Диссертация (1149156), страница 16
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Åñëè ýëåêòðîííîå ñîñòîÿíèå n ñîîòâåòñòâóåò ñâÿçàííîìóýëåêòðîíó, òî òîãäà ñóùåñòâóþò òîëüêî äâà ýëåêòðîíà èç ñïëîøíîãî ñïåêòðà√(d1 è d2 ), âîëíîâûå ôóíêöèè êîòîðûõ ñîäåðæàò ìíîæèòåëü 1/ R â ìàòðè÷íîì ýëåìåíòå. Ñîîòâåòñòâåííî, ìû ïîëó÷àåì:Iu1 nd1 d2 ∼1.R(A.11)Åñëè ýëåêòðîííîå ñîñòîÿíèå n ñîîòâåòñòâóåò íåïðåðûâíîìó ñïåêòðó (îòëè÷íîå îò ñîñòîÿíèÿ d2 ), òîãäà ìàòðè÷íûé ýëåìåíò ñîäåðæèò òðè ýëåêòðîíà èçíåïðåðûâíîãî ñïåêòðà (d1 , d2 è n). Ýòî îçíà÷àåò, ÷òî:Iu1 nd1 d2 ∼1241.R3/2(A.12)Ìû òàêæå äîëæíû ðàññìîòðåòü ñïåöèàëüíûé ñëó÷àé, êîãäà ýëåêòðîííîå ñîñòîÿíèå n ñîâïàäàåò ñ ýëåêòðîíîì d2 . Ìàòðè÷íûé ýëåìåíò Iu1 nd1 d2 ìîæíîçàïèñàòü â âèäå:∫dr1 ψu+1 (r1 )ψd1 (r1 )Iu1 nd1 d2 =×1r1∫r1 ≤R∫dr2 ψn+ (r2 )ψd2 (r2 ) +r2 ≤r1r1 ≤r2 ≤R1dr2 ψn+ (r2 ) ψd2 (r2 ) .r2(A.13)Åñëè âîëíîâàÿ ôóíêöèÿ ψn ñîâïàäàåò ñ ψd2 , òîãäà ïîñëåäíèé ÷ëåí â êâàäðàòíûõ ñêîáêàõ:∫dr2 ψn+ (r2 )r1 ≤r2 ≤R1ψd (r2 ) =r2 2∫r1 ≤r2 ≤R= Nd−221dr2 ψd+2 (r2 ) ψd2 (r2 )r2∫Rdr2(A.14))1 (|gd2 (r2 )|2 + |fd2 (r2 )|2 (A.15),r2r1ãäå àñèìïòîòèêà ôóíêöèé gd2 è fd2 äàåòñÿ âûðàæåíèÿìè (2.34), (2.35).
Èñïîëüçóÿ (A.2)-(A.4), ïîëó÷àåì:∫r1 ≤r2 ≤R11dr2 ψn+ (r2 ) ψd2 (r2 ) =(log(R) + O(R0 )) ,r2R(A.16)ãäå ëîãàðèôìè÷åñêèé ÷ëåí äàåòñÿ ïåðâûì ÷ëåíîì â èíòåãðàëå (A.4). Ïåðâûé÷ëåí â êâàäðàòíûõ ñêîáêàõ â (A.13) ïðîïîðöèîíàëåí 1/R. Ñîîòâåòñòâåííî,äëÿ ñëó÷àÿ, êîãäà ýëåêòðîííîå ñîñòîÿíèå n ñîâïàäàåò ñ ýëåêòðîííûì ñîñòîÿíèåì d2 , ìàòðè÷íûé ýëåìåíò Iu1 nd1 d2 èìååò âèä:Iu1 nd1 d2 ∼log(R).R3/2(A.17)Ðàññìàòðèâàåì òàêæå àñèìïòîòèêó ìàòðè÷íîãî ýëåìåíòà èçëó÷åíèÿ ôîòîíà Aud , êîòîðûé çàäàåòñÿ âûðàæåíèåì (2.67).
