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ID:40632576
大小:1.20 MB
页数:9页
时间:2019-08-05
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1、TheHeatCapacityofaSolid6.1INTRODUCTIONTheinvestigationoftheheatcapacityofsolidsisimportantinthestudyofcondensedmatter.Thisexample,likethediatomiccase,illustratestheshortcomingsofclassicalkinetictheoryandtheneedforstatisticalmechanicsandquantumtheorytoprovideanswersthatagreewithexperimentalre
2、sults.In1819,DulongandPetitobservedthatthespecificheatcapacityat4#1#1constantvolumeofallelementarysolidsisapproximately2.49"10JkilomoleKattemperaturesnearroomtemperature.Thisresultcanbeexplainedbytheprincipleofequipartitionofenergybytreatingeveryatomofthesolidasalinearoscillatorwithsixdegree
3、soffreedomandthenassociatinganenergyof1kTwitheachdegreeoffreedom.Then2c=6(R2)=3R,inagreementwiththeobservationofDulongandPetit.vHowever,additionalmeasurementsshowedthatthespecificheatofsolidsvarieswithtemperature,decreasingtozeroasthetemperatureapproacheszero.Thisbehaviorcannotbeexplainedbyt
4、he“freezing”ofdegreesoffreedomwhenthetemperatureisdecreasedsincethespecificheatvariesgraduallywithtemperatureanddoesnotexhibitabruptjumpsbyany1multipleofR(incontrasttothespecificheatofadiatomicgas).Evenatroomtemperature2thespecificheatcapacitiesofcertainsubstancessuchasberyllium,boron,carbon
5、,andsiliconwerefoundtobemuchsmallerthan3R.Quantumstatisticsisneededtoexplainthesediscoveries.6.2EINSTEIN’STHEORYOFTHEHEATCAPACITYOFASOLIDItwasEinsteinwhodevelopedthefirstreasonablysatisfactorytheoryofthespecificheatcapacityofasolid.HeassumedthatthecrystallatticestructureofasolidcomprisingNat
6、omscanbetreatedasanassemblyof3Ndistinguishableone-dimensionaloscillators(threeoscillatorsforeachatomsincetheatomsofasolidarefreetomoveinthreedimensions).FromEquation(5.8),theinternalenergyofasolidmadeupofNatomsis1$11'U=3Nk"E&+"T),(6.1)%2eE#1(where"istheEinsteintemperaturegivenby"#h$k.Theheat
7、capacityatconstantEEvolumeis2)T#"U)&eEECV=%(=3Nk%(2(6.2)$"T'V$T'(e)ET*1)FortemperaturesverylargecomparedwiththeEinsteintemperature,C"3Nk=3nR.(6.3)VThusthehightemperaturelimitofEinstein’sequationgivesthevalueofDulongandPetit.Thefailureoftheirlawbe
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