Nuclear masses and shell corrections of superheavy elements

1. Ning Wang1, Min Liu1, Xi-Zhen Wu2, Jie Meng3 Nuclear masses and shell corrections of superheavy elements 1 Guangxi Normal University, Guilin, China 2 China Institute of Atomic Energy, Beijing, China 3 Peking University, Beijing, China • Introduction • Macroscopic-microscopic mass models • Shell gaps and shell corrections • Summary & discussions “Interfacing Structure and Reaction Dynamics in the Synthesis of the Heaviest Nuclei” at the ECT*, Trento, Italy, September 1 - 4, 2015

2. Super-heavy nuclei r-process、symmetry energy To predict the ~ 4000 unknown masses based on the ~2353/2438 measured masses G. Audi, M. Wang, et al., Chin. Phys. C 36, 1287 (2012) H. Jiang, N. Wang, et al., Phys. Rev. C91(2015)054302

3. Wang, Liang, Liu, Wu, Phys. Rev. C 82 (2010) 044304 Courtesy of Qiu-Hong Mo Yu. Oganessian. SKLTP/CAS - BLTP/JINR July 16, 2014, Dubna Central position of the island for SHE？ neutrons → Mass models with rms error of ~300-600keV

4. Nuclear mass models Global Local Systematics • Garvey-Kelson • n-p residual • ab initio • Shell model • … • Microscopic • Macro-Micro • Duflo-Zuker • … • AME • CLEAN • RBF • … Non-relativistic & relativistic energy density functional , more fundamental, can describe not only the properties of finite nuclei but also those of neutron stars

5. ☺ Macroscopic-Microscopic： • Strutinsky type: (shell corrections) • Finite range droplet model (FRDM): [M, β, Bf,…] • Extended Thomas-Fermi+SI (ETFSI): [M, EOS, β, Bf,…] • Lublin-Strasbourg Drop (LSD) model: [M, β, Bf,…] • Weizsäcker-Skyrme (WS) formula:[M, β, Rch,…]… … • Others : • Esh from valence-nucleons: Kirson, NPA798 (2008) 29 Dieperink & Isacker, EPJA 42 (2009) 269 • Wigner-Kirkwood method: Centelles, Schuck, Vinas, Anna. Phys. 322 (2007) 363; Bhagwat, et al.,PRC81_044321 • KUTY model: Koura, Uno, Tachibana, Yamada, NPA674(2000)47… …

6. Z N ۞Local mass formulas： • Garvey-Kelson • n-p residual interaction • Isobaric Multiplet Mass Equation • …… • Garvey, Gerace, Jaffe, Talmi, Kelson, Rev. Mod. Phys. 41 (1969) S1 • Barea, Frank, Hirsch, Isacker, et al, Phys. Rev. C 77 (2008) 041304(R) Y. M. Zhao,et al., Phys. Rev. C82-054317; Phys. Rev. C84-034311; Phys. Rev. C85-054303; …

7. Mass predictions from local mass equations by using iterations errors increase rapidly with iterations 1）error of local mass equations, ~100keV2）predicted masses are used in new iteration MeV Morales et al. , Nucl. Phys. A 828 (2009) 113 Morales, et al., Phys. Rev. C 83, 054309 (2011)

8. Image reconstruction techniques Morales, Isacker, Velazquez, Barea, et al., Phys. Rev. C 81(2010)024304 CLEAN deconvolution the aim is to select those Fourier components that best explain the observed patterns of the image

9. Radial Basis Function (RBF) corrections Revised masses leave-one-out cross-validation Wang & Liu, Phys. Rev. C 84, 051303(R) (2011)

10. RBF corrections for different mass models N. Wang and M. Liu, J. Phys: Conf. Seri. 420 (2013) 012057

11. AME2012 Z. M. Niu, et al., Phys. Rev. C 88 (2013) 024325

12. Nuclear mass tables • WS mass tables http://www.imqmd.com/mass/ • HFB mass tableshttp://www-astro.ulb.ac.be/bruslib/nucdata/ • AME2012http://amdc.impcas.ac.cn/evaluation/data2012/ame.html • Compilation of mass measurementshttp://nuclearmasses.org/

13. S. GorielyJ. M. Pearson

14. Why is the difference so large for neutron-rich nuclei ?

15. www.nupecc.org Macroscopic-microscopic mass models 1. Liquid-drop formula ‘semi-empirical mass formula’ of von Weizsäcker in 1935 EOS symmetric Mirror nuclei EOS asymmetric Liang, et al., Nucl. Phys. Rev. 28 (2011)1

16. 2. Liquid Drop Energy of nuclei with sharp surface (at small deformations) Myers & Swiatecki, Nucl. Phys. 81 (1966) 1 Volume term Surface term Coulomb term Parabolic approximation for small deformations

