The first systematic study of the ground-state properties of finite nuclei in the relativistic mean field model Lisheng Geng Research Center for Nuclear Physics, Osaka University School of Physics, Beijing University

2 Long-time collaborators Jie Meng School of physics, Beijing University China Hiroshi Toki Research Center for Nuclear Physics Osaka University, Japan

3 Outline ① A brief review of relevant experimental quantities: nuclear masses, charge radii, 2 + energies, deformations, odd-even effects ② Theoretical framework a.The relativistic mean field (RMF) model b.The BCS method c.Model parameters ③ The first systematic study of over 7000 nuclei a. Comparison with experimental data and other theoretical predictions b. The causes of some discrepancies: the not-well-constrained isovector channel ④ Summary and perspective

4 Up to 1940! Introduction I: Nuclear masses

5 Up to 1948!

6 Introduction I: Nuclear masses Up to 1958!

7 Introduction I: Nuclear masses Up to 1968!

8 Introduction I: Nuclear masses Up to 1978!

9 Introduction I: Nuclear masses Up to 1988!

10 Introduction I: Nuclear masses Up to 1994!

11 Introduction I: Nuclear masses Up to 2004!

12 Introduction II: Charge radius E. G. Nadjakov, At. Data Nucl. Data Tables 56 (1994)

13 Introduction II: Charge radius I. Angeli, At. Data Nucl. Data Tables 87 (2004)

14 Introduction II: Charge radius

15 Introduction III: The energy of the first excited 2+ state S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128

16 Introduction III: The energy of the first excited 2+ state S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128

17 Introduction IV: Nuclear deformation

18 Introduction V: The odd-even effect and pairing correlation

19 Introduction V: The odd-even effect and pairing correlation

20 1.The spin-oribit interaction: i.e. the magic number effect 2.The deformation effect: most nuclei are deformed except a few magic nuclei 3.The pairing correlation: important to describe open-shell nuclei and responsible for the very existences of drip line nuclei The essential ingredients to build a nuclear structure model

21 It was found that without the spin-orbit interaction, only the first three magic numbers can be reproduced: 2, 8, 20 However, if one introduces by hand the so-called spin-orbit potential of the following form: All the magic numbers come out correctly Z=2, 8, 20, 28, 50,82 & N=2,8,20,28,50,82,126 Elementary theory of nuclear shell model, M. G. Mayer and J. Hans D. Jensen, 1956 no spin-orbitspin-orbit Spin-orbit interaction in non-relativistic models Therefore, in all non-relativistic nuclear structure models, a similar form of spin-orbit potential has to be introduced by hand and adjusted to reproduce the experimentally observed magic number effects

22 The relativistic mean field (RMF) model The RMF model starts from the following Lagrangian density: Dirac equationKlein-Gordon equation Scalar and Vector potentials

23 Spin-orbit interaction in the RMF model For a spherical nucleus, the Dirac spinor has the following form: Substitute it into the Dirac equation one obtains the coupled one-order differential equations for the large and small components: By eliminating the small component, one obtains a second-order differential equation for the large component, namely spin-orbit interaction The scalar and vector potentials are of the order of several hundred MeV!

24 1.Spin-orbit interaction 2.Nuclear matter saturation 3.Polarization (spin) observables in nuclear reaction 4.Study of high density and high temperature nuclear matter 5.Connection to QCD 6.Pseudospin symmetry The necessity of a relativistic model “The atomic nucleus as a Relativistic system”, L. N. Savushkin and H. Toki, springer, 2005

25 Non-relativistic calculations: not successful. Coester band Empirical saturation non-relativistic relativistic Bruckner Hartree-Fock Relativistic Bruckner Hartree-Fock calculations: encouraging! relativistic mean field theory The RMF model: parameterized to describe the nuclear matter saturation. Nuclear matter saturation

26 The basis expansion method: Treating the deformation The Dirac wave functions can be expanded by the eigen-functions of an axially-symmetric harmonic oscillator potential more specifically Therefore, solving the Dirac equation is transformed to diagonalizing the following matrix The meson fields can be treated similarly

27 The effect of deformation and pairing Binding energy per nucleon of Zirconium isotopes deformation pairing

28 Extending RMF to incorporate the pairing correlation From RMF to RMF+BCS Total energy: BCS equations: Or gap equation Occupation probability RMFRMF +BCS

29 But for weakly bound nuclei, which are the subjects of present research, it fails. The pairing correlation in weakly bound nuclei The constant-gap BCS method: very successful for stable nuclei A zero-range delta force in the particle-particle channel is found to be useful!

