1. Correlation between alpha-decay energies of superheavy nuclei and effects of symmetry energy Wei Zuo Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou Clustering Aspects in Nuclei KITPC/ITP-CAS, April 1 - April 26, 2013

2. Outline Alpha-decay properties of superheavy nuclei and the effect of symmetry energy Alpha-decay properties of superheavy nuclei and the effect of symmetry energy Symmetry energy around saturation density and in heavy nuclei Momentum-distribution in asymmetric nuclear matter and the three-body force effect

3. Introduction and Motivation Syntheses of superheavy nuclei (SHN) becomes an active and exciting field in modern nuclear physics. Up to now SHN with Z = 104–118 have been synthesized in experiments!

4. Z=107–112, cold-fusion reactions, GSI, Darmstadt S. Hofmann and G. Munzenberg, Rev. Mod. Phys. 72, 733 (2000). Z = 113–118, hot-fusion evaporation reactions, JINR- FLNR,Dubna Yu. Ts. Oganessian et al., PRC 69, 021601(R) (2004); PRC 70, 064609 (2004); PRC 72, 034611 (2005); PRC 74, 044602 (2006); PRC 76, 011601(R) (2007); PRL 104, 142502 (2010). Other new superheavy nuclides: Z=113, Z=114 L. Stavsetra et al., PRL 103, 132502 (2009) (LBNL, USA) Ch. E. Dullmann et al., PRL 104, 252701 (2010). (GSI Darmstadt) P. A. Ellison et al., PRL 105, 182701 (2010). (LBNL, USA) K. Morita et al., J. Phys. Soc. Jpn. 73, 2593 (2004); J. Phys. Soc. Jpn. 76, 045001 (2007) (RIKEN, Japan).

5.  Alpha-decay is closely related to nuclear structure properties, it may provide useful information on nuclear properties such as ground state energies, shell effects, ……  Alpha decay is the most efficient approach to identify new nucleus via the observation of alpha-decay chain  Theoretically, one of the major goals is to predict reliably the half-lives of SHN for the experimental design.  It is extremely important and necessary to obtain an accurate theoretical Q α value for a reliable half-life prediction Alpha-decay properties of heavy nuclei and SHN

6.  The cluster model B. Buck et al, PRL72(1994)1326; R. R. Xu et al, PLB72(2006)322 C. Xu et al, PRC73(2006)041301  The density-dependent M3Y effective interaction P.R. Chowdhury et al, PRC77(2008)044603; G.Samanta et al, NPA789(2007)142 G. L. Zhang et al, NPA823(2009)16  The generalized liqiud drop model G. Royer et al, NPA730(2004)355; J. M. Dong et al, NPA832(2010)198; H.F. Zhang et al, PRC74(2006)017304; C77(2008)054318  The coupled channel approach D.S. Delion et al, PRC73(2006)014315; S. Peltonen et al, PRC75(2007)054301

7. Magic numbers in superheavy region ––– Model dependent: Macroscopic-microscopic models: Z=114, N=184 Moller and Nix,JPG 20, 1681 (1994); Baran et al., PRC 72, 044310 (2005) Skyrme-Hartree-Fock: Z =124, 126 and N =184 Cwiok et al., NPA 211 (1996); Kruppa et al.,PRC 61, 034313 (2000) Relativistic mean field models: Z =120, N =172,184 Bender etal., PRC 60, 034304 (1999);Rutz etal.,PRC 56, 238 (1997); Patra etal.,, NPA 117, (1999)

8. Alpha-decay Q values of superheavy nuclei We proposed a formula to directly calculate the alpha decay energy (Q value) for nuclei with Z ≥ 92 and N ≥ 140 based on the work by Prof. Ren [PRC 77, 064310 (2008)] : Dong, Zuo, Gu, Wang, Peng, PRC 81, 064309 (2010)

9. The standard and average deviations for the 154 heavy and superheavy nuclei: The deviations between the experimental values and the formula for the 154 nuclei.

