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8/17/11
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 Volker Oberacker, Vanderbilt
8/17/11
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 Volker Oberacker, Vanderbilt
8/17/11
TDHF, SLy4 interaction, 3-D lattice (50*42*30 points)
96Zr + 132Sn, Ecm = 250 MeV, b = 4 fm (deep-inelastic)
Elapsed @me (contact to disintegra@on) = 984 fm/c = 3.3* 10-‐21 s
Masses and charges aMer reac@on: A1 = 139.5, Z1 = 54.5 A2 = 88.5, Z2 = 35.5
t = 1072 fm/c t = 696 fm/c
t = 0 fm/c t = 432 fm/c t = 208 fm/c
3 Volker Oberacker, Vanderbilt
8/17/11 Volker Oberacker, Vanderbilt 4
TDHF, SLy4 interaction, 3-D lattice (50*40*30 points)
Animation: 96Zr + 132Sn, Ecm =250 MeV, b = 4 fm
8/17/11
TDHF, SLy4 interaction, 3-D lattice (50*40*30 points)
96Zr + 132Sn, Ecm = 250 MeV, b = 2 fm (capture)
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 Volker Oberacker, Vanderbilt
8/17/11 Volker Oberacker, Vanderbilt 6
TDHF, SLy4 interaction, 3-D lattice (50*40*30 points)
Animation: 96Zr + 132Sn, Ecm =250 MeV, b = 2 fm
8/17/11
Total cross sections for capture (unrestricted TDHF)
7 Volker Oberacker, Vanderbilt
110 120 130 140Ec.m. (MeV)
0
200
400
600
800 (m
b)
124Sn+40Ca
unrestricted TDHF
132Sn+48Ca
€
σcapt = πbmax2
sharp cutoff model
8/17/11
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 132,124Sn + 96Zr (HRIBF exp.) and for 132,124Sn + 48,40Ca (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 Volker Oberacker, Vanderbilt
8/17/11 Volker Oberacker, Vanderbilt 9
Umar and Oberacker, PRC 74, 021601(R) (2006)
Calculation of heavy-ion potential with DC-TDHF method
V(R) contains dynamical entrance channel effects (neck formation, particle transfer, surface vibrations, giant resonances)
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48Ca+132Sn Ecm=140 MeV
8/17/11
heavy-ion potential for 132Sn+48Ca: strong Ecm-dependence
10 Volker Oberacker, Vanderbilt
10 12 14 16 18R (fm)
80
100
120
V(R
) (M
eV)
Ec.m.=125 MeV
132Sn + 48Ca
Point Coulomb
Ec.m.=140 MeV
Ec.m.=117 MeV
DC-TDHF
Ec.m.=114 MeV
Ec.m.=180 MeV
8/17/11
Dynamical Effective Mass from TDHF
DC-‐TDHF TDHF
11 Volker Oberacker, Vanderbilt
8/17/11
mass parameter for 132Sn+48Ca: strong Ecm-dependence
12 Volker Oberacker, Vanderbilt
10 12 14 16 18 20R (fm)
0
5
10
15
M(R
)/µ132Sn + 48Ca
Ec.m. = 117 MeV
Ec.m. = 140 MeV
Ec.m. = 114 MeV
Ec.m. = 180 MeV
DC-TDHF
8/17/11 Volker Oberacker, Vanderbilt 13
Formalism for coordinate-dependent mass: point transformation to constant mass
point transformation to new coordinate with constant reduced mass
integrate last equation
transformed heavy-ion potential, corresponding to constant reduced mass
transformed Hamiltonian
original Hamiltonian
€
V (R) = V ( f −1(R )) ≡U(R )
8/17/11
transformed heavy-ion potential for 132Sn+48Ca:
14 Volker Oberacker, Vanderbilt
10 12 14 16 18R,R (fm)
80
100
120V
(R),U
(R) (
MeV
)132Sn + 48Ca
Point Coulomb
Ec.m.=180 MeVEc.m.=140 MeV
DC-TDHF
Ec.m.=114 MeV
8/17/11
Comparison with phenomenological models
15 Volker Oberacker, Vanderbilt
8 10 12 14 16 18 20R (fm)
60
80
100
120
140V
(R) (
MeV
)
double-folding
132Sn + 48Ca
Point Coulomb
DC-TDHF (Ec.m.=140 MeV)
Bass modelproximity model
8/17/11
Comparison: neutron-rich vs. stable system
16 Volker Oberacker, Vanderbilt
8 10 12 14 16 18 20R (fm)
60
80
100
120
140V
(R) (
MeV
)
Ec.m.=125 MeV
124Sn + 40Ca132Sn + 48Ca
Ec.m.=140 MeV
DC-TDHF
vs.
