固体物理(黄昆) 3－10晶格的状态方程和热膨胀

§3－10晶格的状态方程和热膨胀简谐近似：

(1)在简谐近似的情况下，晶格原子振动可描述为3N个线性独立的谐振子的叠加，各振子间不发生作用，也不交换能量；一、晶体的热膨胀和非简谐效应

(2)晶体中某种声子一旦产生，其数目就一直保持不变，既不能把能量传递给其他声子，也不能使自己处于热平衡状态。用简谐近似理论不能解释晶体的热膨胀和热传导现象。晶体的非简谐效应：微扰项声子间有相互作用能量交换系统达到热平衡

两个声子通过非简谐项的作用，而产生第三个声子。这可以看成是两个声子的相互碰撞，最后产生第三个声子。微扰项声子间的相互作用遵循能量守恒和准动量守恒碰撞前后系统准动量不变，对热流无影响。---正常过程(N过程)；(1)---反常过程(U过程)。(2)二、热膨胀

热膨胀：在不施加压力的情况下，晶体体积随温度变化的现象称为热膨胀。

假设有两个原子，一个在原点固定不动，另一个在平衡位置R0附近作振动，离开平衡位置的位移用表示，势能在平衡位置附近展开：0R0R0（1）简谐近似RU(r)R0两原子间距不变，无热膨胀现象（2）非简谐效应两原子间距增大，有热膨胀现象。由玻尔兹曼统计，原子离开平衡位置的平均位移(c、g均为正常数。)(1)简谐近似：是的奇函数在简谐近似下无热膨胀现象。(2)非简谐效应:在非简谐效应下，有热膨胀现象。推导略线膨胀系数当势能只保留到3次方项时，线膨胀系数与温度无关。若保留更高次项，则线膨胀系数与温度有关。显然，在简谐近似下，g=0，=0。三、晶体的状态方程和热膨胀

由热力学知，压强P、熵S、定容比热CV和自由能F之间的关系为：自由能F(T，V)是最基本的物理量,求出F(T，V)，其他热力学量或性质就可以由热力学关系导出。晶格自由能F1=U(V)F2由统计物理知道：Z是晶格振动的配分函数。频率为i的格波，配分函数为：由晶格振动决定T=0时晶格的结合能若能求出晶格振动的配分函数，即可求得热振动自由能。各谐子相互独立，忽略晶格之间的相互作用能，总配分函数为：

