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1、FailureoftheFreeElectronModelThefree-electronmodelaccountsformanyobservedmetallicproperties,butithassomelimitations:•Itsuggeststhattheelectricalconductivityisproportionaltotheelectroncon-centration,butdivalentmetals(Be,Cd,Zn,etc.)andeventrivalentmetals
2、(Al,In)arelessconductivethanmonovalentmetals(Cu,Ag,Au);•Thefree-electronmodelalwayspredictsanegativeHalleffect,butsomemetalssuchasBe,Cd,andZn(seeTable4.3)havepositiveHallconstants;•ItpredictsasphericalFermisurfaceincontrasttomeasurementswhichoftenindica
3、tethatitisnonspherical.Thesedifficultiescanberesolvedbytakingintoaccounttheinteractionoftheelectronswiththelatticeasexplainedbelow.5.2EnergySpectrainAtoms,Molecules,andSolidsTheenergyspectrumofanelectroninacrystaliscomposedofcontinuousbandsinsteadofthedi
4、scretelevelsfoundinfreeatoms.Aqualitativedescriptionoftheenergyspectrumofanelectronmovinginacrystallinesolidcanbeobtainedbyconsidering:•firstthespectrumofafreeatom,andthenstudy•howthisspectrumismodifiedasatomsareassembledtoformthesolid.Considerlithiumasa
5、nexample:•Inafreelithiumatom,theelectronmovesinapotentialwellasshowninFig.5.1(a).•SolutionoftheSchrødingerequationyieldsaseriesofdisretelevelsdenotedby1s,2s,2p,etc..•Thelithiumatomhasthreeelectrons:twooccupiesthe1sshell(completelyfull),andthethirdoccup
6、iesthe2ssubshell.NowwhathappenswhentwolithiumatomsassembletoformaLi2molecule?•AsillustratedinFig.5.1(b),thepotentialnowbecomesadoublewell,andthespectrumconsistsofasetofdoublets=>•Eachoftheatomiclevels(1s,2s,2p,etc.)hassplitintotwocloselyspacedlevels.Th
7、esplittingcanbeexplainedasfollows:•WhentwoLiatomsarefaraparttheinfluenceofoneatomonanelectronintheotherissufficientlysmallthatitcanbetreatedasaperturbation=>•Theunperturbedlevels(1s,2s,2p,etc.)aredoublydegeneratebecause:=>anelectronina1slevel,forinstance,
8、mayoccupythatlevelineitherelectron.However:•Thisdegeneracyisvalidonlyiftheinteractionbetweentheelectronsiscom-pletelyneglected=>•Whenthisinteractionistakenintoaccount,eachlevelissplitintotwosublevels.Sinceeachmolecularlevelcanaccomodate
