1. Nuclear masses and shell corrections of superheavy elements
Ning Wang1, Min Liu1, Xi-Zhen Wu2, Jie Meng3
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. r-process、 symmetry energy
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. Central position of the island for SHE？
Yu. Oganessian. SKLTP/CAS - BLTP/JINR July 16, 2014, Dubna
Central position of the island for SHE？
neutrons →
Wang, Liang, Liu, Wu,
Phys. Rev. C 82 (2010)
Courtesy of Qiu-Hong Mo
Mass models with rms error of ~ keV

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) 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. n-p residual interaction Isobaric Multiplet Mass Equation ……
۞ 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) (R) N Z
Y. M. Zhao, et al.,
Phys. Rev. C ;
Phys. Rev. C ;
Phys. Rev. C ; …

7. Mass predictions from local mass equations by using iterations
errors increase rapidly with iterations
1）error of local mass equations, ~100keV 2）predicted masses are used in new iteration
MeV
Morales et al. , Nucl. Phys. A 828 (2009) 113
Morales, et al., Phys. Rev. C 83, (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
leave-one-out cross-validation
Revised masses
Wang & Liu, Phys. Rev. C 84, (R) (2011)

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

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

12. Nuclear mass tables WS mass tables http://www.imqmd.com/mass/
HFB mass tables
AME2012
Compilation of mass measurements

13. S. Goriely J. M. Pearson

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

15. 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. Parabolic approximation for small deformations
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)

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 p
Pomorski, Comp. Phys. Comm.174(2006)181
Diaz-Torres, Phys. Lett. B594 (2004) 69
: energy smoothing parameter p : order of Gauss–Hermite

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-micro concept & Skyrme energy density functional
PRC ； PRC ； PRC

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

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

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

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

29. 9 y 13 y 4 y Rms (keV) FRDM HFB24 WS WS4 654 549 525 298 31 30 13 18
Rms error
Rms (keV)
FRDM
HFB24 WS
WS4
To known masses
654
549
525
298
Number of model para. 31 30 13 18

30. Predictive power for new masses AME2012
M(WS3) – M(exp.)
Predictive power for new masses AME2012
rmsD (in keV)
WS3
FRDM
DZ28
HFB17
sigma (M)2149
336
656
360
581
sigma (M)219
424
765
673
648

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, (2014)

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

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

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

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

41. Summary
The rms deviations of mass models with respect to known masses fall to about 200 keV (local) and 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
Quadrupole Deformations
Prolate

45. Comparison of nuclear Quadrupole deformationsWS
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, (R)

49. RMF: Lalazissis, Raman, and Ring, At. Data Nucl
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)
For and , rch = 6.24±0.14 and 6.13±0.16 fm from the α-decay data
WS* results: 6.17 and 6.19 fm
Ni, Ren, Dong, Qian, Phys. Rev. C 87, (2013)

51. Uncertainty of Model predictions
Model errors increase for drip line nuclei

52. Litvinov, Palczewski, Cherepanov, Sobiczewski, Acta Phys. Polo
Litvinov, Palczewski, Cherepanov, Sobiczewski, Acta Phys. Polo. B 45 (2014) 1979

54. Oganessian & Utyonkov, Nucl. Phys. A (2015) (in press)

55. Fission barrier
Z=114, A=298
Sobiczewski, Pomorski, Prog. Part. Nucl. Phys. 58 (2007) 292

56. Comparison of Bf and Esh
For Z= region
Kowal,et al., Phys. Rev. C 82_014303

57. Symmetry energy coefficients of nuclei
Parabolic law for drip line nuclei?
Liu, Wang, Li, Zhang, Phys. Rev. C 82_064306

58. Symmetry energy coefficients of finite nuclei from Skryme energy density functional + ETF

59. Convergence test for the higher order terms

60. N. Wang, M. Liu, H. Jiang, J. L. Tian, Y. M. Zhao, Phys. Rev
N. Wang, M. Liu, H. Jiang, J. L. Tian, Y. M. Zhao, Phys. Rev. C 91, (2015)

61. Influence of the coefficient of fourth order terms
Nbound
Jiang, Wang, et al., Phys. Rev.C 91, (2015)

63. Residual – Mirror corr. reduces rms error by ~10%
with the same mass but with the numbers of protons and neutrons interchanged
Residual – Mirror corr.
reduces rms error by ~10%
charge-symmetry / independence of nuclear force

64. Residual – Wigner corr. of heavy nuclei
(N,Z)
N=Z
K. Mazurek, J. Dudek，et al., J. Phys. Conf. Seri. 205 (2010)