Radu Alexandru GHERGHESCU, Dorin POENARU and Walter GREINER
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1 È Ö Ò Ò Ù Ò Ò Ò ÖÝ ÒÙÐ Ö Ý Ø Ñ Radu Alexandru GHERGHESCU, Dorin POENARU and Walter GREINER IFIN-HH, Bucharest-Magurele, Romania and Frankfurt Institute for Advanced Studies, J W Goethe University Frankfurt am Main, Germany Exciting Physics Symposium Makutsi Safari Farm November 2011, South Africa Pairing influence in binary nuclear systems p.1/26
2 ÇÙØÐ Ò ÓÖÑ Ø ÓÒ Ô ÓÖÑ ØÛÓ¹ ÒØ Ö ÐÐ ÑÓ Ð Ò ÖÝ Ñ ÖÓ ÓÔ ¹Ñ ÖÓ ÓÔ ÔÔÖÓ È Ö Ò ÓÖÖ Ø ÓÒ Å Ø Ò ÓÖ Ò ÝÒ Ñ ÆÙÐ Ö Ò ÖØ È Ë ÖÖ Ö Ò Ô Ò ØÖ Ð Ø ÓÖ ÝÒØ Ó Z ½½ ÓØÓÔ Pairing influence in binary nuclear systems p.2/26
3 ÓÖÑ Ø ÓÒ Ô b P b T a T a P R O T z s O P Å ÝÑÑ ØÖÝ η = (A ½º T A P )/(A T +A P ) Ø Ò ØÛ Ò ÒØ Ö R ¾º ÓÖÑ Ø ÓÒ Ô Ö Ñ Ø Ö Ó Ø ÝÒØ Þ Ô Ö Òص ÒÙÐ Ù b º 0 /a 0 ÓÖÑ Ø ÓÒ Ô Ö Ñ Ø Ö Ó Ø ÚÝ Ö Ñ ÒØ Ì Ö Øµ b º T /a T ÓÖÑ Ø ÓÒ Ô Ö Ñ Ø Ö Ó Ø Ð Ø Ö Ñ ÒØ ÈÔÖÓ Ø Ð µ b º P /a P Pairing influence in binary nuclear systems p.3/26
4 ÓÖÑ Ø ÓÒ Ô A W 96 Zr Er 112 Pd Dy 118 Cd Sm 128 Te Ba 136 Ce (R-R f )/(R t -R f ) Pairing influence in binary nuclear systems p.4/26
5 V DTCSM (ρ,z) = V 1 (ρ,z),v 1 V g1 (ρ,z),v g1 V g2 (ρ,z),v g2 V 2 (ρ,z),v 2 ÓÖÑ ØÛÓ¹ ÒØ Ö Ó ÐÐ ØÓÖ ÔÓØ ÒØ Ð Û Ö V 1 (ρ,z) = 1 2 m oω 2 ρ 1 ρ m oω 2 z1(z +z 1 ) 2 V g1 (ρ,z) = 2V 0 [ 1 2 m oω 2 g(ρ ρ 3 ) m oω 2 g(z z 3 ) 2 ] V g2 (ρ,z) = V 0 V 2 (ρ,z) = 1 2 m oω 2 ρ 2 ρ m oω 2 z2(z z 2 ) 2 Pairing influence in binary nuclear systems p.5/26
6 Ì Û ÓÐ ÔÓØ ÒØ Ð ÑÙ Ø ÑÓÓØ ÐÝ Ò ÓÒØ ÒÙÓÙ ÐÝ ÓÚ Ö Ø Ñ Ò Ô Ó ÓÖÑ Ø ÓÒº ÑÙÐØ Ñ Ò ÓÒ Ð Ú ÖÝ ØÛÓ¹ÔÓØ ÒØ Ð ÒØ Ö Ø ÓÒ Ò Ô Ò ÒØ ÖÓÑ Ø ÓØ Ö ÀÝÔÓØ ØÛÓº Å Ø Ò ÔÓØ ÒØ Ð ÙÖ V g (ρ,z) V 1 (ρ,z) V 2 (ρ,z) V 1 (ρ,z) = V 2 (ρ,z) MPE V g (ρ,z) = V 1 (ρ,z) MPE1 V g (ρ,z) = V 2 (ρ,z) MPE2 Pairing influence in binary nuclear systems p.6/26
