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Studying the Effects of Interplay between Anisotropy and Exchange in Molecular Nanomagnets

Permanent Link: http://ufdc.ufl.edu/UFE0024306/00001

Material Information

Title: Studying the Effects of Interplay between Anisotropy and Exchange in Molecular Nanomagnets
Physical Description: 1 online resource (139 p.)
Language: english
Creator: Datta, Saiti
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anisotropy, electron, exchange, molecular
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: This dissertation presents work on several molecular nanomagnets using both theoretical and experimental approaches to understand the effects of the interplay between anisotropy and exchange in these systems. The molecular nanomagnets presented here can be classified into two different categories: single molecule magnets and antiferromagnetic wheels or grids. Magnetic anisotropy plays a significant role in these compounds and leads to bistability of the projections of the magnetic moment (`up' and `down') at lower temperatures, separated by a barrier. The employment of electron paramagnetic resonance techniques allows the understanding of some basic issues in quantum physics. Results presented for two iron complexes show the importance of easy-axis anisotropy in determining whether a molecule can be treated as a single molecule magnet. The importance of transverse anisotropy can be understood from the results obtained for a mixed valent manganese compound. One may consider a giant spin instead of treating the individual spins of the magnetic centers while interpreting these results using the spin Hamiltonian. For a tetranuclear cobalt complex, however, this is no longer a valid approximation and one needs to consider the generalized spin Hamiltonian to interpret the results. An attempt is made to compare anisotropies in ferromagnetic and antiferromagnetic manganese based triangular units. Results from several theoretical techniques like the projection operator and exact diagonalization methods are then supported with results from experiments. This allows an understanding of the combined role of anisotropy and exchange in determining the barrier in these molecular magnets. Finally, results for an antiferromagnetic grid and a wheel are presented to demonstrate how high-frequency/high-field electron paramagnetic resonance can be a useful technique to observe quantum spin dynamics in this class of molecular nanomagnets.
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 Saiti Datta.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hill, Stephen O.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024306:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024306/00001

Material Information

Title: Studying the Effects of Interplay between Anisotropy and Exchange in Molecular Nanomagnets
Physical Description: 1 online resource (139 p.)
Language: english
Creator: Datta, Saiti
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anisotropy, electron, exchange, molecular
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: This dissertation presents work on several molecular nanomagnets using both theoretical and experimental approaches to understand the effects of the interplay between anisotropy and exchange in these systems. The molecular nanomagnets presented here can be classified into two different categories: single molecule magnets and antiferromagnetic wheels or grids. Magnetic anisotropy plays a significant role in these compounds and leads to bistability of the projections of the magnetic moment (`up' and `down') at lower temperatures, separated by a barrier. The employment of electron paramagnetic resonance techniques allows the understanding of some basic issues in quantum physics. Results presented for two iron complexes show the importance of easy-axis anisotropy in determining whether a molecule can be treated as a single molecule magnet. The importance of transverse anisotropy can be understood from the results obtained for a mixed valent manganese compound. One may consider a giant spin instead of treating the individual spins of the magnetic centers while interpreting these results using the spin Hamiltonian. For a tetranuclear cobalt complex, however, this is no longer a valid approximation and one needs to consider the generalized spin Hamiltonian to interpret the results. An attempt is made to compare anisotropies in ferromagnetic and antiferromagnetic manganese based triangular units. Results from several theoretical techniques like the projection operator and exact diagonalization methods are then supported with results from experiments. This allows an understanding of the combined role of anisotropy and exchange in determining the barrier in these molecular magnets. Finally, results for an antiferromagnetic grid and a wheel are presented to demonstrate how high-frequency/high-field electron paramagnetic resonance can be a useful technique to observe quantum spin dynamics in this class of molecular nanomagnets.
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 Saiti Datta.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hill, Stephen O.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024306:00001


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IwouldliketotakethisopportunitytoacknowledgethedirectandindirectcontributionsofmanypeoplewhohelpedmesailthroughalltheseyearssinceIjoinedtheDepartmentofPhysicsattheUniversityofFloridainFall2004.IamdeeplyindebtedtomyPhDsupervisorDr.StephenHillforhiswillingnesstogivemeachancetoworkintheeldofexperimentalcondensedmatterphysics.AlltheseyearsStevehasbeenawonderfulmentorandasourceofsupport.HehasprovidedmewithfullnancialsupporteversinceIjoinedhislaboratoryinMay2005andthathasenabledmetospendmoretimewithequipmentratherthanteachingintroductoryphysicslaboratories.Hehasalsoencouragedmetoattendseveralconferences,bothinandoutsidethecountry,andthosehelpedmeimmenselytoimprovemypublicspeakingskills.Atthesametime,IalsohadthewonderfulopportunitiestointeractwithsomeofthebestknownpeopleinoureldandIamreallygratefultoSteveforthat.IwouldliketothanktheothermembersofmyPhDcommitteewhomonitoredmyworkandtookeortinreadingandprovidingmewithvaluablecommentsonearlierversionsofthisthesis:Prof.DavidTanner,Prof.PeterHerschfeld,Prof.AmlanBiswasandProf.GeorgeChristou.Iamalsogratefultomycollaborators,Dr.OliverWaldmannandDr.EnriquedelBarcofortheirvaluablesuggestionsduringthiscourseinPhD.IwasalwaysabletolearnsomethingnewasIworkedwiththem.Dr.GeorgeChristou,Dr.DavidN.HendricksonandDr.EuanBrechinhaveprovidedmewithsamplestoworkonforalltheseyears.Iamreallygratefultothemforthat.Aspecialthankstomyrst`teachers'inthelaboratory,SusumuTakahashiandSheng-Chiang(John)Leeforhelpingmestartasanexperimentalphysicist.Theyhavebeenverypatientwithmyclumsiness,attheverybeginning,withtechnicaldrawings,handlingequipmentsandmanymore.Ihavelearnedalotfromthemandandtriedmybesttoutilizethosetogetresultsinlateryears. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1MAGNETICANISOTROPYINMOLECULARNANOMAGNETS ....... 14 1.1Introduction ................................... 14 1.2Environmentandoriginofanisotropy ..................... 16 1.3SpinHamiltonian ................................ 20 1.3.1Single-ionanisotropy .......................... 21 1.3.2Spin-spinanisotropy ........................... 22 1.3.3FormulationofthespinHamlitonian .................. 24 2INSTRUMENTATIONANDEXPERIMENTALTECHNIQUE ......... 28 2.1Introduction ................................... 28 2.2Experimentalsetup ............................... 30 2.2.1Waveguides ............................... 31 2.2.2Coaxiallines ............................... 34 2.2.3Cavity .................................. 35 2.2.4LowfrequencyEPRmeasurements ................... 38 2.2.5MillimeterVectorNetworkAnalyzer .................. 41 2.3Summary .................................... 46 3STUDYINGANISOTROPYINSINGLEMOLECULEMAGNETS ....... 47 3.1Introduction ................................... 47 3.2Ironbasedsinglemoleculemagnets ...................... 48 3.2.1DiscussionofHFEPRresults ...................... 49 3.3Mn9singlemoleculemagnet .......................... 56 3.4Summary .................................... 64 4EFFECTOFANISOTROPICEXCHANGEINSINGLEMOLECULEMAGNETS 65 4.1Introduction ................................... 65 4.2DiscussionofHFEPRresults .......................... 69 4.3Numericalcalculations ............................. 73 4.4Summary .................................... 77 6

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....................... 78 5.1Introduction ................................... 78 5.2Overviewofthemolecularclusters ...................... 79 5.3Projectionoperatortechnique ......................... 83 5.3.1Irreducibletensoroperatorapproach .................. 84 5.3.2AssumptionsandcalculationsforMn3andMn6systems ....... 87 5.4Exactdiagonalizationtechnique ........................ 88 5.5Highfrequencyelectronparamagneticresonancestudies ........... 91 5.5.1Mn3complexes ............................. 91 5.5.2Mn6complexes ............................. 96 5.6Summary .................................... 102 6ANTIFERROMAGNETICMOLECULARWHEELSANDGRIDS ....... 103 6.1Introduction ................................... 103 6.2Mn-[33]grid ................................. 107 6.2.1OverviewoftheMn-[33]grid .................... 107 6.2.2Discussionofexperimentalresults ................... 111 6.3Fe18antiferromagneticwheels ......................... 120 6.3.1DiscussionofHFEPRexperimentalresults .............. 120 6.4Summary .................................... 127 7SUMMARY ...................................... 129 REFERENCES ....................................... 131 BIOGRAPHICALSKETCH ................................ 139 7

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Table page 3-1ComparisonbetweenZFS(cm1)parametersobtainedfromthesestudies(EPR)andthevariousmagneticmeasurementsreportedin[ 48 49 ]. ........... 60 5-1Comparisonofmagneto-structuralparametersfortheMncomplexes.Thetorsionanglesareexpressedindegrees;exchangeparameterJanduniaxialanisotropyparameterD,obtainedfrommagneticstudies,areexpressedincm1. ...... 82 5-2ComparisonbetweenvarioustheoreticalandexperimentalratiosoftheanisotropyassociatedwiththeS=6andS=2Mn3complexes:Disthetotalmolecularanisotropy;Uisthemagnetizationreversalbarrier. ................ 95 5-3ComparisonbetweenvarioustheoreticalandexperimentalratiosoftheanisotropyassociatedwiththeS=12andS=4Mn6complexes:Disthetotalmolecularanisotropy;Uisthemagnetizationreversalbarrier.POT1andPOT2correspondtothetwoschemesusedintheprojectionoperatortechnique;EDT1andEDT2correspondtothestrongandweakexchangelimitsintheexactdiagonalizationtechniqueandMSstandformagneticstudies. ................... 101 6-1Firstline:Best-tresultsfortheZFSsofthetransitionsandthegfactor.Secondline:CalculatedZFSsforthebest-tJ,Dvalues.EnergiesaregiveninunitsofGHz. .......................................... 115 8

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Figure page 1-1Classicationofmolecularnanomagnetsandschematicrepresentationoftheenergybarrier ..................................... 15 1-2Schematicrepresentationofametalioninoctahedralandtetrahedralenvironments 17 1-3Schematicrepresentationofthedorbitals ..................... 18 1-4Schematicrepresentationofdistortionofanoctahedralcomplex ......... 19 1-5SchematicrepresentationoftheenergybarrierforaS=12nanomagnet .... 25 1-6Geometricalrepresentationofthetransverseanisotropyterms .......... 26 2-1TypicalEPRspectrumwiththetransitionsshowninthecorrespondingZeemandiagram ........................................ 29 2-2Schematicrepresentationoftheexperimentalsetup ................ 32 2-3Fielddistributionsofthemodesinthewaveguide,coaxiallineandacylindricalresonantcavity .................................... 36 2-4Schematicoflowfrequencyresonator ........................ 39 2-5Transmissionresultsforthe10GHzresonatorandlowfrequencytemperaturedependenceobtainedforMn-33grid ....................... 40 2-6SchematicrepresentationoftheMVNA ....................... 42 2-7SchematicrepresentationoftheESAoption .................... 44 2-8PictureoftheMVNAsetupwithotherelectronics ................. 45 3-1StructureofFe6andFe7complexes ......................... 49 3-2AngledependentdatafortheFe6complex ..................... 50 3-3FrequencyandtemperaturedependentdatafortheFe6complex ......... 51 3-4SimulatedZeemandiagramforFe6andFe7complexes ............... 53 3-5TemperaturedependentspectrafortheFe7complex ................ 54 3-6FrequencydependentdatafortheFe7complex ................... 55 3-7CorestructureoftheMn9complex ......................... 56 3-8TemperaturedependentdatafortheMn9complex ................. 58 3-9FrequencydependentdatafortheMn9complex .................. 59 9

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........................ 61 3-11HardplanedatafortheMn9complex ........................ 62 4-1MolecularstructureofCo4complex ......................... 68 4-2FrequencydependentresultsforCo4complex ................... 69 4-3TemperaturedependentresultsforCo4complex .................. 71 4-4AngledependentresultsforCo4complex ...................... 72 4-5SimulatedZeemandiagramsobtainedforCo4complex .............. 76 5-1MagneticcoreforMn3complexes .......................... 80 5-2Jahn-TelleraxesinMn3complexes. ......................... 81 5-3SchematicrepresentationsofthemagneticcoreforMn3andMn6systems .... 88 5-4Simulatedenergyleveldiagramusingexactdiagonalizationtechnique ...... 89 5-5Anisotropyratiosasafunctionofexchangeusingexactdiagonalizationtechnique 90 5-6Easyaxistemperaturedependentspectraofcomplex2 .............. 92 5-7Frequencydependentdataofcomplex3 ...................... 93 5-8Temperaturedependentdataofcomplex4 ..................... 94 5-9Temperatureandfrequencydependentdataofcomplex5 ............. 97 5-10Temperatureandfrequencydependentdataofcomplex6 ............. 98 5-11Hardplaneangle,temperatureandfrequencydependentdataofcomplex6 ... 100 6-1Schematicoftheenergyspectrumforaspin5/2system .............. 105 6-2SchematicofquantumtunnelingoftheNeelvector ................ 105 6-3StructureoftheMnbasedgridandadimerofoctanuclearringwithacentralion 107 6-4Simulatedenergydiagramofthegrid ........................ 110 6-5EPRspectrumofthegridandfrequencydependence ............... 113 6-6TemperaturedependenceofEPRspectrumforthegrid .............. 114 6-7Simulatedenergydiagramforthegridandcomparisontoexperimentalresults 117 6-8Simulatedspectrumforthegridandcomparisontoexperimentalresults .... 118 6-9StructureofFe18complex .............................. 121 10

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...................... 123 6-11TemperaturedependentdatafortheFe18wheel .................. 124 6-12FrequencydependentdatafortheFe18wheel .................... 125 6-13SimulatedenergydiagramfortheFe18wheel .................... 128 11

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Thisdissertationpresentsworkonseveralmolecularnanomagnetsusingboththeoreticalandexperimentalapproachestounderstandtheeectsoftheinterplaybetweenanisotropyandexchangeinthesesystems.Themolecularnanomagnetspresentedherecanbeclassiedintotwodierentcategories:singlemoleculemagnetsandantiferromagneticwheelsorgrids.Magneticanisotropyplaysasignicantroleinthesecompoundsandleadstobistabilityoftheprojectionsofthemagneticmoment(`up'and`down')atlowertemperatures,separatedbyabarrier.Theemploymentofelectronparamagneticresonancetechniquesallowstheunderstandingofsomebasicissuesinquantumphysics. Resultspresentedfortwoironcomplexesshowtheimportanceofeasy-axisanisotropyindeterminingwhetheramoleculecanbetreatedasasinglemoleculemagnet.Theimportanceoftransverseanisotropycanbeunderstoodfromtheresultsobtainedforamixedvalentmanganesecompound.OnemayconsideragiantspininsteadoftreatingtheindividualspinsofthemagneticcenterswhileinterpretingtheseresultsusingthespinHamiltonian.Foratetranuclearcobaltcomplex,however,thisisnolongeravalidapproximationandoneneedstoconsiderthegeneralizedspinHamiltoniantointerprettheresults. Anattemptismadetocompareanisotropiesinferromagneticandantiferromagneticmanganesebasedtriangularunits.Resultsfromseveraltheoreticaltechniquesliketheprojectionoperatorandexactdiagonalizationmethodsarethensupportedwithresults 12

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Finally,resultsforanantiferromagneticgridandawheelarepresentedtodemonstratehowhigh-frequency/high-eldelectronparamagneticresonancecanbeausefultechniquetoobservequantumspindynamicsinthisclassofmolecularnanomagnets. 13

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1 ]tosynthesizethesepolynuclearclustersbytreatingmoleculesasbuildingblocks.Ideallyonecanstartwithamagneticcenterandaddonemagneticcenteratatime.Thiscanleadtolargeassembliesofidenticalparticleswhichcanthenbeusedtostudyvariousquantumeects. Themolecularnanomagnetsdiscussedinthisdissertationcanberoughlyclassiedintotwocategories:systemswithferro-orferri-magneticinteractionleadingtoalargemagneticmomentforeachmoleculeandsystemswithdominantantiferromagneticinteractionleadingtononeorveryweakmagneticmoment.Forbothsystemsmultipletransitionmetalionswithunpairedelectronsconstitutethemagneticcore.Intherstcasestrongcouplingbetweentheionsthroughintramolecularexchangeinteractionsleadstoalargemagneticmomentforeachmoleculeandthesecomplexesarecalledsinglemoleculemagnets.Anisotropyplaysasignicantroleinthesesystemsandleadstobistabilityoftheprojectionsofthemagneticmoment(`up'and`down')atlowertemperatures.AschematicrepresentationoftheanisotropybarrierandbistablestatesareshowninFig.1-1. Theidenticationoftherstsinglemoleculemagnet(SMM)wasmadein1993[ 2 ]whenthepolycrystallinepowderofamolecularcomplex[Mn12O12(CH3COO)16(H2O)4]orMn12-acetate,synthesizedbackinthe1980s[ 3 ],wasshowntoexhibitmagnetichysteresisresultingfromslowrelaxationofthemagnetizationoftheindividualmolecules.SincethenseveralSMMshavebeensynthesizedandstudied.Intermolecularexchangeinteractionsinthesecomplexesareratherweakandthereisnolongrangeordering.Therequisitesforsinglemoleculemagnetsare: 14

