1. Study on Sub-barrier Fusion Reactions and Synthesis of Superheavy Elements Based on Transport Theory Zhao-Qing Feng Institute of Modern Physics, CAS

2. Contents Introduction Improved isospin dependent quantum molecular dynamics model Study on dynamics of fusion reactions near Coulomb barrier Production cross sections of the superheavy nuclei based on dinuclear system model Summary

3. 1. Introduction 60s, 20 century, Shell model prediction: “stability island” around Z=114,N=184 Experiments GSI: 110-112 Dubna: 113-116 Riken: 113 IMP: 105, 107 (new nuclei)

4. Theoretical models for the description of superheavy nuclei: Dinuclear system model (Adamian et al. NPA 633 (1998) 409, Li et al. EPL 64(2003)750, Feng et al. CPL 22 (2005) 846) Fluctuation-dissipation model (Aritomo et al. PRC 59 (1999) 796) Nucleon collectivization model (Zagrebaev et al. PRC 65 (2001) 014607) Macroscopic dynamical model (S. Bjornholm and W.J. Swiatecki, NPA 391(1982) 471) Improved isospin dependent quantum molecular dynamics model (Wang et al. PRC69 (2004) 034608), Feng et al. NPA 750 (2005) 232

5. 2. Improved isospin dependent quantum molecular dynamics model Purpose: to study fusion mechanism near Coulomb barrier Improved aspects including: 1. Yukawa term is replaced by introducing density dependent surface term derived from self-consistently Skyrme interaction. (Wang et al. PRC 65 (2002) 064608) 2. Introducing surface symmetry term. (Wang et al. PRC 69 (2004) 034608) 3. Nucleon’s fermionic nature is improved by using phase space constraint method. (M. Papa et al. PRC 64 (2001) 024612)

6. 4. Coulomb exchange term is included in the model. (Wang et al. PRC 67 (2003) 024604) 5. Shell effect is considered in the model. (Feng et al. NPA, 750 (2005) 232 ) 6. Switch function method is introduced in the model, which can effectively prevent some unphysical nucleus emissions in the process of projectile and target appoarchng. (Feng et al. HEP&NP,2005,29(1) 41 )

7. 2.1 Introduction on the improved isospin dependent quantum molecular dynamics model In the improved model, the effective interaction potential energy is denoted as

9. Switch function method is introduced, which can prevent some unphysical nucleons emission. So the surface interaction energy of the system is written as S is called as switch function Taking coefficients must satisfy the continuity of the surface energy and its first derivative! C 0 C 1 C 2 C 3 C 4 C 5 0 0 0 10 -15 6

10. Parameter set in the model  /MeV  /MeV  C sym /MeV  sym /fm 2 g surf /MeV  fm 2 g  /MeV  0 /fm -3 -356.0 303.0 7/6 32.0 0.08 8.0 10.0 0.165 Parameter set taken by Wang et al. The ground state properties, static (dynamical) barriers fusion (capture) excitation function as well as neck dynamical behaviour et al. can be described very well using the improved model.

11. 2.2 Consideration of shell effect in ImIQMD As we know that shell effect is the diversity of shell model (shell structure) and macroscopic model (bulk property). Thus, the shell correction energy can be obtained from the variance of shell levels and uniformed levels, which is written by Using Strutinsky method (NPA 95 (1967) 420), the shell correction energy is written as

12. The smoothed level density is usually given by Gaussian distribution width In the calculation, 3rd-order Laguerre polynomial is used. The Fermi energy is obtained by The shell levels are calculated by using deformed two center shell model. ( R.A. Gherghescu, Phys. Rev. C 67 (2003) 014309) In ImIQMD, the Shell correction energy is denoted by Using canonical equation, the force can be obtained as

13. One can obtain the force of each nucleon derived from the shell correction energy as From energy density functional, we can also know that shell effect mainly embodies the surface of the nucleus! (M. Brack, C. Guet, H.B. Hakansson, Phys. Rep. 123 (1985) 276) Considering the Woods-Saxon distribution form of the nuclear density, it is more self-consistently by denoting the shell correction energy as

14. It is very important to fill these levels in ImIQMD. In our calculation, we label each nucleon according to angular momentum and single particle energies, which are obtained respectively by

15. 3. Dynamical study on fusion reactions near Coulomb barrier Based on improved isospin dependent quantum molecular dynamics model, the static and dynamical Coulomb barrier, fusion/capture cross sections, neck dynamical behaviour et al. are studied systematically. 3.1 Dynamical barrier Here, E pt, E p and E t are the total, projectile and target energy respectively, the kinetic energy part is approximated by using Thomas-Feimi model as

16. The static nucleus-nucleus interaction potential

17. Static barriers, prox. (W.D. Myers et al., PRC 62 (2000) 044610)

18. Dependence on the projectile-target combinations leading to the same compound nucleus formation 258 Rf

19. The static and dynamical interaction potentials calculated by using the ImIQMD for the reaction sytems 40,48 Ca+ 40,48 Ca.

