Systematic study of fusion reactions leading to super-heavy nuclei Ning Wang ( ) Guangxi Normal University Workshop on Synthesis of Super-heavy nuclei, August 10-13, 2012, Lanzhou 1. Introduction 2. Capture cross section 3. Survival probablity W sur 4. Fusion probabilty P CN 5. Summary

1) to study the three parts individually; 2) to estimate the model uncertainty

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 …

I. Capture cross sections with the Skyrme energy-density functional Density distributions of the reaction partners Entrance-channel Coulomb barrier Capture 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)

Woods-Saxon form for densities Search for the minimum of energy by varying densities (R 0p, R 0n, a p, a n ) according to Hohenberg-Kohn theorem 1. Determination of density distributions

E1E1 E2E2 Sudden approximation for density R V.Yu. Denisov and W. Noerenberg, Eur. Phys. J. A15, 375 (2002). 2. Entrance-channel inter-nucleus potential

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. 16 O+ 208 Pb, 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 16 O bombarding on medium mass targets. Wang et al. Sci China G 52, 1554 (2009) suppression enhancement

Deviations from exp. data for 120 reactions For about 70% systems, the deviations are smaller than 0.005, estimated systematic error 18%. N. Wang et al., J. Phys. G: 34 (2007) 1935 rms deviation for (E>B 0 )

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 In the standard HIVAP code: r a =1.153fm

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

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

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) ). Masses of super-heavy nuclei An improved nuclear mass formula WS* 152

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

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

B0B0 BfBf B qf III. Fusion probability E* also influences the results

1) quasi-fission barrier Wang, Tian, Scheid, PRC84, (R) (2011) Yu. Oganessian Wang, et al, PRC77, (2008)

Fusion probability

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

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

Opt. 50 Ti+ 249 Bk 50 Ti+ 249 Cf 54 Cr+ 248 Cm 58 Fe+ 244 Pu Zagrebaev PRC(2008) ~ 50 fb ~ 40 fb ~ 20 fb~ 5 fb Liu-Bao PRC(2011) ~ 600 fb ~ 100 fb Nasirov PRC(2011) ~ 100 fb~ 70 fb Ning Wang PRC(2011) ~ 35 fb ~ 20 fb ~ 5 fb~ 3 fb Nan Wang PRC(2012) ~ 110 fb ~ 50 fb ~ 6 fb~ 4 fb Siwek- Wilczynska PRC(2012) ~ 30 fb ~ 6 fb ~ 1 fb~ 0.1 fb exp. (GSI) < 70 fb < 200 fb (from talk of D. Ackermann)

Summary Models for calculations of capture cross sections, survival probability of compound nucleus and the fusion probability in fusion reactions leading to SHN have been tested 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.

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

Fission fragment yields with driving potential DNS: En-Guang Zhao Shan-Gui Zhou Jun-Qing Li Nan Wang Nuclear engineering Nuclear structure Fission cycle in nuclear astrophysics

Thanks for your attention Some codes and data are available at

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

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

2). Level density parameters In the standard HIVAP code: E d =18.5MeV, r a =1.153fm

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

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