固体物理英文版课件：Chapter 3 Crystal Binding and Elastic Constants

Chapter3CrystalBindingand ElasticConstantsKeypoints:

crystallinebindingvandeWaalsinteraction

MadelungenergycovalentbondmetalbondWhatholdsacrystaltogether?Theattractiveelectrostaticeinteractionbetweenthenegativechargesoftheelectronsandthepositivechargesofthenucleiisentirelyresponsibleforthecohesionofsolid.vandeWaalsionicmetalliccovalentThecohesiveenergy

内聚能:

theenergythatmustbeaddedtothecrystaltoseparateitscomponentsintoneutralfreeatomsatrest,atinfiniteseparation,withthesameelectronicconfiguration.Thelatticeenergy

晶格能

(forioniccrystals):theenergythatmustbeaddedtothecrystaltoseparateitscomponentionsintofreeionsatrestatinfiniteseparation.Generally,thehigherthemeltingtemperature,thehigherthecohesiveenergy,andalsothebiggerbulkmodulus(体弹性模量).Crystalsofinertgases

transparentisolatorweaklyboundlowmeltingtemperaturesmallcohesiveenergyandhighionizationenergycompletelyfilledoutermostelectronshell

fccstructurevandeWaals-LondoninteractionvandeWaals-LondoninteractionTheatom-induceddipolemomentscauseanattractiveinteractionbetweentheatoms.Model:1.considertwoinertgasatomsaslinearharmonicoscillatorstheuncoupledHamiltonian2.considerthesetwoinertgasatomsastwodipolesTheinteractedHamiltonianThetotalHamiltonian:H=H0+H1Diagonalizedbythenormalmodetransformation:thenandsymmetricmodeantisymmetricmodeThefrequency:Theenergy:attractiveinteractionquantumeffectdoesnodependonanyoverlapofthechargedensitiesoftwoatomsRepulsiveinteractionPauliexclusionprinciple:twoelectronscannothavealltheirquantumnumbersequal.Lennard-Jonespotential:DifferentapproximationsofrepulsivepotentialEquilibriumLatticeconstantsneglectthekineticenergytakeLennard-JonespotentialThetotalpotentialenergy:ForfccstructureForhcpstructureTheequilibriumvalueofRisgivenbytheminimumofUtotwhencesameforallelementswithfcc(orhcp)structureCohesiveenergyThecohesiveenergyofinertgascrystalatabsolutezeroandzeropressure:quantumcorrectionThequantumkineticenergyIoniccrystalsIoniccrystalsaremadeupofpositiveandnegativeions.Theionicbondresultsfromtheelectrostaticinteractionofoppositelychargedions.Thetypicalioniccrystals:thesodiumchlorideandthecesiumchloridestructures.Thechargedistributionshaveapproximatelysphericallysymmetric.Electronionizationenergy,electronaffinityenergyandcohesiveenergy(latticeenergy)ElectrostaticorMadelungenergyelectrostaticinteraction q2/r(Long-range)Pauliexclusionprinciple repulsivepotentialexp(-r/) (short-range)NeglectvandeWaalsinteractionTheinteractionbetweeniandjionsMadelungenergy马德隆能ThetotallatticeenergyforNmolecules(2Nions)zisthenumberofthenearestneighborsofanyion.TheMadelungconstantEquilibriumstatethenThetotallatticeenergyTheelectrostatic(Madelung)energyNote:isoftheorderof0.1R0TheMadelungconstantorbetweenbothpositiveornegativechargesbetweenpositivechargeandnegativecharge+Example:1dionslineEwaldmethodforlatticesumcalculations2dsquaredlatticestructureNaClCsClcubicZnS1.7481.7631.6383dlatticeCovalentcrystalsCovalentbond:electronpairorhomopolarbondstrongbondstrongdirectionalpropertiesusuallyformedfromtwoelectrons,onefromeachatomparticipatinginthebond.theelectronsformingthebondtendtobepartlylocalizedintheregionbetweentwoatomsjoinedbythebond.spinsoftheelectroninthebondareantiparallelabR12ra1rb2ra2rb1r12ForthemolecularhydrogenH2Thewavefunctions(notincludingspins)areSymmetricmodeAntisymmetricmodeThewavefunctionstwospinsareThewavefunctionsfortwoelectronsare:andwhereexchangeinteraction(spin-dependentcoulombenergy)PauliprincipleTheelectronconfigurationofC,SiandGetoformcovalentbonds: ns2np2

n(sp)4ContinuousrangeofcrystalsbetweentheionicandthecovalentlimitsMetals

conductionelectronsloweringoftheenergyofthevalenceelectronsinmetalascomparedwiththefreeatominteractionoftheioncorewiththeconductionelectronsweekbondcomparedwithionicbondadditionalbindingfrominnerelectronshellintransitionmetalstendtocrystallizeinrelativelyclosepackedstructure

hcp,fcc,bccetc.HydrogenbondsThehydrogenbondislargelyionicincharacter,beingformedonlybetweenthemostelectronegativeatoms(particularlyF,O,andN)andhydrogen.AtomicradiiDistancesbetweenatomsincrystalcanbemeasuredveryaccuratelybyX-raydiffraction.Norigidsphericalboundaryforanatomself-consistentradiiIoniccrystalradiiAnalysisofelasticstrainsElasticpropertiesviewasahomogeneouscontinuousmediumratherthanasaperiodicarrayofatomsHooke’slaw:strain(应变)stress(应力)nonlinearregion:thestrainissolargethatHooke’slawisnolongersatisfiedSupposetwosetsofaxesfixedorthogonalaxes(beforedeformation)newaxesafterauniformdeformationthenAninfinitesimalstrainbeforedeformationafterdeformationforthedeformationonanatomthedisplacementExpandwithTaylorseriesatR(0)=0WedefinethestraincomponentsDilationThefractionalincreaseofvolumeassociatedwithadeformationiscalledthedilation.ThedilationisgivenbyStresscomponentsTheforceactiononaunitareainasolidisdefinedasthestressninestresscomponents:Xx,Xy,Xz,Yx,Yy,Yz,Zx,Zy,ZzCapitalletterdirectionoftheforcesubscriptnormaltotheplanetowhichtheforceisappliedsixindependentstresscomponentsinthestaticsituationXx,Yy,Zz,Yz,Zx,Xzthedimensionsstresscomponents [force]/[area]or[energy]/[volume]thestraincomponentsaredimensionless.ElasticcomplianceandstiffnessconstantsHooke’slawconverseSij[area]/[force]elasticcomplianceconstant(弹性顺度常量)orelasticconstant(弹性常量)Cij[force]/[area]elasticstiffnessconstant(弹性劲度常量)ormoduliofelasticity(弹性模量)S=C-1Elasticenergydensitywhere1xx,1yy,3zz,4

yz,5

zx,6xy

thenSimilarly,wehaveothercomponents.Theelasticstiffnessconstantsaresymmetrical:21independentelasticstiffnessconstantsElasticstiffnessconstantsofcubiccrystalsthereareonlythreeindependentstiffnessconstantsincubiccrystalduetothesymmetry.C11,C12andC44