Introduc7on: heavy- ion poten7al model for sub- barrier fusion calcula7ons

Transcription

1 Introduc7on: heavy- ion poten7al model for sub- barrier fusion calcula7ons Phenomenological heavy-ion potential 60 Ni + 89 Y point Coulomb potential V (MeV) total heavy-ion potential 0 nuclear Woods-Saxon potential r (fm) 1/9/13 Volker Oberacker, Vanderbilt 1

2 Introduc7on: WKB barrier transmission probability 140 Heavy-ion potential 60 Ni+ 89 Y Coulomb + nuclear Woods-Saxon + centrifugal L= L=20 L=0 E cm =124 MeV WKB barrier transmission probability (see e.g. Shankar, QM, p. 444) T ( E cm ) = exp 2S E cm ( ) [ ] V L (MeV) where the WKB barrier penetration integral is given by r (fm) 1/9/13 Volker Oberacker, Vanderbilt 2

3 Introduc7on: Hill- Wheeler extension of WKB method barrier transmission probability (Hill-Wheeler extension of WKB) T ( E cm ) = In the limit exp( 2S ) >>1 we recover the standard WKB result. 1 [ ( )] 1+ exp 2S E cm barrier transmission coefficient T L Sub-barrier fusion, 60 Ni+ 89 Y, E cm =124 MeV Woods-Saxon potential, WKB / Hill Wheeler partial wave L 1/9/13 Volker Oberacker, Vanderbilt 3

4 Introduc7on: par7al and total fusion cross sec7on 10-2 Sub-barrier fusion, 60 Ni+ 89 Y, E cm =124 MeV Woods-Saxon potential, WKB / Hill Wheeler partial fusion cross section 10-3 σ ( E cm ) = π 2 ( ) ( 2 +1)T 2µE E cm cm sigma_l (mb) total fusion cross section σ tot ( E cm ) = σ ( E cm ) = partial wave L 1/9/13 Volker Oberacker, Vanderbilt 4

5 Theoretical description of heavy-ion fusion at sub-barrier energies! Barrier penetration models (phenomenological heavy-ion potentials) Balantekin and Takigawa, Rev. Mod. Phys. 70, 77 (1998)! Coupled channels calculations (phenomenological heavy-ion potentials) S. Misicu and H. Esbensen, PRL 96, (2006) Ichikawa, Hagino, and Iwamoto, PRC 75, (2007)! TDHF with density-constraint: microscopic calculation of heavy-ion potentials + barrier tunneling Umar and Oberacker, Phys. Rev. C 74, (R) (2006) Umar and Oberacker, Phys. Rev. C 76, (2007) Umar, Oberacker, Maruhn, and Reinhard, Phys. Rev. C 81, (2010) Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) 1/9/13 Volker Oberacker, Vanderbilt 5

6 Density-constrained TDHF calculations microscopic calculation of heavy-ion potentials V(R) and coordinate-dependent mass parameters M(R) barrier tunneling: exact numerical solution of Schrödinger eq. for relative coordinate R with Incoming Wave Boundary Condition Ref: Hagino, Rowley, and Kruppa, Comput. Phys. Commun. 123, 143 (1999) calculate fusion cross sections below and above the barrier 1/9/13 Volker Oberacker, Vanderbilt 6

7 TDHF with Density-Constraint (DC-TDHF) A method to extract internal excitation energy while holding the instantaneous neutron and proton densities constrained. TDHF provides the dynamical densities for calculating V(R) ρ TDHF (r,t) E*(R(t)) quasi-static energy surface E DC (R(t)) ion-ion potential 1/9/13 Volker Oberacker, Vanderbilt 7

8 Calculation of heavy-ion potential with DC-TDHF method #" " #" " Ca+ 132 Sn E cm =140 MeV #" $%!!&"'() #" $%##&'()*!" #" " #"!" #" " #" #" #" " " #" $%#"&'()* #" $%&'(()*+!" #" " #"!" #" " #" V(R) contains dynamical entrance channel effects (neck formation, particle transfer, surface vibrations, giant resonances) 1/9/13 Volker Oberacker, Vanderbilt 8

9 Heavy-ion potential: strong E cm -dependence Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) 132 Sn + Ca 120 DC-TDHF V(R) (MeV) 100 =180 MeV =140 MeV Point Coulomb =125 MeV =117 MeV =114 MeV R (fm) 1/9/13 Volker Oberacker, Vanderbilt 9

10 Dynamical Effective Mass from TDHF TDHF DC- TDHF 1/9/13 Volker Oberacker, Vanderbilt 10

11 Mass parameter: strong E cm -dependence Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) Sn + Ca M(R)/µ 10 5 DC-TDHF = 180 MeV = 140 MeV = 117 MeV = 114 MeV R (fm) 1/9/13 Volker Oberacker, Vanderbilt 11

