【材料课件】第十二章固体材料中电子态－1

1.2.3.4.5.(A)(B)•—————()1—()2()—1.2.3.4.5./10-910-1810510-51071051015(-1m-1)()•Ag—Cu—Al—Fe—•Si—1ppmAsPBAl()1000000•PE—,PP—,()—•SiO2——()1-1m1−Ω=ρσ2.1IdeBroglie1.DeBroglie(1924)(Ep)(νλ)2.•••••100V0.1225nm•TEMωνh==hEλπ02nk=kph=kn0()()[]()[]hprkrr⋅−−=⋅−−=tiAtiAtεωΨexpexp,UeUmhphnm225.120===λ•NiNP(1927)•—TEM(SEM)•2.1II1.I()t,rΨ2.()t,rΨ3.••••()t,rΨ()()()tttP,,,rrrΨΨ∗=()t,rΨ2.1IIII1.II2.(operator)3.•r•pzyxkjirr++==∧∂∂+∂∂+∂∂−=∇−=zyxiikjiphhˆ4.FFf(r,p)=∧F∧∧pr,f2.1IIzyxkjirr++==∧∂∂+∂∂+∂∂−=zyxikjiphˆLrxpzyxLLLkjiL++=zyxLLLˆˆˆˆkjiL++=∧∧∧×=prL()∇−×=hirzyxzyxi∂∂∂∂∂∂−=kji)(h∂∂−∂∂−=xyyxiLZhˆϕddhi−=2.1II1.q2.•()•()ΨΨqF=∧ΦΦϕΦqiLz=−=∧ddh()()hϕϕΦiqCexp=()()ϕΦϕΦ=+π2()()()hhh/π2expexpexpqiiqCiqC⋅=ϕϕ()1π2exp=hqihmq=()ϕϕΦimmeπ21=L,3,2,1,0±±±=m{}{}2.1IIIIII1.IIIF—F()Lz()ϕϕΦ22π21ie=h2=zL2=mzLz()Lz——{}()Lh,2,1,0±±=∈mmLz2.1IIIIII2.III()F()2mmCqFP==mC∫∞∗=τΨψdmmCqmmCmF{m}=mm,F—()2.1III2.III()F∫∞∗=τΨψdqqCF{q}()=∆+=qqqFP~qCq∆2⋅q∫∞=τψΨdqqC(1)——(2)——2.1III1./——2.————3.FFFFτΨΨd∧∞∗∫=FFF2.1III(1)——•——•——•——(2)12C1C222.1——()∂∂−∂∂+∂∂−∂∂+∂∂−∂∂−=∧22222xyyxzxxzyzzyhL∂∂⋅+∂∂∂∂⋅⋅−=∧22222sin1sinsin1ϕθθθθθhL()()ϕθϕθ,,22fLfL=∧()()()ϕΦθΘϕθ⋅=,f0dd222=+ΦϕΦm0sinddsinddsin122=⋅−+⋅ΘθθΘθθθmqa)b)2.1()ϕϕΦimmeπ21={}()Lh,2,1,0±±=∈mmLz——0dd222=+ΦϕΦm0sinddsinddsin122=⋅−+⋅ΘθθΘθθθmqa)b)22hqL=θcos=x()()xg=θΘ()01dd1dd222=−−+−gxmqxgxxπθ≤≤011≤≤−x()()xPxgml=ml()1+=llq()221h+=llLl012…2.1——()()()ϕΦθΘϕθ⋅=,fθcos=x()()xg=θΘ()()lllllxxlxP1dd!212−⋅=()()()()lmlmllmlml1coscosdd!21cos1222−⋅⋅−=++θθθθΘ()()()xPxxxPlmmmmldd122⋅−=l012…()221h+=llLlm≤()θϕcosmlimlmPeN=()ϕθΥ,lm=()()()ϕΦθΘϕθ⋅=,flm≤——2.1——()221h+=llLzzzz()1cos+==llmLZLθhmLz=z2.1.——L2Lz()ϕθ2cossinπ1615d222⋅=−yxϕθ222,2sinπ3215ieY⋅=ϕθ222,2sinπ3215ieY−−⋅=L2()=−22dyx2,22,22121−⋅+⋅YY2.1()221h+=ssS1.—Stern-Gerlachz——2.()1cos+==ssmSsZSθhszmS=2/1=s2/3h=S2/1±=smssmmσσ⋅=∧2243hSssmsmZmSσσ⋅=∧h2/12/121σσ⋅=∧hZS2.1——()h1+=LLPLnmimsizL=∑=niim1zzLLLLmL,1,,1,−+−−=LzhLLzmP=2.1——nmimsizS=∑=nisim1()h1+=SSPSzzzhSSzmP=SSSSmS,1,,1,−+−−=L2.12.Pauli(1)(2)(3)J=|LS|J=LS13.Hund2.1———HundzJ()h1+=JJPJzzzhJJzmP=JJJJmJ,1,,1,−+−−=Lz————z()h1+=JJPJhJJzmP=JJJJmJ,1,,1,−+−−=L2.1(6)ms1/2(5)(4)(3)(2)(1)ms1/221012mFe(Z=26)z—3p6LSJ00z3d6L=2S=2J=4h52=JPh6=SPhhhh2,,0,,2−−=SzPhhLhh4,3,,3,4−−=ZJP2.1IVIV()2.——V1.V(r,t)ΨΨ∧=∂∂Htih()()rrψψEH=∧()()()htiEtnn−⋅=exp,rrψΨ2.1IVΨΨ∧=∂∂Htih()()rrψψEH=∧()()()htiEtnn−⋅=exp,rrψΨ2.()()()tftrrψΨ=,tftddiiψΨhh=∂∂()()()rrψψΨ∧∧∧=⋅=HffHtH,()rVmH+∇−=∧222hEHtff==ψψˆ1dd1ih()()htiECtfn−=exptEffddi=h2.1IV1.