10. On the whole, the formula provides good results for Z=117 isotope chain. Alpha-decay Q values for Z=117 isotope chain

11. In order to predict the α-decay energies of superheavy nuclei more accuracely, a new scheme is proposed according the correlation between the α-decay energies of superheavy nuclei

12. Correlation between alpha-decay energies of neighboring ① Starting from the liquid-drop model, once the decay energy Q 1 of a reference nucleus A Z 1 is known, the Q 2 value of the other nucleus A Z 2 (target nucleus) with the same mass number A can be estimated by: Reference nucleus ： Q 1 Target nucleus ： Q 2

13. ③ The correlation between the Q α values of the nuclei belonging to an isotone chain with a neutron number N is given by ② The correlation between the Q α values of the nuclei belonging to an isotope chain with a proton number Z is given by

14. ④ In general, if one selects ξ=xZ+yN and β as the two independent variables, the relationship between the Q α values of two superhevy nuclei can be written as

15. Alpha-decay energies of superheavy nuclei: comparison between the prediction and the experimental data 380 reference-target combinations

16. Since the Q values of the reference nuclei are taken from the experimental measurements in calculations, the agreement suggests that the experimental data themselves are consistent with each other, which indicates that the experimental observations and measurements of the SHN are reliable to a great extent. The agreement between the experimental and theoretical values has significant importance.

17. For the eight nuclides of elements 116 and 114 ( 290-293 116 and 286-289 114) together with the six nuclei with a neutron number N =174 ( 290 116, 289 115 and 288 114) and N =172 ( 287 115, 286 114 and 285 113), the experimental Q values can be reproduced very accurately that confirms Z =114 and N =172 are most possibly not shell closures for the presently observed superheavy region experimentally. About the shell closures in the region of supperheavy nuclei

18. It is the effect of symmetry energy that primarily enhances the stability against alpha decay with larger neutron number for these synthesized SHN not around shell closures. Due to the inclusion of the effect of symmetry energy, the Q values reduce much more rapidly as N increases, and hence a superheavy element becomes longer- lived against alpha-decay with increasing N. Effect of symmetry energy on the isospin dependence of the Q  values along an isotope chain of SHN

19. Alpha decay half-lives A new approach: estimate the half-life of a nucleus with the help of its neighbors based on some simple formulas. (Based on Royer’s formula) Dong, Zuo, Scheid, NPA, 861, 1 (2011) (Based on VSS formula)

20. 2015-12-9 The two formulas are found to work very well.

21. Applicability of WKB approximation We calculated the barrier penetrability for alpha decay, proton and cluster emission accurately with the recursion formulas by dividing the potential barrier into a sequence of square barriers and the results are compared with those of the WKB approximation.

22. We cut off the barrier at a sufficiently large distance of r 2 = 1000 fm. Classical turning points dividing the potential barrier into a sequence of ‘square’ barriers

23. The wave function u(r) (Ψ(r) = Y lm (θ, ϕ )u(r)/r) of the emitted particle with Q value in these n regions can be written as

24. The wave function outside of the barrier is given by By using the connection condition of wave function, one can deduce the transmission amplitude and reflection amplitude for the nth square barrier:

25. and for the j th (j <n) square barrier:

26. The penetration probability is given by: Relative deviation of penetrability caused by the WKB approximation:

27. WKB method produces relative deviations by about (−40)–(−30)% for alpha decay of heavy and superheavy nuclei, (−40)–(−20)% for proton emission and (−5)–15% for cluster radioactivity. The deviations being nearly constants in each decay mode.

28. Summary  A new idea has been proposed for predicting the Q  values of SHN. A simple formula has been got for describing the correlation between the  -decay energies of the SHN  Our investigation indicates that the reliability of the experimental observations and measurements on these synthesized SHN  Z=114 and N=172 are most probably not shell closures for the presently observed superheavy region experimentally.  The increased stability against alpha-decay for the SHN not around shell closures with larger neutron number, is primarily attributed to the effect of the symmetry energy.

