高等固体物理

ChapterChapter4:CrystalI.VibrationsofCrystalswithMonatomicWhydowestudycrystalConsidertheelasticvibrationsofacrystalwithoneatomintheprimitivecell.Thesolutionisthesimplestinthe[100],[110],[111]propagationdirectionsincubicWhenawavepropagatesalongoneofthesedirections,entireplanesofatomsmoveinphasewithdisplacementseitherparallelorperpendiculartothedirectionofthewavevector.[100],[110],[100],[110],and[111]propagationdirectionsincubiccrystals.I.I.VibrationsofCrystalswithMonatomicWecandescribewithasinglecoordinateusthedisplacementoftheplanesfromitsequilibriumItisone-dimensionalForeachwavevectortherearethreemodes,oneoflongitudinalpolarizationandtwooftransverseI.I.VibrationsofCrystalswithMonatomicAssumethattheelasticenergyisaquadraticfunctionoftherelativedisplacementofanytwopointsinthecrystal;CubicandhigherordertermsmaybeWeconsideronlynearest-neighborinteractions,thenthetotalforceonscomesfromplanesFs=C(us+1-us)+C(us-1-us).(Hooke’sCistheforceconstantbetweennearest-neighborplanesandwilldifferforlongitudinalandtransverseI.I.VibrationsofCrystalswithMonatomicWeregardCasdefinedforoneatomoftheplane,sothatFsistheforceononeatomintheplanes.TheequationofmotionoftheplanesMd2us/dt2=C(us+1+us-1-2us),Misthemassofanatom.Welookforsolutionswithalldisplacementshavingthetimedependenceexp(-it).Thend2us/dt2=-2usus=uexp(isKa)exp(-i-M2us=C(us+1+us-1-2us)I.I.VibrationsofCrystalswithMonatomic-M2us=C(us+1+us-1-2us)Thisisadifferenceequationinthedisplacementsandhastravelingwavesolutionsoftheus±1=uexp(isKa)exp(±iKa)exp(-it),whereaisthespacingbetweenplanesandKistheWehavethedispersionrelationconnectingand2=(2C/M)(1-Theboundaryofthe1stB.Z.liesatK=±I.I.VibrationsofCrystalswithMonatomic BrillouinTheratioofthedisplacementsoftwosuccessiveplanesisgivenby:us+1/us=uexp[i(s+1)Ka]/uexp(isKa)=exp(iKa)Therange-to+forthephaseKacoversallindependentvaluesoftheexponential.1.2(-0.8 4.2(0.2TherangeofindependentvaluesofKisspecified-/a<K/a(the1stBrillouinInthecontinuumlimita0andKmax± BrillouinThedisplacementc waysbedescribedbyawavevectorwithinthe Proof:SupposeKliesoutsidethe zone,butarelatedwavevectorK’=K-2n/alieswithinthezone,wherenisaninteger.Then:us+1/us=exp(iKa)=exp(i2n)exp[i(Ka-=exp(i2n)exp[i(K-2n/a)a]=ThusbysubtractionofanappropriatereciprocallatticevectorfromK,wealwaysobtainanequivalentwavevectorinthe1stzone. BrillouinAttheboundariesKmax=±/a,thesolutionus=uexp(isKa)exp(-it)doesnotrepresentatravelingwave,butastandingwave.us+1/us=exp(iKa)=-1.