8/17/11
Fusion / capture cross section for two spherical nuclei
total fusion cross section
transformed potential
solve Schrödinger equation numerically, with Incoming Wave Boundary Condition (IWBC)
17 Volker Oberacker, Vanderbilt
€
−2
2µd2
dR 2+2( +1)2µR 2
+U(R ) − Ec.m.
⎡
⎣ ⎢
⎤
⎦ ⎥ φ (R ) = 0
Schrödinger equation for transformed radial coordinate
reduced mass
8/17/11
Total fusion cross section for 132Sn+48Ca
18 Volker Oberacker, Vanderbilt
110 120 130 140Ec.m. (MeV)
10-1
100
101
102
103
(mb)
132Sn+48Ca
unrestricted TDHF
DC-TDHFEc.m.- specific potential
potential at Ec.m.=140 MeVpotential at Ec.m.=114 MeV
potential at Ec.m.=180 MeV
8/17/11
Exp. fusion cross sections (HRIBF 2011) J.F. Liang, Proceedings of Fusion 11 conference, St. Malo (France)
19 Volker Oberacker, Vanderbilt
8/17/11
Scaling used for exp. data
20 Volker Oberacker, Vanderbilt
Scaling of fusion cross section
Scaling of center-of-mass energy €
R0 = A11/ 3 + A2
1/ 3
Introduce 2 scaling parameters
€
B0 = Z1Z2 /R0
€
σ →σ /R02
€
Ec.m. →Ec.m. /B0
8/17/11
Fusion cross sections: theory vs. experiment
21 Volker Oberacker, Vanderbilt
0.85 0.9 0.95 1 1.05 1.1 1.15Ec.m. / B0
10-2
10-1
100
101
(mb)
/ R
02
124Sn+40Ca, unrestricted TDHF
132Sn+48Ca, DC-TDHF, U(Rbar)
132Sn+48Ca, exp. (HRIBF)
124Sn+40Ca, exp. (HRIBF)
beta = 0 deg.
Ec.m.-specific potential
Excitation energy E* (DC-TDHF)
8/17/11 22 Volker Oberacker, Vanderbilt
8 10 12 14 16 18 20 22R (fm)
0
20
40
60
E* (M
eV)
Ec.m.=125 MeV
132Sn + 48Ca
Ec.m.=140 MeV
DC-TDHF
Ec.m.=114 MeV
Ec.m.=180 MeV
Ec.m.=111 MeV
TDHF / DC-TDHF for heavy systems with fission competition Comparison of the reactions 132,124Sn + 96Zr
8/17/11 23 Volker Oberacker, Vanderbilt
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*= Ecm + Qgg
8/17/11
132Sn + 96Zr, central collision, mass densities at Rmin Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
24 Volker Oberacker, Vanderbilt
Interaction barrier
8/17/11
Central collision, dynamic quadrupole moment Q20(t) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
25 Volker Oberacker, Vanderbilt
0 2 4 6 8t (10-21s)
0
50
100
150
Q20
(b)
132Sn+96Zr
Ec.m. = 220 MeVEc.m. = 230 MeVEc.m. = 250 MeV
228Th
Ec.m. = 300 MeV
8/17/11
Interaction barriers and heavy-ion potential barriers, dependence on isospin Tz=½(Z-N)
26 Volker Oberacker, Vanderbilt
Reac%on VI [MeV] RI [fm] VF [MeV] RF [fm]
132Sn+96Zr 195 14.77 211.4 (at Ecm=230 MeV) 215.0 (at Ecm=300 MeV)
13.03 12.56
124Sn+96Zr 204 14.05 210.6 (at Ecm=230 MeV) 213.7 (at Ecm=300 MeV)
13.06 12.59
interaction barrier and position
Conclusion: interaction barriers differ substantially, but heavy-ion potential barriers are fairly similar
heavy-ion potential barrier and position
8/17/11
Heavy neutron-rich system: strong Ecm-dependence Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
27 Volker Oberacker, Vanderbilt
