Vibrational-Properties-of-the-Lattice-Valparaiso-University的晶格振动性质的瓦尔帕莱索大学文课件

IV.VibrationalPropertiesoftheLatticeHeatCapacity—EinsteinModelTheDebyeModel—IntroductionAContinuousElasticSolid1-DMonatomicLatticeCountingModesandFindingN()TheDebyeModel—Calculation1-DLatticeWithDiatomicBasisPhononsandConservationLawsDispersionRelationsandBrillouinZonesAnharmonicPropertiesoftheLatticeA.HeatCapacity—EinsteinModel(1907)Havingstudiedthestructuralarrangementsofatomsinsolids,wenowturntopropertiesofsolidsthatarisefromcollectivevibrationsoftheatomsabouttheirequilibriumpositions.mkykxkzForavibratingatom:Classicalstatisticalmechanics—equipartitiontheorem:inthermalequilibriumeachquadratictermintheEhasanaverageenergy,so:ClassicalHeatCapacityForasolidcomposedofNsuchatomicoscillators:Givingatotalenergypermoleofsample:Sotheheatcapacityatconstantvolumepermoleis:ThislawofDulongandPetit(1819)isapproximatelyobeyedbymostsolidsathighT(>300K).Butbythemiddleofthe19thcenturyitwasclearthatCV

0asT0forsolids.So…whatwashappening?EinsteinUsesPlanck’sWorkPlanck(1900):vibratingoscillators(atoms)inasolidhavequantizedenergies[laterQMshowedisactuallycorrect]Einstein(1907):modelasolidasacollectionof3Nindependent1-Doscillators,allwithconstant,andusePlanck’sequationforenergylevelsoccupationofenergyleveln:(probabilityofoscillatorbeinginleveln)classicalphysics(Boltzmannfactor)Averagetotalenergyofsolid:SomeNiftySummingUsingPlanck’sequation:NowletWhichcanberewritten:Nowwecanusetheinfinitesum:Togive:Soweobtain:Atlast…theHeatCapacity!Differentiating:Nowitistraditionaltodefinean“Einsteintemperature”:Usingourpreviousdefinition:Soweobtaintheprediction:LimitingBehaviorofCV(T)LowTlimit:Thesepredictionsarequalitativelycorrect:CV

3RforlargeTandCV

0asT0:HighTlimit:3RCVT/EButLet’sTakeaCloserLook:HighTbehavior:ReasonableagreementwithexperimentLowTbehavior:CV

0tooquicklyasT0!B.TheDebyeModel(1912)DespiteitssuccessinreproducingtheapproachofCV

