IntrToSolidStatePhysicsSolutionKittel

IntroductiontoSolidStatePhysics,8thEditionCharlesKittelCHAPTER1:CRYSTALSTRUCTURE.PeriodicArrayofAtoms.FundamentalTypesofLattices.IndexSystemforCrystalPlanes.SimpleCrystalStructures.DirectImagingofAtomicStructure.NonidealCrystalStructures.CrystalStructureData.CHAPTER2:WAVEDIFFRACTIONANDTHERECIPROCALLATTICE.DiffractionofWavesbyCrystals.ScatteredWaveAmplitude.BrillouinZones.FourierAnalysisoftheBasis.CHAPTER3:CRYSTALBINDINGANDELASTICCONSTANTS.CrystalsofInertGases.IonicCrystals.CovalentCrystals.Metals.HydrogenBonds.AtomicRadii.AnalysisofElasticStrains.ElasticComplianceandStiffnessConstants.ElasticWavesinCubicCrystals.CHAPTER4:PHONONSI.CRYSTALVIBRATIONS.VibrationsofCrystalswithMonatomicBasis.TwoAtomsperPrimitiveBasis.QuantizationofElasticWaves.PhononMomentum.InelasticScatteringbyPhonons.CHAPTER5:PHONONSII.THERMALPROPERTIES.PhononHeatCapacity.AnharmonicCrystalInteractions.ThermalConductivity.CHAPTER6:FREEELECTRONFERMIGAS.EnergyLevelsinOneDimension.EffectofTemperatureontheFermiDiracDistribution.FreeElectronGasinThreeDimensions.HeatCapacityoftheElectronGas.ElectricalConductivityandOhm’sLaw.MotioninMagneticFields.ThermalConductivityofMetals.CHAPTER7:ENERGYBANDS.NearlyFreeElectronModel.BlochFunctions.KronigPenneyModel.WaveEquationofElectroninaPeriodicPotential.NumberofOrbitalsinaBand.CHAPTER8:SEMICONDUCTORCRYSTALS.BandGap.EquationsofMotion.IntrinsicCarrierConcentration.ImpurityConductivity.ThermoelectricEffects.Semimetals.Superlattices.CHAPTER9:FERMISURFACESANDMETALS.ConstructionofFermiSurfaces.ElectronOrbits,HoleOrbits,andOpenOrbits.CalculationofEnergyBands.ExperimentalMethodsinFermiSurfaceStudies.CHAPTER10:SUPERCONDUCTIVITY.ExperimentalSurvey.TheoreticalSurvey.HighTemperatureSuperconductors.CHAPTER11:DIAMAGNETISMANDPARAMAGNETISM.LangevinDiamagnetismEquation.QuantumTheoryofDiamagnetismofMononuclearSystems.Paramagnetism.QuantumTheoryofParamagnetism.CoolingbyIsentropicDemagnetization.ParamagneticSusceptibilityofConductionElectrons.CHAPTER12:FERROMAGNETISMANDANTIFERROMAGNETISM.FerromagneticOrder.Magnons.NeutronMagneticScattering.FerrimagneticOrder.AntiferromagneticOrder.FerromagneticDomains.SingleDomainParticles.CHAPTER13:MAGNETICRESONANCE.NuclearMagneticResonance.LineWidth.HyperfineSplitting.NuclearQuadrupoleResonance.FerromagneticResonance.AntiferromagneticResonance.ElectronParamagneticResonance.PrincipleofMaserAction.CHAPTER14:PLASMONS,POLARITONS,ANDPOLARONS.DielectricFunctionoftheElectronGas.Plasmons.ElectrostaticScreening.Polaritons.ElectronElectronInteraction.ElectronPhononInteraction:Polarons.PeierlsInstabilityofLinearMetals.CHAPTER15:OPTICALPROCESSESANDEXCITONS.OpticalReflectance.Excitons.RamanEffectsinCrystals.EnergyLossofFastParticlesinaSolid.CHAPTER16:DIELECTRICSANDFERROELECTRICS.MacroscopicElectricField.LocalElectricFieldatanAtom.DielectricConstantandPolarizability.StructuralPhaseTransitions.FerroelectricCrystals.DisplaciveTransitions.CHAPTER17:SURFACEANDINTERFACEPHYSICS.SurfaceCrystallography.SurfaceElectronicStructure.MagnetoresistanceinaTwoDimensionalChannel.pnJunctions.Heterostructures.SemiconductorLasers.LightEmittingDiodes.CHAPTER18:NANOSTRUCTURES.ImagingTechniquesforNanostructures.ElectronicStructureof1DSystems.ElectricalTransportin1D.ElectronicStructureof0DSystems.ElectricalTransportin0D.VibrationalandThermalPropertiesofNanostructures.CHAPTER19:NONCRYSTALLINESOLIDS.DiffractionPattern.Glasses.AmorphousFerromagnets.AmorphousSemiconductors.LowEnergyExcitationsinAmorphousSolids.FiberOptics.CHAPTER20:POINTDEFECTS.LatticeVacancies.Diffusion.ColorCenters.CHAPTER21:DISLOCATIONS.ShearStrengthofSingleCrystals.Dislocations.StrengthofAlloys.DislocationsandCrystalGrowth.HardnessofMaterials.CHAPTER22:ALLOYS.GeneralConsideration.SubstitutionalSolidSolutions–HumeRotherbyRules.OrderDisorderTransformation.PhaseDiagrams.TransitionMetalAlloys.KondoEffect.CHAPTER11.Thevectorsˆˆˆ++xyzandˆˆˆ−−+xyzareinthedirectionsoftwobodydiagonalsofacube.Ifθistheanglebetweenthem,theirscalarproductgivescosθ=–13,whence.1cos13901928'10928'−θ==°+°=°2.Theplane(100)isnormaltothexaxis.Itinterceptsthea'axisatandthec'axisat;thereforetheindicesreferredtotheprimitiveaxesare(101).Similarly,theplane(001)willhaveindices(011)whenreferredtoprimitiveaxes.2a'2c'3.Thecentraldotofthefourisatdistancecos60actn60cos303aa°=°=°fromeachoftheotherthreedots,asprojectedontothebasalplane.Ifthe(unprojected)dotsareatthecenterofspheresincontact,then222aca,23⎛⎞⎛⎞=+⎜⎟⎜⎟⎝⎠⎝⎠or2221c8ac;1.633.34a3==11CHAPTER21.ThecrystalplanewithMillerindiceshkisaplanedefinedbythepointsaA1h,a2k,and.(a)Twovectorsthatlieintheplanemaybetakenasa3Aa1h–a2kand13h−Aaa.Buteachofthesevectorsgiveszeroasitsscalarproductwith12hk3=++AGaaa,sothatGmustbeperpendiculartotheplane.(b)Ifistheunitnormaltotheplane,theinterplanarspacingishkAnˆ1ˆh⋅na.But,whence.