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

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

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) 044304 Courtesy of Qiu-Hong Mo Mass models with rms error of ~300-600keV

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

☺ 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 … …

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) 041304(R) N Z Y. M. Zhao, et al., Phys. Rev. C82-054317; Phys. Rev. C84-034311; Phys. Rev. C85-054303; …

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, 054309 (2011)

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

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

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

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

Nuclear mass tables WS mass tables http://www.imqmd.com/mass/ HFB mass tables http://www-astro.ulb.ac.be/bruslib/nucdata/ AME2012 http://amdc.impcas.ac.cn/evaluation/data2012/ame.html Compilation of mass measurements http://nuclearmasses.org/

S. Goriely J. M. Pearson

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

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

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

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

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

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

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

Shell effects are evident at magic numbers

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

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

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

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

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

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

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

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

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

For new masses after 2012

Shell gaps

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

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

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

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

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

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

142 152 162

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.

Thank you for your attention Differences make the world more beautiful

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

Quadrupole Deformations Oblate Quadrupole Deformations Prolate

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

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

Deformation energies

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

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

Nuclear charge radii from WS* model Angeli & Marinova, J. Phys. G: Nucl. Part. Phys. 42 (2015) 055108 For 286114 and 290116, 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, 024310 (2013)

Uncertainty of Model predictions Model errors increase for drip line nuclei

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

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

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

Comparison of Bf and Esh For Z=114-120 region Kowal,et al., Phys. Rev. C 82_014303

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

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

Convergence test for the higher order terms

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, 044308 (2015)

Influence of the coefficient of fourth order terms Nbound Jiang, Wang, et al., Phys. Rev.C 91, 054302 (2015)

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

Residual – Wigner corr. of heavy nuclei (N,Z) N=Z K. Mazurek, J. Dudek，et al., J. Phys. Conf. Seri. 205 (2010) 012034