29. symmetry energy Effective NN interaction in nuclear medium

30. Oyamatsu et al., NPA634(1998)3. Properties of Neutron-rich Nuclei Symmetry energy and the properties of neutron-rich nuclei

31. B. A. Brown, PRL, 85,5296(2000) Symmetry energy and the properties of neutron-rich nuclei Correlation between symmetry energy and neutron skin thinkness

32. R.J.Furnstahl, NPA706(2002)85 Symmetry energy and the properties of neutron-rich nuclei

33. Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006) Neutron star radius and the density-dependence of symmetry energy around saturation density

34. Theoretical Approaches Skyrme-Hartree-Fock Approach Relativistic Mean Field Theory Relativistic Hartree-Fock Variational Approach Green’s Function Theory Brueckner Theory Dirac-Brueckner Approach Effective Field Theory

35. Symmetry energy predicted by SHF and RMF approaches L.W. Chen, C.M. Ko, B.A. Li, Phys. Rev. Lett. 94 (2005) 032701. B. A. Li ， L. W. Chen ， C. M. Ko ， Phys. Rep. 464 （ 2008 ） 113 SHF RMF

36. C. Fuchs and H. H. Wolter, EPJA30(2006)5 Dieperink et al., PRC67(2003)064307. Symmetry energy predicted by various many-body theories Effective field theory DBHF BHF Greens function Variational

37. Symmetry energy around saturation density Around the saturation density, the symmetry energy can be expanded as follows: Slope parameter: Curvature parameter:

38. Correlation among S 0, L & K sym 1. 2. 3. Symmetry energy from M3Y-type of interaction:

39. By performing a least-squares fit with the calculated S 0, L, and K sym using the interactions above, we get a almost model-independent relation for describing the correlation of S 0, L, and K sym Correlation of S 0, L & K sym (1)LNS1, (2)LNS5, (3)MSL0, (4)SIV, (5)SkT4, (6)T6, (7)SkP, (8)SkM*, (9)SkX, (10)PK1, (11)D1S, (12)SLy4, (13)FSUGold, (14)SkMP, (15)SkI5, (16)NLSH, (17)TM1, (18)NL3, (19)NL1, (20)Sk255, (21)DDME1, (22)DDME2, (23)DDM3Y, (24)PC-F1, (25)Ska, (26)SV, (27)QMC, (28)MSkA, (29)SkI2, (30)MSk7, (31)HFB-17, (32)BSk8, (33)BSk17, (34)GM1, (35)GM3, (36)Sk272, (37)v090

40. Correlation between L & K sym : Correlation of L & K sym Dong, Zuo, Gu and Lombardo, Phys. Rev. C 85, 034308 (2012) Correlation between K sym and ΔR np of 208 Pb : Slope parameter L : Curvature parameter K sym :

41. Symmetry energy of finite nuclei In the Skyrme-Hartree-Fock approach, the total energy density functional:

42. Symmetry energy of finite nuclei The density functional for the symmetry : with The total symmetry energy of a finite nucleus

43. Symmetry energy of 208 Pb the ratio of the surface symmetry coefficient to the volume symmetry coefficient

44. The surface part of a heavy nucleus contributes dominantly to its symmetry energy compared to its inner part Distribution of symmetry energy density in 208 Pb Dong, Zuo, Gu, Phys. Rev. C 87, 014303 (2012)

45. The ratio of the surface symmetry coefficient to the volume symmetry coefficient is also determined from the measured alpha-decay energies of 162 heavy and superheavy nuclei The neutron skin thickness in 208 Pb

46. Summary  Based on various mean-field interactions, we obtain a correlation for the symmetry energy at saturation density S 0, the slope parameter L, and the curvature parameter K sym  With the help of the obtained correlation and available empirical information, the density-dependent behavior around the saturation density is determined  The surface region of a heavy nucleus contributes dominantly to its symmetry energy as compared to its inner part  The symmetry energy coefficient and the ratio of the surface symmetry coefficient to the volume symmetry coefficient are calculated

47. Nucleon momentum distributions in nuclear matter  A measure of the strength of the dynamical NN correlations induced by the NN interaction in a nuclear many-body system  Providing desirable information on the depletion of the deeply bound states inside finite nuclei  Understanding the short-range correlations in nuclear medium  Understanding the nature of nucleon-nucleon interactions (tensor force, three-body force, hard core, …..)  Testing the validity of the physical picture of independent particle motion in the mean field theory or the standard shell model  Understanding the properties of neutron stars ( cooling mechanism, nucleon pairing inside neutron stars, transport parameters ……) L. Frankfurt, M. Sargsian, and M. Strikman, Int. J. Mod. Phys. A 23, 2991 (2008) J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep. 25, 83 (1976) A. Rios, A. Polls, and W. H. Dickhoff, Phys. Rev. C 79, 06430 (2009) W. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004) M. Baldo, I. Bombaci, G. Giansiracusa, U. Lombardo, Nucl. Phys. A 530, 135 (1991) R. Subedi, R. Shneor, P. Monaghan et al., Science 320, 1476 (2008)

48. Nucleon momentum distributions and NN correlations Green function theory H. Muther et al., PRC 52, 2955 (1995); T. Alm et al., PRC 53, 2181 (1996); Y. Dewulf et al., PRC65, 054316 (2002); T. Frick et al., PRC 71, 014313 (2005) A. Rios et al., PRC79, 064308 (2009) In-medium T-matrix method P. Bozek, PRC 59, 2619 (1999); 65, 054306 (2002) V. Soma et al., PRC 78, 054003 (2008) Variational Monte Carlo method R. Schiavilla et al., PRL 98, 132501 (2007) Correlated basis function approach S. Fantoni et al., NPA 427, 473 (1984); O. Benhar et al., Phys. Rev. C 41, R24 (1990) Extended BHF method R. Sartor et al., PRC 21, 1546 (1980); P. Grange et al., NPA 473, 365(1987) M. Jaminon et al., PRC 41, 697 (1990); M. Baldo et al., PRC 41, 1748 (1990); C. Mahaux et al., NPA 553, 515 (1993); Kh. S. A. Hassaneen et al., PRC 70, 054308(2004)

49. Nucleon momentum distributions and NN correlations Experiments: the (e, e’p),(e, e’NN), and proton induced knockout reactions P. K. A. de Witt Huberts et al., JPG16, 507(1990); L. Lapikas et al, PRL 82, 4404 (1999); R. Starink et al., PLB 474, 33 (2000); M. F. van Batenburg, Ph.D. thesis, University of Utrecht, 2001; D. Rohe et al. (E97-006 Collaboration), PRL 93,182501 (2004); R. A. Niyazov et al. (CLAS Collaboration), PRL 92, 052303 (2004); K. S. Egiyan et al. (CLAS Collaboration), PRL 96, 082501 (2006); F. Benmokhtar et al. (Jefferson Lab Hall A Collaboration), PRL 94, 082305 (2005); R. Shneor et al. (Jefferson Lab Hall A Collaboration), PRL 99, 072501 (2007); J. L. S. Aclander et al., Phys. Lett. B 453, 211 (1999); A. Tang et al., PRL 90, 042301 (2003); E. Piasetzky et al, PRL 97, 162504 (2006); C. J. G. Onderwater, et al., PRL 81, 2213 (1998); L. A. Riley et al., PRC 78, 011303 (2008); R. Subedi et al., Science 320, 1476 (2008)

50. Experiments at NIKHEF: (e, e’p) reactions on 208Pb (M. F. van Batenburg, Ph.D. thesis, University of Utrecht, 2001) Conclusion: The depletion of the deeply bound proton states is 15%–20% for describing the measured coincidence cross sections Experiments at Jlab: two-nucleon knockout reactions 12C(e, e’pN) [ R. Subedi et al., Science 320, 1476 (2008) ] Only 80% of the nucleons in the 12C nucleus acted independently; about 20% of the nucleons form SRC pairs. For the 20% of correlated pairs, 90±10% are in the form of p-n SRC pairs; 5±1.5% are in the form of p-p SRC pairs

51. Bethe-Goldstone Theory Bethe-Goldstone equation and effective G-matrix → Nucleon-nucleon interaction: ★ Two-body interaction : AV18 (isospin dependent) ★ Effective three-body force → Pauli operator : → Single particle energy : → “Auxiliary” potential : continuous choice Confirmation of the hole-line expansion of the EOS under the contineous chioce (Song,Baldo,Lombardo,et al,PRL(1998))

52. Brueckner Theory of Nuclear Matter

53. Two problems of the BHF approach : 1. T he empirical saturation properties of nuclear matter can not reproduced within the framework of the nonrelativistic BHF approach by adopting realistic two-body forces [Coestor band, Coester et al., PRC1(1970)765] Solution: to include the effect of three-body forces 2. At densities around the saturation density, the predicted optical potential depth is too deep as compared to the empirical value, and it destroy the Hugenholtz-Van Hove (HVH) theorem. Solution: to include the effect of ground state correlations J. P. Jeukenne et al., Phys. Rep. 25 (1976) 83 M. Baldo et al., Phys. Lett. 209 (1988) 135; 215 (1988) 19

