Influence de la différence entre les distributions de protons et de neutrons dans le noyau sur les processus de fusion et de fission

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(1)Influence de la différence entre les distributions de protons et de neutrons dans le noyau sur les processus de fusion et de fission Artur Dobrowolski. To cite this version: Artur Dobrowolski. Influence de la différence entre les distributions de protons et de neutrons dans le noyau sur les processus de fusion et de fission. Physique mathématique [math-ph]. Université Louis Pasteur - Strasbourg I, 2006. Français. �tel-00391715�. HAL Id: tel-00391715 https://tel.archives-ouvertes.fr/tel-00391715 Submitted on 4 Jun 2009. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) UMR7178. . . . . . . .. IPHC 06-007 N° ordre 5082. Thèse. .. Thèse. Influence de la différence entre les distributions de protons et de neutrons dans le noyau sur les processus de fusion et de fission Présentée par. Artur DOBROWOLSKI. Institut Pluridisciplinaire Hubert Curien 23 rue du Loess BP 28 F-67037 Strasbourg cedex 2 Tél. : (33) 03 88 10 62 14 Fax : (33) 03 88 10 62 92 http://wwwires.in2p3.fr/.

(3) IPHC 06-007 No d’ordre 5082. THESE présentée pour obtenir le grade de. DOCTEUR DE L’UNIVERSITE LOUIS PASTEUR DE STRASBOURG Spécialité: Physique Nucléaire Théorique par. Artur DOBROWOLSKI. Titre:. Influence de la diff´ erence entre les distributions de protons et de neutrons dans le noyau sur les processus de fusion et de fission Influence of differences in the proton and neutron distributions on nuclear fusion and fission. Soutenue a` Lublin (Pologne) le vendredi 21 Avril 2006 devant la commission d’examen : Philippe QUENTIN. Président et Rapporteur externe. Jacek DOBACZEWSKI. Rapporteur externe. Jerzy DUDEK. Rapporteur interne. Johann BARTEL. Directeur de thèse. Krzysztof POMORSKI. Co-Directeur de thèse. Andrzej GOZDZ. Examinateur. Wieslaw KAMINSKI. Examinateur. Wieslaw GRUSZECKI. Membre invité. Ryszard TARANKO. Membre invité. Marek ROGATKO. Membre invité.