20. Dependence of dynamical barriers on incident energy

21. Dependence of fusion barrier on projectile neutron number leading to the same element formation

22. 3.2 Neck dynamical behaviour Time evolution of N/Z at neck region for 40,48 Ca+ 40,48 Ca

23. Nucleon transfer in neck region for reaction system 48 Ca+ 238 U N/Z dependence on incident energy at neck region for system 48 Ca+ 208 Pb

24. Neck radius development in the process of neck formation

25. 3.3 The calculation of fusion/capture cross sections Experimental data taken in Refs M. Trotta et al., Phys. Rev. C 65 (2001) R011601 H.A Aljuwair et al., Phys. Rev. C 30 (1984) 1223 Fusion excitation functions for 40,48 Ca+ 40,48 Ca

26. Positive Q value will lead to the enhancement of sub- barrier fusion cross sections 40,48 Ca+ 124,116 Sn, 16,18 O+ 42,40 Ca, 9,11 Li+ 208,206 Pb suggested by V.I. Zagrebaev PRC 67 (2003) 061601R.

27. Capture cross sections for heavy systems Experimental data taken from E.V Prokhorova et al., nucl- exp/0309021 and W.Q Shen et al., Phys. Rev C 36 (1987) 115

28. M. Dasgupta et al., Nucl. Phys A 734 (2004) 148 K. Nishio et al., Phys. Rev. Lett 93 (2004) 162701 M. G. Itkis, Yu. Ts. Oganessian, E. M. Kozulin et al., Proceedings on Fusion Dynamics at the Extremes, Dubna, 2000, edited by Yu. Ts. Oganessian and V. I. Zagrebaev page 93.

29. Preliminary consideration on the calculation of the evaporation cross sections based on ImIQMD Formation probability at excitation energy E* is written as Where E 0 is the critical excitation energy depending on the reaction system,  is the barrier distribution width, we can take it as So the evaporation cross section can be denoted by

30. 4. Production cross sections of the superheavy nuclei based on dinuclear system model In dinuclear system mdoel, evaporation cross section is denoted by Schematic illustration of the fusion process Cap.Q-fission Fission

31. , T(E c.m.,J) is usually taken 0.5. Fusion probability The mass distribution probability P(A 1,E 1,t) is given by master equation which is solved numerically in the model. If only considering the competition of neutron emission and fission, the survival probability W sur with emitting X neutrons can be written as Energy and angular-momentum dissipation are described by Fokker-Planck equation

32. Based on dinuclear system model, the production cross sections of superheavy nuclei in cold fusion reactions are studied systematically. Height of the pocket are 6.39 MeV (0.61) 4.80 MeV (0.56) 1.70 MeV (0.04) 1.71MeV (0.02) In the DNS model, the compound nucleus formation is governed by the driving potential.

33. Production cross sections for asymmetric and nearly symmetric reaction systems, comparison with coupled channel model which has included nucleon transfer and surface vibration is also shown. (V.Yu. Denisov Prog. Part. Nucl. Phys. 46 (2001) 303) Feng, Jin, Fu et al., Chin. Phys. Lett., 22(4), 2005, 846

34. Improvement of dinuclear system model In order to describe correctly the capture process, barrier distribution function method is included in the model. (P.H. Stelson, PLB 205 (1988) 190, V.I. Zagrebaev et al., PRC 65 (2001) 014607) The transmission coefficient is denoted as The barrier distribution function satisfies the normalization condition, which is usually taken as a asymmetric Gaussian distribution form.

35. Here B m =(B 0 +B s )/2, B 0 and B s are the height of the Coulomb barrier and the saddle point respectively. Gaussian distribution function  2 = (B 0 -B s )/2,  1 is less than the value of  2 (usually 2 MeV). V. I. Zagrebaev PRC64 (2001)034606

36. Capture cross section can be reproduced very well by introducing the barrier distribution function method

37. Comparison of calculated evaporation residue cross sections with experimental data for 1,2,3,4 neutron emission Experimental data taken from E.V Prokhorova et al., nucl-exp/0309021 Yu. Ts. Oganessian et al., Phys. Rev. C 64, 054606 (2001).

38. Production cross sections of superheavy nuclei 286-xn 112, 292-xn 114, 296-xn 116 in 48 Ca induced reactions and comparison with Dubna data (Yu.Ts. Oganessian et al., PRC 70 (2004) 064609)

39. Extension to multi-dimension degree of freedom for the driving potential In the DNS model, we only consider the mass asymmetry degree of freedom, so there is some difficulties for describing the mass distribution of quasi-fission or fission, as well as reasonably showing the formation process of the compound nucleus. We need to consider the center of mass distance R and deformation degree of freedom  et al. in the process of the superheavy compound nucleus formation. Y. Aritomo, M. Ohta, NPA 744 (2004) 3

40. 5. Summary The isospin dependent quantum molecular dynamics model is improved by introducing switch function method for the surface term and considering shell effect. Experimental fusion/capture cross sections can be reproduced very well using the improved model. Fusion barrier and neck dynamical behaviour in fusion process are studied systematically. The dinuclear system model is improved by introducing the barrier distribution function method, dynamical deformation is considered in the capture process. Evaporation residue cross sections can be regenerated well for 1n,2n,3n 4n evaporation. Further studies are in progress!