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ClassicationofmolecularnanomagnetsdiscussedinthisdissertationandschematicrepresentationoftheenergybarriershowingthebistablestatesofaSMM. ThetemperaturebelowwhichtherelaxationofthemagnetizationbecomesslowforaSMMisknownastheblockingtemperature.ForthemoststudiedsinglemoleculemagnetMn12-acetate,thebarriertomagnetizationreversalisabout60K[ 4 { 6 ]andthecorrespondingrelaxationtimeisseveralmonthsat2Kand50yearsat1.5K.ThisrecordremainedunbrokenuntiloneoftherecentlydiscoveredMn6complexes,byBrechinandothers,wasshowntoexhibitthehighestbarrierof86Kandhighestblockingtemperatureof4.2Kfrommagneticstudies[ 7 8 ]. Antiferromagneticsystemshoweverdonotshowslowrelaxationofmagnetizationbutexhibitotherinterestingmagneticproperties.PropertiessuchasrotationoftheNeelvectorandspinwavescanalsobeobservedinthesesystems.Thesephenomenaare 15

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Molecularnanomagnetsalsoallowtestingoftheoriessuchasthecoexistenceofquantumandclassicaleectsinmagnets[ 9 ].Eectssuchasquantumtunnelingofthemagnetization[ 10 { 13 ],oscillationsofthetunnelsplitting[ 14 ]havebeenobservedinthesemolecularmagnetsandsincehaveattractedtheattentionofagrowingnumberofresearchers. Thebistablestatesinthelowenergysectorofthesemolecularnanomagnetsoerseveraladvantages.Thesingleelectronspinsinthesestatescanbetreatedasqubits,pavingthewaytowardsquantumcomputing[ 16 { 18 ].Otherwidespreadpotentialtechnologicalapplicationscouldbeinhigh-densityinformationstorageduetotheirlongcoherencetimes,[ 19 20 ]inmagneticcooling(asmagneticrefrigerants)duetothelargemagneticentropyvariationsrelatedtotheirlargespin,[ 21 ]andinbiomedicineasmagneticcontrastagentsinmagneticresonanceimagingandinhyperthermictreatmentoftumors[ 22 23 ]. 16

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Schematicrepresentationofametalioninan(a)octahedraland(b)tetrahedralenvironments. Foranisolatedion,thedorbitalsaredegenerate.Howeverwiththecrystaleld,thedegeneracyisbrokenandthed-orbitalssplitintotwosetswithanenergydierence:orbitalsthatpointbetweenthex-,y-andz-axes(dxy,dxzanddyzorbitals)andtheorbitalsthatpointalongtheseaxes(dz2anddx2y2orbitals).SchematicrepresentationoftheorbitalscanbeseeninFig.1-3.Fortheoctahedralsymmetry,thecoulombicrepulsionsbetweenthenegativelychargedligandsandelectronsoccupyingdz2anddx2y2orbitalsresultinraisedenergiesoftheseorbitalsrelativetothelevelsofthedegeneratecase.Incontrastorbitalsdxy,dxzanddyzareloweredinenergy.Thereverseistrueforthetetrahedralenvironment.Furthermore,theenergysplittinginthetetrahedralcaseisroughlyequalto4/9thofthesplittingintheoctahedralcasesincetheligandelectronsintetrahedralsymmetryarenotorienteddirectlytowardsthed-orbitals.Thesizeofthegapbetweentheorbitalsdependsonseveralfactors,includingtheligandsandgeometryofthecomplex.Foratransitionmetalionwithunlleddorbitals,theorderinwhich 17

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Figure1-3. Schematicrepresentationofthedorbitals. Nowthesecrystaleldenvironmentscansometimesdistorttohaveenergeticallyfavorableenvironment.ThisisknownastheJahn-Tellereectwheretheorbitallydegenerateelectronicstateofanon-linearmoleculearedistortedtoremoveremainingdegeneracy.Forcompletelyfulloremptyorbitals,theoverallenergydoesnotchangeinthisprocess.However,forpartiallylledorbitalsthechangeissignicantwithanetreductioninenergy.Forexample,Mn3+ionsinanoctahedralenvironmentshowthiskind 18

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SchematicrepresentationofJahn-Tellerelongation(b)andcompression(c)ofanoctahedralcomplex(a).Thesplittingsofthedorbitalsarealsoshown. ofbehaviortolowertheoverallelectronicenergy.Iftheoctahedroniscompressedalongthez-axis,theenergyoftheorbitalsassociatedwiththez-axis,dxz,dyzanddz2increaseswhiletheenergyofdxyanddx2y2orbitals,linkedtothexandyaxes,isreduced.Theorbitalstendtomoveinoppositedirectionsiftheoctahedroniselongatedalongthez-axis.AschematicoftheeectisshowninFig.1-4.Thelevelsplittingsareshownforboththeelongationandthecompressioncases. Spin-orbitcouplingisanothereectivewaytoremovedegeneracyfromtheenergystates.Magneticanisotropyisintroducedinthesystemasthismechanismfavorstheorientationofthemagneticmomentofthemoleculealongcertaindirection.Auniaxialanisotropyleadstobistablestates(`up'or`down'),separatedbyanenergybarrier.ForcomplexeswithaJahn-Tellerdistortion,theaxisofdistortionistheanisotropyaxisandthesignoftheanisotropyparameterdependsonwhetherthedistortionisacompressionoranelongation. 19

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Withtheintroductionofuniaxialandg-anisotropywenowfocusontheformulationofaneectiveHamiltonianforthesystem.WewilldiscussthespinHamiltonianfromhereon,ignoringthecontributionfromorbitalmoment. 20

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HerewetrytoformulatethespinHamiltonianforthenanomagnetsfromthepointofanisotropyinthesystem. 24 ].Fortransition-metalions,SOCislargeenoughtogivesignicantanisotropyeects,intheabsenceofanexternaleld(zero-eldsplitting)bymakingcertainallignmentspreferable.AgivenS-multiplet,isdegenerateinzero-eldduetoitssphericalsymmetrybutthedegeneracyisremovedwhenthesymmetryisreduced.Theasymmetryinthiscasecanbeexplainedwithaseriesexpansioninsphericalharmonics[ 25 ],asshowninthespin-Hamiltoniandescribedby: ^H1=D[S2zS(S+1)=3]+E(S2xS2y)(1{1) ThersttermintheHamiltonianisdiagonalinthejS;Mibasis,whilethesecondtermmixesstateswhichdierinMby2.DrepresentstheaxialanisotropytermwhereasErepresentsthesecondordertransverseanisotropytermandthelimitofthevariationofEisgivenby:1=3E=D1=3.ThenegativesignoftheuniaxialZFSparametercorrespondstoaneasy-axistypemagneticanisotropy,wherethehighestjmSjlevellielowestinenergy. 21

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Inpresenceofanexternalmagneticeld,anisotropyisintroducedinthesystemintermsofZeemaninteraction.TheZeemanHamiltonianisgivenby: ^H2=B^S!g~B(1{2) whereBistheBohrmagneton,Bistheexternalmagneticeld,gistheLandetensorandSisthespinoperatorforthemolecule.Formostofthemolecularmagnetsdiscussedinthisdissertation,gisconsideredisotropic.HoweverforthetetranuclearCocomplexdiscussedinCh.4,ananisotropicgtensorhastobetakenintoaccount. Anothereectthatmighthavecontributiontothesingle-ionanisotropyishyperneinteractionortheeectofthemagneticnuclei.Theelectronspin-density,interactionbetweentheelectronspinandthemagneticspinandalsotheinteractionbetweentheorbitalmomentoftheelectronandthenuclearspincangiverisetothiseect.However,theenergiesinvolvedinthiscasearemuchsmallerandthuswouldnotbeconsideredforourstudiesdiscussedinthisdissertation. 22

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ForasystemcharacterizedbyspinsS1andS2,theHamiltoniancannowbewrittenas: ^H3=^S1!J12^S2=J12^S1^S2+^S1!D12^S2+!d12(^S1^S2) (1{4) wherethersttermrepresentsisotropicinteractionwhichremovesthedegeneracyoftheeigenvaluesofS2.Theenergyexpressionforthe(2S+1)degeneratestatesgeneratedonlybythisprocessisgivenby Inthiscasethespinscaneitherbeparallelorantiparallel. Thesecondtermdenotesanisotropicexchangeinteractionbetweenthetwospincenters.InCh.4,wediscusstheroleofanisotropicexchangeinteractioninaCo4complexleadingtolargezeroeldsplittings.Fordipolarinteraction!D12=2B(g1g23(g1~R)(~Rg2))=R3.~Risaunitvectorparalleltothe1-2directionandRthedistancebetweenthetwomagneticcenters.Forapairofspinsthedipolarinteractiontendstoalignthemparalleltoeachother,givingrisetoeasy-axistypeanisotropy,paralleltothelineconnectingthetwocenters.ThemagnitudeofthedipolartermsscalesasR3.Theanisotropiccomponentisnegativewhenthepointdipolarapproximationisvalid.Forallthesinglemoleculemagnetsdiscussedinthisdissertationthecontributionfromthedipolarexchangeinteractionismuchsmallerthantheuniaxialanisotropyandisthusneglected.HoweverforantiferromagneticsystemssuchasMn-[33]griddiscussedinchapter6,thedipolarinteractionisthemainsourceofanisotropy. ThethirdtermgivestheantisymmetricinteractionbetweenthemagneticcentersandiscommonlyknownasDzaloshinsky-Moriainteraction.Spinorbitcouplingleadsto 23

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^Heff=NXiNXjiJij^Si^Sj+NXifDi[S2i;zS(S+1)=3]+E(S2i;xS2i;y)g+NXiNXji!dij(^Si^Sj)(1{6) whereNisthenumberofspincentersofthepolynuclearcluster. Instrongexchangelimit(J>>D),onecanrewritetheHamiltonianusingagiantspinapproach(GSA).Theresultantspinvectorisobtainedthroughthevectorsumofthespinsintheclusters.ThesinglespinHamiltonianintermsoftheso-calledStevensoperatorequivalents[ 26 ]isgivenby ^HGSA=XkkXq=0Bkq^Okq(Sz;S)=D[S2zS(S+1)=3]+E(S2xS2y)+Xk>2kXq=0Bkq^Okq(Sz;S) (1{7) wherethe^Okqsareconstructedofthez-projection,raisingandloweringangularmomentaoperatorsandtheBkqcoecientsaredeterminedfromexperiments.Theoriginofthezeroeldsplittingcomesfromthedirectinteractionbetweenthemagneticdipolesoftheunpairedelectronsforthelowestorderterms,whilethemixingofexcitedstatesintothegroundstateviaSOCgiverisetothesehigherordertermsintheHamiltonianforsystemswithspinS2. Figure1-5representstheenergybarrierforasystemofS=12.ForD<0thelowestenergystatesaregivenmS=12.ThesetwostatesareseparatedbyanenergybarrierU=jDjS2.WewouldbediscussingtheeectofthenegativeuniaxialanisotropytermonSMMsinCh.3andCh.5. 24

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SchematicrepresentationoftheenergybarrierforaS=12nanomagnetatzeroeld. Theseoperatorsareobtainedbyusingtheoperatorequivalentmethodofnitegroupswheretheangularmomentumoperatorshavethesametransformationpropertiesasthesphericalharmonics,requiredintheexpansionofthepotentialduetoacrystalelectriceldoftheappropriatesymmetry.Herethequantummechanicalequivalentofagivensphericalharmonicisexpressedasafunctionofthespinangularmomentumoperator^SwithintheselectedS-multiplet.Forexample,theoperators^O24,^O34and^O66canbeexpressedas: ^O24=1=2[7^S2z(S(S+1)+5);^S2+^S2]+^O34=1=2[^Sz;^S3+^S3]+^O66=1=2(^S6+^S6) (1{8) 25

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Geometricalrepresentationofthetransverseanisotropyterms. Theexpressionscanthenbewrittenintermsof^Sxand^Syoperatorsandnallyintermsofpolarcoordinatesas: ^O24cost2sint2^O34cos(t)(cost23sint2)^O66cost6sint615cost2sint2cos2t Thepolarrepresentationsoftheoperatorsasafunctionoftwheret=02areshowninFig.1-6.ThelobesinthegureclearlyshowthesymmetrytakenintoaccountasthesetermsareincludedintheHamiltonian.Forexample,^O22and^O24showtwo-foldsymmetry,^O34three-foldand^O66six-foldsymmetry.TheeectofthesetransverseanisotropytermsarediscussedindetailsinCh.3. TheeectivespinHamiltonianconsidersallkindsofexchangeinteractionsbetweenparamagneticionswithunlledelectronshellsofthetransitionmetalionslinkedby 26

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TheexperimentalobservablesofmolecularnanomagnetsareusuallywelldescribedbythespinHamiltonian,makingitpossibletoreproducetherelativepositionsoftheenergylevelsconstitutingthespingroundmultipletandanalyzetheadmixtureofthecorrespondingwavefunctions.TheresultsobtainedfromEPRexperimentsanddiscussedinchapter3canbewellanalyzedusingthegiantspinHamiltonianwhereastheeectivespinHamiltonianisusedtoanalyzetheresultsshowninCh.4,Ch.5andCh.6. 27

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Inthischapterwewilldiscusselectronparamagneticresonance(EPR),thetechniqueusedtoconductourstudiesonmolecularmagnets.Electronparamagneticresonanceisthemicrowavebranchofspectroscopythatallowstostudytheinteractionbetweenelectronicmagneticmomentsandtheirenvironmentsandmagneticelds.WehavealreadytalkedabouttheZeemanterminthespinHamiltonianinchapter1.ForasystemwithspinS,theenergylevelsaresplitinto2S+1stateswiththeapplicationofanexternalmagneticeld.Ifthemagneticeldisparalleltothez-axisof!g,theenergydierenceisgivenby E=gzBBzmS(2{1) whereBzistheappliedeldandmSisthedierenceintheprojectionofthespinsmagneticmomentalongthezaxis.Whentheenergydierencematchestheincidentradiationenergy(h),resonanceisobservedandismeasuredbysweepingthemagneticeldataxedfrequency. 28

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Figure2-1. (a)TypicalEPRspectrumobtainedat331GHzand10KforaferromagneticMn6complexwithS=12.ThepeaksarelabeledwithcorrespondingmStransitions.(b)SimulatedZeemandiagramforaS=12systemwitheldappliedalongthez-axis.ThearrowscorrespondtothepeaksobtainedfromEPR. 29

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Unfortunately,oneneedstoovercomeseveraltechnicalchallengestomonitortheEPRresponseofasampleinthepresenceofexternalmicrowaveradiationandmagneticeld. Allthesefactorsweretakenintoaccountwhiledesigningtheexperimentalsetuptoconductmicrowavespectroscopy. 30

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StartingfromtheMaxwell'sequationsgivenby @t(2{2) 31

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SchematicrepresentationoftheexperimentalsetupforhighfrequencyEPRmeasurements.Schematicdiagramsofthevariouscomponentsoftherotatingcavitysystemandtheverticalcavityarealsoshown.Themagneticelddirectionsfordierentmagnetsusedfortheexperimentsareclearlyshowninthegure. 32

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@t(2{3) andusingtheappropriateboundaryconditions,onecanndthethreegeneraltypeswavecongurationsthatpropagatealongthetransmissionlines[ 29 30 ].Forthetransverseelectromagnetic(TEM)wavesHz=Ez=0,fortransverseelectric(TE)wavesEz=0;Hz6=0andfortransversemagnetic(TM)wavesHz=0;Ez6=0.ArectangularwaveguidesupportsTMandTEmodesbutnotTEMwaves.Themagneticeldlinesformclosedloopsparalleltothesurfaceoftheconductorsinarectangularwaveguide.Thestrengthofthemagneticeldisproportionaltotheelectriceld.Nowletusconsiderarectangularwaveguidewithdimensionsaandbwherea>b.Thecharacteristicimpedancesinthiscasearegivenby )1=2[1(fc )1=2[1(fc wherethecut-ofrequencyfcisgivenby 2p a)2+(n b)2(2{5) mandnareintegerswithmdenotingthenumberofhalfcyclevariationsoftheeldsinthex-directionandndenotingthenumberofhalfcyclevariationsoftheeldsinthey-direction.Thecut-ofrequencyrepresentthelowerlimitofthefrequenciesthatwillpropagateforagivenmode.Themodewiththelowestcut-ofrequencyiscalledthefundamentalmode.TE10modeistheminimumpossiblemodethatgivesnonzeroeldexpressionsforrectangularwaveguidesandhenceisthefundamentalmodeofarectangularwaveguidewitha>b.Thecut-owavelengthandtheimpedanceforthismode 33

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)1=2[1( Thustooperateinthedominantmode,the`a'dimensionofthewaveguidemustbeatleastonehalf-wavelengthofthefrequencytobepropagated.Ontheotherhand,thehigh-frequencylimitofarectangularwaveguideisachievedwhenthedimension`a'becomeslargeenoughtoallowoperationinamodehigherthanthatforwhichthewaveguidehasbeendesigned.Adetaileddiscussionofthewaveguideprobecanbefoundelsewhere[ 31 ]. 34

