Introduc7on: heavy- ion poten7al model for sub- barrier fusion calcula7ons 200 160 Phenomenological heavy-ion potential 60 Ni + 89 Y point Coulomb potential V (MeV) 120 80 40 total heavy-ion potential 0 nuclear Woods-Saxon potential -40-80 6 8 10 12 14 16 r (fm) 1/9/13 Volker Oberacker, Vanderbilt 1
Introduc7on: WKB barrier transmission probability 140 Heavy-ion potential 60 Ni+ 89 Y Coulomb + nuclear Woods-Saxon + centrifugal L=40 130 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) 120 110 where the WKB barrier penetration integral is given by 100 8 10 12 14 16 r (fm) 1/9/13 Volker Oberacker, Vanderbilt 2
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 10-2 10-3 10-4 10-5 10-6 10-7 Sub-barrier fusion, 60 Ni+ 89 Y, E cm =124 MeV Woods-Saxon potential, WKB / Hill Wheeler 10-8 0 10 20 30 40 partial wave L 1/9/13 Volker Oberacker, Vanderbilt 3
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) 10-4 10-5 10-6 total fusion cross section σ tot ( E cm ) = σ ( E cm ) =0 10-7 0 10 20 30 40 partial wave L 1/9/13 Volker Oberacker, Vanderbilt 4
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, 112701 (2006) Ichikawa, Hagino, and Iwamoto, PRC 75, 057603 (2007)! TDHF with density-constraint: microscopic calculation of heavy-ion potentials + barrier tunneling Umar and Oberacker, Phys. Rev. C 74, 021601(R) (2006) Umar and Oberacker, Phys. Rev. C 76, 014614 (2007) Umar, Oberacker, Maruhn, and Reinhard, Phys. Rev. C 81, 064607 (2010) Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 1/9/13 Volker Oberacker, Vanderbilt 5
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
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
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
Heavy-ion potential: strong E cm -dependence Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 132 Sn + Ca 120 DC-TDHF V(R) (MeV) 100 =180 MeV =140 MeV Point Coulomb =125 MeV =117 MeV =114 MeV 80 10 12 14 16 18 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 9
Dynamical Effective Mass from TDHF TDHF DC- TDHF 1/9/13 Volker Oberacker, Vanderbilt 10
Mass parameter: strong E cm -dependence Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 15 132 Sn + Ca M(R)/µ 10 5 DC-TDHF = 180 MeV = 140 MeV = 117 MeV = 114 MeV 0 10 12 14 16 18 20 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 11
Transformed heavy-ion potential Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 132 Sn + Ca V(R),U(R) (MeV) 120 100 =180 MeV =140 MeV =114 MeV DC-TDHF Point Coulomb 80 10 12 14 16 18 R,R (fm) 1/9/13 Volker Oberacker, Vanderbilt 12
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
Total fusion cross section: TDHF and DC-TDHF Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 10 3 10 2 132 Sn+ Ca (mb) 10 1 10 0 10-1 unrestricted TDHF DC-TDHF - specific potential potential at =114 MeV potential at =140 MeV potential at =180 MeV 110 120 130 140 (MeV) 1/9/13 Volker Oberacker, Vanderbilt 14
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 0 10 1 10 0 132 Sn+ Ca (mb) / R 0 2 10-1 exp. (HRIBF) DC-TDHF and TDHF Oberacker, Umar, Maruhn, and Reinhard, Phys. Rev. C 85, 034609 (2012) 124 Sn+ 40 Ca 10-2 exp. (HRIBF) TDHF +BCS/LN 1/9/13 0.85 0.9 0.95 1 1.05 1.1 1.15 / B 0 Volker Oberacker, Vanderbilt 15
64 Ni + 132 Sn: heavy-ion interaction potential for deformed + spherical nuclei ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, 014614 160 150 β = 0 o 64 Ni + 132 Sn Δβ = 10 o V(R,β) (MeV) 140 130 120 β = 90 o Heavy-ion potential depends on initial orientation angle β of deformed nucleus 110 8 10 12 14 16 18 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 16
64 Ni + 132 Sn Fusion Cross-Section DC-TDHF theory Umar and Oberacker, ( 2007 ) PRC 76, 014614 Exp. Data (HRIBF, ORNL) J.F. Liang et al., PRL 91, 152701 (2003) PRC 75, 054607 (2007) PRC 78, 047601 (2008) 1/9/13 Volker Oberacker, Vanderbilt 17
16 O + 208 Pb fusion ( 2009 ) Umar and Oberacker, Eur. Phys. J. A 39, 243 V(R),U(R) (MeV) 75 70 65 60 55 50 = 76 MeV = 78 MeV = 90 MeV 16 O + 208 Pb Point Coulomb 6 8 10 12 14 16 18 20 R,R (fm) σ f (mb) 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 16 O + 208 Pb = 76 MeV = 78 MeV = 90 MeV 10-6 65 70 75 80 85 90 95 100 105 110 (MeV) 1/9/13 Volker Oberacker, Vanderbilt 18
64 Ni + 64 Ni fusion ( 2008 ) Umar and Oberacker, Phys. Rev. C 77, 064605 V(R) (MeV) 100 95 90 85 80 75 70 65 (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 60 8 10 12 14 16 R(fm) σ (mb) 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 64 Ni + 64 Ni data DC-TDHF (Core) 85 90 95 100 105 110 (MeV) 1/9/13 Volker Oberacker, Vanderbilt 19
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
Ca + 238 U heavy-ion potential Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, 064607 (2010) V(R) (MeV) 200 180 160 140 120 } Exp. Energies β = 45 o Point Coulomb =250 MeV =220 MeV =200 MeV}DC-TDHF =185 MeV Ca + 238 U 8 10 12 14 16 18 20 22 24 26 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 21
E* (MeV) Ca + 238 U: excitation energy Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, 064607 (2010) 100 90 80 70 60 50 40 30 20 10 Ca + 238 U =250 =220 MeV}DC-TDHF =200 MeV =185 MeV 0 8 10 12 14 16 18 20 22 24 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 22
Ca + 238 U capture cross section Umar, Oberacker, Maruhn & Reinhard, Phys. Rev. C 81, 064607 (2010) 10 3 σ capture (mb) 10 2 10 1 10 0 Ca + 238 U DC-TDHF Experiment: Oganessian et al., J. Phys. G 34 R165 (2007) 10-1 10-2 180 185 190 195 200 205 (MeV) 1/9/13 Volker Oberacker, Vanderbilt 23
70 Zn + 208 Pb heavy-ion potential ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, 014614 V(R) (MeV) 260 250 240 230 220 210 }Exp. Energies 70 208 Zn + Pb Point Coulomb E = 400 MeV c.m. = 350 MeV = 290 MeV}DC-TDHF = 280 MeV = 265 MeV 200 8 9 10 11 12 13 14 15 16 17 18 R (fm) 1/9/13 Volker Oberacker, Vanderbilt 24
Excitation energies at capture point in hot and cold fusion ( 2007 ) Umar and Oberacker, Phys. Rev. C 76, 014614 E* c (MeV) 100 90 80 70 60 50 40 30 20 10 Ca + 238 U (β = 0 o ) Ca+ 238 U (β = 45 o ) Ca + 238 U (β = 90 o ) 70 Zn + 208 Pb 0 180 200 220 240 260 280 300 320 340 360 (MeV) 1/9/13 Volker Oberacker, Vanderbilt 25