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Systematic study of fusion reactions leading to super-heavy nuclei
Ning Wang
Guangxi Normal University
www.ImQMD.com/wangning/
BLTP/JINR-KLFTP/CAS Joint Workshop on Nuclear Physics, Aug. 2-6, 2012, Dubna
1. Introduction
2. capture cross sections
3. Survival probablity Wsur
4. Fusion probabilty PCN
5. Conclusion
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I. Capture cross sections with the Skyrme energy-density functional
Density distributions of the reaction partners
Entrance-channel fusion barrier
Fusion cross sections
Skyrme energy-density functional
Barrier penetration & empirical fusion barrier distribution D(B)
M. Liu, N. Wang, Z. Li, X. Wu and E. Zhao, Nucl. Phys. A 768 (2006) 80
Ning Wang, et al., Phys. Rev. C 74 (2006) 044604
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Woods-Saxon form for densities
Search for the minimum of energy by varying densities (R0p, R0n, ap, an)
according to Hohenberg-Kohn theorem
0 2 4 6 8 10 120.00
0.02
0.04
0.06
0.08
0.10
(f
m-3
)
r (fm)
p n
208Pb
1. Determination of density distributions
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E1E2
Sudden approximation for density
R
V.Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002).
2. Entrance-channel fusion barrier
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8 12 1630
40
50
60
70
80
Vb (
MeV
)
R (fm)
28Si+92ZrB0
R0
3. Fusion (capture) cross section
with
D(B) considers the coupling between the relative motion and other degrees of freedom such as dyn. deform. etc.
16O+208Pb, E=80MeV, ImQMD
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for reactions with nuclei near the beta-stability line but the neutron-shell is not closed
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The fusion excitation functions for a series of reactions with 16O bombarding on medium mass targets.
Wang et al. Sci China G 52, 1554 (2009)
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50 55 60 65 70 75
0.0
0.1
0.2
0.3
50 60 70
0.0
0.1
0.2
70 80 90 100 110
0.0
0.1
0.2
24 28 32 36 40
0.0
0.2
0.4
D (
MeV
-1)
B (MeV)
Dder
Deff
16O+144Sm
(b)
D (
MeV
-1)
B (MeV)
Dder
Deff
16O+154Sm
(d)
D (
MeV
-1)
B (MeV)
Dder
with E=2.5
Dder
with E=4.0
Deff
19F+208Pb
(c)
D (
MeV
-1)
B (MeV)
Dder
Deff
12C+92Zr
(a)
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24 30 36 42 480.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
12C+92Zr
(a)40 50 60 70
0.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
16O+92Zr
(b)75 80 85 90 95 100
0.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
35Cl+92Zr
(d)70 80 90 100
0.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
33S+92Zr
(c)
40 48 56 640.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
16O+112Cd
(e)50 60 70 80 90
0.01
0.1
1
10
100
1000
fus
(mb)
Ec.m.
(MeV)
exp. calc.
16O+144Nd
(f)50 60 70 80 90 100
0.01
0.1
1
10
100
1000
exp. calc.
fus
(mb)
Ec.m.
(MeV)
16O+166Er
(g)
60 80 100 120 140 1600.01
0.1
1
10
100
1000
10000
fus
(mb)
Ec.m.
(MeV)
exp. B.B.Back exp. H.Q.Zhang calc.
16O+232Th
(h)
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75 90 105 1200.01
0.1
1
10
100
1000
10000
20 30 40 50 601
10
100
1000
100 110 120 130 140 1500.01
0.1
1
10
100
1000
100 120 140 160 1800.01
0.1
1
10
100
1000
10000
100 110 120 130 140 1500.01
0.1
1
10
100
1000
105 120 135 150 1650.01
0.1
1
10
100
1000
110 120 130 140 1500.01
0.1
1
10
100
1000
120 140 160 180 2000.1
1
10
100
1000
120 140 160 180 2000.01
0.1
1
10
100
1000
fus (m
b)
Ec.m.
(MeV)
exp. calc.
19F+197Au
(a)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
28Si+28Si
(b)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
28Si+178Hf
(c)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
28Si+198Pt
(d)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
29Si+178Hf
(e)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
30Si+186W
(f)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
31P+175Lu
(g)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
32S+181Ta
(h)
fus (m
b)
Ec.m.
(MeV)
exp. calc.
32S+182W
(i)
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60 70 80 900.01
0.1
1
10
100
1000
fus (
mb)
Ec.m.
(MeV)
exp. calc.
16O+186WDeviations from exp. data for 120 reactions
About 70% systems are less than 0.005, which gives the system error 18%.
