Microscopic (TDHF and DC-TDHF) study of heavy-ion fusion and capture reactions with neutron-rich nuclei INT Program INT- 11-2d Interfaces between structure and reactions for rare isotopes and nuclear astrophysics INT, Seattle (Aug. 15-19, 2011) Speaker: Volker E. Oberacker, Vanderbilt University Co-authors: A. S. Umar, Vanderbilt University J. A. Maruhn, Univ. Frankfurt P.-G. Reinhard, Univ. Erlangen Research supported by: US Department of Energy, Division of Nuclear Physics German BMBF 1
Our main goal: microscopic description of fusion reactions involving exotic neutron-rich nuclei 1. Unrestricted TDHF calculations above the fusion barrier deep-inelastic and capture / fusion reactions 2. Density-constrained TDHF calculations: microscopic calculation of heavy-ion potentials calculate fusion / capture cross sections below and above the barrier 3. Dynamic microscopic study of excitation energy E*(t) 2
96 Zr + 132 Sn, E cm = 250 MeV, b = 4 fm (deep-inelastic) TDHF, SLy4 interaction, 3-D lattice (50*42*30 points) t = 0 fm/c t = 696 fm/c t = 208 fm/c t = 1072 fm/c t = 432 fm/c Elapsed @me (contact to disintegra@on) = 984 fm/c = 3.3* 10-21 s Masses and charges amer reac@on: A 1 = 139.5, Z 1 = 54.5 A 2 = 88.5, Z 2 = 35.5 3
Animation: 96 Zr + 132 Sn, E cm =250 MeV, b = 4 fm TDHF, SLy4 interaction, 3-D lattice (50*40*30 points) 4
96 Zr + 132 Sn, E cm = 250 MeV, b = 2 fm (capture) TDHF, SLy4 interaction, 3-D lattice (50*40*30 points) t = 0 fm/c t = 144 fm/c t = 600 fm/c t = 1200 fm/c t = 1800 fm/c t = 2400 fm/c 5
Animation: 96 Zr + 132 Sn, E cm =250 MeV, b = 2 fm TDHF, SLy4 interaction, 3-D lattice (50*40*30 points) 6
Total cross sections for capture (unrestricted TDHF) 800 unrestricted TDHF (mb) 600 400 124 Sn+ 40 Ca sharp cutoff model 132 48 Sn+ Ca 2 σ capt = πb max 200 0 110 120 130 140 (MeV) 7
2. Density-constrained TDHF calculations microscopic calculation of heavy-ion potentials use with barrier tunneling (Incoming Wave Boundary Condition) calculate fusion (capture) cross sections below and above the barrier results for 132,124 Sn + 96 Zr (HRIBF exp.) and for 132,124 Sn + 48,40 Ca (recent HRIBF exp.) Previous work Umar & Oberacker, PRC 74, 021601(R) (2006) PRC 74, 061601 (R) (2006) PRC 76, 014614 (2007) PRC 77, 064605 (2008) EPJA 39, 243 (2009) Umar, Maruhn, Itagaki & Oberacker, PRL 104, 212503 (2010) 8
Calculation of heavy-ion potential with DC-TDHF method Umar and Oberacker, PRC 74, 021601(R) (2006) #" " #" " 48 Ca+ 132 Sn E cm =140 MeV #" $%!!&"'() #" $%##&'()*!" #" " #"!" #" " #" #" #" " " #" $%#"&'()* #" $%&'(()*+!" #" " #"!" #" " #" V(R) contains dynamical entrance channel effects (neck formation, particle transfer, surface vibrations, giant resonances) 9
heavy-ion potential for 132 Sn+ 48 Ca: strong E cm -dependence 132 Sn + 48 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) 10
Dynamical Effective Mass from TDHF TDHF DC- TDHF 11
15 mass parameter for 132 Sn+ 48 Ca: strong E cm -dependence 132 Sn + 48 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) 12
Formalism for coordinate-dependent mass: point transformation to constant mass original Hamiltonian point transformation to new coordinate with constant reduced mass integrate last equation V (R) = V ( f 1 (R )) U(R ) transformed heavy-ion potential, corresponding to constant reduced mass transformed Hamiltonian 13
transformed heavy-ion potential for 132 Sn+ 48 Ca: 132 Sn + 48 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) 14
Comparison with phenomenological models 140 132 Sn + 48 Ca V(R) (MeV) 120 100 80 60 Point Coulomb double-folding Bass model proximity model DC-TDHF ( =140 MeV) 8 10 12 14 16 18 20 R (fm) 15
Comparison: neutron-rich vs. stable system 140 132 Sn + 48 Ca vs. 124 Sn + 40 Ca V(R) (MeV) 120 100 80 60 DC-TDHF =140 MeV =125 MeV 8 10 12 14 16 18 20 R (fm) 16
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) 17
Total fusion cross section for 132 Sn+ 48 Ca 10 3 10 2 132 Sn+ 48 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) 18
Exp. fusion cross sections (HRIBF 2011) J.F. Liang, Proceedings of Fusion 11 conference, St. Malo (France) 19
Scaling used for exp. data Introduce 2 scaling parameters R 0 = A 1 1/ 3 + A 2 1/ 3 B 0 = Z 1 Z 2 /R 0 Scaling of fusion cross section σ σ /R 0 2 Scaling of center-of-mass energy /B 0 20
Fusion cross sections: theory vs. experiment 10 1 10 0 (mb) / R 0 2 10-1 124 Sn+ 40 Ca, exp. (HRIBF) 132 Sn+ 48 Ca, exp. (HRIBF) 10-2 124 40 Sn+ Ca, unrestricted TDHF beta = 0 deg. 132 48 Sn+ Ca, DC-TDHF, U(Rbar) -specific potential 0.85 0.9 0.95 1 1.05 1.1 1.15 / B 0 21