(c)Forasimplecubiclatticeˆ||=nGG1d(hk)Gh||2|G|=⋅=πAaGˆˆˆ(2a)(hk)=π++AGxyz,whence22222221Ghk.d4a++==πA123113aa022112.(a)Cellvolume3aa02200⋅×=−aaac213ac.2=231212323ˆˆ411(b)23aa0||223ac0021ˆˆ(),andsimilarlyfor,.a3×π=π=−⋅×π=+xˆcyzaabaaaxybb(c)Sixvectorsinthereciprocallatticeareshownassolidlines.Thebrokenlinesaretheperpendicularbisectorsatthemidpoints.TheinscribedhexagonformsthefirstBrillouinZone.3.Bydefinitionoftheprimitivereciprocallatticevectors3323311212331233C(aa)(aa)(aa))(2)|(aaa)||(aaa)|V.BZV(2(2)×⋅×××=π⋅×⋅×=π=πForthevectoridentity,seeG.A.KornandT.M.Korn,Mathematicalhandbookforscientistsandengineers,McGrawHill,1961,p.147.4.(a)Thisfollowsbyforming2122122121exp[iM(ak)]1exp[iM(ak)]|F|1exp[i(ak)]1exp[i(ak)]sinM(ak)1cosM(ak).1cos(ak)sin(ak)−−⋅∆−⋅∆=⋅−−⋅∆−⋅∆⋅∆−⋅∆==−⋅∆⋅∆(b)Thefirstzeroin1sinM2εoccursforε=2πM.Thatthisisthecorrectconsiderationfollowsfrom1zero,asMhisaninteger11sinM(h)sinMhcosMcosMhsinM.22±π+ε=πε+πε�� ��� �125.j1j2j32i(xv+yv+zv)123S(vvv)fej−π=ΣReferredtoanfcclattice,thebasisofdiamondis111000;.444Thusintheproduct123S(vvv)S(fcclattice)S(basis)=×,wetakethelatticestructurefactorfrom(48),andforthebasis1231i(vvv).2S(basis)1e−π++=+NowS(fcc)=0onlyifallindicesareevenorallindicesareodd.Ifallindicesareeventhestructurefactorofthebasisvanishesunlessv1+v2+v3=4n,wherenisaninteger.Forexample,forthereflection(222)wehaveS(basis)=1+e–i3π=0,andthisreflectionisforbidden.321G0033003232200022206.f4r(aGr)sinGrexp(2ra)dr(4Ga)dxxsinxexp(2xGa)(4Ga)(4Ga)(1rGa)16(4Ga).−=ππ−=−=++∫∫0Theintegralisnotdifficult;itisgivenasDwight860.81.Observethatf=1forG=0andf1G4for0Ga1.>>7.(a)ThebasishasoneatomAattheoriginandoneatom1Bata.2ThesingleLaueequationdefinesasetofparallelplanesinFourierspace.Intersectionswithasphereareasetofcircles,sothatthediffractedbeamslieonasetofcones.(b)S(n)=f2(integer)⋅∆π×ak=A+fBe–iπn.Fornodd,S=fA–22fB;forneven,S=fA+fB.(c)IffA=fBtheatomsdiffractidentically,asiftheprimitivetranslationvectorwere1a2andthediffractioncondition1()2(integer).2⋅∆=π×ak23CHAPTER31.2222E(h2M)(2)(h2M)(L),with2L=πλ=πλ.=2.bcc:126U(R)2N[9.114(R)12.253(R)].=εσ−σAtequilibriumand660R1.488=σ,0U(R)2N(2.816).=ε−fcc:126U(R)2N[12.132(R)14.454(R)].=εσ−σAtequilibriumandThusthecohesiveenergyratiobccfcc=0.956,sothatthefccstructureismorestablethanthebcc.660R1.679=σ,0U(R)2N(4.305).=ε−231693.|U|8.60N(8.60)(6.0210)(5010)25.910ergmol2.59kJmol.