54. Improvement in two aspects: 1. Extend the calculation of the effect of ground state correlations to asymmetric nuclear W. Zuo et al., PRC 60 (1999) 024605 2. Include a microscopic three-body force (TBF) in the BHF calculation W. Zuo et al., NPA706 (2002) 418; PRC 74 (2006) 014317

55. Microscopic Three-body Forces Z-diagram Based on meson exchange approach Be constructed in a consistent way with the adopted two-body force---------microscopic TBF ! Grange et.al PRC40(1989)1040

56. Effective Microscopic Three-body Force Effective three-body force → Defect function:  (r 12 )=  (r 12 ) –  (r 12 ) ★ Short-range nucleon correlations (Ladder correlations) ★ Evaluated self-consistently at each iteration  Effective TBF ---- Density dependent  Effective TBF ---- Isospin dependent for asymmetric nuclear matter

57. W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418 EOS of SNM & saturation properties  (fm -3 ) E A (MeV) K (MeV) 0.19–15.0210 0.26–18.0230 Saturation properties: TBF is necessary for reproducing the empirical saturation property of nuclear matter in a non-relativistic microscopic framework.

58. Hole line expansion of mass operator: 1. The lowest-order BHF approximtion J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep. 25 (1976) 83 2. The rearrangement contribution 3. The renormalization contribution

59. Nucleon momentum distributions in ANM The The renormalized M 1 YIN, li, Wang, and Zuo, Phys. Rev. 87 (2013) 014314 The proton and neutron momentum distribution below and above the fermi surfaces The The renormalized M 2

60. In symmetric nuclear matter, the obtained depletion of the hole states deep inside the Fermi sea is roughly 15% at the empirical saturation density. Nucleon momentum distribution in symmetric nuclear matter At low densities around and below the nuclear saturation density, the TBF effect on the predicted momentum distributions is found to be negligibly weak At high densities well above the saturation density, the TBF is expected to induce strong enough extra short-range correlations and its effect turns out to become noticeable. Inclusion of the TBF lead to an enhancement of the depletion of the Fermi sea. YIN, li, Wang, and Zuo, Phys. Rev. 87 (2013) 014314

61. In asymmetric nuclear matter, the neutron and proton momentum distributions turn out to become different Increasing the isospin asymmetry β tends to enhance the depletion of the proton Fermi sea while it reduces the depletion of the neutron Fermi sea The TBF effect on the predicted momentum distributions only becomes sizable at high densities well above the saturation density Proton and neutron momentum distributions in asymmetric nuclear matter

62. At zero momentum, the neutron occupation probability increases while the proton occupation decreases almost linearly as a function of asymmetry. In dense asymmetric nuclear matter, the TBF may lead to an overall reduction of both the neutron and proton occupations below their Fermi seas in the whole asymmetries range The TBF effect on the isodepletion (i.e., the difference of the neutron and proton occupation probabilities) in asymmetric nuclear matter is shown to be quite small in the density region up to two times saturation density Proton and neutron occupations of their lowest momentum sates

63. Summary In asymmetric nuclear matter, increasing the isospin asymmetry tends to enhance the depletion of the proton Fermi sea while it reduces the depletion of the neutron Fermi sea At low densities around and below the nuclear saturation density, the TBF effect on the momentum distributions is negligible. At high densities well above the saturation density, the TBF may lead to an overall enhancement of both the depletion of the neutron and proton Fermi seas in the whole asymmetries range The TBF effect on the isodepletion is shown to be quite small in the density region up to two times saturation density

64. Collaborators: Jian-Min Dong (IMP, CAS,China) Peng Ying (IMP, CAS, China) Jian-Yang Li (IMP, CAS, China) Pei Wang (IMP, CAS, China) Jianzhong Gu (CAIE, China) U. Lombardo (INFN-LNS, Italy) Werner Scheid (Justus-Liebig-University, Germany)

65. Thank you!