由于非线性振动，格波频率i也是宏观量V的函数，所以NeutralclusteranionclusterDopantCsIhC5VCsIhC5VCSiGeSn0.000.951.230.590.000.560.410.000.440.000.060.140.000.351.090.530.001.080.350.000.800.000.060.37StructuralandelectronicpropertiesoftheneutralandtheanionclustersonIn12X(X=C,Si,Ge,Sn)ThegeometricandelectroniccharactersofIn12X(X=C,Si,Ge,Sn)clustersinitsneutralandanionicchargestateshavebeeninvestigatedusingthedensityfunctionalcalculationswiththegeneralizedgradientapproximation.ThegroundstateofIn12X(X=C,Sn)andtheanionclustershavealowsymmetryCsstructureinsteadofanicosahedron.However,theneutralandanionclustersontheSiandGeatomdopedfavoranicosahedralstructure.FortheneutralIn12X(X=C,Si,Ge,Sn)clusters,ourcalculationdemonstratethattheenergygapsbetweenthehighestoccupiedmolecularorbitalandthelowestunoccupiedmolecularorbitalare0.6ev,1.16eV,1.15eV,0.88eVrespectively,andthemagneticmomentsarezero.However,theenergygapsforitsanionclusterare<0.3eV.Theseresultscanbeexplainedbyelectronicshellclosingof2pshell(40e)forneutralclusters,andindicatethatInatomsinclustersaretrivalent.I.INTRODUCTIONSincethediscovery1andmacroscopicproduction2offullerenes,therehavebeenextensiveintereststosearchforothersimilarstableclusters.Thesestableclustersnamed“magic”clusterscouldactascandidatesfornovelcluster-assembledmaterials.Thesphericaljelliummodelpredictsthattheclusterswith2,8,20,40,...valenceelectronshavehigherstabilityduetotheclosureofelectronicshells.3,4Forexample,theAl13having39valenceelectronsbydopingasinglemonovalentatomorbysubstitutinganAlwithatetravalentatomhas40valenceelectronsandcouldgainelectronicstabilitybyclosingthe1s21p61d102s21f142p6shell,asystematicstudyofbinaryaluminum(Al)clustershasbeenperformedbothexperimentallyandtheoretically5-11recently.DopingasingleC,Si,Ge,Sn,orPbatomintoanAl12cluster,thebinaryneutralclusterscouldgaintheelectronicstabilitywithperfecticosahedrastructureand40valenceelectronsclosingtheelectronicshell.Atthesametime,thetheoreticalstudiesfoundaddingorremovingelectronsfromtheneutralclustermightalteritsstructuralandelectronicproperties8,11.Thoughthealuminum(Al)andindium(In)atombelongtothesamegroupintheperiodictableandtheyhavetwos-valenceelectronsandonep-valenceelectron,thereareonlyafewstudiesconcerningindiumclusters.Intheirclusters,theelectronicfeatureischaracterizedonlybyonepelectroninasmallclusterbecauseanenergygapexistsbetweenthesandpbands.Astheclustersizeincreases,thesandpbandshybridizationandthecomposedatomsarecharacterizedastrivalentatoms.Foraluminumclusters,ithasbeenrevealedthatthes/phybridizationoccursaroundn~5bymeasuringtheirionizationpotential(IPs)[12,13].However,forindiumclusters,ithasbeenreportedtheInatomsaremonovalentupton~15bytheirIPsandphotoelectronspectroscopy[14].AtsushiNakajimaetal[15]foundIPsofIn7NaandIn13NaclusterswerehigherthanthoseofIn7andIn13,andtheIPincrementscouldbeexplainedbyelectronicshellclosingofthe1p(8e)and2pshell(40e),whereInatomsintheclustersweremonovalentandtrivalentrespectively.Meanwhile,theIn12Na3-wasfoundbestablegeometricallyorelectronically.Itsstabilitywasexplainedbytheelectronicshellmodelcorrespondingtothe2pshellclosing(40e),whenindiumatomsweretrivalent.ButnoelectronicclosingshellstructurewasobservedinpureInnclustersaroundn=13.TheythoughttheNaatomadditioninducess/phybridizationintheInnclusters.Howeverrecently,MinoruAkutsuetal16studiedtheelectronicpropertiesofSiandGeatomsdopedInclusters,InnSimandInnGem,byphotoionizationspectroscopyoftheneutralsandphotoelectronspectroscopyoftheanions.Theyfoundthereactivityoftheclustershaslocalminimaatn=10and12forInnSiandatn=9,10,and12forInnGeindicatingthattheIn10Si(In10Ge)andIn12Si(In12Ge)werestableelectronicallyorgeometricallyascomparedtoothersizeclusters.AndthechangingsizefrommonovalenttotrivalentforInnclustersdidnotshowobviousevidence,sotheoreticalcalculationstillwasneeded.Toourbestknowledge,thereisnostudyreportedonthegeometricalandelectronicpropertiesofthetetravalentatomdopingIn12clustersbyfar.Therefore,weperformedthedensityfunctionaltheory(DFT)calculationstoexplorethestructuralandtheelectronicpropertiesofIn12XandIn12X-(X=C,Si,Ge,Sn)clusters.II.COMPUTATIONALDETAILSAllthecalculationshavebeenperformedbasingupontheall-electrondensityfunctiontheory(DFT)17withthegeneralizedgradientapproximation(GGA)

