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 www.ImQMD.com/wangning/ BLTP/JINR-KLFTP/CAS Joint Workshop on Nuclear Physics, Aug. 2-6, 2012, Dubna

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) 044604

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)

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

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

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

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)

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%.

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

Fusion-fission：EDF＋HIVAP Wang, Zhao, Scheid, Wu, PRC 77 (2008) 014603

Deviations of calculated evaporation (and fission) cross sections from exp. data for 51 fusion-fission reactions For 68% reactions, the deviations are smaller than 0.0714, Estimated systematic errors of the HIVAP code: 1.85Wsur and Wsur /1.85

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) 044322 WS*: PRC 82 (2010) 044304 WS3: PRC 84 (2011) 014333

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

N=178 WS* N=162 N=178 N=178 WS* WS* Zhang, et al., Phys. Rev. C 85, 014325 (2012)

III. Fusion probability 1) quasi-fission barrier Yu. Oganessian Wang, Tian, Scheid, PRC84, 061601(R) (2011)

Fusion probability

2) Evaporation residual cross sections Mean barrier height PRC84, 061601(R) (2011)

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

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

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.

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）

Thanks for your attention Codes and data are available at：www.ImQMD.com

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

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 …

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

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

Large-angle quasi-elastic scattering Tail of the barrier distribution influences the large-angle quasi-elastic cross sections PRC78, 014607 (2008)

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