7 ÌÓØ Ð À Ñ ÐØÓÒ Ò H DTCSM = 2 2m 0 +V DTCSM (ρ,z)+v Ωs +V Ω 2 Ô Ö Ð ω ρ1 = ω ρ2 = ω 1 ÓÖ ÔÓØ ÒØ Ð V (d) V (d) 1 (ρ,z) = 1 2 (ρ,z) = m 0ω1ρ m 0ω1(z 2 +z 1 ) 2,z 0 V (d) 2 (ρ,z) = 1 2 m 0ω1ρ m 0ω2(z 2 z 2 ) 2,z 0 ÓÒ Ð Þ Ø ÓÒ 1 Φ m (φ) = 2π exp(imφ) R m n ρ (ρ) = ( 2Γ(nρ +1)α 2 1 Γ(n ρ + m +1) )1 2 exp ( α2 1 ρ2 2 )(α 2 1ρ 2 ) m 2 L m n ρ (α 2 1ρ 2 ) Z ν (z) = C ν1 exp C ν2 exp [ α2 1 (z+z 1) 2 2 [ α2 2 (z z 2) 2 2 ] H ν1 [ α 1 (z +z 1 )], z < 0 ] H ν2 [α 2 (z z 2 )], z 0 Pairing influence in binary nuclear systems p.7/26
8 H DTCSM ÓÔ Ö ØÓÖ Ç ÐÐ ØÓÖ ÓÔ Ö ØÓÖ V 1 (ρ,z) = V 1 (ρ,z) V (d) (ρ,z),v 1 V 2 (ρ,z) = V 2 (ρ,z) V (d) (ρ,z),v 2 V g (ρ,z) = V g (ρ,z),v g ls l 2 Ò ËÔ Ò¹ÓÖ Ø ÔÓØ ÒØ Ð { } κ 1 (ρ,z),( V p)s m V ls = { 0 ω 01 } κ 2 (ρ,z),( V p)s m 0 ω 02 V l 2 = { { m 2 0 ω3 01 m 2 0 ω3 02 κ 1 µ 1 (ρ,z),( V p) 2 } κ 2 µ 2 (ρ,z),( V p) 2 },A 1 region,a 2 region,a 1 region,a 2 region Pairing influence in binary nuclear systems p.8/26
9 ËÔ Ò¹ÓÖ Ø ls Ò l 2 ÓÔ Ö ØÓÖ ls 1 2 (Ω+ s +Ω s + )+Ω z s z Ò Ö Ð ÜÔÖ ÓÒ Ô Ò ÒØ ¹ ÓÔ Ö ØÓÖ Ë Ô Ω [ Ω + (v 1 ) = e iϕ V1 (ρ,z) ρ z V 1(ρ,z) z ρ i ρ V 1 (ρ,z) z ] ϕ ] = e [m iϕ 0 ωρ 2 1 ρ z m 0ωz 2 1 (z +z 1 ) ρ i ρ m 0ωz 2 1 (z +z 1 ) ϕ Ω (v 1 ) = e iϕ [ V1 (ρ,z) ρ Ω z (v 1 ) = i ρ z V 1(ρ,z) z ρ + i ρ V 1 (ρ,z) z ] ϕ ] = e [m iϕ 0 ωρ 2 1 ρ z m 0ωz 2 1 (z +z 1 ) ρ + i ρ m 0ωz 2 1 (z +z 1 ) ϕ V 1 ρ ϕ = im 0 ω 2 ρ 1 ϕ Pairing influence in binary nuclear systems p.9/26
10 ËÔ Ò¹ÓÖ Ø ÒØ Ö Ø ÓÒ ÌÓØ Ð Ô Ò¹ÓÖ Ø ÓÔ Ö ØÓÖ V Ωs (v 1 ) = m 0 ω 01 κ 1 {Ωs(v 1 ),(v 1 )} V Ωs (v 2 ) = m 0 ω 02 κ 2 {Ωs(v 2 ),(v 2 )} V Ωs (v g ) = m 0 ω 01 κ 1 {Ωs(v g1 ),(v g1 )} m 0 ω 02 κ 2 {Ωs(v g2 ),(v g2 )} Pairing influence in binary nuclear systems p.10/26