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2)( )1=2ln(ro whereroistheradiusoftheouterconductorandrithatoftheinnerconductor.Theattenuationduetoconductorlosses whereRistheresistanceperunitlength,givenby Copperandstainlesssteelcoaxiallineswereusedtoconstitutetheprobe.Thelengthofeachcoaxiallinewasdeterminedbythetemperatureproleinthecryostat.Thelineswereconnectedtoeachotherwith50impedanceSMA(subminiatureversionA)connectors. [!mnp]TE=1 [!mnp]TM=1 35

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(a)FielddistributionofthedominantmodeTE01inarectangularwaveguide.(b)FielddistributionsofTEMmodeinacoaxialline.(c)FielddistributionofthedominantmodeTE011inacylindricalresonantcavity. wherem,nandpareintegersandxmnandxdmnarethenthrootsofthemthorderBesselfunctionanditsrstderivative,respectively. Abovethecut-ofrequency,thepowerspectrumfollowsLorentzianshape 4(!!0)2+(2)2(2{12) wherethecenterfrequencyf0=!0=2,isthefullwidthathalf-maximum.Thenthemeasureofsharpnessofresponseofthecavitytoexternalexcitation,qualityfactororQ-factorisgivenby 36

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Cavityperturbationtechnique[ 32 ]wherethehighqualityfactorofresonantmodesofthecavity,ensuregoodcouplingoftheradiationtothesampleandcompensateforthesmallllingfactorofthesample,wasthenappliedforEPRmeasurements.Howeverintheprocess,oneislimitedtoworkingatfrequenciesthatcorrespondtothemodesofthecavity.Withtheintroductionofthesampleintothecavity,thecharacteristicsoftheresonancechangeslightlywithashiftoftheresonantfrequencyandachangeinbandwidth. Themicrowavesfromthewaveguideswerecoupledtothecavitythroughtwosmallcouplingholeswithadiameterofroughly=6forafrequencyofapproximately50GHz.Thecouplingholesattheendofthewaveguideactasanelectriceldnodeandcancoupletothemagneticeldinsidethecavityonly.Thecouplingstrengthwasdeterminedbytheratiooftheabsorbedpowerattheresonancefrequencytothereectedpowerjustotheresonance.Thediameterandthicknessoftheaperturesdeterminethecouplingstrength.Whilelargeaperturesensurestrongcouplingandalargedynamicrange,theydecreasethesensitivityandtheQfactorofthecavity.Smallaperturesontheotherhand,limitradiationandalsoensurehighercavityQvaluesandsensitivity.Alsothethicknessofthecouplingholeneedtobe=20toreducethepowerloss. 37

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33 ]. Microstripresonatorshaveseveraladvantagesoverastandardcavity.Theyhaveabetterllingfactorandthesizeoftheresonatorissmallerthanthesizeoftheneededcavityatlowfrequencies.Thesameresonatorcanbeusedfordierentharmonicsof 38

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Schematicoflowfrequencyresonator. theresonancefrequencyasinthecaseofthecylindricalcavity.TheACmagneticeldisconcentratedinasmallervolumenearthesmallerstrip,wherethesamplematerialsareplaced,andtheirresonancefrequenciescanbedesignedtobeintherangeofmanymagneticmaterials.TworesonatorsweredesignedincollaborationwiththegroupofDr.AndrewKent,NewYorkUniversity,tohavearesonancefrequencyofabout10and15GHz.TheresonatorsweremadebyevaporatingcopperontopofGalliumArsenide(GaAs)wafersandthenevaporatingcopperonthebacktomakeagroundplane. Theresonatorisconstructedintwosteps[ 38 ].Atrstthestructureconsistsofacenterconductingstripseparatedfromtwogroundplanesbyadielectricmaterial.Thecongurationallowsthepropagationoftransverseelectromagnetic(TEM)waves,similartothecoaxiallines.Thecharacteristicimpedanceoftheresonatorisgivenby wherewisthelengthofthecenterconductingstripandbisthewidthoftheresonator.Theguidewavelengthisgivenby 39

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(a)Calibrationofs12(transmission)componentforthe10GHzresonator.(b)Lowfrequency(10GHz)temperaturedependenceobtainedfortheantiferromagneticMn-33gridwiththeeldparalleltothehardplaneofthegrid.DetaileddescriptionofthegridcanbefoundinCh.6. whereisthefree-spacewavelengthandkisthedielectricconstantofthematerialthatseparatesthetwogroundplanes.Fora50transmissionlineandk=2.32,thetypicalvaluesofwandbare4.8mmand6.4mmrespectively. Next,twoequalslitsarecutonthecenterstriptoformtheresonatorshowninFig.2-3.Theresonantfrequencyoftheresonatorisdeterminedbythelength,l,oftheresonantstrip.Theresonantconditionisgivenby 40

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Theresonatoristhenplacedinarectangularbrasscavityandthecoaxialleadswereattachedtothegroundplanes.Figure2-4(a)showsthecharacterizationofa10GHzmicrostrip.Severalmodescanbeseenasthefrequencyissweptfrom0-50GHz.Themodeat10GHz,withaQfactorof200,waschosentoperformexperimentsontheantiferromagneticMn-33grid.Adetaileddiscussiononthesamplecanbefoundinchapter6.Thesamplewasplacedatonthecenteroftheresonatorand17TOxfordmagnetwasusedtoperformexperiments.Themagneticeldwasappliedalongthehardplaneofthemolecularmagnet.Figure2-4(b)showsthetemperaturedependenceoftheEPRspectrumobtainedat10GHz.Atthelowesttemperature,severaltransitionsareobservedcorrespondingtotheS=5/2groundstatemultiplet.Asthetemperatureisincreasedfrom2Kto15K,onlyacenterlineisobservedcorrespondingtothetransitionM=1 2!+1 2asexpectedforahalf-integerspinsystem.Thedataclearlyshowthatonecanusethehalf-waveresonatortoobtainlowfrequencyEPRdatawithourexperimentalsetup.However,theprocessoerseverallimitations.Priorinformationonthemagneticaxesofthecrystalisneededtoalignthesampleinthemagneticeld.Also,wearelimitedtoobtaindataforaxedorientationsincetheOxfordmagnetdonotprovideanyrotationalcapability.ThiswasourrstsuccessfulattempttoobtainEPRdataatfrequenciesaslowas10GHzandinfuturewehopetoimproveourinstrumentationtohavebettersensitivities. 34 ].SchematicrepresentationsoftheMVNAwithoutandwithanESAoptionareshowningures2-6and2-7 41

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Figure2-6. SchematicrepresentationoftheMVNA. ThesecondoscillatorsuppliesasecondsignalF2toasecondSchottkydiode,theso-calledharmonicmixer(HM).ThetwofrequenciesFmmandF2arebeattogetheratHMandthebeatfrequencyisgivenbyFbeat=jFmmN1F2j=j(NF1)(N1F2)jwhereN1istheharmonicassociatedwiththesecondSchottkydiode.Thisbeatfrequencyisthen 42

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ToachievealownoiseleveltheharmonicsandthephasesfromboththeSchottkydiodesmustbematchedi.e.N=N1and1=2.FordetectionofthesignalatthevectorreceivertheMHzcomponentofthebeatsignalFr=N(F1F2)mustcorrespondtothepreciseoperatingfrequencies(9.01048828125or34.01048828125MHz)ofthereceiver. Inthiscasebeat=0andthemainoscillatorsendsthephasereferencedirectlytothevectorreceiver.TheschematicrepresentationshowninFig.2-6givesthebasicworkingprincipleoftheMVNAwithoutanyexternalsource.TheMVNAandthediodesoeralargedynamicrange.Forfrequenciesupto100GHz,thetypicaldynamicrangeisabove100dB,therangedecreases90dBat110GHztoroughly60dBat170GHz. Withtheincreaseinharmonicnumber,theoutputpoweroftheharmonicgeneratordecreases.Forfrequenciesabove165GHz,theESAoptionconsistingofaGunndiodeandamulti-harmonicmultiplierwasused.Thisalsoallowtoobtainalargedynamicrangeforfrequenciesabove170GHz.ThetwoGunnoscillatorscalledGunnA1andGunnA2operateinthefrequencyrangefrom69-82.3GHzand82.3-102.5GHz,respectively.WiththeGunndiodesandthemultiharmonicmultiplier,frequenciesupto500GHzandhighercanbeachieved.Themultiharmonicmultipliercanbemechanicallytunedtooptimizetheincomingandtheoutputpowerataparticularfrequency.Variouslterswereusedfordierentfrequencyranges(>138GHz,>235GHz,>345GHz,>460GHz,>560GHz)toremovelowerharmoniccomponentsofthesignal.TheworkingprincipleforbothGunnA1andA2isthesameandconstitutesseveralsteps. TheGunnsourceislockedtothekthharmonicoftherstYIGoscillatoroftheMVNAwithanosetequaltothe50MHz(F0)referenceoscillatorusingaphaselockedloop.ThefrequencyinthiscaseisgivenbyFa=(kF1)F0.Theemittedmicrowavefrequency(Fa)fromtheGunnisthenfedintotheSchottkymultiplier(MU).NonlineareectsfromthediodegenerateharmonicsofthefrequencygivenbyFmm=MFa= 43

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SchematicrepresentationoftheESAoption. (MkF1)(MF0).Toperformanyexperiment,oneoftheharmonicsischosenandtheappropriatelterisused. TheHMdetectorsimilarlycreatesharmonics(N)ofthefrequencyoftheemittedmicrowave(F2)froma2ndlocaloscillatorintheMVNA.ThesefrequenciesinterferewiththeincomingfrequenciesgivingrisetointermediatefrequenciesgivenbyFif=(MFa)(NF2)=(MkF1)(MF0)(NF2).Duetothephaselockedloopthereisaxedfrequencydierencebetweenthefrequenciesemittedbythetwosources.ThuswecanconsiderF1F2=f.NowFif=(MkN)F1+Nf(MF0).InordertoobtainFif=Fr,MkhastobeequaltoN.Then,f=F0=k+Fr=N.Thephasenoiseinthiscaseisgivenbyif=g(N2)=(MkN)1=0. TwomagnetsystemsareavailableattheUniversityofFlorida,Gainesville:anaxialsuperconductingmagnetproducedbyOxfordInstruments[ 35 ]andonetransversesplit-coilsuperconductingmagnetbyQuantumDesign[ 36 ].WhiletheOxfordmagnet 44

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PictureoftheMVNAsetupwithotherelectronics.TheMVNAiscontrolledbythePCcomputershownontheleft.TheMVNAisphaselockedwiththeEIPcounter.Theoscilloscopedisplaystheamplitudeandphaseofthedetectedsignal.OntherightthepowersuppliesandthecontrolelectronicsoftheGunnsystemsareshown. allowamaximumeldof17T,theQDphysicalpropertymeasuringsystem(QDPPMS)isrestrictedto7T.Mosttheresultspresentedinthisdissertationwereobtainedusingthesetwomagnets.TheuseofrotatingcavityintheOxfordmagnetallowustodosingleaxisrotationstudieswhilethesamecavityintheQDsystemallowsfortwoaxisrotationstudies.AsteppermotorassociatedwithQDPPMSallowsustodorotationinxy-planewhilethez-axisrotationisdoneusingtherotationmechanismofthecavityendplate.SomeoftheresultsobtainedforCo4complexwereobtainedusinga9TverticaleldsuperconductingQDPPMSmagnet.AlsosomeoftheresultsfortheantiferromagneticFe18wheels(discussesinchapter6)wereobtainedusingthe33TaxialresistivemagnetattheNationalHighMagneticFieldLaboratory,Tallahassee[ 37 ].Thecryostatsdesignedforallthesemagnetsarebasedon4Hesystem.Adetaileddescriptionoftheexperimentalsetupcanbefoundelsewhere[ 39 { 41 ]. 45

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46

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Theresultspresentedinthischaptercanbefoundinthearticles:EPRcharacterizationofhalf-integer-spinironmolecule-basedmagnets,S.Datta,A.Betancur-Rodriguez,S.C.Lee,S.Hill,D.Foguet-Albiol,R.BagaiandGeorgeChristou,Polyhedron26,2243-2246(2007)(reusedwithpermissionfromElsevier);Diversityofnewstructuraltypesinpolynuclearironchemistrywitha(N,N,O)-tridentateligand,R.Bagai,S.Datta,A.Bentacur-Rodriguez,K.A.Abboud,S.Hill,andG.Christou,Inorg.Chem.,46(11),45354547(2007)(reusedwithpermissionfromAmericanChemicalSociety);Transverseanisotropyinthemixed-valentMnII2MnIII4MnIV3single-moleculemagnet,S.Datta,C.J.Milios,E.Brechin,andS.Hill,J.Appl.Phys.103,07B913(2008)(reusedwithpermissionfromAmericanInstituteofPhysics.). 47

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Fe6crystallizesinthetriclinicspacegroupP-1withtheasymmetricunitcontainingtwoindependentFe6clusters,bothlyingoninversioncenters;sincethetwomoleculesareessentiallysuperimposable,weshowanddiscussthestructureofonlyoneofthemhere.Thecoreconsistsofan[Fe4(3-O)2)]unit(Fe1,Fe1',Fe2andFe2']oneithersideofwhichisattached[Fe(3-O)2)(-OR)]unitcontainingFe3;theOH-ionsareO9andO10ononeside,andtheirsymmetrypartnersontheother.Fe7crystallizesinthemonoclinicspacegroupC2/cwiththemoleculelyingonacrystallographicC2axispassingthroughthecentralFe4atom.ThestructureofboththecomplexesisshowninFig.3-1. 48

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(a)LabeledrepresentationofthestructureofFe6complex.Hydrogenatomsandphenylrings(exceptfortheipsocarbonatoms)havebeenomittedforclarity.TheC2symmetryaxisisapproximatelyvertical.(b)Labeledrepresentationofthecentro-symmetricstructureofFe7complex.Hydrogenatomsandmethylgroupsonpivalategroupshavebeenomittedforclarity.Colorcode:FeIIIgreen;Ored;Nblue;Cgrey. ^H=D^S2z+E(^S2x^S2y)+gB^S~B(3{1) whereEistherhombicZFSparameterand^Sxand^Syarethexandycomponentsofthetotalspinoperator^S.EPRbeingahighresolutionspectroscopictechnique,canbeusedtoinvestigatethecompletespinHamiltonian,whereastsofbulkmagnetizationdataareessentiallyinsensitivetoinclusionoftherhombicEterm. Single-axisangle-dependencestudieswererstperformedtoroughlydeterminetheorientationofeachcrystalinthemagneticeld.Bothcomplexespossesslow-symmetrystructures.Thus,determiningtheprecisesymmetrydirectionsrepresentsahighlycomplextaskrequiringdetailedtwo-axisrotationstudies.However,aswehaverecently 49

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43 { 45 ],inparticular,thesignofD,whichisthecrucialfactorinwhetheraparticularcomplexisaSMM. Figure3-2. PlotoftheHFEPRpeakpositionsfortheFe6complexobtainedfromangledependentstudiesat116GHzand1.4K. Figure3-2displaystheangledependenceoftheeldpositionsofthestrongestEPRtransitionsdeterminedfromeld-sweptspectrarecordedat116GHzand1.4KfortheFe6complex;giventhelowtemperature,thesedatapointsmustcorrespondtotransitionsfromthelowest-lyingmSlevels.Twoseriesofresonancesareobserved(blackandreddatapoints),whichshiftsignicantlyuponrotationoftheeld,thusprovidingtheclearestevidenceforasignicantmagneto-anisotropy.Bothseriesexhibit180operiodicity,withvirtuallyidenticalamplitudes.ThesourceofthetwoserieshasanaturalexplanationfortheFe6complexforwhichtherearetwodierentlyorientedmoleculesintheunitcell.Thus,onenaturallyexpectstwodistinctEPRsignatures,onefromeachorientation.The 50

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Figure3-3. FrequencydependencefortheFe6complexwiththeeldorientedalongoneoftheminimainFigure3-2(191o);theinsetdisplaystemperature-dependentspectraobtainedat106GHz. TodeterminethesignofD,frequency-andtemperaturedependentdatawerecollectedontheFe6complexwiththeeldorientedalongoneoftheminimainFig.3-2(191o).Figure3-3displaysthefrequencydependenceoftheangle-dependentpeakfromFig.3-2,andtheinsetdisplaysrepresentativespectratakenathighertemperatures.Aremarkablefeatureofthefrequency-dependentdataisthatallpeakslieonastraightline,whichextrapolatestoanitefrequencyontheverticalaxis;i.e.,thereisnoevidenceforcurvatureinthedata.AssumingDS1.5cm1(fromreducedmagnetizationmeasurements),onerealizesthatatleasta3TmagneticeldwouldberequiredtoovercometheaxialterminEq.3-1.ThissuggeststhattheZeemaninteractioncommutes 51