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(130) # %&' ( * ) $+* XY)9*\34"5#6+:*A-29*A:CQ3TSTO'm#*A=Lc78#"M* 5#6+)9!a;Lv!*A2RLvI92L'a;aA09"!Lv5*d29*;Ma;"56+!#I'&!)9* Lfm#*;"5L'O#*=Ie09a;:*fL'"BH9I929I+O8*AI9*;"MO'$}!)+*;I6+09"5*=:/10+2}29"M#6Lv6969"5#L'ab)Ie5"512909aA*;2I}!)9*=69"M* m1#095*AaA5#I _o"MM'&.Lv:: #5)9782+*;:q7\L'Z'*A !)+*4LvM!*A7\6+5nI+a;:0+29*CL=M09"M&L'aA*4aA09"MmvLv509"5* !*A"57 B)9ab) L'=5)+ BIRI!)9*8509H+5*;/e09*AI[P5*AaA!#Iu7869"5im#*A-M09H9bL'I[!L':$!)9*829*A&'"57\L !#IHt*;)qLfm1'09" v& !)9*7Lva;"5'5a;'69aC*;I9*A"5O'$8B)9ab)%)qL'lL-29"5*;aglI+q09*AI9a;*4#I\5)9*4M)qL'6t*;U'& Ie09a;:*fL'" qM5'IH9L'"5"M*;"M; XY)+>7\129*A:ubLvZ'*;-L':5}I[5 Lva;a;'09I[=5)9*P&L'aAT5)qLvT5)9*PIe09a;:*fL'" 29*;I+5w$%I9M29*j5)9*>I10+a;:*A09BY: *;"M*;2cB)9*;I!)9*T2+*A&#"M7Lv5#I}YI9aA"5*;L'5*A2 Ht*;a;L'095*=LPa;*A" bL'IR6qLv"M v&BI10+a;:*A#I9-IR5)9*Ie09aA:*fLv"nI[5*;"5#"=7\O#"!Lv5*P!5)9**AI9:L'"MO#I+O50+"M&L'aA*"5*AO#'I wI5)qLvCa;'Ie5* 1 !)+*=M09"M&Lva;*-L'M$1787\*g!"M$!*;"M7 69"M#6N'"M!#IqL':t! I A 6N#!09:Lv!*A2 I}!)9* :/10+2P29"M#6\78129*;:9v& *;/NQ 1@#So'H+bL'I9Î9i LTa;:*fL'"M*;"I[5*;"56+"5*A!Lv!#I[IqLv7\*A:$n5)9*BI9aA"5*;L'5* '&l5)9*8I10+a;:*;L'">*;I9*A"5O'$=L':M2+09*855)9*8I9*;0+5"5#I* aA*;5-B)9ab)RL'696t*fL'"M=I5)9*dIe09a;:*fL'" 509"&L'a;*v XY)+d69"51a;*A58 l)9i *Am#*A" l *fLvZ'*;I+*;2 H[$!)9*cIe09aA:*fLv"8M09"M&Lva;*c!*;I+5'I ps4Pl* afL'Ià;'I9a;:0929*}&"5#7 !)9*L'Htim#* !)+* 3"5#69:*A 129*;:Y"M* M*AaA!P5)9*L'5M097\6!'I`v&409I+&#"M7 29*;I+55*AdL'I92 ^78#"5*Aim#*;" 29!I+O#09M)9*;nHN*gwl*A*;I`69"5'5#IpL'I92pI9*A0+!"M#I29M!"MH90!'I9; ULvC5)9*;"M*A&#"M*=I+*;a;*A55L'"M$c!dIe5"512909aA*j!)9*T&#::iBI9OnI9*A /109L'I[!5*A J!)9abZeI9*;M'&(!)+*=I+*;0+5"5#I5ZeI • t J"5*;:Lv!m#*T* +aA*;M4'&(!)9*>I9*A0+!"M#I29*;I9Mw$ • δ J"5*A:Lv5m#*T2+*Am1Lv5#I}v&u5)9*>29*;I9Mw$}&"M#7 !YIe09a;:*fL'"Y7\LvM!*;"UmvL':09* ρ • ε J5)9**g~*AaA5m#*I10+a;:*;L'"nM09"M&Lva;*%M!09Lv!*A2RHt*Aw *;*;Ip69"M'!'IL'I92 I+*;0+5"5#I 2+M5"5H90 • Σ !#I9A XY)9*I9*A0+!"M#I 5ZeI869"M290+a;*;2 L'%LIqLv!0+"!L':Y!*AI929*AI9aA$ '&j5)9*Ie09aA:*fLv"\7\Lv55*;"d! "5*A7L'IpM$17\78*A5"5acB!) "5*;M6N*AaA8!<69"5'5#I L'I92I+*;0+5"5#I I10+7 HN*A"5A 78*fL'I+85)qLvd&#" "5*;L':M!anIe09aA:*; &'"jB)9ab) N > Z !)9*A"5*d* M!=L%&#"Ma;* B)9ab)<690+5)9*A>5)9*d* a;*A5-'&I9*;0 !"M#I9=#0+>&"5#7 5)9*8H909:Z"5*;O'#II[!c!)+*\M09"M&L'aA*'}XY)9*kl#09:#7 H&#"5aA*Lv551a;Lv!*A2B!) 69"5v!#Id29M!"MH90!'I "5*A2909a;*A5)9u!*;I+29*;I9ag$dH[$ !"$I9OT!TI9a;"M*fL'M*B5)9*Y"!Lv2909^v&t5)9*B69"5v!#I 29!"MH90+5#I~uXY)9^&#"5aA*C&L'aA*;^B5)85)9*B"5*A5bL'I+a;*C#I85)9*C509"&L'a;*BB)9ab)\L':5T!*AI929!>!)9* M$1787\*g!"5ada;#I92+!#I9> *'\B)9*;IR5)9*8I9*;0+5"5#I<5ZeI<291*;=I9'-* f%wIR5)9>aA#I[!* 1-!)9* I9*;0!"5'IMZeI 09I929*A"5!12L'!)9*P29M!L'I9a;* t Ht*Aw *;*AI5)9* :a;Lv!#I9C'&!)9* 6+"5'5#I L'I92 I9*;0!"5'I<29~0+5*;2<509"&L'a;*d69"M q:*;>#"TI'5)9*;"T #"52+ Lv>5)9*d5)9& TB)9ab)>I9*Aa;*A5!Lv"M$! 690+B#I+*j69"5 q:*j'I%!#6}'&G5)9*T'!)+*;";^XY)+U:*fL'29 !P5'7\*j*AI9"5ab)978*;I[B'&G!)9*4509"&L'a;*T:Li$'*;" I I9*;0+5"5#I+jH90+TI+'45%L690+"5* I9*A0+!"M#I:Li$'*;"AjXY)9*PL'Htim#* aA#I9M29*A"!Lv5#I9B$1*A:2 5)qLv!)9* &#"5aA*TB)9ab)Y"M*;56t#I9MH9:*j&#"Y69"M290+a;I+Od5)9*>509"&L'a;*>I9*A0+!"M#I5ZeIYO#m#*AI H[$ 3 Q WE[S t = r JI/Q, . 2. 2/3. 0. 2. 0.