11 Ì ËÅ Ñ ØÖ Ü ÌÓØ Ð Ñ ØÖ Ü Ð Ñ ÒØ i DTCSM j = E (d) osc(n ρ, m,ν) + i V 1 j + i V 2 j + i V g j + i V Ωs j + i V Ω 2 j E osc (d) Ø ÓÒ Ð Þ Ó ÐÐ ØÓÖ Ò Ö Ý Û Ö E (d) osc = ω 1 (2n ρ + m +1)+ ω 1 (ν ) ÓÒ Ó Ø Ò Ø ØÓØ Ð Ò ÖÝ ºÔº º Ò Ö Ý Ð Ú Ð {ǫ ÓÒ Ð Þ Ø ÓÒ k } Ø ÓÖ Ø ÐÙÐ Ø ÓÒ Ó ÐÐ Ò Ô Ö Ò ÓÖÖ Ø ÓÒ º ÒÔÙØ Pairing influence in binary nuclear systems p.11/26
12 {E sp } ǫ i ÒÔÙØ Ø ØÓ ÐÙÐ Ø E shell δu = i ǫ i Ũ Ë ÐÐ ÓÖÖ Ø ÓÒ Ũ ËÑÓÓØ Ò ÑÓÓØ Ð Ú Ð ØÖ ÙØ ÓÒ g(ǫ) ½µ g(ǫ) = 1 ( ) γ ζ ǫ ǫ g(ǫ γ )dǫ = 1 ( ) γ i=1 ζ ǫ ǫi γ Û Ö Ø ÑÓÓØ Ò ÙÒØ ÓÒ ζ(x) = 1 π exp( x 2 ) m a 2k H 2k (x) k=0 ¾µ Ì Ò Ø ÑÓÓØ Ô ÖØ Ñ Ö Ò Ũ = ω = λ g(ǫ)ǫdǫ Pairing influence in binary nuclear systems p.12/26 µ Û Ö Ø ÑÓÓØ ÖÑ Ð Ú Ð ÓÑ ÖÓÑ N e ÓÒ ÖÚ Ø ÓÒº
13 È Ö Ò ÒØ Ö Ø ÓÒ»¾ Ð Ú Ð Ö ÓÙÔ ÍÌ ÓÒÐÝ n Ð Ú Ð ÐÐÓÛ Ò n Ð Ú Ð ÆÓÖÑ ÐÝ Ø ÖÑ Ò Ö Ý ÓÒØÖ ÙØ ØÓ Ø Ô Ö Ò ÒØ Ö Ø ÓÒº ÓÚ Ω Ø ÙØÓ Ò Ö Ý n = n = Ω g Á s Ò /2 = 12/ A ω0 0º ÕÙ Ø ÓÒ Ë 0 = k f ǫ k λ (ǫk λ) k i 2 G = k f (ǫk λ) k i 1 k i = Z/2 n+1, k f = Z/2+n, 2 G 2 g( λ)ln( 2Ω ) Pairing influence in binary nuclear systems p.13/26 ËÙÔÔÓ Ø Ø Ô Ö Ò ØÖ Ò Ø G Ø Ñ ÓÖ ÙÒ ÓÖÑ ØÖ ÙØ ÓÒ
14 È Ö Ò ÒØ Ö Ø ÓÒ ÓÒ ÕÙ Ò Ó Ø Ô Ö Ò ÓÖÖ Ð Ø ÓÒ Ø Ð Ú Ð ÐÓÛ Ø ÖÑ Ò Ö Ý ÓÒÐÝ Ô ÖØ ÐÐÝ ÐÐ Û Ð Ø Ó ÓÚ Ø ÖÑ Ò Ö Ý Ö Ô ÖØ ÐÐÝ ÑÔØݺ Ö Á {ǫ k } Ö Ø Ì ËÅ Ò Ð Ô ÖØ Ð Ò Ö Ø Ò v 2 k = [1 (ǫ k λ)/e k ]/2 Ò u 2 k = 1 v 2 k Û Ö Ø ÕÙ ¹Ô ÖØ Ð Ò Ö Ý E k = (ǫ k λ) Û Ö λ Ò Ö Ø ÓÐÙØ ÓÒ Ó Ø Ë Ý Ø Ñ Ó ÕÙ Ø ÓÒ º Pairing influence in binary nuclear systems p.14/26
15 È Ö Ò ÒØ Ö Ø ÓÒ δp = p p Ì Ô Ö Ò ÓÖÖ Ø ÓÒ Ø Ö Ò ØÛ Ò Ø Ô Ö Ò ÓÖÖ Ð Ø ÓÒ Ò Ö ÓÖ Ø Ö ÔÖ ÒØ Ð Ú Ð ØÖ ÙØ ÓÒ Ö Ø p = k f k=k i 2v 2 kǫ k 2 Z/2 ǫ k 2 G k=k i Ò ÓÖ Ø ÓÒØ ÒÙÓÙ Ð Ú Ð ØÖ ÙØ ÓÒ p = ( g 2 )/2 = ( g s 2 )/4 ØÓ ÐÐ ÓÖÖ Ø ÓÒ Ø Ô Ö Ò ÓÖÖ Ø ÓÒ ÓÙØ Ó Ô Ò ÓÑÔ Ö ÇÒ Ò δp = δp Ñ ÐÐ Öº p +δp Ò n δe δu+δpº = Pairing influence in binary nuclear systems p.15/26