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Figure3-4(a)displaysasimulationoftheZeemandiagramforaSMMwithS=5,i.e.,withD<0.Ascanclearlybeseen,thetransitionfromthelowest-lyingmSleveloccursatthelowesteld;theexcited-statetransitionsalloccurathighereld.ThisagreesqualitativelywiththedatainFig.3-3.Therefore,wecanconcludethatDisnegativeandthattheFe6complexisaSMM.TheinterceptonthefrequencyaxisinFig.3-3(66.4GHz)thencorrespondstotheZFSbetweenthegroundandrstexcitedstates.IfoneassumesthatS=5,thenD=-0.25(1)cm1,whichisinreasonableagreementwiththevaluefromthemagnetizationts(D=-0.28(3)cm1).Becauseoftheuncertaintyinthepreciseorientationoftheeldrelativetotheeasyaxis,wecannotquoteaprecisevalueforg;themainpurposeoftheHFEPRmeasurementswastounambiguouslydeterminethesignofD,whichwassuccessfullyachieved. Single-axisrotationexperimentsfortheFe7complexwerenotabletolocatetheaxialdirection(presumably,therotationplanewasinclinedsignicantlywithrespecttothemagneticzaxisofthemolecule).Nevertheless,wewereabletolocatetheplaneperpendiculartotheaxialdirection(xyplane)frommeasurementssimilartothoseshowninFig.3-2.Thus,allofthetemperature-andfrequency-dependentstudieswerecarriedoutwiththeeldalignedwithinthemagneticxyplaneoftheFe7molecule.Onlyasinglemolecularspecieswasanticipatedforthecomplex,makinginterpretationofthedatamorestraightforward. Furthermore,thiscomplexexhibitssharperEPRpeaks,asevidentfromFig.3-5,whichshowsthehigh-eldxy-planespectraobtainedatdierenttemperaturesandafrequencyof197GHz.ComparisonofthedatainFigure3-5withthesimulatedZeemandiagraminFig.3-4(b)revealsthattheFe7complexcannotbeaSMMbecauseitsD 52

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(a)SimulatedZeemandiagramforaspinS=5systemwithD<0withthemagneticeldparalleltothezaxis.Theredlines(labeledatod)correspondtothetransitionsshownintheinsetofFig.3-3.(b)SimulatedZeemandiagramforaspinS=5/2systemwithD>0withthemagneticeldparalleltothexyplane.Theredlines(labeledatoc)correspondtothetransitionsshowninFig.3-5. 53

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TemperaturedependentspectrafortheFe7complexat197GHzwiththeDCmagneticeldappliedwithintheeasy(xy)plane. valueispositive.AscanbeseenfromFig.3-4(b),uponreductionofthetemperature,thestrongerEPRpeaksshouldbeobservedatthelowesteldsforaneasy-planemagnet(D>0)whentheeldisappliedparalleltotheeasy(xy)plane;thisisexactlywhatisseeninthedata.IfthecomplexwereaSMM,theintensitiesofthevetransitions(labeleda-einFig.3-5)wouldbereversed. Figure3-6displaystheresultsofamultifrequencystudyfortheFe7complex,withtheeldappliedwithintheeasyplane;thetemperaturewas20K.Fits(solidcurves)tothepositionsoftheEPRpeakswereperformedviaexactdiagonalizationofEq.3-1.ItisveryclearfromFig.3-6thatthedatalieonaseriesoflinesthatarenotevenlyspacedandexhibitsignicantcurvatureatlowfrequenciesandelds.Thesetrendsareacharacteristicofxy-planespectraobtainedforasystemwithasignicantuniaxialanisotropy(both 54

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Easy-planepeakpositionsfortheFe7complexplottedversusfrequencyat20K.ThesolidlinesaresimulationsusingtheZFSparametersgiveninthemaintext. positiveandnegativeD),duetothecompetitionbetweentheorthogonalZeemanandZFS(D^S2z)interactions.Inotherwords,thedatadisplayedinFig.3-6providefurtherconrmationthattheeldisinthexyplaneand,whencombinedwiththetemperaturedependenceinFig.3-5,alsoconrmthepositivesignofD.ThetassumesanS=5/2groundstateandyieldsvaluesofg=2.0andD=+0.62cm1.Thisvalueagainagreesreasonablywellwiththatfromthereducedmagnetizationstudies(D=+0.77(7)cm1).Thesomewhatlowervalueofgobtainedfromthereducedmagnetizationtsisduetothelowerprecisionavailablefrombulkmagnetizationtsandalsototheassumptionofaxialanisotropyinthelatter;HFEPRdataaremorereliable.Infact,thebestttotheHFEPRdatarequiredtheinclusionofarhombicZFSanisotropy,E0.067cm1.Thisisnotunexpected,giventhelowsymmetryofthemolecule.OurestimateofErepresents 55

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46 ]whichisconsiderablylowerthanthatforthecomplex.Thiscanbeattributedtotheirdierentstructuralarrangementsleadingtothedierencesinsingle-ionanisotropyandspin-spinanisotropy. 47 ].Thesenanosizedmagneticmaterialsdisplaymagnetizationhysteresisandquantumtunnelingofmagnetization(QTM)[ 88 89 ]suggestingthattheymayonedayndapplicationsininformationstorageandpossiblyquantumcomputation. Here,wepresentsingle-crystalhigh-frequencyelectronparamagneticresonance(HFEPR)studiesofamixedvalentMn2IIMn4IIIMn3IVcomplex. Figure3-7. CoreoftheMn9molecule.Colorcode:Mnpink;Ored;Nblue;Cgrey.Hydrogenatomsareomittedforclarity. 56

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48 49 ].Goodsizedblackcrystalswereobtainedforsingle-crystalHFEPRmeasurements.Themetallicskeletonofthecomplexcanbethoughttocomprisetworings:asmaller[Mn3IVO]10+trianglewithina[Mn4IIIMn2IIO6]4+hexagon(thechargeiscompensatedbytheligands).Atrstsight,themagneticcoreappearstohaveapseudothreefoldtopology.However,closerinspectionoftheMnvalencestatesontheouterhexagonrevealamuchlowersymmetry[ 49 ].AlloftheMnionsareindistortedoctahedralgeometrieswiththeJahnTellerelongationoftheMn3+ionslyingalmostperpendiculartotheplaneofthe[Mn4IIIMn2IIO6]4+hexagon.Thecomplexcrystallizessuchthattherearetwosymmetry-equivalent,butdierentlyorientedmoleculesintheunitcellwhosemagneticeasyaxesareapproximatelyperpendiculartoeachother.ThestructureofthemagneticcoreofthemoleculeisshowninFig.3-7. HFEPRexperimentswereperformedonasinglecrystalatvarioustemperaturesandfrequenciesfrom50to200GHzwiththedcmagneticeldappliedalongdierentcrystallographicdirections.Thespectrawereobtainedatxedfrequenciesandtemperatureswhilevaryingthestrengthofthedcmagneticeld.Detailsoftheexperimentaltechniquecanbefoundinchapter2. Single-axisrotationstudieswererstperformedtoroughlydeterminetheorientationofthecrystalinthemagneticeld.Figure1showstemperaturedependentspectraobtainedat120GHz,withtheeldorientedreasonablyclose(30o)totheeasyaxisassociatedwithoneofthetwositesintheunitcell.Theintensitiesofthelowesteldpeaksdecreaseuponincreasingthetemperature.Thiscanbeexplainedassuminganegativeuniaxialanisotropy(D<0).TheappearanceoftwosetsofpeaksinFig.3-8indicatesthat,inadditiontothetwodierentmolecularorientations,thereexistinequivalentMn9specieswithslightlydierentZFSparameters.WelabelthestrongerpeaksAandtheweakeronesB.Peaks1,2,3,4,and5correspondtothefollowingne-structuretransitions:mS=17 2!15 2;15 2!13 2;13 2!11 2;11 2!9 2and 57

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TemperaturedependentEPRspectraobtainedat120GHzwiththeeldorientedreasonablyclose(30o)totheeasyaxisofoneofthetwomolecularorientations.Eachsetofne-structuresisfurthersplitintopeakslabeledAandB,correspondingtoinequivalentmolecularspecieswithslightlydierentZFSparameters(D).Seemaintextforexplanationofnumbering. 2!7 2,respectively,wheremSrepresentsthespinprojectionalongtheeasy(z)axisofthecrystal. Figures3-9(a)and3-9(b)displaythepositionsoftheobservedEPRpeaksplottedversusfrequencyforspeciesAandB,andforthesameeldorientationasthedatadisplayedinFig.3-8.ThesolidcurvesweresimulatedusingthefollowingspinHamiltonian(Eq.3-2),containingonlyaxialZFSparameters ^H=D^S2z+B04[35^S4z30S(S+1)^S2z]+gB~B^S(3{2) ThesimulationsassumeS=17 2,andbestoverallagreementwiththedataisobtainedwithD=-0.24cm1(D=-0.25cm1)forspeciesA(speciesB),B04=+6.68106cm1

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Frequencydependenceofthepeakpositionsassociatedwiththetwospecies:(a)Aand(b)B.Datawereobtainedat5KforthesameeldorientationasinFig.3-8.ThesolidlinesaresimulationsbasedonEq.3-2,usingtheparametersgiveninthemaintext. 59

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50 ].AscanbeseenfromTable3-1,theobtainedaxialparametersagreeverywellwithpreviousmagneticandspectroscopicmeasurements[ 48 49 ]. Table3-1. ComparisonbetweenZFS(cm1)parametersobtainedfromthesestudies(EPR)andthevariousmagneticmeasurementsreportedin[ 48 49 ]. ZFSFDMRSINSMagnetization-SQUIDDFTEPR(A)EPR(B) D-0.247(5)-0.249(5)-0.29(3)-0.258-0.235-0.24-0.25B04/1064.600(1)7(4)6.76.7 Rotationaboutasingleaxisguaranteeseld-alignmentinthehardplane,althoughtheorientationoftheeldwithinthehardplaneisnotknown.Detailedstudies(notshown)allowidenticationofoneorotherofthehardplaneorientationsfromtheangledependenceofthepeakpositions(see[ 44 ]).Figure3-10(a)displaystemperaturedependent52GHzspectraforoneofthesehard-planeorientations.TheAandBpeaksareagainobserved,correspondingtothetwospecies.ThereversedorderingofAandB(seeFig.3-8)isconsistentwithEq.3-2.PeakslabeledA1',A3',andA5'(likewisefortheBpeaks)correspondtothefollowingne-structuretransitions:mS=17 2!15 2;13 2!11 2and9 2!7 2,respectively,wheremSnowrepresentsthespinprojectionalongthe(high)magneticeldquantizationaxis.Theloweldportionofthegure(eldsbelowA5')iscomplicatedbyabsorptionsduetotheothermolecularorientation. Wenowarguethatfourthandhigher-ordertransverseZFSinteractionsarenecessaryinordertoaccountforthesespectra.ItiswelldocumentedthatHFEPRmeasurementswithB?cprovideinformationconcerningtransverseterms[ 50 ].Inthefollowinganalysis,weconstraintheaxialterms(DandB04)onthebasisofthesimulationsinFig.3-8.Densityfunctionaltheory(DFT)calculationspredictthatMn9possessesarhombohedralZFSparameterE=0.035cm1[ 49 ].Wendthatitisimpossibletoobtainagreementbetweenourresultsandsimulationsincludingonlythisinteraction[E(^S2x^S2y)],as 60

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(a)TemperaturedependentEPRspectraobtainedat52GHzwiththeeldinthehardplaneofoneofthemolecularorientations.Thenestructuresplitting(AandBpeaks)canagainbeclearlyseen(refertomaintextforexplanationofnumbering).Atthehighesttemperature,additionalpeaksappear(labeledX)whichweattributetoexcitedspinmultiplets.(b)Frequencydependenceofthe7KhardplanepeakpositionsassociatedwithspeciesA[thedashedcurvecorrespondstothedatain(a)];theorientationoftheeldwithinthehardplaneisnotknown.ThecurvescorrespondtovarioussimulationsbasedonEq.3-2withtheinclusionofarhombictermE.Seemaintextforexplanation. 61

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Hard-planedata(fromFig.3)andsimulationsbasedonEq.(1),withtheadditionalinclusionofthehigher-ordertransverseinteractions(a)B24^O24and(b)B66^O66(actualfunctionsgiveninthegures).BothsimulationsagreereasonablywellwiththeexperimentaldatausingtheaxialZFSparametersdeterminedfromthesimulationsinFig.3-9,alongwiththefollowingparameters:(a)B24^O24=8.4105cm1;(b)B66^O66=8.4107cm1;and=0forboth(a)and(b).Seemaintextfordetailedexplanation.ThebluecurvescorrespondtothesplittinginthegroundstatemS=17/2doublet. 62

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Itturnsoutthat,withtheexceptionofA1',allEPRpeaksarereasonablyclosetothepositionsonewouldexpectforextremelyweaktransversesecondorderanisotropy(or=45o).Incontrast,A1'isshiftedconsiderablytohigherelds.Itisonlypossibletomimicitsbehaviorusinghigherordertransverseterms,asillustratedinFig.3-11.Infact,onecanobtaingoodagreementwiththehard-planespectraforseveraldierentparametersets.ExamplesaredisplayedinFig.3-11involvingpurelyB24^O24andB66^O66.Thecoecientsaregiveninthecaptions.Interestingly,B66^O66givesexcellentagreement,whereastermsthatonemightexpecttoworkwell,suchasB34^O34,donotgivegoodagreement.Inreality,itislikelythatthetransverseHamiltonianinvolvesadmixturesofalloftheseinteractions,reectingthepseudothreefold,[ 49 ]albeitlowsymmetryofthemolecule.Onlydetailedmultihigh-frequencymeasurementsperformedasafunctionoftheeldorientationwithinthehardplanecanresolvethisissue,whichwouldbegreatlycomplicatedbythemultiplespecies,orientations,andtheoveralllowsymmetryofthiscomplex.Nevertheless,thepresentmeasurementsserveausefulpurpose,hintingatthesignicantfourth(andhigher-order)anisotropythatlikelyresultsasaconsequenceofSmixingbroughtaboutbylow-lyingexcitedspinstates[ 42 ].Indeed,thespectrainFig.3-11(a)clearlyshowfeatures(labeledX)associatedwiththepopulationoflow-lying 63

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2spinstates.OnenalpointtonotefromFig.3-11(a)isthehugetunnelsplittingofthelowest-lyingmS=17 2doublet,whichisclearlyvisibletothenakedeyedowntolowelds.ThissuggeststhataB24^O24interactionwouldcauseveryfasttunnelinginthisMn9complex,whichisnotfoundexperimentallyand,therefore,seemstobeunphysical.Again,thishintsattheimportanceofmultiplehigh-ordertransverseZFSinteractionsthatcanaccountforboththeEPRdatapresentedhereandtheslowmagnetizationdynamicsinthequantumregime.Wealsonotethatinternaldipolarandhyperneeldsmustbeimportantforzero-eldQTMinthesehalfintegerSMMs. 64

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Theresultspresentedinthischaptercanbefoundinthearticle:AnisotropicexchangeinatetranuclearCoIIcomplex,J.Liu,S.Datta,E.Bolin,J.Lawrence,C.C.Beedle,E.C.Yang,P.Goy,D.N.HendricksonandS.Hill,Polyhedrondoi:10.1016/j.poly.2008.10.064(2008)(reusedwithpermissionfromElsevier). TherehavebeenseveralattemptstoproduceSMMsusingCoII,whichisknowntoexhibitstrongspin-orbitcouplingincomparisontoMnII-IV,FeIIIandNiII,fromwhichthevastmajorityofknownSMMshavebeenrealized.TheearliestreportinvolvedaCo4clusterverysimilartotheonethatformsthebasisforthepresentinvestigation[ 51 ].Inthisstudy,magnetichysteresiswasobservedbelowabout1K,whichwasalsoweaklyeld-sweep-ratedependent-theclassicsignatureofaSMM.Furthermore,magnetizationmeasurementshintedatasizeablespingroundstateandanegativeaxialzero-eldsplitting(ZFS)parameter,againsuggestingthatthisCo4complexisaSMM.Sincethen,afewadditionalexamplesofhomometallicpolynuclearCoIIclustershavebeenreportedwhichshowsignaturesofSMMbehavior[ 52 { 56 ].However,nospectroscopicdatahavebeenreportedinsupportoftheseassignments. Duringthepastfewyears,severalpapershavebeenpublishedshowingextensiveinvestigationsofafamilyoftetranuclearNiIISMMswiththegeneralformula[Ni(hmp) 65