(131) &

(132) &

(133)

(134) B)9*;"M* r 8!)9* aA#I9M!L'I[}"!Lv2909'&>Ie09a;:*fL'"%7\LvM!*;" J !)9*I10+a;:*;L'"%$7878*A!"$ *AI9*;"MO'$ a;1* 8a;*;I[ 9L'I92 Q 5)9*T*AN*;ag!m#*T50+"M&L'aA*TM!~I+*;5YaA*8a;*AI[YXY)9*TI+a;:0+5'I%'&G!)9*4*;:*AaA5"5 M!Lv!aT*;I9*A"5O'$%&'"Y69"5'5#I9-Q*A/e0qL': !8L'Ht#0+ c Z /A B!) c = 3e /5r ≈ 0.7322 M eV S "5*A2909a;*AC5)9Y&#"Ma;* JI L'I92c:*;L'29Y! 3 JI − c ZA Q VeS t= r . 2 Q _q09"M5)9*;"Ba;'I9529*;"5Lv!#I9Y:*;L'2}!P!)+*=78129 qa;Lv!#I%'&(!)+*=2+*;I9#78I9Lv!#" '&o5)9Y* 69"5*A55#I 3 JI − c ZA Q F#S . t= r 2 Q + JA XY)94*;/e0qLv5#IaA#"5"M*;ag!:$ 69"M*;29aA!!)qLv4B)9*;I &#"4L'I09I9ab)9L'"5O#*A2I10+a;:*A09 Q !*;I+29j!%A*;"M L'::N!)9*=* a;*A5jI+*;0+5"5#I9L'"M*=6+095)9*A2!)+*;II[!d!)9*>M09"M&L'aA*%Q t −→ t = 2/3R I S l_q#"CL Ie09a;:*;09C'&oL'"5H9!"5L'"M$\5)qLv6N*T!)+*=*A/e0qLv!#Ic&#" t Y'&o5)9*T&#"57 Q W[S t = t˜ + t¯, B)9*;"M* (B /B ) 3 JI − c ZA Q F?#S t¯ = r , 0. 1. 0. 0. 2. 1/3. 1 12 1. −1/3. 1 12 1 9 4. −1/3. 1. 2. 0. −1/3. max. 2. 0. 1 12 1. Q+. 0. −1/3. vol surf 9 −1/3 JA Bsurf 4. Q Wr[S B)9*;"M*-I%!)9*=L'Htim#* v˜ U!)+*=2+*Am1Lv5#I%&"M#7 5)9*T*;:*;ag!"5'MbL !aT6t'5*;I[!L': v 69"512909aA*;2H[$cL 09I9&#"578:$}2+M5"5H90+!*A2ab)qLv"5O#* Ze B)9:* v˜ B!)9*nmvL':09*='& v˜ #I!)9* M09"M&Lva;* LvI92 v¯˜ B!)9* 509"&L'a;*\Lfm#*A"!L'O#*8'& v˜ 8XY)9*\7\L'aA"5#Ma;#69adH9I92+I9O}*AI9*;"MO'$I!)9*834"M#69:*An 129*;:^>5)9*;I * 69"M*;5M*;2I}!)9*T&#::iBI9O-&#"57 .F20 3 e vs − v¯˜), t˜ = r0 (˜ 8 Q s. s. 1 1 B(A, Z, def ) = − a1 + J δ¯2 − K ε¯2 + M δ¯4 A 2 2 9 J 2 ¯2 2/3 + a2 + δ A Bsurf + a3 A1/3 Bcurv 4 Q +c1 Z 2 A−1/3 BCoul − c2 Z 2 A1/3 Br (def ) − c5 Z 2 Bw (def ) c3 Z 2 c4 Z − − √ + EWig , 3 A 2. Q WD[S B)9*;"M* B B L'I92 B * 69"5*A5C"5*;M6N*AaA5m#*A:$%5)9*>29*A&'"57\Lv!#I%29*A6N*AI929*;I+a;*;'&(!)9* kl#09:#7 Hn*AI9*;"MO'$PLvI92n!)+*U509"&L'a;*la;09"m'L !09"M*U*;I9*A"5O'$' XY)9*l&0+I9aA5#I9 B (def ) L'I92 B (def ) L'"5*B"M*;56t#I9MH9:*B"5*;M6N*AaA5m#*A:$P&#" 5)9*CI9#I1J1)+#7\'O#*;I9*A#09ab)qL'"MO#*B29!"MH90+5#IPI9529*CL'I92\Lv !)9*>M09"M&L'aA*T!)9*>Ie09a;:*;09A XY)+* O#I9*A"U!*;"M7 E Y'&(!)9*T&#"M7 . 0 Q [S E (A, Z) = −10MeV · exp(−42|I|/10). Coul. surf. curv. r. Wig. Wig. w.

