microscopic (tdhf and dc-tdhf) study of heavy-ion fusion ......8/17/11% our main goal: microscopic...

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)


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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


8/17/11 Volker Oberacker, Vanderbilt 4


Animation: 96Zr + 132Sn, Ecm =250 MeV, b = 4 fm

8/17/11


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




Animation: 96Zr + 132Sn, Ecm =250 MeV, b = 2 fm

8/17/11

Total cross sections for capture (unrestricted TDHF)


110 120 130 140Ec.m. (MeV)

0

200

400

600

800 (m

b)

124Sn+40Ca

unrestricted TDHF

132Sn+48Ca

€

σcapt = πbmax2

sharp cutoff model

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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)



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 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

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Dynamical Effective Mass from TDHF

DC-‐TDHF TDHF


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mass parameter for 132Sn+48Ca: strong Ecm-dependence


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


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:


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


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


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.

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Fusion / capture cross section for two spherical nuclei

total fusion cross section

transformed potential

solve Schrödinger equation numerically, with Incoming Wave Boundary Condition (IWBC)


€

−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


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

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Exp. fusion cross sections (HRIBF 2011) J.F. Liang, Proceedings of Fusion 11 conference, St. Malo (France)


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Scaling used for exp. data


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

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Fusion cross sections: theory vs. experiment


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


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)


Interaction barrier

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Central collision, dynamic quadrupole moment Q20(t) Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)


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)


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

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Heavy neutron-rich system: strong Ecm-dependence Oberacker, Umar, Maruhn & Reinhard, PRC 82, 034603 (2010)


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)


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


Capture and deep-inelastic cross sections Unrestricted TDHF and DC-TDHF


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)


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


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)


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


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


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


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


microscopic (tdhf and dc-tdhf) study of heavy-ion fusion ......8/17/11% our main goal: microscopic...

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