(Ref.18)usingtheDMol3package19.TheBecke-Lee-Yang-Parr(BLYP)correlationexchangefunctionisadoptedwithDNPbasisset.BLYPfunctionisthecombinationoftheexchangefunctionexploitedbyBecke20andthecorrelationfunctionexploitedbyLee,Yang,andParr21.DNPbasisfunctions,comparabletoGaussian6-31G**sets,arethedoublenumericalatomicorbitalaugumentedbypolarizationfunctions:i.e.,functionswithangularmomentumonehigherthanthatofthehighestoccupiedorbitalinthefreeatom.TheelectronicstructureisobtainedbysolvingtheKohn-Sham(KS)equations22self-consistentlyusingthespinunrestrictedscheme.Self-consistentfieldproceduresweredonewithaconvergencecriterionof10−6a.u.ontheenergyandelectrondensity.GeometryoptimizationswereperformedusingtheBroyden-Fletcher-Goldfarb-Shanno(BFGS)algorithm23withaconvergencecriterionof10-3.onthedisplacementand10-5.ontheenergy.III.RESULTSANDDISCUSSIONSOneofthemostimportantthingsinstudyingclustersistodetermineitsequilibriumgeometry.FortheneutralandanionIn12X(X=C,Si,Ge,Sn)clusters,threepossiblegeometricconfigurationswithIh,C5v,Cssymmetryshowninfig.1areconsidered.ThethreeconfigurationsarethefavorablegeometriesoftheneutralandthechargedAl12X(X=C,Si,Ge,Sn,Pb)clusters.Theicosahedronisthetetravalentatomsittingatthecenter,whiletheC5vstructureisthetetravalentatomadsorbedonthevertex.Intable.1,wetabulatedtherelativevalueofthecalculatedbindingenergiesforthestructuresasshowninFig.1intheneutralandanionstates.ForIn12X(X=C,Sn)anditsanions,wefoundtheCsgeometryisground-statestructures,whiletheicosahedrongeometryforIn12Snanditsanionisnearlyenergeticallydegenerate(ΔE=0.06eV).ThestructurewithIhsymmetrywaslowest-energystructureofAl12Snchargedcluster8,10.Meanwhile,ourcalculationshowsthattheicosahedraconfigurationisthegroundstatesofIn12X(X=Si,Ge)anditsanionclusters.ThoughthedifferenceofthebindingenergybetweentheIhandCsstructurefortheIn12Snanditsanionclusterisverylitter,theCsgeometryhasthehigherbindingenergy(0.35eV~0.60eV)fortheSiandGeatomdopingIn12clusterseitherintheneutralorintheanionsthantheIhgeometry.TheaverageIn-SnandIn-InbondlengthsintheCsstructureforIn12Snare3.507Åand3.295ÅÅÅforIn12Sn-respectively.WhiletheaverageIn-CandIn-InbondlengthsintheCsstructureforIn12Care2.464Åand3.315ÅÅÅforIn12C-.Fortheseclusters,therearetwelveX(Si,Ge,Sn)-InbondsintheIhstructures,whiletheCsstructureshaveonlyfiveX-Inbonds.ThesingleSiandGeatomssitatthecenteroftheicosahedron.IntheseclusterswithIhÅÅforIn12SiandIn12ÅÅÅÅforIn12SiandIn12ÅÅforitsanion.FromtheSitoGe,thesizeofclustersisincreasingfortheIn12SiandIn12Geclusters,whichmayattributetothesizeofatomdopedsittingatthecenteroftheclusters,andsuggestsalsothebondingstrengthisweakerandweaker.Andthesizeoftheclustersforneutralclusterissmallerthanitsanion,whichindicatesthenetchargehasanobviouseffectonthestructureofSiandGedopingIn12clusters.InFig.2,Weshowthedensityofstates(DOS)ofthegroundstatesstructures.Thedensityofstatesiscalculatedbybroadeningthebandenergyofeachorbitalwiththeline-shapeLorentzianfunction.TheDOSforthegroundstatesofneutralclustersandtheiranionclustersareshownforadirectcomparison.Fromtheresults,wecouldseethatthegroundstatesstructurewiththesamesymmetryhadthesimilarstatesdensity(DOS).Forexample,theIn12SiandIn12GeclusterswithIhsymmetry,thelocationsandwidthsoftheirpeaksarealmostsame.FortheIn12CandIn12SnwiththesameCssymmetry,theirDOSarealmostsimilar,exceptthatthepeakwidthoftheIn12CislargerthantheclusterdopedSn,whichmayattributetothelocationoftheatomsdopedinPeriodicTable.AsonemovedowntothePeriodicTable,theelementbecomesincreasinglymoremetallicandlesselectronegative,andthesizeislargemoreandmore.ComparingtheDOSoftheanionclusterswiththeirneutralclusters’,wecouldseetheenergylevelsofelectroncouldbealteredbyaddinganelectron.Forexample,thefirsthighpeakfromtheleft,itislocalizedataround-3.8eVintheneutralstateforIn12SiandIn12GehavingthesameIhstructure.However,itisshiftedtoaround-4.4eVintheanionclusters.Takingawayanelectronincreasestheenergylevelofthevalenceelectron.XiLietal10hasputforwardthattheelectronegativityandthesizeofatomsbetwoimportantfactorsinthestructureevolutionfortheAl12X-speciesfromX=CtoPb.TheythoughtthattheAl12C-favoredtheCsstructurebecauseofthehighelectronegativity,andtheCsstructurewasthegroundstateofAl12Snforitslargesize.Inourcomputation,thegroundstatesofIn12CandIn12SnfavoringCsstructuresatisfyapparentlythereason.Thetotalelectrondensityoftheneutralclustersisshowninfig.3.WecouldseetheelectrondistributionofCinIn12CislitterthanotherIndiumcluster,whilethedifferenceexistshardlyinIn12Sn.InthePeriodicTable,theSnantInatomisproximateandthustheirelectronegativityaresamenearly.ButtheelectronegativityofCatomishigherthanthem,whichresultsinthedifferenceofthetotalelectrondensity.Infig.3,weshowalsothetotalelectrondensityoftheneutralIn12SiandIn12GeclusterswithsameIhstructure.TheelectrondistributionofInatomsinIn12SiandIn12Geclustersissameduetotheirhighsymmetry,WealsocarriedouttheHOMO-LUMObandgap,andfoundtheneutralclusterduetotheclosedelectronicshellpredictedbythejelliummodelhadalargerbandgapthantheiranionclusters.Theyare0.6,1.16,1.15,0.88eVforC,Si,Ge,Sndopedclusters,respectively.Fromtheseresults,wecouldseethestructuraleffectsonthebandgap.ThestablestructuresforIn12SiandIn12Gearetheicosahedron,andtheirbandgapsarearound1.15eV.However,thebandgapoftheCsstructuresthatarethegroundstatesofIn12CandIn12Sn,aresmallerthantheclusterswithIhstructure,whichindicatestheIhstructurewithhighsymmetrywillbekineticallymorestablethantheCsstructure.Comparingtheneutralclusters,thebandgapoftheiranionisallsmaller(<0.3eV).Sotheneutralclustersareverystablekinetically,andconsistentwithcompletelythejelliummodelfor40electrons.AndtheHOMO-LUMOgapsoftheneutralIn12X(X=Si,Ge)havingclosedelectronicstructureswereabout0.8eVreportedbyMinoruAkutsu20.Forhavingclosedelectronicstructure,wealsoobtainfromthespinsthatarezeroforallneutralclusters.However,thespinoftheIn12clusteris2,whichindicateitstwoelectrons(2p2)areunpaired.DuetothespinoftheneutralclustersofIn12X(X=C,Si,Ge,Sn),theyshouldhavetheclosed1S21p61d102S21f142P6electronicshell.SowesuggestthattheIndiumatombetrivalentintheIn12X,whichisconsistentwiththeresultsobtained