−=ε=××=×=Thiswillbedecreasedsignificantlybyquantumcorrections,sothatitisquitereasonabletofindthesamemeltingpointsforH2andNe.4.WehaveNaNa++e–5.14eV;Na+eNa–+0.78eV.TheMadelungenergyintheNaClstructure,withNa+attheNa+sitesandNa–attheCl–sites,is2102128e(1.75)(4.8010)11.010erg,R3.6610−−−α×==××or6.89eV.HereRistakenasthevalueformetallicNa.ThetotalcohesiveenergyofaNa+Na–pairinthehypotheticalcrystalis2.52eVreferredtotwoseparatedNaatoms,or1.26eVperatom.Thisislargerthantheobservedcohesiveenergy1.13eVofthemetal.WehaveneglectedtherepulsiveenergyoftheNa+Na–structure,andthismustbesignificantinreducingthecohesionofthehypotheticalcrystal.5a.2nAqU(R)N;2log2Madelungconst.RR⎛⎞α=−α==⎜⎟⎝⎠Inequilibrium2n02n1200UnAqnN0;RRRR+⎛⎞∂α=−+==⎜⎟∂α⎝⎠A,qand200Nq1U(R)(1).Rnα=−−31b.()()220000021UU(RR)URRR,2R∂δ=+δ+∂bearinginmindthatinequilibriumR0(UR)0.∂∂=22n233320000Un(n1)A2q(n1)q2NNRRRRRR2+⎛⎞⎛⎛⎞∂+α+α=−=−⎜⎟⎜⎜⎟∂⎝⎠⎝⎠⎝20q⎞α⎟⎠Foraunitlength2NR0=1,whence0222220422200R0RUqU(n1)qlog2(n1);CRRR2RR⎛⎞∂α∂−=−==⎜⎟∂∂⎝⎠.6.ForKCl,λ=0.34×10–8ergsandρ=0.326×10–8Å.FortheimaginedmodificationofKClwiththeZnSstructure,z=4andα=1.638.ThenfromEq.(23)withxR0ρwehave2x3xe8.5310.−−=×BytrialanderrorwefindorRx9.2�,0=3.00Å.TheactualKClstructurehasR0(exp)=3.15Å.Fortheimaginedstructurethecohesiveenergyis2200αqpUU=1,or=0.489RRq⎛⎞⎜⎟⎝⎠inunitswithR0inÅ.FortheactualKClstructure,usingthedataofTable7,wecalculate2U0.495,q=−unitsasabove.Thisisabout0.1%lowerthancalculatedforthecubicZnSstructure.Itisnoteworthythatthedifferenceissoslight.7.TheMadelungenergyofBa+O–is–αe2R0perionpair,or–14.61×10–12erg=–9.12eV,ascomparedwith–4(9.12)=–36.48eVforBa++O.ToformBa+andO–fromBaandOrequires5.19–1.5=3.7eV;toformBa++andOrequires5.19+9.96–1.5+9.0=22.65eV.ThusatthespecifiedvalueofR0thebindingofBa+O–is5.42eVandthebindingofBa++Ois13.83eV;thelatterisindeedthestableform.8.From(37)wehaveeXX=S11XX,becauseallotherstresscomponentsarezero.By(51),11111211123S2(CC)1(CC).=−++Thus22111211121112Y(CCC2C)(CC);=+−+further,alsofrom(37),eyy=S21Xx,whenceyy2111121112xxeeSSC(CC)σ===−+.9.ForalongitudinalphononwithK||[111],u=v=w.32221144124412111244[C2C2(CC)]K3,orvK[(C2C4C3)]ρρ=+++==++Thisdispersionrelationfollowsfrom(57a).10.Wetakeu=–w;v=0.Thisdisplacementistothe[111]direction.Shearwavesaredegenerateinthisdirection.Use(57a).11.Let12xxyyee=−=ein(43).Then22111124441112211221112UC(ee)Ce[(CC)]e=+−=−2sothat22n233320000Un(n1)A2q(n1)q2NNRRRRRR2+⎛⎞⎛⎛⎞∂+α+α=−=−⎜⎟⎜⎜⎟∂⎝⎠⎝⎠⎝20q⎞α⎟⎠istheeffectiveshearconstant.12a.Werewritetheelementaij=p–δij(λ+p–q)asaij=p–λ′δij,whereλ′=λ+p–q,andδijistheKroneckerdeltafunction.Withλ′thematrixisinthe“standard”form.Therootλ′=Rpgivesλ=(R–1)p+q,andtheR–1rootsλ′=0giveλ=q–p.b.Seti[(K3)(xyz)t]0i[..]0i[..]0u(r,t)ue;v(r,t)ve;w(r,t)we,++−===asthedisplacementsforwavesinthe[111]direction.Onsubstitutionin(57)weobtainthedesiredequation.Then,by(a),onerootis221112442pqK(C2C4C)3,ρ=+=++andtheothertworoots(shearwaves)are22111244K(CCC)3.