Excitation energy E* (DC-TDHF) 60 132 Sn + 48 Ca E* (MeV) 40 20 DC-TDHF =180 MeV =140 MeV =125 MeV =114 MeV =111 MeV 0 8 10 12 14 16 18 20 22 R (fm) 22
TDHF / DC-TDHF for heavy systems with fission competition Comparison of the reactions 132,124 Sn + 96 Zr Unrestricted TDHF above the barrier: capture cross sections DC-TDHF (below and above barrier): interaction barriers (nuclear surfaces touch) microscopic heavy-ion potentials, fusion barriers deep-inelastic and capture cross sections investigate E* at capture point, compare to E*= E cm + Q gg 23
132 Sn + 96 Zr, central collision, mass densities at R min Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) Interaction barrier 24
Central collision, dynamic quadrupole moment Q 20 (t) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) 150 132 96 Sn+ Zr Q 20 (b) 100 50 = 220 MeV = 230 MeV = 250 MeV = 300 MeV 228 Th 0 0 2 4 6 8 t (10-21 s) 25
Interaction barriers and heavy-ion potential barriers, dependence on isospin T z =½(Z-N) interaction barrier and position heavy-ion potential barrier and position Reac%on V I [MeV] R I [fm] V F [MeV] R F [fm] 132 Sn+ 96 Zr 195 14.77 211.4 (at E cm =230 MeV) 215.0 (at E cm =300 MeV) 13.03 12.56 124 Sn+ 96 Zr 204 14.05 210.6 (at E cm =230 MeV) 213.7 (at E cm =300 MeV) 13.06 12.59 Conclusion: interaction barriers differ substantially, but heavy-ion potential barriers are fairly similar 26
Heavy neutron-rich system: strong E cm -dependence Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) 240 132 Sn + 96 Zr 220 DC-TDHF V(R) (MeV) 200 180 160 140 120 doublefolding Point Coulomb = 220 MeV = 230 MeV = 250 MeV = 300 MeV 10 12 14 16 18 20 R (fm) 27
132 Sn+ 96 Zr (solid lines) vs. 124 Sn+ 96 Zr (dashed lines) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) 220 132,124 Sn + 96 Zr 210 DC-TDHF V(R) (MeV) 200 190 = 210 MeV = 220 MeV = 230 MeV = 250 MeV = 300 MeV 10 12 14 R (fm) 28
Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF 1000 132 Sn+ 96 Zr all L L max =87 (mb) 100 L max =51 DC-TDHF: deep-inel. DC-TDHF: capture TDHF: capture 10 200 220 240 260 (MeV) 29
1000 Comparison of exp. capture-fission cross section to theor. capture cross section 132 Sn + 96 Zr L max =87 Theory: Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) (mb) 100 L max =51 theor. interpretation of low-energy data: quasi-elastic and deep-inelastic rather than capture-fission 10 DC-TDHF: capture DC-TDHF: deep-inel. exp: capture-fission 200 220 240 260 (MeV) exp. data (HRIBF): Vinodkumar et al., PRC 78, 054608 (2008) 30
Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF 1000 124 Sn+ 96 Zr all L L max =77 (mb) 100 DC-TDHF: deep-inel. DC-TDHF: capture TDHF: capture 10 200 220 240 260 (MeV) 31
Comparison of exp. capture-fission cross section to theor. capture cross section 1000 124 Sn+ 96 Zr L max =77 Theory: Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) (mb) 100 exp. data (HRIBF): Vinodkumar et al., PRC 78, 054608 (2008) DC-TDHF: capture DC-TDHF: deep-inel. exp: capture-fission 10 200 220 240 260 (MeV) 32
Excitation energy E* (DC-TDHF) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010) E* vs. internuclear distance R solid lines: 132 Sn + 96 Zr dashed lines: 124 Sn + 96 Zr E* at capture point vs. E cm 100 100 E* (MeV) 90 80 70 60 50 40 30 132,124 Sn + 96 Zr DC-TDHF = 220 MeV = 230 MeV = 250 MeV = 300 MeV E c *(MeV) 80 60 40 20 E*= +Q gg 132 Sn+ 96 Zr (DC-TDHF) 124 Sn+ 96 Zr (DC-TDHF) 20 10 0 200 250 300 (MeV) 0 10 12 14 16 18 20 R (fm) 33
Conclusions Microscopic description of heavy-ion reactions (DC-TDHF) only input: Skyrme N-N interaction, no adjustable parameters heavy-ion interaction potential V(R), cross sections for capture and deep-inelastic reactions dynamic excitation energy E*(t) applied to reactions of 132,124 Sn + 96 Zr, 132,124 Sn + 48,40 Ca (HRIBF) V(R) shows strong E cm dependence, significant changes in width of potential barrier investigated E* at capture point, compared to E*= E cm + Q gg In Future: systematic study of heavy-ion potential barriers vs. T z =(Z-N)/2 excitation energy division between the two fragments 34
Open Problems and Challenges Improved energy-density functionals parameter fits should include deformed and neutron-rich nuclei Include pairing (TDHF does not, densities may be unrealistic) a) very challenging: TDHFB code for 2 nuclei on 3-D lattice (exists only for 1 nucleus in time-dep. external field) b) simplification: start from TDHFB equation, derive formalism for TDHF with time-dep. BCS-pairing amplitudes u k (t) c) easy: use frozen BCS pairing amplitudes of projectile and target 35