()()rrψψEH=∧remH02202π42ε−∇−=∧h∂∂+∂∂∂∂+∂∂∂∂=∇2222222sin1sinsin111ϕθθθθθrrrrrψψεψϕθθθθθErerrrrrm=−∂∂+∂∂∂∂+∂∂∂∂−0222222202π4sin1sinsin1112h()()()ϕθϕθψ,,,frgr=Efgfgreffrgrgrrrfm=−∂∂+∂∂∂∂+−0222222202π4sin1sinsin1dddd2εϕθθθθθh202222202π4sin1sinsin11dddd12Errefffrgrrgm=−∂∂+∂∂∂∂+−εϕθθθθθh2.1IV1.()()rrψψEH=∧()()()ϕθϕθψ,,,frgr=()()ϕθϕθ,,ˆ22fqfLh=()()ϕθϕθ,,lmYf=()()1,122+=+=llqllLh202222202π4sin1sinsin11dddd12Errefffrgrrgm=−∂∂+∂∂∂∂+−εϕθθθθθh20222202π4ˆ1dddd12ErrefLfrgrrgm=−−−εhh0π42ˆ1dddd120220222=++−ErremfLfrgrrgεhhqfLfErremrgrrg==++22202202ˆ1π42dddd1hhε()1π42dddd1202202+=++llErremrgrrgεh()01π42dddd12022022=+−++grllEremrgrrrεh2.1IV2eV6.13nEn−=1.()()rrψψEH=∧()()()ϕθϕθψ,,,frgr=()01π42dddd12022022=+−++grllEremrgrrrεhzE0rEmx208h−=()()xLnareNxRllnlnarnlnl12020++−=()()xLxxLnl1llln+++++=121212dd()()xlnn1nlxlnexxexL−++++⋅=dd()1−≤nl2.1IV1.zr0()()rrrRrrrrPnlddd,22=+()0dd=rrPz()()()ϕθϕθψ,,,lmnlnlmYrRr=()()1,ddsindd22π00π02*=⋅=∫∫∫∫∞∞ϕθϕθθτψψlmnlYrRrr()()1ddsin,d2π020π022==∫∫∫∞ϕθθϕθlmnlYrrrRz—()()2,,ϕθϕθlmYP=ϕθθΩddsind=()()22rrRrPnl=2.1VV()111()()ttPNijNiij,,,,,,,,,,,,,,ˆ2121rrrrrrrrrrj⋅⋅⋅⋅⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅⋅⋅⋅⋅ΨΨ()()tqtPNjiNiij,,,,,,,,,,,,,,ˆ2121rrrrrrrrrrj⋅⋅⋅⋅⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅⋅⋅⋅⋅ΨΨ()()tqtPNjiNiij,,,,,,,,,,,,,,ˆ212212rrrrrrrrrrj⋅⋅⋅⋅⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅⋅⋅⋅⋅ΨΨ()tNji,,,,,,,21rrrrr⋅⋅⋅⋅⋅⋅⋅⋅⋅=Ψ2.1V∧ijP113.FermiBoseFermiBose()tNji,,,,,,,21rrrrr⋅⋅⋅⋅⋅⋅⋅⋅⋅Ψ()tNij,,,,,,,21rrrrr⋅⋅⋅⋅⋅⋅⋅⋅⋅=ΨsFermiA()()ttPNjiNiij,,,,,,,)1(,,,,,,,ˆ2121rrrrrrrrrrj⋅⋅⋅⋅⋅⋅⋅⋅⋅−=⋅⋅⋅⋅⋅⋅⋅⋅⋅ΨΨ()()ttNijNji,,,,,,,,,,,,,,2121rrrrrrrrrr⋅⋅⋅⋅⋅⋅⋅⋅⋅−=⋅⋅⋅⋅⋅⋅⋅⋅⋅ΨΨ2.1V4.FermiPaulizE1E212()()()()()[]()()()()221221111221221121212/,rrrrrrrrrrψψψψψψψψΨ=−=Az()212,1αα=X()()22,11221βαβα+=X()2121212,1ββαα=AX()212,1ββ=Xz()()()()()()[]()()()()()2121212/2,1,2211211221221121αψαψαψαψααψψψψΨ=−=rrrrrrSAX()()2,1,21ASXrrΨE1E212()()()()()[]()()()()221221111221221121212/,rrrrrrrrrrψψψψψψψψΨ=−=A()()2211rrψψz12E1E2E1E2()z12E1E2()()1221rrψψz12(E1E2)()()()()111222222111,,,,,,,,zyxgzyxfzyxgzyxf≠z2(E1E2)H2——LCAOz12H1szH2()−⋅=0a30aexp1araiiπψ()−−+++−∇+∇−=∧12abb12a2b1a02222102111111π42rRrrrremHεh()+−∇+∇−=∧2ba102222102I11π42rremHεh()+−∇+∇−=∧1ba202222102II11π42rremHεh()()()()()12212,1baba/ψψψψΨ±=AS()III/2,1ΨΨΨ±=AS21*21*ddddττΨΨττΨΨ∫∫∫∫∞∞∧==HEEH2——LCAOzH2()−−+++−∇+∇−=∧12abb12a2b1a02222102111111π42rRrrrre