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57 ].Akeyndingofthisworkwasthefactthatcrystalsofthehigh(S4)symmetry[Ni(hmp)(dmb)Cl]4complexcontainednosolvatemolecules,resultinginexceptionallyhighqualityhigh-frequencyelectronparamagneticresonance(HFEPR)spectra[ 58 ].Furthermore,HFEPRmeasurementsforaZnanalogdopedwithsmallquantitiesofNiIIenabledevaluationofthesingle-ionZFStensors[ 59 ].ThecombinedstudiesprovidedimportantinsightsconcerningthephysicsofSMMs[ 42 ].Thesuccessoftheseinvestigationsthusprovidedmotivationforinvestigatingtwoisostructuralcompounds:[Co(hmp)(dmb)Cl]4(complex1)and[Zn1xCox(hmp)(dmb)Cl]4(complex2). Single-crystalHFEPRstudiesof[Zn1xCox(hmp)(dmb)Cl]4werereportedpreviously[ 60 ],demonstratingthatthegroundstateoftheCoIIionscanbemodeledasaneectivespinS'=1/2Kramersdoublet.SuchamodelignorestheupperKramerslevelsassociatedwiththeS=3/2CoIIions.Thestronganisotropyisinsteadparameterizedviaananisotropicg-tensorassociatedwiththeeectiveS'=1/2groundstateKramersdoublet.TheHFEPRstudiesofcomplex2establishedahugeg-anisotropy(gz=7.8andgx;y2.0)oftheeasy-axistype.Itwasalsofoundthattheindividualeasy-axisdirectionsaretiltedawayfromthecrystallographiccdirectionby58o.Thehugeg-anisotropyisclosetothemaximumexpectedforanoctahedralCoIIcomplex,suggestingaverysignicantZFSseparatingthetwoKramersdoublets.However,itwasnotpossibletodeterminethemagnitudeofthisZFSonthebasisoftheHFEPRstudiesofcomplex2. TheclearsuccessinmodelingtheHFEPRspectrumoftheCoIIionsincrystalsofcomplex2aseectivespin-1/2particlesbegstheobviousquestionastowhethersuchasimpleapproachmaybeextendedtothepolynuclearcomplex1.Firstandforemost,wenotethatarigoroustreatmentofthespin-orbitcouplingiscomputationallychallenging(theHamiltonianmatrixhasdimensionsof2073620736),andalsorequiresfarmoreinformationconcerningtheCoIIZFSthanobtainedfromHFEPRstudiesofcomplex2.Thus,insomesense,thecoupledS'=1/2descriptionoerstheonlyrealisticstarting 66

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Theargumentsgivenaboveignoreonecrucialdetail:inthecaseofthecoupledCo4system,eachCoIIionexperiencesanexchangeeldduetoitsneighboringionsinthecluster.However,theprojectionmethodcompletelyignoresthisexchangecoupling.Thesolutiontothisproblemturnsouttoberelativelysimple,andinvolvesamultispinHamiltonianconsistingofjustthepairwiseinteractionsbetweentheCoIIionstogetherwiththeindividualZeemaninteractions.Theanisotropicspin-spininteractionsarethenparameterizedintermsofanexchangetensor,!Jij,andtheHamiltonianmaybeexpressedas: ^H=Xi
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Figure4-1. StructureoftheCo4molecule.Colorcode:Coblue;Ored;Nblue;Cgrey;Clgreen.Hydrogenatomsareomittedforclarity. Crystalsofcomplex1werepreparedaccordingtoasimilarproceduretothepreviouslyreported[Co(hmp)(MeOH)Cl]4complex[ 51 ];adetailedaccountofthesynthesiscanbefoundelsewhere.Goodsized,darkredcrystalsformintheshapeofasquare-basedpyramid,reectingtheS4symmetryofthestructure.ThestructureofthemoleculeinFig.4-1.Onecaneasilyidentifytheprincipalcrystallographicdirectionsonthebasisofthecrystalmorphology,thec-axisbeingorientedperpendiculartothesquarebaseofthepyramid,i.e.inthedirectiondenedbytheapexofthepyramid.Allofthedatapresentedinthispaperwereobtainedwiththemagneticeldappliedparalleltothecrystalc-axis,whichalsocorrespondstotheS4axis.ThesamplewasmountedwithitssquarebaseattachedwithvacuumgreasetoaatcopperdiskattheendoftheHFEPRprobe/cavity. 68

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Figure4-2. A2Dfrequencyverseseldplotrepresentingthepositionsofthestrongerresonancesobservedfromthefrequencydependencemeasurementswiththeeldparalleltothec-axis.Thesolidcurvesarepurelyguidestotheeye. 69

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60 ]. RepresentativetemperaturedependentmeasurementsareshownfortwodierentfrequenciesinFig.4-3.Multiplepeaksareobservedforboththefrequencies.Thesignal-to-noisediminishesathigherfrequenciesduetothereduceddynamicrangeofthehigh-frequencyspectrometer.Nevertheless,clearresonancesareobservableatthefrequenciesshown,andeachspectrumishighlyreproducible.Giventhatthetemperatureisratherlow(muchlessthanthemicrowavequantum,whichrangesfrom12to34K),onemayassumethatalloftheobservedHFEPRintensityinvolvesexcitationsfromthegroundstateoftheCo4complex.Fromthisgure,onecanconcludethattheresonancesobservedat268GHzcorrespondtoexcitationsfromthegroundstate.Incontrast,at129GHz,alltransitionsexceptforthelowesteldpeakappeartooriginatefromexcitedstates,duetovanishingintensityasT!0.ThecorrespondingdatapointsinFig.4-2have,therefore,beencoloredred/bluesoastodistinguishthemfromthestrongground 70

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Temperaturedependencemeasurements(B//c)fortwodierentfrequencies;thetemperaturesandfrequenciesaregiveninthegures. statetransitions.AparticularlyinterestingobservationfrombothguresisthefactthatthisCo4complexisalmostEPRsilentateldsabove1Tforintermediatefrequenciesrangingfrom250-400GHz.Thisfrequencygapappearstoseparatethelow-andhigh-gresonancebranches. ThekeyndingfromtheseexperimentsistheclearobservationofsignicantZFSassociatedwiththelow-lyingspinstatesofcomplex1.AnotherimportantobservationisthefactthatitiscompletelyimpossibletotthedatainFig.4-2usingagiantspinapproximation.Basedonsuchamodel,oneexpectsthe2Dplotfortheeasy-axis(Figure 71

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Figure4-4. Resultsfromangledependentmeasurementsperformedat117GHz.Thebluelinescorrespondtothez-axiswhereastheredlinecorrespondtotheab-plane. Manyaspectsofthedatahintstronglyatcompetinginteractions/symmetries,suchthatneitherthetotalspin,S,noritsprojection,mS,aregoodquantumnumbers.Firstofall,theobservationofmultiplegroundstateresonancessuggestsanabsenceofsimpleselectionrules.Thiscontraststhegiant-spincase,whereSandmSaregoodquantumnumbersandthemS=1EPRselectionrulepermitsonlyonepossibletransitionfrom 72

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ItshouldbenotedthateachspinmultipletexhibitsnoZFSinthissimpleisotropiccase.Furthermore,theEPRselectionruleforbidsS=1transitions.Thus,allresonancebranchesshoulddependlinearlyontheeld,andintersectthe2Dfrequency/eldplot(Figure4-2)attheorigin.Infact,justasingleEPRpeakwouldbeobserved(seered 73

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^H=JXi
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Giventhepresenceofexchangeanisotropy,onewouldalsoexpectantisymmetricexchangeforthismolecule,whichdoesnotpossessacenterofinversion.InclusionofsuchaninteractionwouldimmediatelyresultinadditionalS-mixingandfurtherrelaxationoftheS=0EPRselectionruletotheextentthatonecouldexpecttoobservestronginter-spinmultiplettransitions,i.e.multiplegroundstateresonancesspanningawidefrequencyrange(seeredandgreenarrowsinFig.4-5(c)).InclusionofanantisymmetricterminEq.4-1addsverylittleintermsofcomputation.Nevertheless,forthesakeofsimplifyingthediscussion,wedonotincludesuchaninteractioninthepresentwork.However,wenotethat,withtheconsiderationofS-mixing,Fig.4-5(c)beginstoembodymanyofthecharacteristicsthatwouldaccountqualitativelyfortheexperimentalobservations.Nevertheless,inspectionofFig.4-2revealsmanydierentZFSspanningtherangefromabout65GHzupto650GHz.However,thereremainsignicantzero-eldgapsinFig.4-5(c).Thesegapscanbelledtosomeextentifonerelaxestheassumptionofasingleexchangeparameter.Indoingso,onenallyarrivesatFig.4-5(d),whichconsiderstwoexchangeconstants,consistentwiththeS4symmetryoftheCo4cluster,i.e.J1=250GHzandJ2=+50GHz,withthesameanisotropyconstant,,forboth.Ifwedenote~Ji;j=[^Siz^Sjz+(^Six^Sjx+^Siy^Sjy)]thenthespinHamiltonianinthiscaseisgivenby ^H=J1(~J1;2+~J3;4)+J2(~J1;3+~J1;4+~J2;3+~J2;4)+XiB~Bgi^Si(4{3) 75

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SimulatedZeemandiagramsobtainedforcomplex1onthebasisofEq.(1)(alsoEq.(2)),usingvariousdierentparametersets:(a)representstheisotropiccase;(b)addsanaxialexchangeanisotropy;(c)includestheeectofatilting(58o)ofthesingle-iong-tensorsawayfromtheaxialdirection;and(d)considerstwoinequivalentexchangeparameterswithinthecluster.Seemaintextfordetailsofthecalculations(includingvaluesforthevariousparameters),andanin-depthdiscussionofthesediagrams.ThearrowsrepresentpossibleHFEPRtransitions. Fig.4-5(d)representsacrudeattempttoaccountqualitativelyandsemi-quantitativelyformanyofthefeaturesobservedintheHFEPRexperimentsofcomplex1.However,theexactnumbersusedforthesimulationsshouldnotbetakentooseriously.OurintentistoshowthatEq.4-1containsessentiallyallofthephysicsneededtoaccountfortheEPRdataand,possibly,thehysteresismeasurements.Theparameterswerechosensothat(a)theZFSbetweenthegroundandrstexcitedstateroughlymatchesthesmallestZFSintheexperiment,i.e.67GHz,and(b)thefullspectrumspansaround650GHz,whichwas 76

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77

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Theresultspresentedinthischaptercanbefoundinthearticle:AcomparativeEPRstudyofhigh-andlow-spinMn6single-moleculemagnetsS.Datta,E.Bolin,R.Inglis,C.J.Milios,E.K.BrechinandS.Hill,Polyhedrondoi:10.1016/j.poly.2008.12.025(reusedwithpermissionfromElsevier). kT,whereU=jDjS2isthebarrierheighttothemagnetizationreversalandDisthemagneticanisotropyparameter.Forpossibleapplicationsofmolecularmagnets,therelaxationtimehastobeatleast15yearsatroomtemperature[ 65 ].Henceoneneedstondthecriteriontoraisethebarrierheightandtheblockingtemperatureandforthatitisnecessarytounderstandtheroleofexchangeinteractionsandanisotropyinthesemolecularunits.InthischapterwefocusourstudyonMnbasedtriangularmagneticunitsanddiscusstheroleofthesetwoimportantfactorsinraisingthebarrierheight. TherstMnbasedtriangularcomplexwasobtainedwitharelativelysmallligand-inducedstructuraldistortion,switchingtheexchangebetweenMnIIIatomsinaoxide-centeredtriangularunitfromantiferromagnetictoferromagnetic.Thisleadtoahigh-spin(S=6)groundstate[ 66 67 ]ascomparedtopreviouslyobtainedantiferromagneticcomplexeswithlow-spingroundstatesofS=2[ 68 69 ].TheseMnIIItriangularunitscanalsobecoupledtogethertobuildlargerstructures,leadingtoevenlargerspinvalues.IntherecentlydiscoveredMn6family[ 7 ],twoofthesetriangularunitscoupleferromagneticallygivingamaximumpossiblespingroundstateofS=12.TheexchangeinteractionswithinthemetaltriangledependontheindividualMn-O-N-Mntorsionalangles.Thepairwiseinteraction(MnMn)switchesfromantiferromagneticto 78

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OurinteresthereistostudytheinterplaybetweenexchangeandanisotropyinthissituationandtostudytheeectofthechangeinSontheresultantbarrierandSMMbehavior.ThisisparticularlyimportantinlightofseveralrecentpapersmakingpredictionsastohowUshoulddependonS[ 70 71 ].However,resultsfrommagneticstudieswereusedtoreachtheconclusionsandthatinvolvedobtainingonlyeectivevaluesforthebarrier.Toobtainpreciseandaccuratevaluesfortheuniaxialanisotropyandthebarrieroneneedstoconsiderresultsfromspectroscopy.Inourapproachtoaddressthisissue,westartbylookingatthesingleionanisotropytensorsassociatedwiththeionsintheMn3clustersandthenextendtheapproachtothesixparticlecase.Wealsosupportourcalculationswithextensiveanddetailedhighfrequencyelectronparamagneticresonance(HFEPR)studies. 67 72 ].InthischapterwewouldbediscussingtheresultsobtainedforthreeMn3complexeswithS=6:[Mn3O(O2CMe)3(mpko)3](ClO4)3CH2Cl2(complex1),[Mn3O(O2CEt)3(mpko)3](ClO4)1.2CH2Cl24H2O(complex2),[MnIII3O(Et-sao)3(ClO4)(MeOH)3](complex3)andaS=2cluster[Mn3O(sao)3(2-benz)(H2O)(py)3](complex4). Complex1and2crystallizeinthemonoclinicspacegroupP21/candshowsimilarcorestructures[ 67 72 ].Thecrystalstructureconsistofa[Mn3O(O2CMe)3(mpko)3]+cation,ClO4anion,andlatticeCH2Cl2molecules.ThecationconsistsofthreeMnIIIatomsinatriangulararrangementbridgedbyacentraloxygenatom.ThethreeMeCO2orEtCO2groupslieononesideoftheMn3plane,andthethreeoximategroupsontheother.TheMnatomsarenear-octahedral,andtheMnIIIcentersexhibitaJahn-Teller 79

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Figure5-1. MagneticcoreforMn3complexes.Colorcode:Mnpurple,Nblue,Ored.OneoftheMn-N-O-Mntorsionalanglesareclearlyshowninthegure. Complex3wassynthesizedusingthesamemethodofstructuraldistortion[ 73 ].TheaxialcarboxylateandsolventligandsarereplacedwithfacecappingtripodalligandsmaintainingthestructuralintegrityofthemagneticcorebutincreasingitsdistortionandgreatlyenhancingitsSMMproperties(seetable5-1).ThiscomplexcrystallizesinthetrigonalspacegroupR-3anditsstructurecomprisesanoxo-centeredMnIII3trianglewiththeMn-N-O-Mntorsionalangle()nowsignicantlyenhancedascomparedtocomplexes1and2.TheJahn-TelleraxesareperpendiculartotheMn3planeforthiscomplex.ThisisshowninFig.5-2(b). Complex4crystallizesingroupP-3andthecorecomprisesthecommon[MnIII3O]7+oxo-centredtrianglebutwithrelativeplanarcongurationcomparedtothecomplexes 80

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74 ].Thisleadstoantiferromagneticinteractionintheclusterandreducesthespinto2. Figure5-2. Jahn-TelleraxesinMn3.Theangle()betweentheJahn-TelleraxisofeachMnIIIatomandtheMn3planeareapproximately60o(a)and90o(b).Onlythemagneticcoreisshowninthegure.AdditionalOatomsareomittedforclarity. Figure5-1clearlyshowsthemagneticcorefortheMn3complexesandhighlightsthetorsionangle()betweenMnMnpairs.AcomparisonofvariousparametersfortheMn3complexesisgivenintable5-1.ThetorsionangleshasdirectinuenceonthetheexchangeparameterJ.ItcanbenotedthatthevaluesofandJdonotchangesignicantlyforcomplexes1and2butforcomplex3changeinissignicant.TheorientationoftheJahn-TelleraxesinuencetheuniaxialanisotropyparameterDandthusincreasingbarrierheightaswell.AstheJT-axesbecomeperpendiculartotheMn3plane,thevalueofDincreasesdramaticallyascanbeseenfromtheresultsobtainedfromthemagneticstudies.Now,eventhoughonecantreattheseMn3systemsastoymodelstounderstandtheroleoftheinuenceofbothanisotropyandexchangetheymightnotbeidealsystemsforourstudy.ThepresenceofsignicantfrustrationintheMn3trianglesmakesitdiculttogetresultsfromHFEPRthatcanbeanalyzedtogetreliable 81

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Table5-1. Comparisonofmagneto-structuralparametersfortheMncomplexes.Thetorsionanglesareexpressedindegrees;exchangeparameterJanduniaxialanisotropyparameterD,obtainedfrommagneticstudies,areexpressedincm1. MoleculeComplexSTorsionangleExchangeD Mn65442.4,25.5,29.7+1.20,-1.9561243.1,39.1,34.9+1.63-0.43 TherecentlydiscoveredMn6complexes[ 7 8 ]consistoftwoferromagneticallycoupled[MnIII3O]7+triangularunits.Deliberate`twisting'ofthe(-Mn-N-O-)ringinducestheswitchofexchangeinteractionresultinginthestabilizationoftheferromagnetichighspin(S=12)groundstate.HFEPRmeasurementswereperformedonsingle-crystalsofMnIII6O2(Me-sao)6(O2CCPh3)2(EtOH)4(complex5)andMnIII6O2(Et-sao)6(O2CPh(Me)2)2(EtOH)6(complex6).Whilebothcomplexeshavemonoclinicstructures,thedominantintra-triangleexchangeincomplex5isantiferromagneticandthemoleculeshavealowspingroundstateofS=4,whereaslargeMn-N-O-Mntorsionanglesincomplex6resultsinthepairwiseexchangebecomingincreasinglyferromagnetic,resultinginaspinS=12groundstate.TheJahn-Teller(JT)axesareapproximatelyparallelandperpendiculartothe[Mn3O]7+planes.AcomparisonofthetorsionanglesandtheexchangeparametersforthetwoMn6complexesareshownintable5-1. OurnextgoalistoseehowtheanisotropychangesinbothMn3andMn6complexesastheexchangeinteractionchangesfromanti-ferromagnetictoferromagnetic,startingwiththeoreticalanalysisandthenfollowingupwithexperimentalresults. 82