16 236 Pu 118 Ag Ag Ü ÑÔÐ 236 ÈÙ E (MeV) shell corr. p shell corr. n shell corr. p+n pair corr. p+n shell + pair corr (R-R i )/(R t -R i ) 236 Ô Ö Ò ÓÖÖ Ø ÓÒ ÓÖ ÈÙº Ì ºÔº º Û Ö Ó Ø Ò Û Ø Ø Ë ÐÐ Ò ÑÓ Ðº ÐÐ ØÛÓ¹ ÒØ Ö Pairing influence in binary nuclear systems p.16/26
17 Å ÖÓ ÓÔ Ò Ö Ý E C = 2π 3 ρ e zmax dz zmax dz F C (z,z ) z min z min Ì Ñ ÖÓ ÓÔ Ò Ö Ý Ø ÙÑ Ó ÓÙÐÓÑ E C Ò Ù Û E Y Û Ö F C (z,z ) ¹ Ô Ô Ò ÒØ Ò ÖÝ Ù ÓÒ ÓÒ ÙÖ Ø ÓÒ E C = 2π 3 (ρ2 e1f C1 +ρ 2 e2f C2 +2ρ e1 ρ e2 F C12 ) Û Ö ρ e1 Ò ρ e2 Ö Ø Ö Ò Ø Ò E Y = 1 4πr0 2 [c s1 F EY1 +c s2 F EY2 +2(c s1 c s2 ) 1/2 F EY12 ] Ì ØÓØ Ð ÓÖÑ Ø ÓÒ Ò Ö Ý E def = E C +E Y +δu+δp Ì ÙÖ c si = a s (1 κi 2 i) where I i = (N ix Z ix )/A ix Pairing influence in binary nuclear systems p.17/26 Ò Ö Ý ÓÒ Ø ÒØ
18 ÓÖÑ Ø ÓÒ Ú Ö Ð ÚÓÐÙØ ÓÒ χ T χ P [ ) ] R RkT 2 = χ T0 +(χ 0 χ T0 )exp ( R t R f k T [ ) ] R RkP 2 = χ P0 +(χ Pf χ P0 )exp ( R t R f k P Î Ö Ø ÓÒ Ó Ø Ñ Ü Ö Ø Ó χ T = b T /a T Ò χ P = b P /a P Û Ö χ Pf = χ P0 i P 10 (χ 0 χ P0 ) b P χ T χ P = b P (k T,k P,i P ;R) = χ T (k T,k P,i P ;R) = χ P (k T,k P,i P ;R) Ö Ú Ö Ð ÙÒØ ÓÒ Ó R Pairing influence in binary nuclear systems p.18/26
19 Å Ø Ò ÓÖ Ò ÝÒ Ñ Ö Ò Ò ÑÓ Ð Ì Ò ÓÖ ÓÒØÖ Ø ÓÒ ÐÓÒ R B(R) = B bp b P ( dbp dr ) 2 db P dχ T +2BbP χ T dr dr +2B db P b P χ ) P dr 2 dχ T +2BχT χ P ( 2B bp R db P dr +B dχt χ T χ T dr ( 2B χt R dχ T dr +B dχp χ P χ P dr dr dχ P dr + dχ P dr + ) 2 +2BχP R dχ P dr +B RR È Ò ØÖ Ð ØÝ P ÓÖ Ú Ò Ù ÓÒ Ô Ø (fus) P = exp( K ov ) Û Ö K ov (b P,κ T,κ P ;R) = 2 (fus) [2B(R) bp,κ T,κ P E def (R) bp,κ T,κ P ] 1/2 dr Pairing influence in binary nuclear systems p.19/26
