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Systematics of fusion probability in reactions leading to super-heavy nuclei
Ning Wang ( 王宁 )
Guangxi Normal University
www.imqmd.com 2011 Dec., Beijing
N. Wang www.Im（王宁）QMD.com
Introduction
Calcualtions of capture cross sections
Survival probablity Wsur
Fusion probabilty PCN
N. Wang www.Im（王宁）QMD.com
2010
6 new superheavy elements
48 new isotopes
Yu.
Oga
nes
sian
20
10118
116
115
114
113
112
111
110
109
108
107
106
105
104
T1/2= 320d
249Bk +48Ca
117
Yu. Oganessian “SHE in JINR” 109-th Session of the SC of JINR, Feb. 17-18, 2011, Dubna
???
N. Wang www.Im（王宁）QMD.com
S.Hofmann, GSI
2003-2008, 7477hours / 312days, 2 events
Theoretical support for these very time-consuming and extremely-expensive experimentsis vital in choosing the optimum target-projectile-energy combinations, and for the estimation of cross sections.
50Ti, 54Cr, 58Fe, 64Ni, 70Zn
+ 208Pb
48Ca ： 1 克 20 万美金 Cf ： 1 克 10 亿美元
N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
I. Capture
II. Decay
III. Formation
# Coulomb barrier
# Barrier distribution (~2005 -- 2007)
# Deformation & dynamics … (~2000 -- present)
# Fission barrier
# Masses & shell corrections (~2009 -- present)
# Mass distributions …
# Quasi-fission barrier
# Potential energy surface (~2005 、 2011…)
# Dynamics …
N. Wang www.Im（王宁）QMD.com
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
N. Wang www.Im（王宁）QMD.com
Kinetic Nuclear Coulomb
Skyrme energy-density functional
M. Brack, C. Guet, H.-B. Hakanson, Phys. Rep. 123, 275 (1985).
Skyrme force SkM*
N. Wang www.Im（王宁）QMD.com
Spherical symmetric Fermi functions
Search for the minimum of energy by varying densities (R0p, R0n, ap, an)
according to Hohenburg-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
N. Wang www.Im（王宁）QMD.com
E1E2
Sudden approximation for density
R
V.Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002).
2. Entrance-channel fusion barrier
N. Wang www.Im（王宁）QMD.com
withDetermination of D(B) is the key of this approach
8 12 1630
40
50
60
70
80
Vb (
MeV
)
R (fm)
28Si+92ZrB0
R0
D(B) to empirically take into account the coupling between the relative motion and other degrees of freedom such as deformation etc.
3. Fusion (capture) cross section
Min Liu, Ning Wang, et al., Nucl. Phys. A 768 (2006) 80
N. Wang www.Im（王宁）QMD.com
for reactions with nuclei near the beta-stability line but the neutron-shell is not closed
N. Wang www.Im（王宁）QMD.com
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)
N. Wang www.Im（王宁）QMD.com
60 70 80 900.01
0.1
1
10
100
1000
fus (
mb)
Ec.m.
(MeV)
exp. calc.
16O+186WDeviations from experimental 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 偏差
N. Wang www.Im（王宁）QMD.com
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)
N. Wang www.Im（王宁）QMD.com
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)
N. Wang www.Im（王宁）QMD.com
The fusion excitation functions of a series of reactions with 16O bombarding on medium mass targets.
N. Wang www.Im（王宁）QMD.com
Large-angle quasi-elastic scattering
PRC 78, 014607 (2008)
N. Wang www.Im（王宁）QMD.com
S. G. ZhouTail of barrier distribution influences the large-angle quasi-elastic cross sections
N. Wang www.Im（王宁）QMD.com
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
N. Wang www.Im（王宁）QMD.com
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
N. Wang www.Im（王宁）QMD.com
Obtained deviations with different models for 51 fusion-fission reactions
N. Wang, M. Liu and Y. Yang, Sci. China G 52, 1554 (2009)
N. Wang www.Im（王宁）QMD.com
2). Level density parameters
In the standard HIVAP code: Ed=18.5MeV, ra=1.153fm
N. Wang www.Im（王宁）QMD.com
Radius parameter ra in the level density parameter
N. Wang www.Im（王宁）QMD.com
Optimal value of the radius parameter ra
68% reactions have a value smaller than 0.0714,
Estimated systematic errors of the HIVAP code: 1.85Wsur and Wsur /1.85
N. Wang www.Im（王宁）QMD.com
A reliable nuclear mass model is needed for studies on synthesis of super-heavy, nuclear astrophysics and nuclear symmetry energy, etc.
“Mass Olympics” Trento 2008
WS : PRC 81 (2010) 044322
WS*: PRC 82 (2010) 044304
WS3: PRC 84 (2011) 014333
WS*3). Masses of super-heavy nuclei
N. Wang www.Im（王宁）QMD.com
Ning Wang, et al., Phys. Rev. C 77 (2008) 014603
Fusion-fission ： EDF ＋ HIVAP ER cap CN surP W
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N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
N. Wang www.Im（王宁）QMD.com
M. Itkis ， Dubna
1) Driving potential
2) Quasi-fission
3) Calculations of evaporation residual cross sections
III. Fusion probability PCN (preliminary)
N. Wang www.Im（王宁）QMD.com
Y. Oganessian, 109th Session of the JINR Scientific Council, 2010, Dubna.
2) quasi-fission barrier
Bqf
Esh
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N. Wang www.Im（王宁）QMD.com
Mean barrier height
3) Evaporation residual cross sections
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N. Wang www.Im（王宁）QMD.com
Uncertainty at E>Bm : 1.18 (capture) x 1.85 (Wsur) x 2 (PCN) = 4.4
N. Wang www.Im（王宁）QMD.com
Collected by Junlong Tian
N. Wang www.Im（王宁）QMD.com
A. K. Nasirov, et al., Phys. Rev. C 84, 044612 (2011)
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Results fromNan Wang ( 王楠 )
http://www.phys-hutc.net/conference/2011-09-trans/pdf/ 王楠 .pdf
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N. Wang www.Im（王宁）QMD.com
Zagrebaev, et al., arXiv:1105.1659v1
For 4n channel
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Opt. 50Ti+249Bk 50Ti+249Cf 54Cr+248Cm 58Fe+244Pu
Nasirov(2009)(2011)
~ 100 fb
~ 10 pb ~ 70 fb
~ 6 fb
Liu & Bao(2011)
~ 600 fb ~ 100 fb
Nan Wang(2011)
~1000 fb ~ 200 fb ~ 40 fb ~ 30 fb
Zagrebaev(2008)
~ 50 fb ~ 40 fb ~ 20 fb ~ 5 fb
This work(2011)
~ 35 fb ~ 20 fb ~ 5 fb ~ 3 fb
N. Wang www.Im（王宁）QMD.com
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 are established step by step.
Coulomb barrier, fission barrier and quasi-fission barrier play important role for the calculations of three parts.
More precise calculations for masses, fission barrier and the study on dynamics of fusion and fission processes are still required.
N. Wang www.Im（王宁）QMD.com
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）
N. Wang www.Im（王宁）QMD.com
Thanks for your attention
I am very grateful to Prof. Shan-Gui Zhou ( 周善贵 ) for many discussions and valuable suggestions
This work was supported by Alexander von Humboldt Foundation and National Natural Science Foundation of China
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