VibrationalPropertiesoftheLattice的晶格振动性质

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1、ElectronicPropertiesofMetalsTheFreeElectronGas(FEG)ModelPropertiesoftheFEG:FermiEnergyEFandDensityofStatesN(E)PlasmaOscillationsandPlasmonsHeatCapacityoftheFEGA.TheFreeElectronGasModelHavingstudiedthestructuralarrangementsofatomsinsolids,andthethermalandvibrationalpropertiesofthelattice,wenowconsi

2、dertheelectronicpropertiesofsolidsintermsofaverysimplemodel.PlotU(x)fora1-Dcrystallattice:Simpleandcrudefinite-square-wellmodel:Canwejustifythismodel?Howcanonereplacetheentirelatticebyaconstant(zero)potential?UU=0AssumptionsoftheFEGModel1.Metalshavehighelectricalconductivityandnoapparentactivation

3、energy,soatleastsomeoftheirelectronsare“free”andnotboundtoatoms4.ThereforemodelthebehaviorofthefreeelectronswithU=0insidethevolumeofthemetalandafinitepotentialstepatthesurface.Assumeeachatomhasn0freeelectrons,wheren0=chemicalvalence.Assumethatresistancecomesfromelectronsinteractingwithlatticethrou

4、ghoccasionalcollisions:2.CoulombpotentialenergyofpositiveionsU1/risscreenedbyboundelectronsandisweakeratlargedistancesfromnucleus3.ElectronswouldhavelowestU(highestK)nearnuclei,sotheyspendlesstimenearnucleiandmoretimefarfromnucleiwhereUisnotchangingrapidly.B.PropertiesoftheFEG:FermiEnergyandDensi

5、tyofStatesSolutionshave1.Wavefunctions:2.Energies:Time-independentSchrödingerEquation:WithU=0:Travelingwaves(planewaves)Parabolicenergy“bands”EkxPropertiesoftheFEGThisshowsthatthevolumeink-spacepersolutionis:ByusingperiodicboundaryconditionsforacubicsolidwithedgeLandvolumeV=L3,wedefinethesetofallo

6、wedwavevectors:Andthusthedensityofstatesink-spaceis:Justasinthecaseofphonons!SincetheFEGisisotropic,thesurfaceofconstantEink-spaceisasphere.ThusforametalwithNelectronswecancalculatethemaximumkvalue(kF)andthemaximumenergy(EF).kxkzkyFermisphereFermiWaveVectorandEnergyAndthemaximumenergyiseasilyfound

7、:Takingintoaccountthespindegeneracyoftheelectron,NelectronswillbeaccommodatedbyN/2energystates,so:ThesequantitiesarecalledtheFermiwavevectorandFermienergyinhonorofEnricoFermi,who(alongwithArnoldSommerfeld)didthem