(135) &

(136) &

(137)

(138) iE XY)9*=mvL'"ML'H+:*; δ¯ LvI92 ε¯ L'"M*>5)9*-Lim'*;"5L'O#*=/109L'I[!5*A δ ε m'*;"B!)9*>*g~*AaA!m#*=509"&L'a;* Σ L'I92 L'"5*T*A/e0qL':.!% ZA f (def ) I+ Q# S , δ¯ = . 1. 3 c1 2/3 16 Q J −1/3 A Bsurf + 94 Q. i 1h − 2a2 A−1/3 f (def ) + Lδ¯2 + C1 Z 2 A−4/3 g(def ) , K ci. XY)9*-aA*8a;*AI[! a;L'I Ht*>* 69"M*;5M*;2 LvB&'::iB ε¯ =. Q #S. 3 e2 , 5 r0 c21 1 18 + , c2 = 336 J ! K 5 b c3 = c1 , 2 r0 c1 =. . 5 c4 = c1 4 1 c5 = c1 64. . 3 2/3 , 2π ! c21 . Q. L'I92c!)9*>aA*\aA*AIe5B*;I[!*A"5I+O85)9*>*;/e;uQ 1FE[SJ QfE[Sl)qLfm#*>!)9*T&#::iBI+O-mvL':0+*; .F 0 . QiE[S. a2 = 20.69 M eV K = 240.0 M eV L = 100.0 M eV r0 = 1.18 f m J = 36.8 M eV Q = 17.0 M eV e2 = 1.440 M eV · f m. YX )9*>34"5'69:*gY7\129*A:~a;'Ie!L'I+UI9I+*4&"5*A*j6qL'"5L'7\*g!*A"5lB)9ab)}l*A"5*>L'2M0+M!*A2}!P* 6t*;"M 7\*AI[bL'::$jZeI9iBIPI=5)qLv(!78*^7\L'5M*; LvI92 qM5#InHqLv"5"5*;"M;(XY)9*laA#7 H9IqLv!#I>'&+5)9*U,./10+2 34"5'6` 2+*;:4Q5*A*%5*AaA5#I`#S>L'I92 5)9*%3"5#69:*AP 2+*;:29*;Ma;"5HN*A2pL'Ht m'*B)9ab)69"5*A5*;"m#*A #I9:$}!)9*-7\#C"5*;:*AmvL'I[4&*;Lv!0+"5*;'&^Ht'!)!)9*A5*=wl}Lv6969"5#L'ab)9*;4C69"5*A5*;I[5*;2 I!)9*-I9* 1 509H+5*;ag!#I b = 1.0 f m. " * " %&'( #% ) $ +* KI+a;*5)9*2+*fLR'&&% *A!)+*L'I92 <*;A-,'abZv*;" !)+*I9aA:09M#I '&TmvL'"5#09\I9*A !*A"578I !)9* Lfm#*;"5L'O#*bJ1*;I9*A"5O'$8* +6+"5*;M5#I%)qL'Ht*;*;I69"5#6t#M*;2H+0+ 5)9*CHqL'5aBa;#I9aA*;6+l'&.L>ab)qL'"MO#*;2\:/10+2 . . .