byAtsushiNakajimaetal23.ConclusionsWehaveperformedadensityfunctionalcalculationsonthestructuresandelectronicpropertiesofIn12X(X=C,Si,Ge,Sn)anditsanionclusterswiththegeneralizedgradientapproximation.TheCsstructurehasahighestbindingenergyfortheIn12X(X=C,Sn)andtheiranionclustersinsteadofanicosahedron.However,thehighestbindingenergystructuresaretheicosahedronfortheIn12X(X=Si,Ge)andtheiranionclusters.Foralltheneutralandanionclusters,theC5vstructuresoptimizedfromanicosahedronwithavertexcappedbyatetravalentatomhavethelowestbindingenergy.Thedifferenceofthebindingenergyisonly0.06eVbetweentheIhstructureandtheCsstructureforIn12Snanditsanion.FromtheDosoftheneutralandtheanionclusters,theadditionalchargeandthestructureaffecttheelectronicpropertiesoftheseclusters.Duetotheclosedelectronicshellfortheneutralcluster,thebandgapsbetweenthehighestoccupiedmolecularorbitalandthelowestunoccupiedmolecularorbitalare0.6ev,1.16eV,1.15eV,0.88eVfortheIn12X(X=C,Si,Ge,Sn)respectively.However,thebandgapsfortheanionclusterare<0.3eV.Moreover,wefindthespinsoftheneutralclustersarezero,whichfavorsmuchmorethe2pshellclosing.TheresultofthespinsandthebandgapsshowstheIndiumatominneutralclustersistrivalent.Fig.1TheschematicfiguresforthestateconsideredofIn12X(X=C,Si,Ge,Sn)anditsanionclusters.ThetetravalentatomsitsatthecenterofIhstructure,whiletheC5vstructuresarethetetravalentatomadsorbedonthevertex.ThetetravalentatomontheoutsideofIn12clustersformsCsstructure.Table1.ThecalculatedbindingenergiesineVforIn12X(X=C,Si,Ge,Sn)andtheiranionclusters.Fig.2ThedensityofstatesofthegroundstructuresforIn12XandIn12X-(X=C,Sn,Si,Ge)clusters.The(a)isfortheneutralclusters;the(b)isforthenegativelychargedclusters.ThedensityiscalculatedbybroadendingthebandenergyoftheoccupiedorbitalandtheunoccupiedorbitalseparatelywiththeLorentzianfunction.(a)(b)Fig.3theelectrontotaldensityof(a)theneutralIn12CandIn12SnclusterswithsameCsstructure,(b)theelectrontotaldensityoftheneutralIn12SiandIn12GeclusterswithsameIhstructure.TheatomsdopedarelocatedattheleftsideoftheCsstructures.TheSiandGeatomsarelocatedthecenterofthecage.ReferenceH.W.Kroto,J.R.Heath,S.C.O’Brien,R.F.Curl,andR.E,Smalley,Nature(London)318,162(1985).W.Kratschmer,L.D.Lamb,K.Fostiropoulos,andD.R.Huffman,Nature(London)347,354(1990).3.W.D.Knight,K.Clemenger,W.A.deHeer,W.A.Saunders,M.Y.Chou,andM.L.Cohen,Phys.Rev.Lett.52,2141(1984).W.A.deHeer,Rev.Mod.Phys.65,611(1993).,L.,Phys.Rev.B2002,65,153404.H.Kawamata,;Y.Negishi,;A.Nakajima,;,Chem.Phys.Lett.2001,337,255.D.E.Bergeron,A.W.Castleman,N.,S.N.Kahanna,Chem.Phys.Lett.2005,415,230.G.ChenandY.Kawazoe,J.Chem.Phys.2007,126,014703.K.H.YoungandJ.Jaehoon,J.Chem.Phys.2004,121(17),8500.X.Li,andL.S.Wang,Phys.Rev.B65,153404(2002).S.F.LiandX.G.Gong,Phys.Rev.B70,075404(2004).12.K.E.Schriver,,E.C.Honea,R.L.Whetten,.64,1990,2539.13.J.L.Persson,,H.P.Cheng,R.S.Berry,Chem.Phys.Lett.186,1991,215.14.M.Gausa,G.GantefÖr,H.O.Lutz,K.H.Meiwes-Broer,Int.J.Massspectrum.IonProcesses,102,227,1990.15.N.Atsushi,H.Kuniyoshi,S.Tsuneyoshi,N.Takashi,T.Tetsuya,etal,J.Phys.Chem.1993,97,86-90.16.A.Minoru,K.Kiichirou,A.Junko,M.Ken,M.Masaaki,andN.Atsushi,J.Phys.Chem.A2007,111,573-577.17.,D.Sánch