1. Systematic study of fusion reactions leading to super-heavy nuclei
1. Introduction
2. capture cross sections
3. Survival probablity Wsur
4. Fusion probabilty PCN
5. Conclusion
Ning Wang
Guangxi Normal University
BLTP/JINR-KLFTP/CAS Joint Workshop on Nuclear Physics, Aug. 2-6, 2012, Dubna

3. I. Capture cross sections with the Skyrme energy-density functional
Density distributions of the reaction partners
Skyrme energy-density functional
Entrance-channel fusion barrier
Barrier penetration & empirical fusion barrier distribution D(B)
Fusion cross sections
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)

4. 1. Determination of density distributions
according to Hohenberg-Kohn theorem
Woods-Saxon form for densities
Search for the minimum of energy by varying densities (R0p, R0n, ap, an)

5. 2. Entrance-channel fusion barrier
Sudden approximation for density
V.Yu. Denisov and W. Noerenberg,
Eur. Phys. J. A15, 375 (2002).

6. 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

7. for reactions with nuclei near the beta-stability line but the neutron-shell is not closed

8. 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)

12. Deviations from exp. data for 120 reactions
N. Wang et al., J. Phys. G: 34 (2007) 1935
rms deviation for (E>B)
About 70% systems are less than 0.005, which gives the system error 18%.

13. 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

14. Fusion-fission：EDF＋HIVAP
Wang, Zhao, Scheid, Wu, PRC 77 (2008)

16. Deviations of calculated evaporation (and fission) cross sections from exp. data for 51 fusion-fission reactions
For 68% reactions, the deviations are smaller than ,
Estimated systematic errors of the HIVAP code: 1.85Wsur and Wsur /1.85

17. WS*
3). Masses of super-heavy nuclei
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)
WS*: PRC 82 (2010)
WS3: PRC 84 (2011)

18. Alpha decay energies of super-heavy nuclei have been predicted
rms ~ 248 keV to 46 Qa of SHN

19. N=178
WS*
N=162
N=178
N=178
WS*
WS*
Zhang, et al., Phys. Rev. C 85, (2012)

20. III. Fusion probability
1) quasi-fission barrier
Yu. Oganessian
Wang, Tian, Scheid, PRC84, (R) (2011)

21. Fusion probability

22. 2) Evaporation residual cross sections
Mean barrier height
PRC84, (R) (2011)

23. Uncertainty at E>Bm : 1.18 (capture) x 1.85 (Wsur) x 2 (PCN) = 4.4

24. Opt. ~ 50 fb ~ 40 fb ~ 20 fb ~ 5 fb ~ 600 fb ~ 100 fb ~ 70 fb ~ 35 fb
50Ti+249Bk
50Ti+249Cf
54Cr+248Cm
58Fe+244Pu
Zagrebaev PRC(2008)
~ 50 fb
~ 40 fb
~ 20 fb
~ 5 fb
Liu-Bao PRC(2011)
~ 600 fb
~ 100 fb
Nasirov PRC(2011)
~ 70 fb
Ning Wang PRC(2011)
~ 35 fb
~ 3 fb
Nan Wang PRC(2012)
~ 110 fb
~ 6 fb
~ 4 fb
Siwek-Wilczynska PRC(2012)
~ 30 fb
~ 1 fb
~ 0.1 fb

25. 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.

26. Zhu-Xia Li、Xi-Zhen Wu、Kai Zhao （李祝霞） （吴锡真） （赵凯）
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）

27. Thanks for your attention
Codes and data are available at：

29. Skyrme energy-density functional
Skyrme force SkM*
Kinetic
Nuclear
Coulomb
M. Brack, C. Guet, H.-B. Hakanson, Phys. Rep. 123, 275 (1985).

33. I. Capture II. Decay III. Formation # Coulomb Barrier (Skyrme EDF)
# Barrier Distribution
# Deformation & Dynamics … (ImQMD)
I. Capture
II. Decay
III. Formation
# Fission Barrier
# Masses & Shell corrections (mass formula)
# Fission Fragment Yields … (DNS)
# Quasi-fission barrier
# Potential energy surface
# Dynamics …

34. 1). Fission barrier Nuclei Cohen-Swiatecki Sierk Dahlinger MWS
244Pu
256No

35. 2). Level density parameters
In the standard HIVAP code: Ed=18.5MeV, ra=1.153fm

36. Large-angle quasi-elastic scattering
Tail of the barrier distribution influences the large-angle quasi-elastic cross sections
PRC78, (2008)

37. Tail of barrier distribution influences the large-angle quasi-elastic cross sections
S. G. Zhou