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^H=Xi
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whereRkistheWignerrotationmatrixandspansthek-thirreduciblerepresentationoftherealorthogonalrotationgroupSO(3).Letusstartwiththegeneralcaseofnspinsinanexchangecoupledcluster.ThecouplingschemeisgivenbyS1+S2=~S1,~S1+S3=~S2,...~Sn1+Sn=S;m1+m2=~m2,~m2+m3=~m3,....,~mn1+mn=Msandthespinstatescanbedenotedas where~Sisthefullset~Sifori=1,...,n-1.ThenthegeneralizedeectivespinHamiltoniancanbeexpressedintermsofirreducibletensoroperator's(ITOs)as[ 75 ]: ^H=Xk1k2:::knX~k1~k2:::~kn1XkqC(k)q(k1k2~k1:::~kn1kn)^T(k)q(k1k2~k1:::~kn1kn)(5{5) whereC(k)q(k1k2~k1:::~kn1kn)areparametersrelatedtothephysicalpropertiesoftheclusterand^T(k)q(k1k2~k1:::~kn1kn)istheqthcomponentoftheITO^S(ki)qi(i)S(ki)qiofrankkwhose 84

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^T(k)q(k1k2~k1:::~kn1kn)=(:::n^T(k1)(S1)^T(k2)(S2)o(~k1)^T(k3)(S3)(~k2):::^T(kn)(Sn))(k)q(5{7) whereisthetensorproductand~kiaretheranksoftheintermediateITO'sobtainedusingthecouplingscheme.Thematrixelementsof^T(k)qoperatorscanbecalculatedusingtheWigner-Eckarttheorem: whereD~SpSp^T(ki)q(k1k2~k1:::~kn1kn)~SSEisthereducedmatrixelementofT(k)andC~S~MSMkqisaClebsch-Gordoncoecient[ 76 ].Thereducedmatrixelementcanbewrittenintermsofthereducedmatrixelementsoftheindividualspinofthecluster: Qn1i=1nh~ki+1ih~Si+1ih~Spi+1io1=2DSi^S(ki)SiE8>>>><>>>>:~kiki+1~ki+1~Spi~Si+1~Spi+1~Si~Si+1~Spi+19>>>>=>>>>; Now,underniterotations,thetransformationoftheenergylevelsisrepresentedbythepointgroupofthecluster.Theprojectionoperatormethodallowsonetolinkthestates 85

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Thus,foranyoperator^RofthepointgroupG,theprojectionoperatorisoftheform ^R~SSMsE=P~SpD~S~Sp(^R)~SpSMsE. Theprojectionoperatoroperatingonthestateoftheirreduciblerepresentationreproducesthestatewithnonzeroeigenvalues.Hencetheprojectioncoecientscanbewrittenintermsofreducedmatrixelementsasfollows: UsingtheproceduresuggestedbyBenciniandGatteschi[ 62 ]onecanwritethecoecientsforadinuclearclusteras wherec+andccanbewrittenas (2S+3)(2S1)S(S+1) (2S+3)(2S1)S(S+1) 86

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wheredi(S1S2S12S3S),i=1,2,3,denotethethreecoecientsobtainedforthetrimercase,di(S1S2S12),i=1,2,werethecoecientsobtainedwiththeferromagneticcouplingofS1andS2.Thecoecientsdi(S12S3S),i=1,2,wereobtainedwiththeferromagneticcouplingofthethirdspintotheintermediatespinS12. WeunderstandthatsomeofourapproximationsmightoversimplifytheproblembuttoobtainasimplecorrespondencebetweenthemolecularDvalueandthesingle-ionanisotropyDS,webelievethatthesimplicationsarenecessary. Usingtheabovementionedapproximations,fortheMn3systems,onecanthenobtainDS=6=3DS/11fortheferromagneticcaseandDS=2=69DS/49fortheantiferromagneticcase,theratiobeing[DS=6 87

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SchematicrepresentationsofthemagneticcoreforMn3andMn6systems. Figure5-2displaystheschematicrepresentationofthemagneticcoreforboththeferromagneticandtheantiferromagneticcasesforMn3andMn6systems. UsingthesameapproachfortheMn6systems,thecalculationfortheferromagneticcase(S=12)becomestrivial,andoneobtainsDS=12=3 23DS=0.13DS.Theantiferromagneticcase(S=4)isnotastrivial,withtheresultsdependingonthecouplingscheme:forscheme1,(((((S1+S2)+S4)+S5)-S3)-S6),oneobtainsDS=4=0.84DS,i.e.DS=12=DS=4=0:155;andforscheme2,(((((S1+S2)-S3)+S4)+S6)-S6),oneobtainsDS=4=0.7DS,i.e.DS=12=DS=4=0:186. 88

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Figure5-4. EnergyleveldiagramsimulatedforthreeferromagneticallycoupledS=2particlesusingexactdiagonalizationtechnique.Onlythelowestfewlevelsareshowninthegure.representstheenergydierencebetweenthelowesttwolevels. TheplotshowsallthespinmultipletsobtainedusingthegeneralizedspinHamiltonian.Theinsetshowsthelevelsforthegroundstatemultiplet.Theanisotropyandthebarrierheightcannowbeobtainedusingtheexpressions=(2S1)DandU=jDjS2respectively.However,resultsobtainedusingthismethodareexactonlyinthestrongexchangelimit.Theresultsbecomelessaccurateonweakexchangelimit.Thegroundstatemultipletarenolongerisolatedandthemixingoftheexcitedstatesintothegroundstateleadtohigherorderterms.Wedonotconsidertheeectofhigherordertermswhilecomparingthedierentmethods.Buteventhenonecanobtainreasonableagreement(within10%)betweenthetheoreticalandexperimentalresults. 89

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PlotoftheratiosDS=6=DS=2andDS=12=DS=4versusJ/Dobtainedfromthematrixdiagonalizationcalculations TheHilbertspacefortheMn3SMMismuchsmaller([(2S+1)3]2=125125)ascomparedto(108108),forMn12-acetate[ 78 ]andcanbeeasilycalculatedinanymodernPC.Forthestrongexchangelimit(J>>D),theratioDS=6=DS=2turnsouttobe0.194,whichisclosetothevalueobtainedpreviouslyusingtherecurrencerelationshipsintheprojectionoperatortechnique. HoweverforthesixparticlecasethesizeoftheHilbertspaceis[(2S+1)6]2=1562515625andtogettheeigenvaluesoneneedtousethesparsematrixalgorithm.Inthestrongexchangelimit,theratioofanisotropiesturnsouttobeDS=12=DS=4=0.160,whichdiersby3%and16%fromtheratiosobtainedusingschemes1and2fortheantiferromagneticcase,respectively.Figure5-5displaysthesimulatedresultsforbothMn3andMn6systemsassumingasingleexchangeconstant(Jij=J,foralliandj). 90

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42 79 ]. Previousstudiesontheferromagneticcomplex1haveprovidedthezeroeldsplitting(ZFS)parametersinthegiantspinHamiltonian[ 67 72 ] ^H=D^S2z+gB^SH+E(^S2x^S2y)+B04^O04(5{10) wheretheaxialZFSparameterD=-0.3cm1,fourthorderZFSinteractionB04=-3105cm1,secondordertransverseanisotropyE0:015cm1andg=2andS=6. Similarstudieshavebeenperformedonsinglecrystalsofcomplex2.Thiscomplexhastwodierentorientationsofthemoleculesinthecrystal,leadingtotwomagneticeasy-axesorientations,similartocomplex1.Figure5-6showstemperature-dependencespectraobtainedat137GHzforcomplex2withtheeldappliedapproximatelyparallel 91

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Figure5-6. Temperaturedependentspectraofcomplex2,obtainedat137GHzforeasyaxisorientation. Multi-frequencydataobtainedfromthestudiesoncomplex3suggestanaxialZFSparameterD=-0.685cm1,fourthorderZFSinteractionB04=-4105cm1,g=2andS=6.Figure5-7showsthebestsimulationofthefrequencydependentdataforcomplex3,withthemagneticeldparalleltotheeasyaxisofthecomplex. 92

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Energydierencediagramconstructedfromfrequency-dependentmeasurementsfortheeasy-axisorientationforcomplex3atatemperatureof15K.Thesolidlinesarethesimulationsusingtheparametersgiveninthemaintext. HFEPRstudiesonMn3complexeswithS=2howeverhavebeendicultduetothepresenceoflowlyingexcitedstate,asaresultoffrustrationinthesystem.Moreover,weakexchangeinteractioninthemagneticcorecausesthelowlyingexcitedstatestobeevenclosertothegroundstateandhencetheidenticationofthestatesbecomesextremelycomplicated.WehavetriedHFEPRexperimentsonseveralcomplexeswithS=2andhaveobtainedonlyabroadpeakattheg=2positioninmostcases.Thebestresultswereobtainedfromexperimentsperformedonsinglecrystalsofcomplex4. Figure5-8showstemperaturedependentdataobtainedat52GHzforthehardplaneorientation.ThetransitionsobservedarenotonlyrelatedtotheS=2multipletbutalsoduetothepresenceoflowlyingexcitedstatesinthemolecularcluster.Itisextremely 93

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Temperaturedependentspectraofcomplex4,obtainedat52GHzforhardplaneorientation.TheblackarrowsdenotetranstionswithinS=2multipletwhereasthebluearrowsdenotetransitionsfromlowlyingexcitedstates. hardtotthefrequencydependentdataforthehardplaneorientationbutaroughestimationgivesavalueofD=-3.7cm1,withsignicantrhombicanisotropyterms.Thisisduetothefactthatthegiantspinapproximationisnolongeravalidapproximationundertheseconditionsandoneneedtousethesingleionmodeltogetabetterestimateoftheanisotropyparameters.HenceitistrickytocomparetheratioofDS=6=DS=2fromexperimentalresultswithcalculatedornumericalresults. Forcomplexes1&2,DS=6obtainedfromexperimentalresultsarecloseandDS=6=DS=2turnouttobe0.08&0.097,respectively.Forcomplex3theratioDS=6=DS=2is0.1864.Theratioobtainedfromthemagnetizationdataforcomplexes1&2is0.1whichisclosertothevaluesobtainedfromHFEPRmeasurementsbutforcomplex3(-0.95 94

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Table5-2. ComparisonbetweenvarioustheoreticalandexperimentalratiosoftheanisotropyassociatedwiththeS=6andS=2Mn3complexes:Disthetotalmolecularanisotropy;Uisthemagnetizationreversalbarrier. RatioComplexProj.Op.ExactDiag.Mag.StudiesHFEPR Insummary,itcanbeclearlynotedthatthestructuraldistortionsimposedbythebridgingmpko-groupsinuencethemolecularDvalueofthecomplexes.Thesingle-ionMnIIIanisotropyaxes(zaxes)i.e.,theJahn-Tellerelongationaxes(JT-axes),areatanangle(60.7oforcomplex1and59.6oforcomplex2)withtheMn3planeandthemolecularanisotropyaxis(i.e.theeasyaxis)isperpendiculartotheMn3plane.AstheJTaxesbecomeparalleltothez-axis,Dincreasessignicantly,asisshownincomplex3.ThishasbeenpreviouslydemonstratedbybyHFEPRandmagnetizationtsofthe[Mn4O3(O2CR)4(dbm)3]familyofSMMswithS=9/2,wheredbm-istheanionofdibenzoylmethane[ 80 81 ]andfromastudybyinelasticneutronscatteringofthe[Mn4O3X(O2CMe)3(dbm)3]familyofSMMs,whereXstandsforF-,Cl-,orBr-[ 82 83 ]. WiththeJT-axesbeingparalleltothez-axesthediscrepanciesbetweentheexperimentalresultsandourcalculatedresultsfromprojectionoperatortechniqueandalsoforstrongexchangelimitinthematrixdiagonalizationprocessbecomesmallerasinthecaseofcomplex3.Nowifweconsiderthechangeinbarrierheight,weseethatforcomplexes1&2thebarrierremainsalmostthesamewhereasthereisasignicantincreaseinbarrierheight(1.7times)forcomplex3.ThusitcanbeconcludedthattheapplicationofthestrategyofalteringthesignoftheexchangeinteractionswithintriangularunitcanleadtoadditionalnewexamplesofSMMswithhigherbarrierheight. 95

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MeasurementsperformedontheS=4complexgivethefollowingspinHamiltonianparameters:D=1:27(2)cm1andB04=+1:3(3)104cm1;gwasnotwelldeterminedbecausetheeldwasonlyapproximatelyalignedwiththeeasyaxis(within15o).Figure5-9(b)showsthebestsimulationofthefrequencydependentdataforthelowspincomplex.InadditiontothemainpeakscorrespondingtotheS=4state(blacksquares),anextraseriesofpeaksarealsoplotted,asindicatedinthegurebyblackcirclesandthedashedline.Wenotethatitisnotpossibletosimultaneouslysimulatebothsetsofpeaks,evenforspinvaluesdierentfromS=4.Wecanthusconcludethattheextrapeaksresultfromalow-lyingexcitedstate.ThesizeablepositiveB04valueisalsoconsistentwiththisobservation.Aswehaveshownpreviously,higherorderterms(4th,6th,etc..)arecausedbyad-mixingofexcitedspinstatestothegroundstate[ 42 ]andalsothesignoftheaxial4thordertermusuallyreectsthesignoftheexchangewithinthecluster[ 79 ]. 96

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(top)Temperaturedependentspectraofcomplex5,obtainedat285GHzforeasyaxisorientation.(bottom)Energydierencediagramconstructedfromfrequency-dependentmeasurements.Thesolidanddashedlinesarethesimulationsusingtheparametersgiveninthemaintext. Similarstudiesontheferromagneticcomplex6showclearnegativeuniaxialanisotropy,asevidentfromthetemperature-dependentspectraobtainedat331GHz,withtheeldappliedapproximatelyparalleltoitseasy-axis.Athighertemperaturesvedistincttransitions(C1:ms=12!11,C2:ms=11!10,C3:ms=10!9,C4:ms=9!8,C5:ms=8!7)areobservedwithinthemagneticeldlimitofourexperimentalsetup.ItcanalsobenotedfromFig.5-10(a)thateachofthepeaks 97

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(top)Temperaturedependentspectraofcomplex6,obtainedat331GHzforeasyaxisorientation.(bottom)Energydierencediagramconstructedfromfrequency-dependentmeasurements.Thesolidlinesarethesimulationsusingtheparametersgiveninthemaintext. exhibitseveralnestructuresonthehigh-eldshoulders,whichpossiblysigniesweakdisorderinthecrystalresultinginadistributionofmicroenvironments,i.e.adistributionintheuniaxialanisotropyparameterD[ 58 ]. Multi-frequencyexperimentsthenallowprecisedeterminationofthespinHamiltonianparametersfromthesimulationsfortheeasy-axisdirection(Fig.5-10(b)).ThehighspinsystemcanbebestdescribedwithD=0:360(5)cm1,B04=5:7(5)106cm1

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84 ]. AsasidenoteitcouldbenotedthatthehardplanespectraofthehighspincomplexshowssymmetryeectinthetransverseeldEPRspectra,similartowhatwasreportedpreviouslyforMn12-acetate[ 85 ].PreviousexperimentsshowedthatMn12-acetateexhibitunusualselectionrulesforfrequenciesbelow90GHz,fortransverseeldspectraforrotationoftheeldoutofthehardplane.Astheeldisrotatedawayfromthehardplane,theEPRintensityoscillatesbetweentwoseriesofresonancesexcitedfrom`even'and`odd'spinstates.Theseselectionrulesareextremelysensitivetotheeldorientationforanglesclosetothehardplane.Thiswasalsoclearlyobservedfromtheangledependentmeasurementspreformedat67GHzandat10Koncomplex6,asshowninFig.5-11(a).Rotationwasperformedaroundthehardplaneorientation(inbetweenthetworedtraces)in0:5osteps.Resonancesfrom`even'and`odd'spinstateswithintheS=12manifoldarelabeledandrespectively.Alsoseveralothertransitionscouldbeobservedat10K.Webelievethisisduetoalowlyingexcitedstatemanifold.Thisisalsoconrmedfromtemperaturedependentandmulti-frequencystudies. Figure5-11(b)clearlyrepresentsthehardplanetemperaturedependenceforatypicalsinglemoleculemagnetwiththespectralintensityshiftingtowardsloweldasthetemperatureisincreased.Onlytransitionsareobservedforthisorientationwith10representingtransitionsfrom10!9,8from8!7and6from6!5.Fortemperatureshigherthan7K,anextrasetoftransitionsareobservedbetween10and8andalsobetween8and6,whichwebelievearetransitionsfromS=11manifold. Figure5-11(c)showsthebestttothetransverseeldtransitionsfromsimulations.ThesolidlinesandtheblacksquaresarettothetransitionsfromS=12manifold 99

