1. Systematics of fusion probability in reactions leading to super-heavy nuclei Ning Wang ( ) Guangxi Normal University www.imqmd.com 2011 Dec., Beijing

2. N. Wang www.ImQMD.com Introduction Calcualtions of capture cross sections Survival probablity W sur Fusion probabilty P CN

3. N. Wang www.ImQMD.com 2010 6 new superheavy elements 48 new isotopes Yu. Oganessian 2010 118 116 115 114 113 112 111 110 109 108 107 106 105 104 T 1/2 = 320d 249 Bk + 48 Ca 117 Yu. Oganessian SHE in JINR 109-th Session of the SC of JINR, Feb. 17-18, 2011, Dubna ???

4. N. Wang www.ImQMD.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. 50 Ti, 54 Cr, 58 Fe, 64 Ni, 70 Zn + 208 Pb 48Ca 1 20 Cf 1 10

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6. 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 …

7. N. Wang www.ImQMD.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

8. N. Wang www.ImQMD.com KineticNuclearCoulomb Skyrme energy-density functional M. Brack, C. Guet, H.-B. Hakanson, Phys. Rep. 123, 275 (1985). Skyrme force SkM*

9. N. Wang www.ImQMD.com Spherical symmetric Fermi functions Search for the minimum of energy by varying densities (R 0p, R 0n, a p, a n ) according to Hohenburg-Kohn theorem 1. Determination of density distributions

10. N. Wang www.ImQMD.com E1E1 E2E2 Sudden approximation for density R V.Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002). 2. Entrance-channel fusion barrier

11. N. Wang www.ImQMD.com with Determination of D(B) is the key of this approach 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

12. N. Wang www.ImQMD.com for reactions with nuclei near the beta-stability line but the neutron-shell is not closed

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14. Deviations 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

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17. The fusion excitation functions of a series of reactions with 16 O bombarding on medium mass targets.

18. N. Wang www.ImQMD.com Large-angle quasi-elastic scattering PRC 78, 014607 (2008)

19. N. Wang www.ImQMD.com S. G. Zhou Tail of barrier distribution influences the large-angle quasi-elastic cross sections

20. N. Wang www.ImQMD.com II. Survival probability W sur with HIVAP The sensitive parameters: 1. fission barriers (Liquid-drop barriers, Sierks barriers…) 2. level density parameters (Fermi gas model, angular-momentum and shape-dependent) 3. masses shell corrections and particle separation energies

21. N. Wang www.ImQMD.com 1). Fission barrier Nuclei Cohen-Swiatecki Sierk Dahlinger MWS 244 Pu 4.16 5.17 3.95 4.13 256 No 1.74 1.44 1.02 1.19

22. N. Wang www.ImQMD.com Obtained deviations with different models for 51 fusion-fission reactions N. Wang, M. Liu and Y. Yang, Sci. China G 52, 1554 (2009)

23. N. Wang www.ImQMD.com 2). Level density parameters In the standard HIVAP code: E d =18.5MeV, r a =1.153fm

24. N. Wang www.ImQMD.com Radius parameter r a in the level density parameter

25. N. Wang www.ImQMD.com Optimal value of the radius parameter r a 68% reactions have a value smaller than 0.0714, Estimated systematic errors of the HIVAP code: 1.85W sur and W sur /1.85

26. N. Wang www.ImQMD.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

27. N. Wang www.ImQMD.com Ning Wang, et al., Phys. Rev. C 77 (2008) 014603 Fusion-fission EDF HIVAP

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34. M. Itkis Dubna 1) Driving potential 2) Quasi-fission 3) Calculations of evaporation residual cross sections III. Fusion probability P CN (preliminary)

35. N. Wang www.ImQMD.com Y. Oganessian, 109th Session of the JINR Scientific Council, 2010, Dubna. 2) quasi-fission barrier B qf E sh

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37. Mean barrier height 3) Evaporation residual cross sections

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39. Uncertainty at E>B m : 1.18 (capture) x 1.85 (W sur ) x 2 (P CN ) = 4.4

40. N. Wang www.ImQMD.com Collected by Junlong Tian

41. N. Wang www.ImQMD.com A. K. Nasirov, et al., Phys. Rev. C 84, 044612 (2011)

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43. Results from Nan Wang ( ) http://www.phys-hutc.net/conference/2011-09-trans/pdf/.pdf

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45. Zagrebaev, et al., arXiv:1105.1659v1 For 4n channel

46. N. Wang www.ImQMD.com Opt. 50 Ti+ 249 Bk 50 Ti+ 249 Cf 54 Cr+ 248 Cm 58 Fe+ 244 Pu 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

47. N. Wang www.ImQMD.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.

48. N. Wang www.ImQMD.com China Institute of Atomic energy Zhu-Xia Li Xi-Zhen Wu Kai Zhao ( ) ( ) ( ) Institute of Theoretical Physics (CAS) En-Guang Zhao Justus-Liebig-Univ. Giessen Werner Scheid Guangxi Normal Univ. Min Liu Anyang Normal Univ. Jun-Long Tian

49. N. Wang www.ImQMD.com Thanks for your attention I am very grateful to Prof. Shan-Gui Zhou ( ) for many discussions and valuable suggestions This work was supported by Alexander von Humboldt Foundation and National Natural Science Foundation of China