固体物理13-18参考答案

18.1k(x)a3(1)()sin(2)()cos(3)()()(4)()(1)()kkkllklxxxxiaaxfxlaxfxlaππψψψψ∞=−∞∞=−∞===−=−−∑∑f(x−la)4.32221(),()20,(1)mbxnanabxnabVxnabxnabω−−−≤≤+=−+≤≤−a=4b12011()LikxikxVeVxedxLL−=∫222111[()]2nabikxikxnabVNembxnaedxLLω+−−=−−∫222[()]2nabnabNVmbxnadxLω+−=−−∫xnaLNaξ−=222221[]296bbaVmbdmaωξξω+−=−=∫20011()()()ninaaiKaVneVdeVdaaπξξξξξξ⋅⋅==∫∫()2221,()20mbbbVxbbωξξξ−−≤≤+=+≤≤−,2222222211()()2()2inbabinbabVnembdamebdaπξπξωξξωξξ⋅+−⋅+−=−=−∫∫22221()2ibabmVebdaπξωξξ+−=−∫42222()2ibabmVebdaπξωξξ+−=−∫112gEV=222gEV=4.1222(,)4coscosUxyUxyaaππ=−,,aaππ()()()()22222222,,,,,,,,2222(,)2coscosixyixyixyixyaaaaaaaaUxyUxyxyaaaaUeeeeππππππππππππ−−−−=−++−=−+++:22222222,,,,,,,aaaaaaaaππππππππ−−−−−U,0,,kaaππ=0123()0(,)222220,(0,),(,0),(,),nnKKkkaaKKKKaaaaππππππ−======i:0,0,0,−U()1()()mikKrkmmraKeN+Ψ=Ω∑i123(0),(),(),()aaKaKaK123()()12()31()[(0)()()()]ikKrikKrikrkikKrraeaKeaKeNaKe+++Ψ=++Ω+iiii0()'()()kkkHrHrErΨ+Ψ=Ψ012300010002030()()()()()()()0KKKKaEEUaKEEUaKEEUaKEEUφφφφ−++−++−++−+=x,01230*0*0*0*,,,KKKKφφφφ0102031012132021233031320123012310231203()(0)()()()0(0)()()()()0(0)()()()()0(0)()()()()0KKKKKKKKKKKKKKKKKKKKKKKKEEaUaKUaKUaKUaEEaKUaKUaKUaUaKEEaKUaKUaUaKUaKEEaK−−−−−−−−−−−−−+++=+−++=++−+=+++−=123(0),(),(),()aaKaKaK012311213221233313200000KKKKKKKKKKKKKKKKKKEEUUUUEEUUUUEEUUUUEE−−−−−−−−−−−=−−0000000000000EEUEEUUEEUEE−−−−=−−−−()4400EEU−−=E0000,,,EUEUEUEU++−−2U15.1EF=EK/2−|UK|+∆K=2π/a(1,0,0)EK/2K/2UKK(1)∆0(2)0∆2|UK|22m∆(3)∆2|UK|ρ1ρ224KmUπ(1)∆0EK/2−|UK|+∆EK/2−|UK|EK/2K/2EK/2−|UK|K/2EK/2|UK|K/20∆K/2(2)0∆2|UK|,EK/2−|UK|EFEK/2+|UK|.,,:(π/a,ky,kz)22222222yzkkFkkUUEmamaππ++−=−+∆=2222yzmkk∆+=22m∆,(3)2|UK|∆2|UK|EF=EK/2−|UK|+∆EK/2+|UK|.,()22222221212,2222kkkkUkUkkUmmm−=+−=()()2222221122222212122,4kkkaamkkUππρρππρρπ=+=+−=−=19.1∆k0∆k0ħ2/2m=1(1)∆k0=VG/k0k0VG(2)k0k0+∆k02πVG()()()1202212121200242GkmkkVρρππρρπρρρρπ≈=−=+−=⋅∆=022GGmkVV∆==()2212242GGmVVππρρπ−==4.4sES(k)0()()ssikRisRNearestEkJJReε−⋅==−−∑s()sEk0()()ssikRsssRNearestEkJJReε−⋅==−−∑,,00,,,0,222222,,00,,,0,222222,,,,00,,,0,222222,,00,,,0,222222aaaaaaaaaaaaaaaaaaaaaaaa−−−−−−−−−−−−12022saaRijk=++120()()ssikRsssRNearestEkJJReε−⋅==−−∑()()ssrrϕϕ−=s*1()()[()()]()}0sisiJJRRUVdϕξξξϕξξ==−−−∫()sJR1()sJJR=01()ssikRssRNearestEkJJeε−⋅==−−∑022saaRijk=++()022()2sxyzsxyaakRkikjkkijkakRkk⋅=++⋅++⋅=+()2(cossin)(cossin)2222xysaikkikRyyxxeekakakakaii−+−⋅==−−——1201()4(coscos22coscoscoscos)2222ysxsyxzzkakaEkJJkakakakaε=−−++01(0,0,0)12SkEJJε==−−012(,0,0)4SkEJJaπε==−+116J8,,,,222222,,,,222222,,,,,222222,,,,222222aaaaaaaaaaaaaaaaaaaaaaaa−−−−−−−−−−−−222saaaRijk=++120()()ssikRsssRNearestEkJJReε−⋅==−−∑()()ssrrϕϕ−=s*1()()[()()]()}0sisiJJRRUVdϕξξξϕξξ==−−−∫()sJR1()sJJR=01()ssikRssRNearestEkJJeε−⋅==−−∑222saaaRijk=++()222()2sxyzsxyzaaakRkikjkkijkakRkkk⋅=++⋅++⋅=++()2(cossin)22(cossin)(cossin)2222xyzsaikkkikRxxyyzzkakaeeikakakakaii−++−⋅==−×−−——801()8coscoscos222ysxzskakakaEkJJε=−−01(0,0,0)8SkEJJε==−−012(,0,0)8SkEJJaπε==−+116J4.7aNa(1)sES(k)(2)(3)sT=0K4.13sES(k)222(1)0,,01)4(12cos)21(2),0)212cos23(3),0,0)44(cos2cos)212(4)0,,,01)24(coscosyzxsxyzsxyzszxysXkkkaELkkkaEKkkkaEWkkkaaEπΓµµεβγµππΓµµεβγµππΓµµεβγµπµπππΓµµµεβγµπ===≤≤=−−+===≤≤=−−===≤≤=−−+===≤≤=−−11coscos)22µπµπµπ+−12SEsKKxKyKz5.12271()coscos288Ekkakama=−+a(1)(2)k(3)1kE0,(0)0==2222712,()(coscos2)88kEaamamaππππ==−+=222()(0)EEEamaπ∆=−=2k21()1(),()(sinsin2)4dEkvkvkkakadkma==−32222222224*/,(coscos2)8EEmakaakakkma∂∂==−∂∂*1coscos22mmkaka=−0,*2kmm==km2,*3maπ==−5.22.5Å102V/m107V/m12tk=T=2dkFdt=()dkqEvkdt==//2()/TaatvkEqππ===17.1)1(∆−−=ankπ222(1)2(1)nnnnnnnnnnTVTVTVETVTVTV±+++∆+=+−−∆−17.2sES(k)sH(0,1/2,0)W(1/2,1/4,0)sES(k)01()8coscoscos222ysxzskakakaEkJJε=−−01(0,0,0)8SkEJJε==−−012(,0,0)8SkEJJaπε==−+116J22*2221/2xxxEmkJa∂==∂22**221122yyzzmmJaJa==2***212xxyyzzmmmJa===−2***212xxyyzzmmmJa===sES(k)01()4(coscos22coscoscoscos)2222ysxsyxzzkakaEkJJkakakakaε=−−++01(0,0,0)12SkEJJε==−−0122(,0,0)(,,0)4SkkEJJaaaπππε===−+116J22*2221/2xxxEmkJa∂==∂22**221122yyzzmmJaJa==4.10f.c.cBZ1414LΓL2111(,,)222aπ3LaπΓ=()1/323Fknπ=f.c.cb.z()1/3235.43Fknaaππ==≈35.4na=34anaf.c.c3335.448(1)0.35nxxaaax==−⋅+⋅⇒=0.35:0.655.3kSnrAn2nnASqB=()dkdkdrqvkBqBdtdtdt=−×⇒=−×rkqB∆=∆kSnrAnB∆r∆k2nnASqB=5.4(1)··(2)B=1T1B∆212211122FFFSqqBBBNSNSkSππππ∆=−====5.5(1)(2)(3)Bα,β,γ(4)*qBmω=1/2222*123123mmmmmmmαβγ−++=19.1(1)(2)(3)19.2a(1)(2)(3)(4)(5)123322322aaaijaaaijack=+=−+=1232232232bijaabijaabkcπππππ=+=−+=19.3