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计算材料学第三章原子间相互作用势赵纪军三束实验室，物理学院&高科技研究院Email:zhaojj@dlut.edu.cn，电话：847097483-2.金属键合的经验势描述计算材料学第三章•Themainphysicalpointtomodelisthatbondsbecomeweakerwhenthelocalenvironmentbecomesmorecrowded(consequenceofthePauliprinciple).•ThepotentialsmustbeabletoreproducetheenergydifferencebetweenFCC,HCPandBCC.描述金属键合的基本要求镶嵌原子方法(embeddedatommethod,EAM)EAM的物理图像镶嵌原子方法的物理思想起源于Friedel提出的原子嵌入能概念，即：原子的凝聚能主要取决于该原子所占据位置的局域电子密度。后来也和密度范函理论建立了联系。ionionionionionionionionionionElectrondensityρEAM框架下内聚能的一般形式镶嵌原子势的内聚能表述电子密度和镶嵌函数0.000.010.022.03.04.05.06.0R(0.1nm)ρ(R)(103e/nm3)Ag←Cu→Au↓(a)0.000.010.022.03.04.05.06.0R(0.1nm)ρ(R)(103e/nm3)Ni→Pd↑Pt←(b)原子的电子密度函数“Training”ofparameters拟合EAM参数采用的典型实验数据EAM在金属中的成功应用基于镶嵌原子思想的各种经验势•Effectivemediumtheory(EMT):Phys.Rev.B26,2875(1982).•EAMpotential:Phys.Rev.Lett.50,1285(1983).•Finnis-Sinclair(FS)potential:Philos.Mag.A50,45(1984).•Gluepotential:Phys.Rev.Lett.57,719(1986).•Sutton-Chen(SC)potential:Philos.Mag.Lett.61,139(1990).•Tight-binding(TB)potential:Phys.Rev.B48,22(1993).基于镶嵌原子的思想,先后发展出了描述原子间相互作用势的有效介质理论(EMT)、镶嵌原子方法(EAM)、紧束缚势(TB)、二级矩近似(SMA)和胶体模型(Gluemodel)等。它们的区别在于镶嵌能的非线性表述不同或是镶嵌密度的测度不一样。¾EAM和SC势比较适用于没有成键取向结构的密堆金属；¾TB和FS势更适用于具有体心立方结构的过渡金属；¾EMT势的势函数计算往往过于复杂。•Energydifferencesareeasiertocalculatethanenergies.Chooseareferencesystemwithaknownenergy,andconcentrateontheenergydifference.Originalideal:J.K.Nørskov,Phys.Rev.B26,2875(1982).Effectivemediumtheory(EMT)•Referencesystem:anatominaFCCcrystal.Thelatticeconstantofthereferencecrystalshouldmatchthedensityoftheactualsystem.Useameasureforthelocalelectrondensityasthemap.Eref:theenergyofthereferencesystem;EASisthecorrection.Effectivemediumtheory(EMT)•Theenergyofanatom(andthusitsforce)isafunctionofthepositionoftheatomsinalocalneighbourhood.•Lessonsfromquantummechanicsarebuiltintothefunctionalform.K.W.Jacobsen,M.PuskaandJ.K.Nørskov,Surf.Sci.366,394(1996);Phys.Rev.B35,7423(1987).•Bluelines:Electrondensityfromblueatoms.•Greenline:Sumofcontributionsfromblueatoms.Thisistheembeddingdensityofredatom.Effectivemediumtheory(EMT)Effectivemediumtheory(EMT)Effectivemediumtheory(EMT)PhonondispersionrelationforAl.Lines:EMT,dots:experimentPhilos.Mag.A50,45(1984).Finnis-SinclairpotentialTheFSpotentialincorporatesthebandcharacterofmetalliccohesionandallowstoobtaincorrectvaluesforthevacancyformationandcohesiveenergy.Itreproducesthelatticeconstants,bulkmodulusandsurfaceenergyaccurately.Gluepotential~ercolessi/potentials/In1983-1986,Ercolessietal.developedthegluemodel,aformulationcontainingadensity-dependentmany-bodyterminadditiontousualtwo-bodyinteractions.Thistermallowstomimicthegluingcharacterofthecohesionduetoconductionelectronsinmetals:ionshavealowenergyaslongastheyareimmersedintheelectronsea,whiletheexactpositionofneighbouringionsisrelativelyunimportant.ThisledustowriteaHamiltonianwhereashort-rangeddensityfunctionisattachedtoatoms,sothatforeachatominthesystemwecancomputeaneffectivecoordinationdefinedasthesumofallthedensitycontributionscomingfromneighboringatoms.Theenergyofthisatomwillthendependnon-linearlyuponthiseffectivecoordination.Thenon-linearityoftheenergydependenceuponcoordinationessentiallymodelsthisphysicalfact,ultimatelyaconsequenceofPauli'sprinciple:thestrengthofindividualbondsdecreasesasthelocalenvironmentbecomesmorecrowded.Incontrast,withtwo-bodypotentialsthestrengthofindividualbondsdoesnotdependontheenvironment.Thisfeatureiscrucialtocapturethephysicsofbondinginmetals.F.Ercolessi,M.ParrinelloandE.Tosatti,Philos.Mag.A58,213(1988)GluepotentialforAuGluepotentialforAuGluepotentialforAuLatticeparameterofAuasfunctionoftemperature.Line:MDsimulationusinggluepotential;Dots:experimentGluepotentialforAuPhonondispersionofAu.Lines:gluepotential;dots:experimentsForcematchingmethod~ercolessi/forcematching.htmlSutton-ChenpotentialTheSutton-Chenpotentialprovidesareasonabledescriptionofvariousbulkproperties,withanapproximatemany-bodyrepresentationofthedelocalizedmetallicbonding.However,itdoesnotincludeanydirectionalterms,whicharelikelytobeimportantfortransitionmetalswithpartiallyoccupieddshells.Philos.Mag.Lett.61,139(1990).•C:adimensionlessparameter•ε:aparameterwithdimensionsofenergy,•α:latticeconstant,•m,n:positiveintegerswithnmTight-bindingpotentialFriedelmodelofdbandEF=W(Z-5)/10R.P.Gupta,Phys.Rev.B23,6265(1981).•Cohesiveenergyduetodelectron•BandwidthWcomputedfrom2rdmomentofdensityofstatefordband•μ2canbecalculatedfromsumoverintersitehoppingintegral•RepulsiveenergyraisedfromselectronTotalenergyTight-bindingpotentialF.Cleri,V.Rosato,Phys.Rev.B48,22(1993).Tight-bindingpotentialFCC-CuHCP-Co陈难先的逆问题方法陈难先的逆问题方法Q.Xie,N.X.Chen,Phys.Rev.B51,15856(1995).EAM的局限性EAM的修正－MEAMMEAM可以用于合金、共价键合等金属中常用的EAM势总结•“Sandia”(NationalLab.)EAMbyDaw,Baskes,Foiles:[S.M.Foiles,Phys.Rev.B32,3409(1985)];[S.M.Foiles,M.I.Baskes,andM.S.Daw,Phys.Rev.B33,7983(1986)];[M.S.Daw,S.M.Foiles,andM.I.Baskes,Mat.Sci.Rep.9,251-310(1993)]–themostpopularpotential,provideagooddescriptionofAu,Cu,Pd,Ag,Pt,Ni,Alaswellassomeofthealloys.Thefunctionsdonothaveananalyticalform,aregivenasatableofd