Microscopic (TDHF and DC-TDHF) study of heavy-ion fusion and capture reactions with neutron-rich nuclei
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1 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 , 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
2 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
3 96 Zr 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 (contact to disintegra@on) = 984 fm/c = 3.3* s Masses and charges amer reac@on: A 1 = 139.5, Z 1 = 54.5 A 2 = 88.5, Z 2 =
4 Animation: 96 Zr Sn, E cm =250 MeV, b = 4 fm TDHF, SLy4 interaction, 3-D lattice (50*40*30 points) 4
5 96 Zr 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
6 Animation: 96 Zr Sn, E cm =250 MeV, b = 2 fm TDHF, SLy4 interaction, 3-D lattice (50*40*30 points) 6
7 Total cross sections for capture (unrestricted TDHF) 800 unrestricted TDHF (mb) Sn+ 40 Ca sharp cutoff model Sn+ Ca 2 σ capt = πb max (MeV) 7
8 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, (R) (2006) PRC 74, (R) (2006) PRC 76, (2007) PRC 77, (2008) EPJA 39, 243 (2009) Umar, Maruhn, Itagaki & Oberacker, PRL 104, (2010) 8
9 Calculation of heavy-ion potential with DC-TDHF method Umar and Oberacker, PRC 74, (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
10 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 R (fm) 10
11 Dynamical Effective Mass from TDHF TDHF DC- TDHF 11
12 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 R (fm) 12
13 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
14 transformed heavy-ion potential for 132 Sn+ 48 Ca: 132 Sn + 48 Ca V(R),U(R) (MeV) =180 MeV =140 MeV =114 MeV DC-TDHF Point Coulomb R,R (fm) 14
15 Comparison with phenomenological models Sn + 48 Ca V(R) (MeV) Point Coulomb double-folding Bass model proximity model DC-TDHF ( =140 MeV) R (fm) 15
16 Comparison: neutron-rich vs. stable system Sn + 48 Ca vs. 124 Sn + 40 Ca V(R) (MeV) DC-TDHF =140 MeV =125 MeV R (fm) 16
17 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
18 Total fusion cross section for 132 Sn+ 48 Ca Sn+ 48 Ca (mb) unrestricted TDHF DC-TDHF - specific potential potential at =114 MeV potential at =140 MeV potential at =180 MeV (MeV) 18
19 Exp. fusion cross sections (HRIBF 2011) J.F. Liang, Proceedings of Fusion 11 conference, St. Malo (France) 19
20 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
21 Fusion cross sections: theory vs. experiment (mb) / R Sn+ 40 Ca, exp. (HRIBF) 132 Sn+ 48 Ca, exp. (HRIBF) Sn+ Ca, unrestricted TDHF beta = 0 deg Sn+ Ca, DC-TDHF, U(Rbar) -specific potential / B 0 21
22 Excitation energy E* (DC-TDHF) Sn + 48 Ca E* (MeV) DC-TDHF =180 MeV =140 MeV =125 MeV =114 MeV =111 MeV R (fm) 22
23 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
24 132 Sn + 96 Zr, central collision, mass densities at R min Oberacker, Umar, Maruhn & Reinhard, PRC 82, (2010) Interaction barrier 24
25 Central collision, dynamic quadrupole moment Q 20 (t) Oberacker, Umar, Maruhn & Reinhard, PRC 82, (2010) Sn+ Zr Q 20 (b) = 220 MeV = 230 MeV = 250 MeV = 300 MeV 228 Th t (10-21 s) 25
26 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 (at E cm =230 MeV) (at E cm =300 MeV) Sn+ 96 Zr (at E cm =230 MeV) (at E cm =300 MeV) Conclusion: interaction barriers differ substantially, but heavy-ion potential barriers are fairly similar 26
27 Heavy neutron-rich system: strong E cm -dependence Oberacker, Umar, Maruhn & Reinhard, PRC 82, (2010) Sn + 96 Zr 220 DC-TDHF V(R) (MeV) doublefolding Point Coulomb = 220 MeV = 230 MeV = 250 MeV = 300 MeV R (fm) 27
28 132 Sn+ 96 Zr (solid lines) vs. 124 Sn+ 96 Zr (dashed lines) Oberacker, Umar, Maruhn & Reinhard, PRC 82, (2010) ,124 Sn + 96 Zr 210 DC-TDHF V(R) (MeV) = 210 MeV = 220 MeV = 230 MeV = 250 MeV = 300 MeV R (fm) 28
29 Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF Sn+ 96 Zr all L L max =87 (mb) 100 L max =51 DC-TDHF: deep-inel. DC-TDHF: capture TDHF: capture (MeV) 29
30 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, (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 (MeV) exp. data (HRIBF): Vinodkumar et al., PRC 78, (2008) 30
31 Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF Sn+ 96 Zr all L L max =77 (mb) 100 DC-TDHF: deep-inel. DC-TDHF: capture TDHF: capture (MeV) 31
32 Comparison of exp. capture-fission cross section to theor. capture cross section Sn+ 96 Zr L max =77 Theory: Oberacker, Umar, Maruhn & Reinhard, PRC 82, (2010) (mb) 100 exp. data (HRIBF): Vinodkumar et al., PRC 78, (2008) DC-TDHF: capture DC-TDHF: deep-inel. exp: capture-fission (MeV) 32
33 Excitation energy E* (DC-TDHF) Oberacker, Umar, Maruhn & Reinhard, PRC 82, (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 E* (MeV) ,124 Sn + 96 Zr DC-TDHF = 220 MeV = 230 MeV = 250 MeV = 300 MeV E c *(MeV) E*= +Q gg 132 Sn+ 96 Zr (DC-TDHF) 124 Sn+ 96 Zr (DC-TDHF) (MeV) R (fm) 33
34 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
35 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
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