12 Transformed heavy-ion potential Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) 132 Sn + Ca V(R),U(R) (MeV) =180 MeV =140 MeV =114 MeV DC-TDHF Point Coulomb R,R (fm) 1/9/13 Volker Oberacker, Vanderbilt 12

13 Fusion / capture cross section for two spherical nuclei total fusion cross section Schrödinger equation for transformed radial coordinate # 2 % $ 2µ d 2 dr + 2 ( +1) 2 & +U(R ) E 2µR 2 c.m. ( φ (R) = 0 ' reduced mass transformed potential solve Schrödinger equation numerically, with Incoming Wave Boundary Condition (IWBC) 1/9/13 Volker Oberacker, Vanderbilt 13

14 Total fusion cross section: TDHF and DC-TDHF Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) Sn+ Ca (mb) unrestricted TDHF DC-TDHF - specific potential potential at =114 MeV potential at =140 MeV potential at =180 MeV (MeV) 1/9/13 Volker Oberacker, Vanderbilt 14

15 Fusion cross sections: theory vs. experiment Scaling parameters: R 0 = A 1 1/ 3 + A 2 1/ 3 B 0 = Z 1 Z 2 /R Sn+ Ca (mb) / R exp. (HRIBF) DC-TDHF and TDHF Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, (2012) 124 Sn+ 40 Ca 10-2 exp. (HRIBF) TDHF +BCS/LN 1/9/ / B 0 Volker Oberacker, Vanderbilt 15

16 64 Ni Sn: heavy-ion interaction potential for deformed + spherical nuclei ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, β = 0 o 64 Ni Sn Δβ = 10 o V(R,β) (MeV) β = 90 o Heavy-ion potential depends on initial orientation angle β of deformed nucleus R (fm) 1/9/13 Volker Oberacker, Vanderbilt 16

17 64 Ni Sn Fusion Cross-Section DC-TDHF theory Umar and Oberacker, ( 2007 ) PRC 76, Exp. Data (HRIBF, ORNL) J.F. Liang et al., PRL 91, (2003) PRC 75, (2007) PRC 78, (2008) 1/9/13 Volker Oberacker, Vanderbilt 17

18 16 O Pb fusion ( 2009 ) Umar and Oberacker, Eur. Phys. J. A 39, 243 V(R),U(R) (MeV) = 76 MeV = 78 MeV = 90 MeV 16 O Pb Point Coulomb R,R (fm) σ f (mb) O Pb = 76 MeV = 78 MeV = 90 MeV (MeV) 1/9/13 Volker Oberacker, Vanderbilt 18

19 64 Ni + 64 Ni fusion ( 2008 ) Umar and Oberacker, Phys. Rev. C 77, V(R) (MeV) (β 1 = 0 o, β 2 = 90 o ) (β 1 = 90 o, β 2 = 90 o ) (β 1 = β 2 = 0 o ) 64 Ni+ 64 Ni R(fm) σ (mb) Ni + 64 Ni data DC-TDHF (Core) (MeV) 1/9/13 Volker Oberacker, Vanderbilt 19

20 Heavy-ion fusion leading to superheavy element Z=112 spherical + spherical: cold fusion (E* small) spherical + deformed: hot fusion (E* large) 1/9/13 Volker Oberacker, Vanderbilt 20

21 Ca U heavy-ion potential Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, (2010) V(R) (MeV) } Exp. Energies β = 45 o Point Coulomb =250 MeV =220 MeV =200 MeV}DC-TDHF =185 MeV Ca U R (fm) 1/9/13 Volker Oberacker, Vanderbilt 21

22 E* (MeV) Ca U: excitation energy Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, (2010) Ca U =250 =220 MeV}DC-TDHF =200 MeV =185 MeV R (fm) 1/9/13 Volker Oberacker, Vanderbilt 22

23 Ca U capture cross section Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, (2010) 10 3 σ capture (mb) Ca U DC-TDHF Experiment: Oganessian et al., J. Phys. G 34 R165 (2007) (MeV) 1/9/13 Volker Oberacker, Vanderbilt 23

24 70 Zn Pb heavy-ion potential ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, V(R) (MeV) }Exp. Energies Zn + Pb Point Coulomb E = 400 MeV c.m. = 350 MeV = 290 MeV}DC-TDHF = 280 MeV = 265 MeV R (fm) 1/9/13 Volker Oberacker, Vanderbilt 24

25 Excitation energies at capture point in hot and cold fusion ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, E* c (MeV) Ca U (β = 0 o ) Ca+ 238 U (β = 45 o ) Ca U (β = 90 o ) 70 Zn Pb (MeV) 1/9/13 Volker Oberacker, Vanderbilt 25