(139) . . . .

(140) &

(141) &

(142)

(143) fV 29"5'6"M*;7\L'I9\m'Lv:2~ % #"M5)"M*;78I92+I9OL }!)9}6N#I[}!)qLvLv:"5*;L'2+$I iD['EC:: L'I92 )9*;*A:*A"aA#I9aA:0929*A2 #I 5)9*<H9L'5v&-5)9*_*A"578jO[L'c782+*;: . E 0T!)qL LàA09"MmvLv509"5* 29*;6t*;I+29*;I[U5*;"57 6+"5#6t#"M5#I9L':+! A 5)9#0+:2* M Id!)+*:/e092eJ129"5'6P*;I9*A"5O'$d&09I9ag!#IqL': XY)9*naA09"MmvLv509"5*=!*A"57ULv4:L !*;"B!0929*;2 I .W?20 qB)9*;"M* 5C7\L'O#I+!0+29*TUL'4L'2 M09!*A2!8!)9* * 6t*;"57\*AI[bL': qM5#I1J1HqLv"5"5*;"U)+*;O')e5BZ1I+ BILvY!)9LvY!7\*v KI9aA*n!)+* !"!Lv295#IqLv:.:/e09229"M#6 7829*A:uB!)9#04!)+*P* 69:a;409M*P'&L%a;0+"MmvLv!0+"5* J *;I9*A"5O'$!*;"M7 6t*;"&#"578*;2Lv:"5*;L'2+$ /e095* l*A:: ~#I9* 7\Lf$* +6t*;ag>!)qLv4!)9*PI9a;:095#I'&50+ab) LM09"M&Lva;* Jea;09"m'L !09"M* !*A"57 69:Lf$"!Lv5)9*;"8L"5#:*c'&jaA#"5"M*;aA5#I~ 8!09"MI98#0+d!)qL \M09ab) L 7\129 qafLv5#I4.#H[m1#0+uLvI92=$1*;:2+G5CLCMO#I9 qafLvIeG7\6+"5im#*A7\*AIeGv&5)9* :/e0921J129"M#6j7829*A: &#"57n09:L+nwI<6qLv"M!a;09:L'" N!)9*dI9*g "; 7 ;P2+*Am1Lv5#I hδM i = 0.698 * LvTa;#786qL'"M*;2! * B5)9I\!)9*j5"!L'29!#IqL':tL'696+"5[L'ab) qL'I92}!)+*TI9*A q5M#IeJ1HqL'"M"5*A"U"; 7WA hδM i = 0.732 29*Am1Lv!#I &#"CIe09a;:*;GB5) Z > 70 hδV i = 0.88 * Na;#786qL'"M*;2! hδV i = 5.58 * .Wr20 s MbLv"M!I9OT6t#I[o&'"o!)9*BLvIqL':$1MoB)9ab)d:*fLv29o!j5)9*YI+a;:0+5'I '&ta;09"mvLv!09"M*Y!*;"M7\o !)9*Y *;::qZ1I+ BI8!"5L'295#I9L':+:/e092eJ129"5'6 Ie09a;:*fL'"7\L'5* 69"M*;M5'I'& $#*A"5lL'I92 1B v5*;abZe Q K ,G3TS. V 0BXY)9C* 69"M*;5M#IlL'/10+!*=50+a;a;*A5&09: I "5*A69"512909aAI9O8!)9*=Ie09a;:*fL'"7L'M5*A H90+-=>ZeI9iBIR5)qLv-I:O')e>Ie09aA:*A= m'*;"M*;M57\Lv!*A-5)9* 955#I HqL'"5"M*A"T)9*;O#)[!=He$096 !\L'Ht#0+- * XY)9*nK w,(3HqL'"M"5*A"5CL'"M*nL':58)9O')9*;"Y!)9L'I5)9#M*n*AmvL':0qLv5*;2He$K*;"MZ .WD20qB!)+IP!)9* 09Z LfULiJe&#:2+I9O>7\L'a;"M#5aA#69aY78129*;:oXY)9*,.09H9:I\K1!"5L'5Ht#09"MOn34"M#6Q,(K+3TS &#"57n09:Ln&#"U5)9*>H9I+29I9OP*AI9*;"MO'$ . r 0G)qL'Y5)9*;"M*A&#"M*>5)9*T&#::iBI9On&'"57 . 1/3. B. B. B(Z, N, def ) = avol (1 − κvol I 2 ) A. + asurf (1 − κsurf I 2 ) A2/3 Bsurf (def ). + acur (1 − κcur I 2 ) A1/3 Bcur (def ) 3 e2 Z 2 Z2 − Econgr , + B (def ) − C Coul 4 5 r0ch A1/3 A. Q fVeS B)9*;"M* B (def ) B (def ) B (def ) L'"5*8* 69:aA!:$ afLv:a;0+:Lv5*;2<&'"=5)9*a;L'5*\'&U!)+* 7\129 q*;2 _q09I9I[$#J::C6qL'"5L'78*A!*A"5CI s4696t*;I92+ NXY)+*P,(K+36qL'"5L'78*A!*A"5C:M!*A2Ht*;: Lv"5* L'2M09!*A2!<L'::Y6+"5*;M*;I[!:$`Z1I+ BI #?'#* 6t*;"57\*AI[bL':BH9I929I+O*AI9*;"MO#*A\!L'Z'*AI&"5#7 !)9* bL'H+:*; . 0+0+5I+O4!)9* 7\a;"M#5aA#69aa;'"5"5*AaA5#I9(&"5#7 . #