0asT0,theEinsteinmodelisclearlydeficientatverylowT.Whatmightbewrongwiththeassumptionsitmakes?•3Nindependentoscillators,allwithfrequency•Discreteallowedenergies:DetailsoftheDebyeModelPieterDebyesucceededEinsteinasprofessorofphysicsinZürich,andsoondevelopedamoresophisticated(butstillapproximate)treatmentofatomicvibrationsinsolids.Debye’smodelofasolid:•3Nnormalmodes(patterns)ofoscillations•Spectrumoffrequenciesfrom=0tomax•Treatsolidascontinuouselasticmedium(ignoredetailsofatomicstructure)ThischangestheexpressionforCVbecauseeachmodeofoscillationcontributesafrequency-dependentheatcapacityandwenowhavetointegrateoverall:#ofoscillatorsperunitEinsteinfunctionforoneoscillatorC.TheContinuousElasticSolidWecandescribeapropagatingvibrationofamplitudeualongarodofmaterialwithYoung’smodulusEanddensitywiththewaveequation:forwavepropagationalongthex-directionBycomparisontothegeneralformofthe1-Dwaveequation:wefindthatSothewavespeedisindependentofwavelengthforanelasticmedium!iscalledthedispersionrelationofthesolid,andhereitislinear(nodispersion!)groupvelocityD.1-DMonatomicLatticeBycontrasttoacontinuoussolid,arealsolidisnotuniformonanatomicscale,andthusitwillexhibitdispersion.Considera1-Dchainofatoms:Inequilibrium:Longitudinalwave:MaForatoms,p=atomlabelp=1nearestneighborsp=2nextnearestneighborscp=forceconstantforatomp1-DMonatomicLattice:EquationofMotionThus:Fortheexpectedharmonictravelingwaves,wecanwritexs=sa=positionofatomsNowweuseNewton’ssecondlaw:Or:So:Nowsincec-p=cpbysymmetry,1-DMonatomicLattice:Solution!Theresultis:Thedispersionrelationofthemonatomic1-Dlattice!Oftenitisreasonabletomakethenearest-neighborapproximation(p=1):Theresultisperiodicinkandtheonlyuniquesolutionsthatarephysicallymeaningfulcorrespondtovaluesintherange:DispersionRelations:Theoryvs.ExperimentIna3-Datomiclatticeweexpecttoobserve3differentbranchesofthedispersionrelation,sincetherearetwomutuallyperpendiculartransversewavepatternsinadditiontothelongitudinalpatternwehaveconsidered.Alongdifferentdirectionsinthereciprocallatticetheshapeofthedispersionrelationisdifferent.Butnotetheresemblancetothesimple1-Dresultwefound.E.CountingModesandFindingN()Avibrationalmodeisavibrationofagivenwavevector(andthus),frequency,andenergy.Howmanymodesarefoundintheintervalbetweenand?#modesWewillfirstfindN(k)byexaminingallowedvaluesofk.ThenwewillbeabletocalculateN()andevaluateCVintheDebyemodel.Firststep:simplifyproblembyusingperiodicboundaryconditionsforthelinearchainofatoms:x=sax=(s+N)aL=Nass+N-1s+1s+2Weassumeatomssands+Nhavethesamedisplacement—thelatticehasperiodicbehavior,whereNisverylarge.First:findingN(k)Thissetsaconditiononallowedkvalues:Sotheseparationbetweenallowedsolutions(kvalues)is:independentofk,sothedensityofmodesink-spaceisuniformSinceatomssands+Nhavethesamedisplacement,wecanwrite:Thus,in1-D:Next:findingN()Nowfora3-DlatticewecanapplyperiodicboundaryconditionstoasampleofN1xN2xN3atoms:N1aN2bN3cNowweknowfrombeforethatwecanwritethedifferential#ofmodesas:Wecarryouttheintegrationink-spacebyusinga“volume”elementmadeupofaconstantsurfacewiththicknessdk:N()atlast!AverysimilarresultholdsforN(E)usingconstantenergysurfacesforthedensityofelectronstatesinaperiodiclattice!Rewritingthedifferentialnumberofmodesinaninterval:Wegettheresult:ThisequationgivestheprescriptionforcalculatingthedensityofmodesN()ifweknowthedispersionrelation(k).WecannowsetuptheDebye’scalculationoftheheatcapacityofasolid.F.TheDebyeModelCalculationWeknowthatweneedtoevaluateanupperlimitfortheheatcapacityintegral:•3independentpolarizations(L,T1,T2)withequalpropagationspeedsvg•continuous,elasticsolid:=vgk•maxgivenbythevaluethatgivesthecorrectnumberofmodesperpolarization(N)Ifthedispersionrelationisknown,theupperlimitwillbethemaximumvalue.ButDebyemadeseveralsimpleassumptions,consistentwithauniform,isotropic,elasticsolid:N()intheDebyeModelFirstwecanevaluatethedensityofmodes:Nextweneedtofindtheupperlimitfortheintegralovertheallowedrangeoffrequencies.Sincethesolidisisotropic,alldirectionsink-spacearethesame,sotheconstantsurfaceisasphereofradiusk,andtheintegralreducesto:Giving:foronepolarizationmaxintheDebyeModelSincethereareNatomsinthesolid,thereareNuniquemodesofvibrationforeachpolarization.Thisrequires:TheDebyecutofffrequency

Giving:NowthepiecesareinplacetoevaluatetheheatcapacityusingtheDebyemodel!Thisisthesubjectofproblem5.2inMyers’book.Rememberthattherearethreepolarizations,soyoushouldaddafactorof3intheexpressionforCV.Ifyoufollowtheinstructionsintheproblem,youshouldobtain:AndyoushouldevaluatethisexpressioninthelimitsoflowT(T<<D)andhighT(T>>D).DebyeModel:Theoryvs.Expt.Universalbehaviorforallsolids!Debyetemperatureisrelatedto“stiffness”ofsolid,asexpectedBetteragreementthanEinsteinmodelatlowTDebyeModelatlowT:Theoryvs.Expt.QuiteimpressiveagreementwithpredictedCV