Ìû ïîëàãàåì, ÷òî âñå âîëíîâûå ôóíêöèè ýëåêòðîíà íîðìèðîâàíû íà åäèíèöó ïî ñôåðå ðàäèóñà R. Åñëè125îáà ýëåêòðîíà (d è u) ÿâëÿþòñÿ ñâÿçàííûìè, òîãäà ìàòðè÷íûé ýëåìåíò Audíå çàâèñèò îò R:Aud ∼ R0 .(A.18)Åñëè îäèí èç ýëåêòðîíîâ (d èëè u) ÿâëÿåòñÿ ñâÿçàííûì, à äðóãîé ýëåêòðîíïðèíàäëåæèò ñïëîøíîìó ñïåêòðó, òîãäà:Aud ∼1.R1/2(A.19)Åñëè îáà ýëåêòðîíà ïðèíàäëåæàò ñïëîøíîìó ñïåêòðó, òî:Aud ∼1261.R(A.20)Ïðèëîæåíèå BÈíòåãðèðîâàíèå ïî óãëàìÈíòåãðèðîâàíèå ïî íàïðàâëåíèþ èìïóëüñà íà÷àëüíûõ ýëåêòðîíîâ â (2.63)ìîæíî ïðåäñòàâèòü àíàëèòè÷åñêè. Òîëüêî äâóõýëåêòðîííàÿ âîëíîâàÿ ôóíêöèÿ íà÷àëüíîãî ñîñòîÿíèÿ çàâèñèò îò íàïðàâëåíèÿ èìïóëüñà. Ìû ðàññìàòðèâàåì ñëåäóþùèé èíòåãðàë: (ñì.
âûðàæåíèÿ (2.63), (2.55))∫I ==11∗∗(r4 )} (B.1)(r3 )ψpµdνp √ det{ψpµp1 =1/2 (r1 )ψpµ =−1/2 (r2 )} √ det{ψpµp3 =1/2p1p3 =−1/222∫1 ∑∗∗dνp (−1)−1/2+µp1 (−1)−1/2+µp3 ψpµp1 (r1 )ψpµ (r2 )ψpµ(r3 )ψpµ(r4 )(B.2)p3p1p32µ µp1 p3Îäíîýëåêòðîííàÿ âîëíîâàÿ ôóíêöèÿ ñ îïðåäåëåííûì ìîìåíòîì è ïîëÿðèçàöèåé ìîæåò áûòü ðàçëîæåíà ïî íàáîðó âîëíîâûõ ôóíêöèé ñ îïðåäåëåííûìóãëîâûì ìîìåíòîì è ÷åòíîñòüþ. Èñïîëüçóÿ (2.50), ìû ïîëó÷àåì:∫∫1 ∑I =dνpdε1 dε2 dε3 dε42µ µ∑(−1)−1/2+µp1 (−1)−1/2+µp3j1 j2 j3 j4 l1 l2 l3 l4 m1 m2 m3 m4∗×apµp1 ,ε1 j1 l1 m1 apµp1 ,ε2 j2 l2 m2 apµp3 ,ε3 j3 l3 m3 a∗pµ ,ε4 j4 l4 m4p3p1 p3×ψε1 j1 l1 m1 (r1 )ψε2 j2 l2 m2 (r2 )ψε∗3 j3 l3 m3 (r3 )ψε∗4 j4 l4 m4 (r4 ) .(B.3)Êîýôôèöèåíòû apµp ,εjlm ìîãóò áûòü çàïèñàíû â âèäå (2.51), (2.52).