ρ=−+13.Setu(r,t)=u0ei(K·r–t)andsimilarlyforvandw.Then(57a)becomes2222011y44yz1244xy0xz0u[CKC(KK)]u(CC)(KKvKKw)ρ=+++++0andsimilarlyfor(57b),(57c).Theelementsofthedeterminantalequationare3322221111x44yz121244xy131244xzMCKC(KK)M(CC)KK;M(CC)KK.ρ;=++−=+=+andsoonwithappropriatepermutationsoftheaxes.Thesumofthethreerootsof2ρisequaltothesumofthediagonalelementsofthematrix,whichis(C11+2C44)K2,where2222xyz2221231144KKKK,whencevvv(C2C),ρ=++++=+forthesumofthe(velocities)2ofthe3elasticmodesinanydirectionofK.14.Thecriterionforstabilityofacubiccrystalisthatalltheprincipalminorsofthequadraticformbepositive.Thematrixis:C11C12C12C12C11C12C12C12C11C44C44C44Theprincipalminorsaretheminorsalongthediagonal.ThefirstthreeminorsfromthebottomareC44,C442,C443;thusonecriterionofstabilityisC44>0.ThenextminorisC11C443,orC11>0.Next:C443(C112–C122),whence|C12|<C11.Finally,(C11+2C12)(C11–C12)2>0,sothatC11+2C12>0forstability.34CHAPTER41a.Thekineticenergyisthesumoftheindividualkineticenergieseachoftheform2S1Mu.2Theforcebetweenatomssands+1is–C(us–us+1);thepotentialenergyassociatedwiththestretchingofthisbondis2s11C(uu)2s+−,andwesumoverallbondstoobtainthetotalpotentialenergy.b.Thetimeaverageof222S11MuisMu.24Inthepotentialenergywehaves1uucos[t(s1)Ka]u{cos(tsKa)cosKasin(tsKa)sinKa}.+=−+=−⋅+−⋅ss1Thenuuu{cos(tsKa)(1cosKa)sin(tsKa)sinKa}.+−=−⋅−−−⋅Wesquareandusethemeanvaluesovertime:221cossin;cossin0.2<>=<>=<>=Thusthesquareofu{}aboveis22221u[12cosKacosKasinKa]u(1cosKa).2−++=−Thepotentialenergyperbondis21Cu(1cosKa),2−andbythedispersionrelation2=(2CM)(1–cosKa)221thisisequaltoMu.4Justasforasimpleharmonicoscillator,thetimeaveragepotentialenergyisequaltothetimeaveragekineticenergy.2.WeexpandinaTaylorseries2222ssu1uu(sp)u(s)papa;x2x⎛⎞∂∂⎛⎞+=+++⎜⎟⎜⎟∂∂⎝⎠⎝⎠Onsubstitutionintheequationofmotion(16a)wehave2222p22p0uuM(paC)tx>∂∂=Σ∂∂,whichisoftheformofthecontinuumelasticwaveequationwith412122pp0vMpaC−>=Σ.3.FromEq.(20)evaluatedatK=πa,thezoneboundary,wehave2122Mu2Cu;Mv2Cv.−=−−=−Thusthetwolatticesaredecoupledfromoneanother;eachmovesindependently.At2=2CM2themotionisinthelatticedescribedbythedisplacementv;at2=2CM1theulatticemoves.202000p0p0sinpka24.A(1cospKa);Mpa2AsinpkasinpKaKM1(cos(kK)pacos(kK)pa)2>>=Σ−∂=Σ∂−−+WhenK=k0,20p0A(1cos2kpa),KM>∂=Σ−∂whichingeneralwilldivergebecausep1.