T3dependenceforAr!(noblegassolid)(SeeSSSprogramdebyetomakeasimilarcomparisonforAl,CuandPb)G.1-DLatticewithDiatomicBasisConsideralineardiatomicchainofatoms(1-DmodelforacrystallikeNaCl):Inequilibrium:M1aM2M1M2ApplyingNewton’ssecondlawandthenearest-neighborapproximationtothissystemgivesadispersionrelationwithtwo“branches”:-(k)0ask0 acousticmodes(M1andM2moveinphase) +(k)maxask0opticalmodes (M1andM2moveoutofphase)1-DLatticewithDiatomicBasis:ResultsThesetwobranchesmaybesketchedschematicallyasfollows:gapinallowedfrequenciesopticalacousticInareal3-Dsolidthedispersionrelationwilldifferalongdifferentdirectionsink-space.Ingeneral,forapatombasis,thereare3acousticmodesandp-1groupsof3opticalmodes,althoughformanypropagationdirectionsthetwotransversemodes(T)aredegenerate.DiatomicBasis:ExperimentalResultsTheopticalmodesgenerallyhavefrequenciesnear=10131/s,whichisintheinfraredpartoftheelectromagneticspectrum.Thus,whenIRradiationisincidentuponsuchalatticeitshouldbestronglyabsorbedinthisbandoffrequencies.AtrightisatransmissionspectrumforIRradiationincidentuponaverythinNaClfilm.Notethesharpminimumintransmission(maximuminabsorption)atawavelengthofabout61x10-4cm,or61x10-6m.Thiscorrespondstoafrequency=4.9x10121/s.IfinsteadwemeasuredthisspectrumforLiCl,wewouldexpectthepeaktoshifttohigherfrequency(lowerwavelength)becauseMLi<MNa…exactlywhathappens!H.PhononsandConservationLawsCollectivemotionofatoms=“vibrationalmode”:Quantumharmonicoscillator:Energycontentofavibrationalmodeoffrequencyisanintegralnumberofenergyquanta.Wecallthesequanta“phonons”.Whileaphotonisaquantizedunitofelectromagneticenergy,aphononisaquantizedunitofvibrational(elastic)energy.Associatedwitheachmodeoffrequencyisawavevector,whichleadstothedefinitionofa“crystalmomentum”:Crystalmomentumisanalogoustobutnotequivalenttolinearmomentum.Nonetmasstransportoccursinapropagatinglatticevibration,sothelinearmomentumisactuallyzero.Butphononsinteractingwitheachotherorwithelectronsorphotonsobeyaconservationlawsimilartotheconservationoflinearmomentumforinteractingparticles.PhononsandConservationLawsLatticevibrations(phonons)ofmanydifferentfrequenciescaninteractinasolid.Inallinteractionsinvolvingphonons,energymustbeconservedandcrystalmomentummustbeconservedtowithinareciprocallatticevector:Schematically:Comparethistothespecialcaseofelasticscatteringofx-rayswithacrystallattice:PhotonwavevectorsJustaspecialcaseofthegeneralconservationlaw!I.BrillouinZonesoftheReciprocalLatticeRememberthedispersionrelationofthe1-Dmonatomiclattice,whichrepeatswithperiod(ink-space):1stBrillouinZone(BZ)2ndBrillouinZone3rdBrillouinZoneEachBZcontainsidenticalinformationaboutthelatticeWigner-SeitzCell--ConstructionForanylatticeofpoints,onewaytodefineaunitcellistoconnecteachlatticepointtoallitsneighboringpointswithalinesegmentandthenbisecteachlinesegmentwithaperpendicularplane.TheregionboundedbyallsuchplanesiscalledtheWigner-Seitzcellandisaprimitiveunitcellforthelattice.1-Dlattice:Wigner-Seitzcellisthelinesegmentboundedbythetwodashedplanes2-Dlattice:Wigner-Seitzcellistheshadedrectangleboundedbythedashedplanes1stBrillouinZone--DefinitionTheWigner-Seitzcellcanbedefinedforanykindoflattice(directorreciprocalspace),buttheWScellofthereciprocallatticeisalsocalledthe1stBrillouinZone.The1stBZistheregioninreciprocalspacecontainingallinformationaboutthelatticevibrationsofthesolid.Onlythevaluesinthe1stBZcorrespondtouniquevibrationalmodes.Anyoutsidethiszoneismathematicallyequivalenttoavalueinsidethe1stBZ.Thisisexpressedintermsofageneraltranslationvectorofthereciprocallattice:1stBrillouinZonefor3-DLatticesFor3-Dlattices,theconstructionofthe1stBrillouinZoneleadstoapolyhedronwhoseplanesbisectthelinesconnectingareciprocallatticepointtoitsneighboringpoints.Wewillseetheseagain!bccdirectlatticefccreciprocallatticefccdirectlatticebccreciprocallatticeI

J.AnharmonicPropertiesofSolidsTwoimportantphysicalpropertiesthatONLYoccurbecauseofanharmonicityinthepotentialenergyfunction:ThermalexpansionThermalresistivity(orfinitethermalconductivity)ThermalexpansionIna1-DlatticewhereeachatomexperiencesthesamepotentialenergyfunctionU(x),wecancalculatetheaveragedisplacementofanatomfromitsT=0equilibriumposition:I

ThermalExpansionin1-DEvaluatingthisfortheharmonicpotentialenergyfunctionU(x)=cx2gives:Thusanynonzero<x>mustcomefromtermsinU(x)thatgobeyondx2.ForHWyouwillevaluatetheapproximatevalueof<x>forthemodelfunctionNowexaminethenumeratorcarefully…whatcanyouconclude?independentofT!Whythisform?OnthenextslideyoucanseethatthisfunctionisareasonablemodelforthekindofU(r)wehavediscussedformoleculesandsolids.Doyouknowwhatformtoexpectfor<x>basedonexperiment?LatticeConstantofArCrystalvs.TemperatureAboveabout40K,wesee:Usuallywewrite:=thermalexpansioncoefficientThermalResistivityWhenthermalenergypropagatesthroughasolid,itiscarriedbylatticewavesorphonons.Iftheatomicpotentialenergyfunctionisharmonic,latticewavesobeythesuperpositionprinciple;thatis,theycanpassthrougheachotherwithoutaffectingeachother.Insuchacase,propagatinglatticewaveswouldneverdecay,andthermalenergywouldbecarriedwith