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(a)ExperimentalEPRspectraobtainedat10Kand67GHz,asafunctionoftheeldorientation(in0:5oincrements)relativetotheeasyaxisofasinglecrystalsampleofcomplex6;=90olieswithinthetworedtraces.(b)Temperaturedependenthardplanespectraobtainedat67GHz.Thearrowsdenotetransitionsfromexcitedstates.(bottom)FrequencydependenceofthemainEPRpeaksat10Kfortheeldappliedparalleltohardplane.ThesolidlinesandblacksuarescorrespondtotransitionsfromtheS=12manifold,whereasdottedlinesandblackcirclescorrespondtothatfromtheS=11manifold.ThesolidlinesaresimulationsyieldingtheZFSparametersgiveninthemaintext. 100

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Nowcomingbacktoourcomparisonbetweenthelow-spinandthehigh-spincomplexes,itcanbeclearlyseenfromtheexperimentalresultsthatthefactorjDjSremainsalmostconstant(5.1cm1forcomplex5and4.3cm1forcomplex6)asSincreasesforthesetwoMn6complexesbutthebarrierheight,U,increasesalmostthreetimes.Ourresultsfromprojectionoperatortechniqueandnumericalcalculationshowevershowmuchsmallerincreaseinbarrierheight(1.395and1.674timesforschemes1and2respectively).Thiscouldberelatedtothefactthatwehaveconsideredseveralapproximationsandalsonotincludedthe4thorderuniaxialanisotropytermtogettheresults.AlsoforthelowspincomplexstrongmixingoftheS=4statewithlow-lyingexcitedstatescouldpossiblyleadtoanadditionalsuppressionofthebarrier.Howevertheresultsfrommatrixdiagonalizationprocessclearlyshowthatasweapproachtheweakexchangelimit(JD)thediscrepancybecomesmuchsmallerandthechangeinbarrierapproachestheresultsfromHFEPRandmagneticmeasurements(2.72times).Asummaryoftheresultsisgivenintable5-3. Table5-3. ComparisonbetweenvarioustheoreticalandexperimentalratiosoftheanisotropyassociatedwiththeS=12andS=4Mn6complexes:Disthetotalmolecularanisotropy;Uisthemagnetizationreversalbarrier.POT1andPOT2correspondtothetwoschemesusedintheprojectionoperatortechnique;EDT1andEDT2correspondtothestrongandweakexchangelimitsintheexactdiagonalizationtechniqueandMSstandformagneticstudies. RatioPOT1POT2EDT1EDT2HFEPRMS ThusweshowexperimentallythatUdoesincrease,andthatitgoesupmorethanS0butlessthanS2.Alsoapreviousligandeldanalysisoncomplex6andadierentS=4complex[MnIII6O2(sao)6(O2CH)2(MeOH)4](similarmolecularstructure)showsthe 101

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86 ].Itcanbenotedthatthisresultisalmostcomparabletoourresultsinthestrongexchangelimit.HencealltheseresultsstronglydisagreewiththerecentpredictionthatthebarrierheightmaygoasS0asthespinvalueincreases,thusimplyingthatUdoesnotchangeintheprocess[ 70 71 ].OurndingsnallydosuggestthattheideaofmaximizingSbyswitchingthesignoftheexchangecouplinginpolynucleartransitionmetalcomplexesisareasonablestrategyforobtainingbetterSMMs. 102

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Portionsoftheworkpresentedinthischaptercanbefoundinthefollowingarticle:Directobservationofmixingofspinmultipletsinanantiferromagneticmolecularnanomagnetbyelectronparamagneticresonance,S.Datta,O.Waldmann,A.D.Kent,V.A.Milway,L.K.ThompsonandS.Hill,Phys.Rev.B76,052407(2007)(Copyright(2007)bytheAmericanPhysicalSociety).Figure2fromthearticle:O.Waldmann,Phys.Rev.B71,094412(2005)isreusedinFig.6-4withpermission.Copyright(2005)bytheAmericanPhysicalSociety. 92 ],isthearchetypeofantiferromagneticrings.RingswithanevennumberofmagneticcentersN=6;8;10;12and18havebeenreportedandtheirpropertiesanalysed([ 93 { 99 ]).Beyondbeingofinteresttotesttheoreticalmodels,theinterestintheselowdimensionalringsincreasedwhenitwassuggestedthattheymightbegoodcandidatesforobservingquantumcoherenceassociatedwiththeoscillationoftheNeelvector[ 100 ].ThesesystemscanbewelldescribedbythefollowingspinHamiltonian ^H=JNXi=1^si^si+1+gBBNXi=1^si+DNXi=1^s2i;z(6{1) where^siisthespinoperatoratsiteiwithspinquantumnumbers,(^sN+1^s1),Jisthenearestneighborexchange,BthemagneticeldandD<0thesingleionanisotropydirectedalongtheringaxis. 103

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^n=1 Atzeroeld,theclassicalgroundstatespincongurationofthewheelshowsalternatingorderwithspinspointingalongez.Thesetwostates,withtheNeelvectornalongez,labeledj"iandj#i,aredegenerateinenergyandareseparatedbyanenergybarrierofheightNDs2.However,thesearenottheenergyeigenstatesofthesystem. Forsmallmagneticanisotropy,thelowestlyingexcitationsarecharacterizedbya(quantized)rotationoftheNeelvector,whichinthequantummechanicalenergyspectrumappearsasalowlying`rotational'band,theLbandhavingminimalenergyforeachvalueofthetotalspinS=0;1;2:::(seeFig.6-1).TheenergieswithinabandincreasequadraticallywithSfollowingtheLanderuleE(S)/S(S+1)[ 98 101 { 104 ].Hencetheanalogywitharotationalband.TherotationmodesathigherenergiesaredenotedastheEband,whichcorrespondtoquantizedspinexcitations(spinwavesormagnons),asexpectedinantiferromagneticspinsystems[ 98 101 102 ].Aschematicrepresentationoftheenergyspectrumforaspin5/2systemfeaturingtheLandEbandsisshowninFig.6-1. Inthequantummechanicalpicturethelowenergyregionofthewheelconsistsofagroundstate(j"i+j#i)=p Astatepreparedinj"iatatimet=0willoscillatebetweenj"iandj#iatafrequencyof=h,whereisthetunnelsplitting[ 105 106 ].AllNspinsinthewheeltunnelsimultaneouslyalongwiththeNeelvectorthroughthepotentialbarrier,governed 104

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Schematicoftheenergyspectrumforsspin5/2system.LandEbandsareclearlyshowninthegure. Figure6-2. SchematicofquantumtunnelingoftheNeelvector. 105

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TheexperimentaldetectionofQTNVhasseveralchallengestoovercome.Highsensitivity,lowtemperaturesandstrongmagneticeldsaremandatoryforexperiments.TechniquessuchasmagnetizationandsusceptibilitymeasurementsprobeonlytheaveragespinprojectionofthemoleculewhichbysymmetryremainsunalteredupontunnelingoftheNeelvector(NV).Withanuncompensatedsublatticespininwhichthegroundstatetunnelsplittingisstilllarge,tunnelingoftheNVleadstoalargesignalintheacsusceptibilityforsmallmagneticelds.AntiferromagneticringswithuncompensatedspinsallowthedetectionofQTNVviathe`sensorspin'whichreectsthedynamicsoftheNeelvector.HighfrequencyEPRcanalsobeemployedtoobservethelongsoughtafterquantumtunnelingoftheNeelvector. AnotherimportantfeatureofmolecularringsisthediscretenessoftheenergylevelsduetothenitenumberofspinsNinthesystem.ApplicationofanexternalmagneticeldliftsthedegeneracyoftheMSlevels,thusinducinglevelcrossings(LC)betweenthegroundstateandtheexcitedstates.Also,thedeterminationoftheLCeldsfromeld-dependentmeasurements,directlyyieldstheenergiesE(S0+1)-E(S0),E(S0+2)-E(S0),etc.,ofthenexthigherspinlevels.Thisparticularmeasurementofenergiesoftenenablesaprecisedeterminationoftheexchangecouplingconstantsinthecluster.Furthermore,intheproximityofthecriticaleldsforlevelcrossings,severalinterestingphenomenacanbeobserved,suchaslevelrepulsions,[ 135 ]andS-mixingeects[ 136 137 ]inadditiontoquantumtunnelingoftheNeelvector. Inthischapterwefocusourattentiontotwoexamplesofthisclassofnanomagnets:aMnbasedgridandanFebasedwheel. 106

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6.2.1OverviewoftheMn-[33]grid 107 { 109 ].ThisworkfocusesonasupramolecularMn(II)-[33]grid. SinglecrystalsofMn-[33],[Mn9(2POAP-2H)6](ClO4)63:57MeCNH2O,werepreparedasreportedin[ 110 ].TheycrystallizeinthespacegroupC2/c.Thecation[Mn9(2POAP-2H)6]6+exhibitsaslightlydistortedS4molecularsymmetry;withtheC2axisbeingperpendiculartothegridplane.TheaveragedistancebetweentheMn(II)ionsis3.93A;thesmallestdistancebetweenclustersislargerthan8A.Thecrystallographicunitcellconsistsoftwomagneticallynonequivalentsets,eachagroupoffour,twistedby31degreesaroundtheC2axis.Thegridexhibitsuniaxialmagneticbehavior;theuniaxialzandmolecularC2symmetryaxescoincide.Thisisduetotheplanarstructureofthegrid. Figure6-3. StructureoftheMn-[33]gridconsideredequivalenttoastructureconsistingofoctanuclearringandasingleion. 107

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111 112 ]. ThetotaldimensionoftheHilbertspaceishugeforthegrid(10077696).However,thedimensioncanbereducedbyapproximatingthetotallatticeintotwosublattices(consideringoctanuclearmodelwhereN=8),namely TheapproximationallowsustodescribethelowestlyingenergystatesoftheringverywellasnowtheinternalspinstructurebecomesclassicalduetothedominantHeisenbergexchangeterm[ 98 ].WiththisapproximationtheeectiveHamiltonianisgivenby 108

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ForJR=JC=5KandDR=DC=0:14KthesimulatedenergyspectrumasafunctionofmagneticeldisshowninFig.6-4(denotestheanglebetweenthemagneticeldandthezaxis). Severalimportantpointscanbenotedfromthesimulatedenergydiagram. Theligand-eldtermsallowmixingofspinlevelswithjSj=0;1;2[ 62 113 ].ThezeroeldsplittingofeachmultipletisduetoS=0.jSj=1allowsmixingoflevelsinvolvedinlevelcrossings.Thisresultsinarepulsionofthetwostateswhenevertheycomecloseand,asaresult,energygapsopenupatthelevelcrossingeldsforcertainorientationsofthemagneticeld.ThehighlightedregioninFig.6-1showsthistrendasthemagneticeldistiltedawayfromtheeasyaxistothehardplane.jSj=2mixingispronouncedonlyinexcitedstates. InthelevelcrossingregiononlytwostatesjS;SiandjS+1;S1iarethermallypopulatedatlowtemperaturesandtheenergyspectrumcanbedescribedbythefollowingHamiltonian 109

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Simulatedenergyspectrumvsmagneticeldfordierentorientationsofthemagneticeld.ThegroundstatebelongstoS=5/2multiplet,thenexthigherstatescorrespondtoS=7/2andS=9/2multiplet. 110

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100 114 ].TheNeelvectorforthegridisdenedasN=SASB+S9. ForEPRstudies,theNeelvectortunnelingcorrespondstothetransitionbetweenthegroundstateandtherstexcitedstateatmagneticeldsabove7T.Calculationsshowsthatthistransitionhassignicantintensity.Thustheenergygap(B)canbeidentiedasthetunnelinggapduetoQTNV(foreldsbeyondtherstlevelcrossing).Thecontributionfromthehigherlyingstatesisminimalformagneticeldsinbetweenlevelcrossingswhere(B)ismaximum. WithahighlysensitivetechniquesuchasEPR,wehavetriedtofocusourstudyinthisregiontoobservetheNeelvectordynamicsofthesupramolecular[33]grid. InthestudieswithmagneticeldalongtheuniaxialzaxiswehaveobservedunambiguouslyatransitionbetweentheS=5=2groundstateandtherstexcitedstate 111

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(1)Todate,themostdirectevidenceforS-mixinghasinvolvedmeasuringenergyspacingsinEPRandINSspectra,andthencomparingthesetocalculations[ 19 42 ].Inthepresentcase,however,spinmixingisevidencedbymeasuringanotherwiseforbiddentransition,i.e.,bydirectlyprobingtherelevantpropertyofthewavefunctions,andnotjustviatheenergyspectrum.Thisis,hence,themostclear-cutexperimentaldemonstrationofspinmixing. (2)Mn-[33]belongstoagrowingclassofantiferromagneticmoleculesthathaveattractedsignicantthoreticalandexperimentalinterest.ItsmagneticexcitationscanbeverywelldescribedbythesimplegenericHamiltonianH(2),wheretheexchangeandanisotropypartsarewelldescribedbythetwoparametersJandDonly.ThepresentEPRresultsconrmthesendingsandprovidetext-bookqualityinsightsintothephysicsofmagneticmoleculesandtheeectsofS-mixing. HFEPRdatawererecordedfortwosamplesoftheMn-[33]gridusingasensitivecavityperturbationtechnique(asdescribedinchapter2)withthemagneticeldappliedalongthezaxis. Figure6-5displaysthe62GHzEPRspectrumobtainedat1.4K.FivetransitionslabeledPn(n=1;2;3;c;d)areclearlyobserved.PcistheM=1 2!+1 2signalexpectedforahalf-integerspinsystem.Thetemperaturedependenceofthe52GHzEPRspectrumisshowninFig.6-6.Uponincreasingthetemperature,additionaltransitionsappear,e.g.,P4,P5,andtherichnestructurearoundPc,whichisofnointerestinthiswork.Above20K,onlyacentrallineisobserved,whichisexpectedduetothethermalpopulationofalargenumberofspinmultipletswithnegligibleZFS.InFig.6-5(b),wedisplaythe 112

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(a)ComplexEPRspectrumat1.4Kand62GHz.ThedipsinamplitudecorrespondtoEPR;thesenseofphaserotationthrougheachdipdierentiatesM=1transitions.TheMn-[33]gridisshownintheinset.(b)FrequencydependenceoftheresonanceeldsBnoftheindicatedpeaks;thesolidlinesrepresentbest-ts(seetext). 113

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127 ].ThissuggeststhatthelargeosetforPdiscausedbyexchange,i.e.,Pdisanintermultiplettransition.Furthermore,previousmagneticandINSmeasurementsdeterminedthezero-eldgapbetweentheS=5 2groundandrstexcitedS=7 2statestobe23010GHz[ 116 117 ],thussuggestingthatPdindeedcorrespondstotheS=5 2!7 2transition. Figure6-6. TemperaturedependenceoftheEPRspectrumat52GHz;severalofthetransitionshavebeenlabeled.TheinsetshowsthetemperaturedependenceoftransitionPd. 114

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118 ].A(B)denotesthesublatticeofcorner(edge)spins[seeFigure5-2(a)],i.e.,^SA=^s1+^s3+^s5+^s7(^SB=^s2+^s4+^s6+^s8).Thedimensionof^H2isonly2646,permittingatrueleast-squaresttothedata.WeobtainJ=4:76(4)KandD=0:144(2)K,inverygoodagrementwithpreviousexperiments[ 116 117 ].ThecalculatedZFSvaluesarecomparedtoexperimentinTable 6-1 ;theagreementisexcellent. Table6-1. Firstline:Best-tresultsfortheZFSsofthetransitionsandthegfactor.Secondline:CalculatedZFSsforthebest-tJ,Dvalues.EnergiesaregiveninunitsofGHz. Fromageneralpoint-of-view,ourmodelforMn-[33]appearsover-simplied.Evenifoneassumesidealsymmetry,theexchangeandanisotropyparametersneednotbeidenticalforallMnIIionsintheMn-[33]grid.Indeed,evidenceforslightvariationsintheexchangeconstantswasinferredfromINSstudies[ 117 ].However,novariationwasfoundfromthepresentmeasurements,orfromthermodynamicstudies[ 116 ].Thisisbecausetherelevantenergiesaregovernedbytheaverageoftheexchangeconstants,henceonlyasingleJisneeded.Similarargumentsapplytotheanisotropyparametersand,infact,noevidenceforavariationhasbeenreportedsofar.Consequently,thegenericHamiltoniangiveninEq.6-3withjustthethreeparametersJ,D,andg,capturesall 115