Σ5.ByanalogywithEq.(18),22s1ss2s1s22s1ss2s1s2iKa122iKa12MdudtC(vu)C(vu);MdvdtC(uv)C(uv),whenceMuC(vu)C(veu);MvC(uv)C(uev),and−+−=−+−=−+−−=−+−−=−+−2iK1212iKa21212(CC)M(CCe)0(CCe)(CC)M−+−−+a=−++−212212ForKa0,0and2(CC)M.ForKa,2CMand2CM.==+=π=6.(a)TheCoulombforceonaniondisplacedadistancerfromthecenterofasphereofstaticorrigidconductionelectronseais–e2n(r)r2,wherethenumberofelectronswithinasphereofradiusris(34πR3)(4πr33).Thustheforceis–e2rR2,andthe42forceconstantise2R3.TheoscillationfrequencyDis(forceconstantmass)12,or(e2MR3)12.(b)Forsodiumandthus23M410g−×�8R210cm;−×�104612D(510)(310)−−××�(c)Themaximumphononwavevectorisoftheorderof10131310s−×�8cm–1.Ifwesupposethat0isassociatedwiththismaximumwavevector,thevelocitydefinedby0Kmax3×105cms–1,generallyareasonableorderofmagnitude.7.Theresult(a)istheforceofadipoleepuponadipolee0u0atadistancepa.Eq.(16a)becomes2P233p>0(2M)[(1cosKa)(1)(2epa)(1cospKa)].=γ−+Σ−−Atthezoneboundary2=0ifPP3p>01(1)[1(1)]p−+σΣ−−−=0,orif.Thesummationis2(1+3p3[1(1)]p1−σΣ−−=–3+5–3+)=2.104andthis,bythepropertiesofthezetafunction,isalso7(3)4.ThesignofthesquareofthespeedofsoundinthelimitKaisgivenbythesignof1<<p32p>012(1)pp,−=σΣ−whichiszerowhen1–2–1+3–1–4–1+=12σ.Theseriesisjustthatforlog2,whencetherootisσ=1(2log2)=0.7213.43CHAPTER51.(a)Thedispersionrelationism1|sinKa|.2=WesolvethisforKtoobtain,whenceand,from(15),1mK(2a)sin()−=2212mdKd(2a)()−=−D().Thisissingularat=2212m(2La)()−=π−m.(b)ThevolumeofasphereofradiusKinFourierspaceis,andthedensityoforbitalsnear304K3(43)[()A]Ω=π=π−32120is,provided<33320D()=(L2)|dd|(L2)(2A)()πΩ=ππ−0.ItisapparentthatD()vanishesforabovetheminimum0.2.Thepotentialenergyassociatedwiththedilationis23B11B(VV)akT22∆.ThisisB1kT2andnotB3kT2,becausetheotherdegreesoffreedomaretobeassociatedwithsheardistortionsofthelatticecell.Thusand24724rms(V)1.510;(V)4.710cm;−−<∆>=×∆=×3rms(V)V0.125∆=.Now,whence.3aaVV∆∆rms(a)a0.04∆=3.(a),wherefrom(20)foraDebyespectrum2R(h2V)−<>=ρΣ11−Σ,whence213DdD()3V4v−=∫=π323v22DR3h8<>=πρ.(b)Inonedimensionfrom(15)wehave,whenceD()Lv=π1dD()−∫divergesatthelowerlimit.Themeansquarestraininonedimensionis22201(Rx)Ku(h2MNv)K2<∂∂>=Σ=Σ223DD(h2MNv)(K2)h4MNv.==4.(a)Themotionisconstrainedtoeachlayerandisthereforeessentiallytwodimensional.ConsideroneplaneofareaA.ThereisoneallowedvalueofKperarea(2πL)2inKspace,or(L2π)2=A4π2allowedvaluesofKperunitareaofKspace.ThetotalnumberofmodeswithwavevectorlessthanKis,with=vK,222N(A4)(K)A4v.