Ïîñëå127èíòåãðèðîâàíèÿ ïî ýíåðãèÿì ïîëó÷àåì:1I =2((2π)3pϵ)2 ∑ ∫∑dνp(−1)−1/2+µp1 (−1)−1/2+µp3µp1 µp3j1 j2 j3 j4 l1 l2 l3 l4 m1 m2 m3 m4×eiφϵj1 l1 ανp µp1 ,j1 l1 m1 eiφϵj2 l2 ανp µp1 ,j2 l2 m2 e−iφϵj3 l3 αν∗ p µp3 ,j3 l3 m3 e−iφϵj4 l4 αν∗ p µ ,j4 l4 m4p3∗∗(r4 ) .(r3 )ψϵj×ψϵj1 l1 m1 (r1 )ψϵj2 l2 m2 (r2 )ψϵj3 l3 m34 l4 m4(B.4)Ðàññìîòðèì îòäåëüíî ñëåäóþùèé èíòåãðàë:I1 =∑ ∫dνp (−1)−1/2+µp1 (−1)−1/2+µp3µp1 µp3×ανp µp1 ,j1 l1 m1 ανp µp1 ,j2 l2 m2 αν∗ p µp3 ,j3 l3 m3 αν∗ p µp3 ,j4 l4 m4.(B.5)Èñïîëüçîâàíèå (2.52) äàåòI1 =∑ ∫µp1 µp3×Cl4 21j4 m4dνp∑l1l1l122322(m,µ)C(m,µ)C(ml3 , µ3 )Cj11 ml11l22jmj12 23 m3ml1 ml2 ml3 ml4 µ1 µ2 µ3 µ4(ml4 , µ4 ) × Yl1∗ml1 (νp )Yl2∗ml2 (νp )Yl3 ml3 (νp )Yl4 ml4 (νp )×(−1)1+µp1 +µp3×[η + (µ1 )v µp1 (νp )][η + (µ2 )v µp1 (νp )](B.6)×[η + (µ3 )v µp2 (νp )]∗ [η + (µ4 )v µp2 (νp )]∗ .Ïîñëå èíòåãðèðîâàíèÿ ïî íàïðàâëåíèþ èìïóëüñà è ñóììèðîâàíèÿ ïî ïðîåêöèÿì [14], ïîëó÷àåì:I11 ∑=Π(j1 , j2 , j3 , j4 , l1 , l2 , l3 , l4 )4πKk×(−1)l1 +j2 +1/2 (−1)l3 +j4 +1/2l1 l2 Kl3 l4 KK j2 j1K j4 j3 × 1/2 l l 1/2 l l 0 0 00 0 01234j1 j2j3 j4×CKk(m1 m2 )CKk(m3 m4 ) .(B.7)Çäåñü èñïîëüçîâàíû 3j -ñèìâîëû, 6j -ñèìâîëû [14] èΠ(j1 , j2 , .
. . , jn ) =√√√2j1 + 1 2j2 + 1 . . . 2jn + 1 .128(B.8)Ñîîòâåòñòâåííî, (B.4) ìîæíî çàïèñàòü â âèäå)2∑(2π)31 ∑Π(j1 , j2 , j3 , j4 , l1 , l2 , l3 , l4 )pϵ4πj1 j2 j3 j4 l1 l2 l3 l4 m1 m2 m3 m4Kkl1 l2 Kl3 l4 K K j2 j1 K j4 j3 × 1/2 l l 1/2 l l 0 0 00 0 012341I =2(j1 j2j3 j4×(−1)l1 +j2 +1/2 (−1)l3 +j4 +1/2 CKk(m1 m2 )CKk(m3 m4 )×eiφϵj1 l1 +iφϵj2 l2 −iφϵj3 l3 −iφϵj4 l4(B.9)∗∗×ψϵj1 l1 m1 (r1 )ψϵj2 l2 m2 (r2 )ψϵj(r3 )ψϵj(r4 ) .3 l3 m34 l4 m4Ýòî âûðàæåíèå ìîæåò áûòü çàïèñàíî â âèäå:()2∑(2π)3 ∑I =AKkϵjl l1 ϵj2 l2 ϵj3 l3 ϵj4 l4pϵKk j1 ≤j2 ,l1 ≤l2 ,j3 ≤j4 ,l3 ≤l4∑ jj1 2×N12CKk(m1 m2 ) det{ψϵj1 l1 m1 (r1 )ψϵj2 l2 m2 (r2 )}m1 m2×N34∑j3 j4∗∗CKk(m3 m4 ) det{ψϵj(r3 )ψϵj(r4 )} ,3 l3 m34 l4 m4(B.10)m3 m4√ãäå N = 1/2 äëÿ ýêâèâàëåíòíûõ ýëåêòðîíîâ è N = 1/ 2 äëÿ íåýêâèâàëåíòíûõ; n = (jlm). Êîýôôèöèåíòû ïðåäñòàâëÿþòñÿ â âèäå:AKkϵjl l1 ϵj2 l2 ϵj3 l3 ϵj4 l4 =1Π(j1 , j2 , j3 , j4 , l1 , l2 , l3 , l4 )4πl1 l2 Kl l K 3 4×0 0 00 0 0 K j2 j1 K j4 j3 × 1/2 l l 1/2 l l 123(B.11)4×(−1)l2 +j2 +1/2 (−1)l4 +j4 +1/2(B.12)×eiφϵj1 l1 +iφϵj2 l2 −iφϵj3 l3 −iφϵj4 l4 .(B.13)129Ëèòåðàòóðà[1] M.
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