(144) 09L'I92n5)9*laA#I9O#"M09*;I+a;*Y*;I+*;"5Ov$ E *;!7\Lv5*;2cI . 0 # biWV#D[ P * κ # 'Fr# + a #xiFD[? [?P * κ # 1@'D'E#r a #E+Fr' [d * κ # WE[?''V a # #F ?'#-&7 C # +FDir+= * r XoLvH t 'uL'"!Lv7\*g!*;"MB'&u5)9*=,(K+378129*;: surf. cur. Coul. congr. vol. vol. sur. sur. cur ch 0. cur 4.

(145) &

(146) &

(147)

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(149). . . . !#"%$&'(*)+$&!#,*".-. q_ #"d*AN*;aA5m'* Ie09aA:*;'I1J1Ie09a;:*;#II[!*A"!L'ag!'I9P'&45)9*KZ[$"M7\*w$16N*c!)9*c!vbL':Y*AI9*;"MO'$`'&TL Ie09a;:*;09CBL &09I9ag!#IqL': Q #S E = E ρ (~r), τ (~r), J~ (~r) d r , '&95)9*l:a;L':e29*;I9M5*; ρ (~r) v5)9*lZeI9*A5al*AI9*;"MO'$=29*;I+55*A τ (~r) L'I92n5)9*lMjafL'::*A2-M69I #"5H+ 29*;I+55*A J~ (~r) . fV 0 Z. . q. q. q. q. . 3. q. q. ρq (~r) =. X ν,s. τq (~r) =. X. |ϕν (~r, s, q)|2 nqν ~ ν (~r, s, q)|2 nq |∇ϕ ν. Q i[S B)9*;"M*l!)+*l50+H95aA"56 q={n, p} 29*AI9'5*;o5)9*lIe09aA:*;'I ab)qL'"MO#*lbLv5* ϕ (~r, s, q) L'"M* 5)9*lMI9O#:* 6qL'"!aA:*=ULfm#* &0+I9aA5#I9B!) '"5H9bL':(L'I92 M69I /e0qLvIe5097 Ie097nHN*A"5 ν L'I+2 s "M*;56t*;ag!m#*;:$ L'I92 n HN*AI9O!)+*<aA#"5"M*;56t#I92+I9O aAa;096qL !'IIe097 Ht*;"M Q*;/e0qL':T5<#" Ì 5)9*<6+09"5* 4L'"!"M*;* J1_1abZ%a;L'5* q#" v &o6qLv"5I9O aA#"5"M*;:L !'I9YL'"M*=I9a;:0929*A2Sg wI-!)+*a;L'5*'&+H9"5#Zv*;I=!78* Je"5*Am'*;"M!L':eM$1787\*g!"$ 6qL'"M5aA09:L'"M:$*;I9aA#09I[!*A"5*A2 IT!)+* a;L'5* '&C"5'!Lv!I9OIe09aA:*; !)9*}*AI9*;"MO'$29*AI95w$29*;6t*;I+29IpL'2+295#Io#Ip'!)9*A" :1afLv:l/e0qL'I[!!*; .fD U 20Y509ab) L'8!)9*a;09"M"5*;I[}2+*;I9Mw$ ~j(~r) LvI92 5)9*56+I1Jem'*;ag!#"\29*;I9Mw$ ρ~(~r) wI5)9* ν,s. J~q (~r) = (−i). X. ν,s,s0. ~ ν (~r, s, q) × hs0 |~σ |si nq , ϕ∗ν (~r, s0 , q)∇ϕ ν ν. q ν. 2 ν.