20 Ö Ò Ò Ñ Ø Ò ÓÖ Ö Ò Ò ÑÓ Ð Ì Ò ÓÖ ÓÒØÖ Ø ÓÒ ÐÓÒ R Ö ÙÐØ Ò Ð Ö ÓÖÑ B ε = 2 2 νµ ν V DTCSM / ε µ µ V DTCSM / ε ν (E ν +E µ ) 3 (u ν v µ +u µ v ν ) 2 Û Ö V DTCSM = V 2osc DTCSM +V ls +V l 2 Ò Ø º Ôº º Ö Ø Ì ËÅ Û Ú ÙÒØ ÓÒ º µ = DTCSM(n ρ,n z,m ρ,z,φ) Pairing influence in binary nuclear systems p.20/26
21 È Ë ÓÖ 292 ½¾¼ E def (MeV) R-R f 120 Sn+ 172 Yb Sn+ Er A Ce+ Nd 146 Nd+ 146 Nd Pairing influence in binary nuclear systems p.21/26
22 120 ËÒ ½¾¼ Ë ÐÐ Ò Ô Ö Ò ÓÖ 292 ½¾¼ E shp, Pp, Ep (MeV) Pp E shp Pp + E shp E shn, Pn, En (MeV) Pn E shn Pn + E shn P E sh, P, E (MeV) E shp + Esh (R-R i )/(R-R t ) Pairing influence in binary nuclear systems p.22/26
23 ÓÐ Ù ÓÒ ÒÒ Ð ÓÖ 292 ½¾¼ Ê Ø ÓÒ E b Šε log 10 P Ê Ø ÓÒ E b Šε log 10 P 146 Æ 146 Æ º ¹ º¾ 120 ËÒ 172 º ¹ º¼ 118 ËÒ 174 º ¹ º À º ¹ º ½ Pairing influence in binary nuclear systems p.23/26
24 È Ë ÓÖ 282 ½¾¼ E def (MeV) R-R f 120 Cd+ 162 Hf Te+ Dy A Ce+ Nd 140 Nd+ 142 Nd Pairing influence in binary nuclear systems p.24/26
25 ÓÐ Ù ÓÒ ÒÒ Ð ÓÖ 282 ½¾¼ Ê Ø ÓÒ E b Šε log 10 P Ê Ø ÓÒ E b Šε log 10 P 140 Æ 140 Æ º ¹ º À º¼ ¹ º 112 È 170 Ï º ½ ¹ º Pairing influence in binary nuclear systems p.25/26
26 ÈÓ Ð ÓÒÐÙ ÓÒ Ô Ð Þ Ò ÖÝ Ñ ÖÓ ÓÔ ¹Ñ ÖÓ ÓÔ ÑÓ Ð ÔÔÐ ØÓ ÐÙÐ Ø Ø ÓÖÑ Ø ÓÒ Ò Ö Ýº Å Ø Ò ÓÖ Ò ÑÙÐØ Ñ Ò ÓÒ Ð Ñ Ò Ñ Þ Ø ÓÒ Ó Ø ÓÒ ÒØ Ö Ð Ù ØÓ Ó Ø Ò ÏÃ Ô Ò ØÖ Ð Ø º ÓÖ ÓÖ ½¾¼ ½¾¼ Æ Æ ¹ º¾ ¹ º Æ 146 logp 140Æ 142 logp 120 ËÒ 172 logp ¹ º À logp ¹ º À logp ¹ º ½ 112 È 170 Ï logp ¹ º Pairing influence in binary nuclear systems p.26/26
F(jω) = a(jω p 1 )(jω p 2 ) Û Ö p i = b± b 2 4ac. ω c = Y X (jω) = 1. 6R 2 C 2 (jω) 2 +7RCjω+1. 1 (6jωRC+1)(jωRC+1) RC, 1. RC = p 1, p
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