17. Parabolic approx. for the deformation energies Skyrme energy density functional+ ETF2 Parabolic approximation can significantly reduce the CPU time

18. The values of g1 and g2 can be obtained by known masses Nuclear surface diffuseness and its isospin dependence result in the deformation energies complicated

19. Isospin dependence of the surface diffuseness  Deformation dependence of the symmetry energy coefficients of nuclei Skyrme energy density functional+ ETF2 Mo, Liu, Chen, Wang, Sci. China - Phys. Mech. Astron. 58 (2015) 082001

20. Deviations from experimental data Myers & Swiatecki, Nucl. Phys. 81 (1966) 1 Lunney, Pearson, Thibault, Rev. Mod. Phys. 75 (2003) 1021

21. Shell effects are evident at magic numbers

22. 3. Strutinsky shell correction Strutinsky & Ivanjuk, Nucl. Phys. A255 (1975) 405 Pomorski, Comp. Phys. Comm.174(2006)181 p : energy smoothing parameterp : order of Gauss–Hermite Diaz-Torres, Phys. Lett. B594 (2004) 69

23. Woods-Saxon potential symmetry potential Cwoik, Dudek, et al., Comput. Phys. Commun. 46(1987) 379

24. 4. Weizsäcker-Skyrme mass formula Liquid drop Deformation Shell Residual Residual：Mirror 、pairing 、Wigner corrections... Macro-microconcept& Skyrme energy density functional PRC81-044322； PRC82-044304； PRC84-014333

25. symmetry potential Isospin dependence of model parameters • Symmetry energy coefficient • Symmetry potential • Strength of spin-orbit potential • Pairing corr. term WS3：Phys.Rev.C84_014333

26. 5.Isospin dependence ofsurface diffuseness Neutron-rich N. Wang, M. Liu, X. Z. Wu, and J. Meng, Phys. Lett. B 734 (2014) 215

27. WS Potential energy surface around ground state deformations By setting different initial values, one can find the lowest energy are considered

28. Determination of model parameters： Simulated annealing global minimum Greedy algorithmlocal minimum

29. Rms error 4y 13 y 9y

30. M(WS3) – M(exp.) Predictive power for new massesAME2012

31. For new masses after 2012

32. Shell gaps

33. New magic numbers Wienholtz, et al., Nature 498 (2013)346

34. Shell structure in heavy and super-heavy nuclei 108 Mo, Liu, Wang, Phys. Rev. C 90, 024320 (2014)

35. N=16 Shell corrections WS* Emic (FRDM): ground state microscopic energy

36. KSO = -1 KSO = 1 FRDM WS* Xu & Qi, Phys. Lett. B724 (2013) 247 WS4, Phys. Lett. B 734 (2014) 215

38. Symmetry energy coefficient and symmetry potential I=(N-Z)/A NPA818 (2009) 36 Wang & Liu, PRC81, 067302

39. Influence of potential parameters on the shell corrections Radius of potential Surface diffuseness Depth of potenital Symmetry potential Spin-orbit potential

40. 142 152 162

41. Summary • The rms deviations of mass models with respect to known masses fall to about 200 keV (local) and 300-600 keV (global) . • For super-heavy nuclei and drip line nuclei, model uncertainty increases rapidly. Isospin dependence of model parameters (such as symmetry potential and spin-orbit potential) influences the new magic numbers and shell corrections of SHE. • The shell gap is a sensitive quantity to test mass models. WS formula indicates N=142, 152, 162, 178; Z=92, 100, 108, 120 could be sub-shell closure in super-heavy region, in addition to traditional magic numbers Z=114, N=184.

42. Thank you for your attention Differences make the world more beautiful

43. Discussions • Nuclear deformations and radii • Uncertainty of model predictions • Fission barrier • Symmetry energy coefficients • Other corrections • … …

44. Quadrupole Deformations Oblate Prolate

45. Comparison of nuclear Quadrupole deformations WS FRDM Zhu Li，Bao-Hua Sun

46. Comparison of nuclear Octupole deformations deformations can also be included in the WS calculations

47. Deformation energies

48. Rms charge radii N. Wang, T. Li, Phys. Rev. C88, 011301(R)

49. RMF: Lalazissis, Raman, and Ring, At. Data Nucl. Data Tables 71, 1 (1999)

50. Nuclear charge radii from WS* model Angeli & Marinova, J. Phys. G: Nucl. Part. Phys. 42 (2015) 055108 For286114 and290116, rch =6.24±0.14 and 6.13±0.16 fmfrom the α-decay data WS* results: 6.17 and 6.19 fm Ni, Ren, Dong, Qian, Phys. Rev. C 87, 024310 (2013)