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Thecalculatedlow-energy,low-eldspectrumforMn-[33]isdisplayedinFigure6-7(a).ItconsistsofaS=5 2groundstateandarstexcitedS=7 2multiplet;thesearefurthersplitintothreeandfourMsublevels,respectively. Inzeroeld,theM=5 2(7 2)levelsoftheS=5 2(S=7 2)multipletlielowestinenergyduetotheeasy-axisanisotropy.TheMsublevelssplitinmagneticeldduetotheZeemaninteraction.TheM=5 2!3 2,M=3 2!1 2,andM=1 2!3 2transitionswithintheS=5 2multipletgiverisetoP1,P2,andP5,respectively.Hence,theZFSforP2andP5shouldbeequivalentinmagnitude,butoppositeinsign,inagreementwithobservation(seeTable 6-1 ).PeaksP3andP4correspondtotheM=7 2!5 2andM=5 2!3 2transitionswithintheS=7 2multiplet.Asalreadynoted,PdcorrespondstothetransitionbetweentheM=5 2leveloftheS=5 2multipletandtheM=7 2leveloftheS=7 2multiplet.Theobservedtemperaturedependence(Figure6-6)isperfectlyconsistentwiththeseassignments.AnothercomparisonofexperimentalandsimulatedresultsareshowninFig.6-8.Thepositionsofthetransitionsfor62GHzseemtoagrreperfectlyforboththeresults.ForthespectrumobtainedfromHFEPRstudies(Fig.6-8(b)),thetransitionPdisnotclearlyseeninthegivenscaleasthemaximumintensityofthespectrumisconcentratedintransitionPc. IntheinsettoFigure6-7(b),wedisplaytheeld-orientation-dependenceofBdatseveralfrequencies.Agreementbetweenexperiment(opensquares)andtheory(solidcurves)isverygood,includingtheopposingtrendsatlowerandhigherfrequencies/elds.Thisbehaviorcanbetracedtothelevelrepulsion(S-mixing)betweentheS=7 2,M=7 2levelandvariousS=5 2statesastheeldistiltedawayfromthesymmetrydirectionoftheMn-[33]grid.Assuch,thisnon-linearfrequencydependenceofBdrepresentsevidenceforquantumspin-stateoscillations[ 19 ]. 116

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(a)Calculatedenergydiagram(relativetotheS=5 2,M=5 2groundstate)forBkz;severaltransitionsareindicated.(b)CalculatedEPRspectraat62GHzandtwotemperatures.TheinsetshowstheangledependenceofBdatthreefrequencies;experimentaldataarerepresentedbyopensquares,andsimulationsbysolidlines. 117

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(a)SimulatedspectrumforBkzat62GHz.(b)HFEPRspectrumobtainedat62GHz.Severaltransitionsareindicated. Accordingtotheabove,PdviolatestheS=0selectionruleforEPR.However,labelinglevelsbyScanbemisleadingbecauseofstrongS-mixing.Infact,Sisnolongeragoodquantumnumber,thoughweretainthenotationasamatterofconvenience.Inordertorigorouslycheckthepeakassignments,weperformedafullsimulationoftheEPRspectrum.Resultsfor62GHzareshowninFigure6-7(b)attwotemperatures.Theagreementwithexperimentisonceagainexcellent.Mostimportantly,thecalculationsconrmthesignicantintensityofPd.Hence,themixingofspinmultipletsisstrong 118

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2;5 2i+0:1625j7 2;5 2i0:0343j9 2;5 2i+:::,and0:9922j7 2;7 2i0:1213j9 2;7 2i+0:0270j11 2;7 2i+:::(withanjS;Minotationforthebasisfunctions).Hence,thegroundstatehas16%ofS=7 2admixedtoit,or2.6%insquared(intensity)units. Inthestrongexchangelimit(jJj>>jDj),thezero-eldenergieswithineachmultipletareexpectedtoscalewithMasM2.However,thisbehaviorisstronglyperturbedinMn-[33],withtheM=3 2and1 2statesevenoccurringoutofsequencefortheS=7 2multiplet.AstandardapproachinvolvestheuseofaneectivespinHamiltonian,^HS=D^S2z+B04^O04,todescribeeachspinmultiplet.ForaS=5 2state,deviationsfromtheexpectedM2behaviormaybecapturedbytheB04parameter(forlargerS,higherordertermssuchasB06^O06maybeimportantalso).ItistheS-mixingthatgivesrisetosignicantB04values(andhigherorderterms),asnotedinpreviousworks[ 42 121 { 123 ].TheratiojB04=Dj,therefore,servesasameasureofthedegreeofS-mixing.FortheS=5 2multipletinMn-[33]itis6:6104,andforS=7 2itis29104.Forcomparison,welistthevaluesofjB04=Djforsomeofthelowestlyingstatesofseveralothermolecularclusters,whichareconsideredtoshowS-mixing:0:2104forFe4(S=5);0:5104forMn12(S=10);2104forNi4(S=4);and5104fortheS=2excitedstateoftheferricwheelCsFe8[ 4 6 14 42 122 124 { 126 ].Apparently,Mn-[33]showsthestrongestS-mixing.Inprinciple,exchangeconstantsmaybedeterminedindirectlyusingEq.6-2onthebasisofdeviationsfromtheexpectedM2behavior.However,thisworksonlyforthesimplestsystems,asrecentlydemonstratedforaNi4SMM[ 42 ].Inthepresentwork,theexchangesplittingwasdeterminedbydirectspectroscopyviaEPR. ItisapparentthattheeectsofS-mixingbecomemoreimportantastheseparationbetweenspinmultipletsdecreases.Thisargumentcan,however,bemisleading.Forexample,theS=10groundandrstexcitedS=9multipletsoverlapinMn12,yet 119

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2and7 2statesarewellseparatedinenergy(seeFigure6-7).Evidently,otherfactorsmustbeimportanttoo,suchasthespatialsymmetriesoftherelevantspinwavefunctions[ 121 ].Therefore,oneshouldbecarefulinjudgingtheimportanceofS-mixingsolelyonthebasisoftheseparationofspinlevels,orindeedontheratioofJandD.Understandingthisissueisofgreatimportanceintermsofthedesignoffuturemolecule-basedmagnets. 129 { 134 ].Thecyclicstructureinthesemolecularringshaveadominantantiferromagneticcouplingbetweenthenearestneighborsandweakeranisotropicinteractions. Herewearegoingtodiscussstudiesonsinglecrystalsofanewfamilyof[Fe18(pd)12(pdH)12(O2CR6)(NO3)6](NO3)6ferricwheels,constitutingantiferromagneticringsof18Fe3+ions[ 128 ].ThestructureofthecomplexisshowninFig.6-9.ThecompoundcrystallizesaslargeyellowcubiccrystalsinrhombohedralspacegroupR3.Asingle-crystalX-raydiractionstudyrevealsthatallofthemoleculesinthecrystalareidenticalandhavethesameorientation.Thegroundstateofthespin-compensatedwheelisasingletwithtotalspinS=0. 120

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StructureofFe18complex.Colorcode:Fegreen;Ored;Nblue;Cgrey.Hydrogenatomsareomittedforclarity. Severaltransitionswereobservedatbothlow(below5T)andhigheldsatthelowesttemperatureofourexperimentalsetup(1.7KatUFand1.3KatNHMFL).At0oseveralhigheldtransitionsareclearlyobserved.Asthemagneticeldisrotatedfrom0oto200oinstepsof10o,thehigheldtransitionsslowlydisappearandat60o(showninblueinFig.6-9(a))noneofthehigheldtransitionsareobserved.Thetransitionsslowlyreappearwithsignicantintensitiesat180o(showninred)andthetwofoldpatternseemstocontinue.Clearlythetwoextremafromtheangledependenceare60oand150o(showninblue).Forbetterclaricationonlytheangledependencearound150oisshowninFig. 121

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Temperatureandfrequencydependentstudieswereperformedattheorientationwherethehigheldfeatureshavethestrongestintensity.Atthelowesttemperatures,onewouldnotexpecttoobservetransitionsatloweldssincethegroundstateofthecomplexisS=0.However,anumberoftransitionsareobservedformagneticeldsbelow5T.Thiswouldthenimplythepresenceoflowlyingexcitedstatesinthecomplex,whichmaketheinterpretationoftheloweldresultsrathercomplicated. Figure6-11(a)showsthetemperaturedependentdataspectraobtainedat73GHz.Asthetemperatureisincreased,theintensitiesofallthetransitionsdecreaseindicatingthatthepeaksobservedatthelowesttemperaturearegroundstatetransitions.Interestingly,thetransitionsupto15Tclearlyshowalternatephaserotation,similartothepatternsobservedinthegridbefore.ThisisclearinFig.6-12(a),showingthetypicalspectrumobtainedat91GHz.At73GHz,thetransitionsaremoreevenlyspacedbutfollowthesamepattern.Atlowelds(<15T),thecavitytransmissionexhibitssharpminima(invertedpeaks)typicalofstandardEPRsignalsduetomagneticdipoletransitionsbetweendiscreteenergylevels.However,athigherelds,thesepeaksevolveintomore-or-lessevenlyspacedoscillations(Fig.6-11(b)).Thesefeaturestendtodisappearattemperaturesabove8K. ThefrequencydependenceplotinFig.6-12(b)displaystypicalcharacteristicsofastandardAFMwheel.Thezero-eldsplitstatesdependlinearlyonfrequencyduetotheZeemanterm.Above4Tthepatternseemtobeclearenoughtoidentifyalternatetransitionswithoppositeslopes.Thedottedlinesdrawnonthegureareroughguidestotheeyes. Tounderstandtheoscillatorybehaviorathighelds,weproposethatthisbehaviorisassociatedwithquantumoscillationsofthelow-energydensityofstates(densityofenergylevels)whichresultsinamodulationofthemicrowavelossesinthecavity.Thedensity 122

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(a)Angledependentdatafrom0oto200oat91GHzand1.3K.Thebluelinesshowthetwoextremawhiletheredlinesdenotethepositionwithstrongesthigheldtransitions.(b)Thepositionsofthetransitionsplottedasafunctionofangleforbetterclarication.Thesolidlinesinthegurearettothedata. 123

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(a)Temperaturedependencespectraobtainedat73GHz,fortheorientationwithstrongesthigheldtransitions.(b)Temperaturedependencespectraobtainedat91GHz,fortheorientationwithstrongesthigheldtransitions.Onlyhigheld(above5T)spectrashowninthegure. 124

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(a)Typicalspectrumobtainedat91GHzand2K.Someofthehigheldtransitionsshowclearchangeinphase.(b)Frequencydependenceofmagneticeldpositionsofthehigheldtransitions.Below15T,severaltransitionsappeartohavenegativeslopeswhileabove15Tallthetransitionshavepositiveslope.Linesdrawntothegurearenotttothedata. 125

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Todescribethelow-temperaturebehaviorofFe18oneneedtosolvethemodelforthe18atomsexactly.ThesizeoftheHilbertspaceishuge(dimension1014)andoneneedstoconsiderseveralapproximationstomakeitacomputationallysolvableproblem.Westartwithreducingthedimensionbyapproximatingthetotallatticeintotwosublattices(similartowhatwedidforthegrid).Thesub-latticesaregivenbyspinsS1=Pi=even^SiandS1=Pi=odd^SiandeachhavealengthofNs/2=45/2.TheeectiveHamiltoniancanthenbewrittenas: ^H12=j^S1^S2+d(S21;z+S22;z)+gBXi^SB(6{5) ThedimensionoftheHamiltoniannowis2116,whichpermitssolutionbyexactnumericaldiagonalization.Theparametersjanddwerechosentobej=-5.1Kandd=0.021K.Thevalueswereobtainedusingasemi-classicalapproachandareinagreementwithotherexperiments[ 138 ].Figure6-13(a)showsthesimulatedenergyspectrumforthemagneticeldalongc-axis.InzeroeldthegroundstateisS=0andwithapplicationofanexternalmagneticeldthegroundstateprogressivelyswitchestoS=1,S=2,etc.,atcorrespondingcriticalvaluesoftheeld.TheredarrowsinthegureroughlycorrespondtothepositionsofthesevenpeaksseeninFig.6-11(b)at91GHz.HoweverdatainFig.6-11correspondtoanorientationwhichis30oawayfromthec-axisandthusinsteadofobservingsharptransitionsweseetransitionswithmuchbroaderline-width.Fromtheangledependentstudiesitwasclearthatforeldappliedalongtheab-planenohighereldtransitionswereobserved.Thereasonforthatcanbeunderstoodbylookingatgure6-13(b).Foreldappliedalongab-plane,anenergygapoftheorderof150GHzappear 126

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Inconclusionthisstudyprovideanicespectroscopicevidenceofthesequenceoflevelcrossingsinwheels(theotherspectroscopicevidencebeingtheinelasticneutronstudiesonCr7Niby[ 19 ]).Thesimulatedenergydiagramwithatwosublatticemodelqualitativelyexplainhigheldoscillatorybehaviorasseenfromtheexperimentalresults. 127

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(a)Simulatedenergydiagramforeldalongthec-axis(a)andforeldalongtheab-plane(b).Theenergyoftheloweststatewassettozeroateacheld. 128

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Thischaptergivesasummaryoftheworkpresentedinthepreviouschaptersofthisdissertation.Thisdissertationisfocusedonstudyingtheeectsofanisotropyinmolecularmagnetswithlowtemperature,higheld,highfrequencymagneticresonancespectroscopicstudies. Chapter1isanintroductiontomolecularmagnetsingeneral.Weexplainedthesourcesandimportanceofanisotropytothemagneticbehaviorofbothsinglemoleculemagnetsandantiferromagnticsystemssuchaswheelsandgrids.WealsodiscussedthebasictheoreticalfoundationsfortheoriginofthespinHamiltonianthatcanbeusedtomodelthesesystems.Whilethegiantspinapproximationisusedforsystemswheretheisotropicexchangeinteractionismuchlargerthantheanisotropicinteractions,thegeneralizedeectivespinHamiltonianprovideinformationforexchangecoupledclusters. Chapter2outlinestheexperimentaltechniquesandequipmentthatweusedtoexperimentsonthemolecularmagnets.Wediscussedthewaveguides,cavitiesandourmaininstrumentthatactsasdualmicrowavesourceandnetworkanalyzerforthehigh(>50GHz)frequencyEPRsetup.Resonatorforthelow(<50GHz)frequencyEPRtechniquewasthendiscussedindetails. InChapter3,wepresenttheresultsobtainedfromstudiesonthreemolecularclustersanddiscusstheroleofanisotropyintheirmagneticproperties.TherstsectionisbasedonresultsfromtwoFe-basedcomplexeswhereweshowtheeectofuniaxialanisotropyindeterminingwhetheramoleculecanbelabeledasaSMM.InthenextsectionwetalkaboutaMn-basedSMMwhereweshowtheroleoftransverseanisotropy,speciallytheeectofhigherorderterms,inuencingthemagneticproperties. InChapter4,weconsidertheeectsofanisotropicexchangeinteractionsbetweentheeectivespinS'=1/2KramersionswithinthetetranuclearCocomplex.Ourmodelcanaccountqualitativelyandsemi-quantitativelyformanyfeaturesobservedinHFEPR 129

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InChapter5wediscusstheroleofexchangeandanisotropyinmodifyingthebarrierheightinsinglemoleculemagnetsinMn-basedtriangularcomplexes.ThespinvalueisincreasedinbothMn3andMn6complexesbydeliberatestructuraldistortionoftheligand.Wediscussthetheoreticalresultsobtainedfromprojectionoperatortechniqueandmatrixdiagonalizationprocess.Thecalculatedresultsarethensupportedwithexperimentalresults.Theresultsprovideaclearevidenceofincreaseinbarrierheightasthespinswitchesfromlowspintohighspin. Chapter6presentsadierentclassofmolecularnanomagnets,namelyantiferromagneticwheelsandgrids.Westartwithamoredetailedintroductionanddiscussthevariousinterestingfeaturesshownbywheelsandthegrids.AdetailedstudyoftheMn-33gridandtheFe18wheelclearlyshowthatEPRcanbeausefultechniquetoobservespin-mixingandlevel-crossingsinthecomplexeswherethestrongexchangelimitislongeragoodapproximation. 130

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SaitiDattawasborninRampurhat,asmalltowninthestateofWestBengal,India.ShespentearlypartofhereducationindierentpartsofthestateandnallygraduatedfromKrishnaBhabiniNariSikshaMandir,Chandannagar(atownnearbyCalcuttawithpre-independenceFrenchheritage).ShejoinedPresidencyCollege,Calcuttaforherbachelor'sdegreeinphysicsandgraduatedin2001.ShethenmovedtoUniversityofPune,Puneforhermaster'sdegreeandobtainedherdegreein2003withspecializationinquantumeldtheory.HernalyearprojectwasstudyingfractaldimensionofFeynman'spathinquantummechanics.Shecontinuedtoworkonaprojecton`discretebreathers'beforemovingtotheUniversityofFlorida,Gainesvillein2004.Trainedasatheoreticalphysicistsofarinhercareer,shedecidedtoswitchtoexperimentalcondensedmatterphysicstohavemore`handson'experience.ShejoinedDr.StephenHill'slabinthesummerof2005andhavesincehadtheopportunitytolearnhowtousemicrowavespectroscopytounderstandsomeofthebasicquestionsrelatedtoquantummechanics.Shereceivedherdegreein2009andhopestocontinueherquestinrecentfuture. 139