=ππ=π2ThedensityofmodesofeachpolarizationtypeisD()=dNd=A2πv2.Thethermalaveragephononenergyforthetwopolarizationtypesis,foreachlayer,DD200AU2D()n(,)d2d,2vexp(h)1=τ=πτ−∫∫==dwhereDisdefinedby.IntheregimeDDND()=∫D>>τ=,wehave3222x02AxUdx.2ve1τ≅π−∫=51Thustheheatcapacity.2BCkUT=∂∂τ(b)Ifthelayersareweaklyboundtogether,thesystembehavesasalinearstructurewitheachplaneasavibratingunit.Byinductionfromtheresultsfor2and3dimensions,weexpectC.ButthisonlyholdsatextremelylowtemperaturessuchthatTDlayervNLτ<<==,whereNlayerListhenumberoflayersperunitlength.5.(a)FromthePlanckdistributionxx111n(e1)(e1)coth(x2)222<>+=+−=,where.ThepartitionfunctionBxhkT=x2sxx2x1Zeee(1e)[2sinh(x2)]−−−−=Σ=−=−andthefreeenergyisF=kBTlogZ=kBTlog[2sinh(x2)].(b)With(∆)=(0)(1–γ∆),theconditionbecomesF0∂∂∆=B1Bhcoth(h2kT)2∆=γΣondirectdifferentiation.Theenergyisjustthetermtotherightofthesummationsymbol,sothatBnh<>U(T)∆=γ.(c)Bydefinitionofγ,wehave,orVδ=−γδVdlogdlogV=−δ.But,whence.DθdlogdlogVθ=−γ52CHAPTER61.Theenergyeigenvaluesare22khk.2mε=Themeanvalueoverthevolumeofasphereinkspaceis22222FF2hkdkk3h3k.2mkdk52m5⋅<ε>==⋅=ε∫∫ThetotalenergyofNelectronsis0F3UN5.=⋅ε2a.Ingeneralp=–∂U∂Vatconstantentropy.Atabsolutezeroallprocessesareatconstantentropy(theThirdLaw),sothat0pdUdV=−,where0F3UN5=ε23223h3NN52mV⎛⎞π=⎜⎟⎝⎠,whence0U2p3V=⋅.(b)Bulkmodulus200002UdUUUUdp222210BVVdV3V3VdV3V3V9V⎛⎞⎛⎞=−=−+=⋅+=⎜⎟⎜⎟⎝⎠⎝⎠0.(c)ForLi,223120113112U3(4.710cm)(4.7eV)(1.610ergeV)V52.110ergcm2.110dynecm,−−−−=××=×=×whenceB=2.3×1011dynecm–2.Byexperiment(Table3.3),B=1.2×1011dynecm–2.3.Thenumberofelectronsis,perunitvolume,()01ndD()e1ε−µτ=εε⋅,+∫whereD(ε)isthedensityoforbitals.Intwodimensions202m1ndhe1m(log(1e)),h(ε−µ)τ−µτ=επ+=µ+τ+π∫wherethedefiniteintegralisevaluatedwiththehelpofDwight[569.1].4a.Inthesunthereare335724210101.710−××�nucleons,androughlyanequalnumberofelectrons.Inawhitedwarfstarofvolume61932834(210)310cmπ××3theelectronconcentrationis572832810310cm.310−××Thus223272074Fh11(3n)101010ergs,or3.10eV.2m222−−ε=π⋅(b)ThevalueofkFisnotaffectedbyrelativityandisn13,wherenistheelectronconcentration.Thus3FFhckhcε��n.(c)Achangeofradiusto10km=106cmmakesthevolume4×1018cm3andtheconcentration3×1038cm–3.Thus(Theenergyisrelativistic.)27101348F10(3.10)(10)2.10erg10eV.−−ε5.Thenumberofmolespercm3is81×10–33=27×10–3,sothattheconcentrationis16×1021atomscm–3.Themassof