(astandingwave:alternateatomsoscillateinoppositephase).ThisisequivalenttoBraggreflectionofx-Kmax=±/asatisfiestheBraggcondition2dsin=n:=/2,d=a,K=2/,n=1,sothatPhysical2.2.GroupThetransmissionvelocityofawavepacketisthegroupvelocity,givenasvg=d/dK,orvg=gradKthegradientofthefrequencywithrespecttoK.Thisisthevelocityofenergypropagationinthemedium.1D thisiszeroattheedgeofthezonewhereK=Herethewaveisastandingwave(zeronettransmi-ssionvelocity).2.2.Group3.3.LongWavelengthWhenKa«1weexpandcosKa1-(Ka)2/2,sothatthe 2=(C/M)K2a2Thefrequencyisdirectlyproportionaltothewavevectorinthelongwavelengthlimit.v=/K==(C/M)1/2aexactlyasinthecontinuumtheoryofelasticwaves.4.4.DerivationofForceConstantsfromInmetalstheeffectiveforcesmaybeofqui range.Couplingshavebeenfoundbetweenplanesofatomsseparatedbyasmanyas20planes. eralizationofthedispersionrelationtonearestplanes C(1-cospKa) Cp=(-Ma/2)B.ZdKKgivestheforceconstantatrangepa,forastructurewithamonatomicbasis.PleasereadPage103forthedetailproofII.II.TwoatomsperprimitiveThephonondispersionshowsnewfeaturesincrystalswithtwoormoreatomsperprimitivebasis.DiamondstructurewithtwoatomsintheprimitiveForeachpolarizationmodeinagivenpropagationdirectionthedispersionrelation(K)developstwobranches,knownastheacousticalandopticalbranches:LA&TAmodes,LO&TOmodes.Iftherearepatomsintheprimitivecell,thereare3pbranchestothedispersionrelation.II.II.TwoatomsperprimitiveIftherearepatomsintheprimitivecell,thereare3pbranchestothedispersionrelation.3acousticalbranches&3p-3opticalbranches.(1dimension:12dimension:Forexample,Ge&KBr,2atoms,6branches:oneLA,oneLO,TwoTA,andtwoTO.patomsintheprimitivecell,Nprimitivecells,pNatoms,total3pNdegrees,3Nacousticalmodesand(3p-3)Nopticalmodes.II.II.TwoAtomsperprimitiveII.II.TwoAtomsperprimitiveConsideracubiccrystalwhereatomsofmassM1lieononesetofplanesandatomsofmassM2lieonplanesinterleavedbetweenthoseofthe Assume:Letaistherepeatdistanceofthelatticeinthedirectionnormaltothelatticeplanes;Considerthateachplaneinteractsonlywithitsnearest-neighborplanesandthattheforceconstantsareidenticalbetweenallpairsofnearest-neighborII.II.TwoAtomsperprimitiveII.II.TwoAtomsperprimitiveTheequationsofmotionaregivenM1d2us/dt2=C(vs+vs-1-2us),M2d2vs/dt2=C(us+1+us-2vs).Welookforasolutionintheformofatravelingwave,withdifferentamplitudesuandvonalternateplanes:us=uexp(isKa)exp(-it);vs=vexp(isKa)exp(-it);II.II.TwoAtomsperprimitiveII.II.TwoAtomsperprimitiveII.II.TwoAtomsperprimitiveFortheopticalbranchatK=0,weu/v=-M2/Theatomsvibrateagainsteachother,buttheircenterofmassisfixed.Wemayexciteamotionofthistypewiththeelectricfieldofalightwave,ifthetwoatomscarriesoppositecharges.Theatoms(andtheircenterofmass)movetogether,asinlongwavelengthacousticalvibrations(acousticalThereisafrequencygapatKmax=±/aII.II.TwoAtomsperprimitive zationofElasticTheenergyofalatticevibrationis zed.Thequantumofenergyiscalledaphononin withthephotonoftheelectromagneticwave.ElasticwavesincrystalsaremadeupofTheenergyofanelasticmodeofangularfrequencywhenthemodeisoccupiedbynphonons.ħ/2isthezeropointenergyofthemode. zationofElasticConsiderthestandingwavemodeofu=u0cosKxcostuisthedisplacementofavolumeelementfromitsequilibriumpositionatxinthecrystal.Thekineticenergydensity(du/dt)2/2=V2u20sin2t/4,(Volumeintegral)ThetimeaveragekineticenergyisV2u20/8=(1/2)(n+1/2)ħ zationofElasticThesquareoftheamplitudeisu20=4(n+1/2)ħ/V,thisrelatesthedisplacementinagivenmodetophononoccupancynoftheSincetheenergyofaphononmustbepositive,soisAnopticalmodewithclosetozeroiscalledasoftmode(ferroelectriccrystals).IV.IV.PhononAphononofwavevectorKwillinteractwithparticlessuchasphotons,neutron,andelectronsasifithadamomentumħK.Thephysicalmomentumofacrystalp=M(d/dt)usWhenthecrystalcarriesaphononp=M(du/dt)exp(isKa)=M(du/dt)[1-exp(iNKa)]/[1-exp(iKa)],p=M(du/dt)exp(isKa)=M(du/dt)[1-exp(iNKa)]/[1-exp(iKa)]=0(K=±2r/Na,risaninteger),TheonlyexceptionisuniformmodeK=0,forwhichusequalu,sothatAphononactsasifitsmomentumwereħK,calledcrystalTheelasticscatteringofanx-rayphotonbyacrystalisernedbythewavevectorselectionruleIV.IV.PhononIfthescatterringofthephotonisinelastic,withthecreationofaphononofwavevectorK,thenthewavevectorselectionrule IfaphononKisabsorbedintheprocess,weV.V.InelasticScatteringbyPhonondispersionrelations(K)areoftendeterminedbytheinelasticscatteringofneutronswiththeemissionorabsorptionofaphonon.Thewidthofthescatteredneutronbeamgivesinformationbearingonthelifetimeofphonons.Thescatteringofaneutronbeambyacrystallatticearedescribedbythewavevectorselectionrule:andbytherequirementofconservationofV.V.InelasticScatteringbyKisthewavevectorofthephononcreated(+)orabsorbed(-)intheprocess,andGisanyreciprocallatticevector.ForaphononwechooseGsuchthatKliesintheBrillouinzone.Ifħ2k2/2Mnisthekineticenergyoftheincidentneutron,ħ2k’2/2Mnistheenergyofthescatteredneutron,thentheconservationofenergyisV.V.InelasticScatteringbyħ2k2/2Mn=ħ2k’2/2Mn±ħwhereħistheenergyofthephononcreated(+)orabsorbed(-)intheprocess.MnisthemassoftheV.V.InelasticScatteringbySummarySummaryofChapterThequantumunitofacrystalvibrationisaphonon.Iftheangularfrequencyis,theenergyofthephononisħ.WhenaphononofwavevectorKiscreatedbytheinelasticscatteringofaphotonorneutronfromwavevectorktok’,thewavevectorselectionrulethaternstheprocessk=k’+K+GwhereGisareciprocallatticeSummarySummaryofChapterAllelasticwavescanbedescribedbywavevectorsthatliewithinthe Brillouinzoneinreciprocalspace.Iftherearepatomsintheprimitivecell,thephonondispersionrelationwillhave3acousticalphononbranchesand3p-3opticalphononbranches.Allelasticwavescanbedescribedbywavevectorsthatliewithinthe1stB.Z.Inreciprocallattice.格简谐近似下，晶体中的所有原子都参与的振动用格波来描述。1、声学波和光学波的区主要是根据长波极限的性质来命名声学波：相邻原子都是沿着同一方向振动。长波极限下原胞内两种原子的运动完全一致，振幅和位相都没有差别，代表原胞质心的振动。光学波：同一原胞内相邻不同原子的振动相反。长波极限下，代表原胞内两个原子的相对振动，而质心保持不变。2、格波的数目支数：色散关系决定。可能有简并情况(横波、纵波）。格波数目：即振动状态数目。可以根据数和格波支数来计算。简约区中的总数等于晶体的原胞数，晶格振动格波的总数等于晶体的度数。晶体维数：d(可取晶体中原胞数声学波支数d光学波支数：d(s-声学波数目：光学波数目：d(s-总格波数（晶体 度数 补充确定晶格振动谱的实验方光光子受晶格的非弹性散射来测定。中子（或光子）与晶格的相互作用即中子（或光子）与晶体中声子的相互作用。