10 12 14 16 18 20R (fm)
120
140
160
180
200
220
240V
(R) (
MeV
)
DC-TDHF
Ec.m. = 220 MeV
132Sn + 96Zr
Point Coulomb
Ec.m. = 230 MeVEc.m. = 250 MeVEc.m. = 300 MeV
double-folding
8/17/11
132Sn+96Zr (solid lines) vs. 124Sn+96Zr (dashed lines) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
28 Volker Oberacker, Vanderbilt
10 12 14R (fm)
190
200
210
220V
(R) (
MeV
)
DC-TDHF
132,124Sn + 96Zr
Ec.m. = 300 MeV
Ec.m. = 230 MeVEc.m. = 250 MeV
Ec.m. = 220 MeVEc.m. = 210 MeV
Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF
8/17/11 29 Volker Oberacker, Vanderbilt
200 220 240 260Ec.m. (MeV)
10
100
1000
(mb)
132Sn+96Zr
DC-TDHF: deep-inel.DC-TDHF: captureTDHF: capture
Lmax=51
Lmax=87
all L
Comparison of exp. capture-fission cross section to theor. capture cross section
8/17/11
exp. data (HRIBF): Vinodkumar et al., PRC 78, 054608 (2008)
30 Volker Oberacker, Vanderbilt
theor. interpretation of low-energy data: quasi-elastic and deep-inelastic rather than capture-fission
200 220 240 260Ec.m. (MeV)
10
100
1000
(mb)
132Sn + 96Zr
DC-TDHF: capture
exp: capture-fissionDC-TDHF: deep-inel.
Lmax=87
Lmax=51
Theory: Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF
8/17/11 31 Volker Oberacker, Vanderbilt
200 220 240 260Ec.m. (MeV)
10
100
1000
(mb)
124Sn+96Zr
DC-TDHF: deep-inel.
TDHF: capture DC-TDHF: capture
all L
Lmax=77
Comparison of exp. capture-fission cross section to theor. capture cross section
8/17/11
exp. data (HRIBF): Vinodkumar et al., PRC 78, 054608 (2008)
32 Volker Oberacker, Vanderbilt
200 220 240 260Ec.m. (MeV)
10
100
1000
(mb)
124Sn+96Zr
exp: capture-fission
DC-TDHF: captureDC-TDHF: deep-inel.
Lmax=77
Theory: Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
Excitation energy E* (DC-TDHF) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)
E* vs. internuclear distance R solid lines: 132Sn + 96Zr dashed lines: 124Sn + 96Zr
8/17/11
E* at capture point vs. Ecm
33 Volker Oberacker, Vanderbilt
10 12 14 16 18 20R (fm)
0
10
20
30
40
50
60
70
80
90
100
E* (M
eV)
DC-TDHFEc.m. = 220 MeV
132,124Sn + 96Zr
Ec.m. = 230 MeV
Ec.m. = 250 MeVEc.m. = 300 MeV
200 250 300Ec.m. (MeV)
0
20
40
60
80
100
E c*(M
eV) E*=Ec.m.+Qgg
132Sn+96Zr (DC-TDHF)124Sn+96Zr (DC-TDHF)
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 132,124Sn + 96Zr, 132,124Sn + 48,40Ca (HRIBF) V(R) shows strong Ecm dependence, significant changes in width of potential barrier
investigated E* at capture point, compared to E*= Ecm + Qgg
In Future: systematic study of heavy-ion potential barriers vs. Tz=(Z-N)/2 excitation energy division between the two fragments
8/17/11 34 Volker Oberacker, Vanderbilt
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 uk(t) c) easy: use frozen BCS pairing amplitudes of projectile and target
8/17/11 35 Volker Oberacker, Vanderbilt
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