfrequencydynamicsoforientationalglasses.Can.J.Chem,66:908,1988. [34] J.DeWall,R.M.Dimeo,andP.E.Sokol.Slowdiffusionofmolecularhydrogeninzeolite13x.J.LowTempPhys.,129:171,2002. [35] J.Eckert,J.M.Nichol,J.Howard,andF.R.Trouw.Adsorptionofhydrogeninca-exchangedna-azeolitesprobedbyinelasticneutronscatteringspectroscopy.J.Phys.Chem.,100:10646,1996. [36] S.Gasiorowicz.QuantumPhysics.JohnWileyandSons,NewYork,1974. [37] R.L.Liboff.IntroductyQuantumMechanic.AdisonWesleyLongman,Inc.,Massachusetts,1998. [38] J.P.Coulomb,N.Flouquet,N.Dufau,P.Llewellyn,andG.Andre.Structuralanddynamicpropertiesofconnedethaneinalpo4-5modelmicroporousaluminophosphate:Doesthepredictedquasi-(1d)phasetransitionexist?Mi-croporousandMesporousMaterials,101:271,2007. [39] C.Kittel.IntroductiontoSolidStatePhysics7thEd.7.JohnWileyandSons,NewYork,1996. [40] C.Martin,J.P.Coulomb,andM.Ferrand.Directmeasurementofthetranslationalmobilityofdeuteriumhydridemoleculesconnedinamodelmicroporousmaterial:Alpo4-5zeolite.Europhys.Lett.,36:503,1996. 80

PAGE 81

BIOGRAPHICALSKETCH YuJiwasbornonFeb.18,1983inChangchun,ChinaastheonlychildofYunxiaLiuandFaJi.After12yearsofelementryandmiddleschool,shespent4yearsinFudanUniversityinShanghaiforherbachelordegreeinphysics.In2006,shegraduatedfromundergraduateschoolandwentUniversityofFloridaforherPh.D.degree.ShejoinedN.S.Sullivan'sgroupin2007summerforNMRstudy.ShemarriedTaoChen,acomputerengineer,in2008andhadasonEricChenin2010.ShereceivedherPh.D.fromtheUniversityofFloridainthespringof2012. 81