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(156) #)+I !)9*A#"5*A 7 o56t*;"&#"57 29*;I+5w$mvL'"5Lv!#IqL':oafL':a;09:Lv!#I9 GB)9*A"5*%!)+*%:1afL':^2+*;I9M!*; L'"M*85)9*dm'Lv"5L !'IqL':(/109L'I[!5*A; I9*85)+#09:2 u)9i *Am'*;" oZ'*;*A6 I78I92 !)9Lv uL'T!)9* ρ (~r) ]^XB_ &0+I9aA5#IqLv: Lv"5*}'&C5*A7\a;:L'5MafLv: IqL !09"M* ^!)9*}2+*;I9Mw$ &09I+aA!#I9 ρ (~r) #HbL'I9*;2pL' L"5*A509:P'&Y!)+*%m'Lv"5L !'IqL':^69"MaA*;2909"M*}afL'I'I9:$<HN*5*;78aA:L'M5afL':I I9Lv!09"M* ^W*vv&Y!)9* :/10+2 29"5#6w$16N*v=XuL'ZeI+O}I[!}Lva;a;'09I[T!)qLvjIM09ab)L%69"51a;*A5j!)+*P6qL'"M5aA:*=I10+7 HN*A"5 N L'I92 Z 5)9#0+:2HN*TaA#I95*A"Mm'*;2 t#I9*>afLvI&#"M7 09:Lv!*45)9*TmvL'"5Lv!#IqL':t69"5I9a;69:*4I%!)+*>&'"57 δ Q ir[S {E[ρ (~r), ρ (~r)] − λ ρ (~r) − λ ρ (~r)} d r = 0 δρ B5)p5)9*c,GL'O#"5L'I9O#*}7 0+:!69:*;"M λ L'I+2 λ !*;I9M09"5*}5)9*ca;#I9M*;"m'L !'I'&4I+*;0+5"5#I`L'I92 69"5v!#IcIe097 Ht*;"A XY)9u29*;I9Mw$=mvL'"5Lv!#IqL':#6+"5#H9:*;7 )qLvoHt*;*;IPM#:m'*;2 I-w j29~*A"5*;I[uULf$1 I=!)9* 6qL' *;!)9*A"H[$d"M*;5':m1I9OT5)9*C]0+:*;"wJ,GL'O#"5L'I9O#*Y*A/109Lv!#I .W# 1vE20"5*A509:!I9OT&"5#7*A/~Qir[So#"H[$ afL'"M"M$1I+O8'0+Y!)9*TmvL'"MLv5#IqLv:ta;L':aA09:L !'I%IcL'I"5*AM5"5ag!*A2M09H9M6qL'a;*='&(&09I9ag!'I9BL'2qL'6!*;2 !j5)9*U69"M#H9:*A7 ' *'oHt*;I9Oj'&tM*;78a;:L'5Ma;L':eIqLv!0+"5* #&"M*;*Uv&NM)9*;::+#Ma;::L !'I9uIn!)+*UIe09a;:*fL'" . q. q. (T F ) q. 2 2/3. q. 5/3 q. q. q. q. q. Z. n. p. n. n n. p. p p. 3.

(157) . . &

(158).

(159) . & . .