1.Singularityindensityofstates(a)ThedensityofmodesforamonatomiclinearlatticeThedensityofmodesisConsideringthenearestneighborinteractions,wehavethedispersionrelationforamonatomiclinearlattice:(b)ThedensityofmodesforanopticalbranchSupposetatanopticalbranchhasthedispersionrelationnearK=0inthreedimension:Thedensityofmodes:For<0,thenumberofmodes:For0,thenumberofmodesN=0,thereforeD()=0.2.Rmsthermaldilationofcrystalcell(a)TherootmeansquarethermaldilationV/VForsodium,B71010erg/cm3,a4108cm.AtroomtemperatureT=300K.Thepotentialenergyofaprimitivecellassociatedwiththedilationis

(b)Therootmeansquarethermaldilationa/a3.Zeropointlatticedisplacementandstrain(4.29)showsthesquareoftheamplitudeofalatticewave:AtT=0,thesystemisinthegroundstate,n=0.thenThemeansquaredisplacementofanatomisthesumof<ui2>overallindependentlatticemodes.(a)ZeropointlatticedisplacementIntheDebyeapproximation,Then(b)OnedimensionlatticeInonedimension,D()=L/v.ThenWhence1and<R2>divergeforaone-dimensionlattice.ThemeansquarestrainInonedimension,D()=L/v,andfortheDebyeapproximation=vK.Theni.e.themeansquarestrainisfinite.4.Heatcapacityoflayerlattice(a)HeatcapacityintheDebyeapproximationfortwodimensionlatticeinlowtemperature.ConsideratwodimensionlatticewiththeareaLL.ThereisoneallowedvalueofKperareaof(2/L)2inKspace.ThetotalnumberofmodeswiththewavevectorlessthanKisIntheDebyeapproximation,=Kv.ThenThedensityofmodesThethermalenergywhere Atlowtemperature,xDThen(b)Supposeinmanylayerstructureadjacentlayersareveryweaklyboundtoeachother.Thesystemcanbeapproximatelytreatedasthisway:insidetheplanes,theatomsvibrateasa2dsystem;alongthenormaldirectionoftheplanes,theplanesvibrateasa1dsystem.Theheatcapacityisthesumover2dcontributionsand1dcontributions.ThetotalnumberofmodeswiththewavevectorlessthanKisIntheDebyeapproximation,=Kv.ThenThedensityofmodesFor1dvibrations,ThethermalenergyAtlowtemperature,xDThenForsuchamanylayerstructure,theinteractionbetweenlayersismuchsmallerthanthecouplinginsidelayer.Hencethe1dDebyetemperatureismuchlowerthanthe2done.Thusthelineartemperaturedependenceofheatcapacityholesonlyatextremelylowtemperature.Atextremelylowtemperature,theheatcapacityofsuchanmanylayerstructureis5.Gruneisenconstant(a)Thepartitionfunctionofaphononmodeoffrequencyis:Thefreeenergyofaphononmodeoffrequencyis:(b)Considerthevolumedilation.Thefreeenergyofthecrystalmaybewrittenaswhereisthefractionalvolumechange,andBisthebulkmodulus.

AssumethevolumedependenceofKis/=,whereisaconstantwhichissupposedindependentofthemodeK.ThenFisminimumwithrespecttowhenThe