N. Wang et al., J. Phys. G: 34 (2007) 1935
rms deviation for (E>B)
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II. Survival probability Wsur with HIVAP
The sensitive parameters:
1. fission barriers (Liquid-drop barriers, Sierk’s barriers…)
2. level density parameters (Fermi gas model, angular-momentum and shape-dependent)
3. masses shell corrections and particle separation energies
In the standard HIVAP code: ra=1.153fm
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Wang, Zhao, Scheid, Wu, PRC 77 (2008) 014603
Fusion-fission : EDF + HIVAP
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For 68% reactions, the deviations are smaller than 0.0714,
Estimated systematic errors of the HIVAP code: 1.85Wsur and Wsur /1.85
Deviations of calculated evaporation (and fission) cross sections from exp. data for 51 fusion-fission reactions
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A reliable nuclear mass formula is crucial for a description of the properties and production cross sections of super-heavy nuclei
WS : PRC 81 (2010) 044322
WS*: PRC 82 (2010) 044304
WS3: PRC 84 (2011) 014333
WS*3). Masses of super-heavy nuclei
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Alpha decay energies of super-heavy nuclei have been predicted
rms ~ 248 keV to 46 Qa of SHN
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Zhang, et al., Phys. Rev. C 85, 014325 (2012)
N=178
WS*
N=178
WS*
N=162 N=178
WS*
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1) quasi-fission barrier
Wang, Tian, Scheid, PRC84, 061601(R) (2011)
III. Fusion probability
Yu. Oganessian
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Fusion probability
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2) Evaporation residual cross sections
Mean barrier height PRC84, 061601(R) (2011)
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Uncertainty at E>Bm : 1.18 (capture) x 1.85 (Wsur) x 2 (PCN) = 4.4
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Opt. 50Ti+249Bk 50Ti+249Cf 54Cr+248Cm 58Fe+244Pu
ZagrebaevPRC(2008) ~ 50 fb ~ 40 fb ~ 20 fb ~ 5 fb
Liu-BaoPRC(2011) ~ 600 fb ~ 100 fb
NasirovPRC(2011) ~ 100 fb ~ 70 fb
Ning WangPRC(2011) ~ 35 fb ~ 20 fb ~ 5 fb ~ 3 fb
Nan WangPRC(2012) ~ 110 fb ~ 50 fb ~ 6 fb ~ 4 fb
Siwek-WilczynskaPRC(2012)
~ 30 fb ~ 6 fb ~ 1 fb ~ 0.1 fb
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Conclusion and discussion
Methods for calculations of capture cross sections, survival probability of compound nucleus and the fusion probability in fusion reactions leading to super-heavy nuclei have been established and checked step by step.
Coulomb barrier, fission barrier and quasi-fission barrier play important roles for the calculations of three parts.
More precise calculations for masses, fission barriers, fission fragment yields and the study of fusion dynamics are still required.
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China Institute of Atomic energy :Zhu-Xia Li 、 Xi-Zhen Wu 、 Kai Zhao (李祝霞) (吴锡真) (赵凯)
Institute of Theoretical Physics (CAS) : En-Guang Zhao (赵恩广)
Justus-Liebig-Univ. Giessen : Werner Scheid
Guangxi Normal Univ. Min Liu (刘敏)
Anyang Normal Univ. Jun-Long Tian (田俊龙)
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Thanks for your attention
Codes and data are available at : www.ImQMD.com
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Kinetic Nuclear Coulomb
Skyrme energy-density functional
M. Brack, C. Guet, H.-B. Hakanson, Phys. Rep. 123, 275 (1985).
Skyrme force SkM*
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I. Capture
II. Decay
III. Formation
# Coulomb Barrier (Skyrme EDF)
# Barrier Distribution
# Deformation & Dynamics … (ImQMD)
# Fission Barrier
# Masses & Shell corrections (mass formula)
# Fission Fragment Yields … (DNS)
# Quasi-fission barrier
# Potential energy surface
# Dynamics …
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1). Fission barrier
Nuclei Cohen-Swiatecki Sierk Dahlinger MWS 244Pu 4.16 5.17 3.95 4.13 256No 1.74 1.44 1.02 1.19
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2). Level density parameters
In the standard HIVAP code: Ed=18.5MeV, ra=1.153fm
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Large-angle quasi-elastic scattering
PRC78, 014607 (2008)
Tail of the barrier distribution influences the large-angle quasi-elastic cross sections
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S. G. ZhouTail of barrier distribution influences the large-angle quasi-elastic cross sections