30 The pairing correlation in weakly bound nuclei Yadav and Toki, Mod. Phys. Lett A 17 (2002) d 3/ g7/2 4s1/2 3p3/2 S.P.E [MeV] The state dependent BCS method can describe weakly bound nuclei properly

31 Resonant states exist due to centrifugal barriers. 1g7/2 (0.55 MeV) this barrier traps 1g7/2

32 The state-dependent BCS method: extremely important!!!! Self-consistent description of spin-orbit interaction: RMF (1980-) Deformation effect: basis expansion method (1990-) Proper pairing correlation: state-dependent BCS method (2002-) Spherical case: Yadav and Toki, MPLA (2002) Sandulescu, Geng and Toki, PRC (2003) Deformed case: Geng and Toki, PTP (2003), NPA (2004) The advantage of the state-dependent BCS method: 1.Effective: valid for all nuclei 2.Numerically simple: systematic study possible

33 Stars: twelve nuclei used in DD-ME2 Nuclear matter: saturation density, binding energy per nucleon symmetry energy compression modulus Finite nuclei: Binding energy Charge radius Model parameters of the mean-field channel Free parameters in the RMF model: the sigma meson mass, the sigma-nucleon, omega-nucleon, rho-nucleon couplings, the sigma non-linear self couplings (2) and the omega non-linear self coupling. In total, there are 7 parameters.

34 The effective force TMA: Parameter values of TMA Saturation properties of SNM  To describe simultaneously both light and heavy nuclei  To simulate the nuclear surface effect

35 The pairing strength and the cutoff energy are determined by fitting experimental one- and two-nucleon separation energies of a large number of nuclei ! Model parameters of the pairing channel

36 Quaqrupole-constrained calculation and the true ground-state The potential energy surfaces of 14 N=116 isotones Z=58 Z=64 Z=71

37 Z=64Z=66 Z=68Z=70 Model predictions: Binding energy per nucleon

38 Z=64 Z=66 Z=68Z=70 Model predictions: Two neutron separation energy

39 Model predictions: Deformation Z=64 Z=66 Z=68 Z=70

40 Model predictions: Charge radius Z=64Z=66 Z=68Z=70

41  First, we want to construct a mass table for all the nuclei throughout the periodic table, which could be used in astrophysical studies and could be compared with other non-relativistic predictions.  Second, for those nuclei that we have experimental data, we want to know, to what extent, the RMF+BCS model can describe them.  Finally, through such a study, we hope to know the limitations of the current RMF model and how to further improve it. The first systematic study: Motivation current RMF ? Nature!

42 The first systematic study: Statistics  The pairing correlation properly treated: the state-dependent BCS method  Axial degree of freedom included: Quadrupole constrained calculation performed for each nucleus, i.e. the potential energy surface of each of the 6969 nuclei is obtained, to ensure that the absolute energy minimum is reached.  The blocking of nuclei with odd numbers of nucleons properly treated  6969 nuclei, even and odd, compared to two previous works about 2000 even-even, no pairing; about 1000 even-even, the constant gap BCS method

43 The sigma is about 2.1 MeV--a small deviation compared to the nuclear mass of the order of several hundred or thousand MeV. Somewhat inferior to FRDM and HFB-2.  about 10 free parameters (FRDM 30, HFB-2 20)  only 10 nuclei to fit our parameters (FRDM 1000, HFB ). In this sense, the predictions of FRDM and HFB-2 are not really predictions. Nuclear mass: theory vs. experiment Experimental data divided into three groups: Group I: experimental error not limited, 2882 nuclei Group II: experimental error less than 0.2 MeV, 2157 nuclei Group III: experimental error less than 0.1 MeV, 1960 nuclei

44 Experimental data divided into four groups: Group I: experimental error not limited, 2790 nuclei Group II: experimental error less than 0.2 MeV, 1994 nuclei Group III: experimental error less than 0.1 MeV, 1767 nuclei Group IV: experimental error less than 0.02 MeV, 1767 nuclei One-neutron separation energy: theory vs. experiment Our results become comparable to those of FRDM and HFB-2 for one- neutron separation energies, which are more important in studies of nuclear structure

45 Most deviations are in the range of minus 2.5 MeV and plus 2.5 MeV The largest overbinding is seen around (82,58) and (126,92) Underbindings are observed in several regions, which might indicate possible shape coexistence, i.e. occurrence of triaxial degree of freedom. Nuclear mass: theory vs. experiment

46 Nuclear mass: How about other effective forces? NL3 G. A. Lalazissis, S. Raman, and P. Ring, At. Data Nucl. Data Tables 71 (1999)1-40 TMA NL3

47 Z=58Z=92 Strongly deformed prolate and oblate shapes coexist in 8<Z<20 and 28<Z<50 regions. Anomalies seen at Z=92 and Z=58: many nuclei with these proton numbers are spherical Nuclear deformation: Theoretical predictions

Z RMF FRDM HFB The rms deviation for 523 nuclei over 42 isotopic chains is only fm! Nuclear charge radii: theory vs. experiment (I)

49 The agreement is particularly good for Z between 40 and 70 Nuclei with less protons are generally underestimated. Nuclei with more protons are generally overestimated. Nuclear charge radii: theory vs. experiment (II)

50 The shell closure at Z=58 is comparable to that at Z=50, and the shell closure at Z=92 is even larger that that Z=82. Therefore, we conclude that the spurious shell closures at Z=58 and Z=92 are the reasons behind the observed anomalies Discrepancies at (82,58) and (92,126): Spurious shell closures?

51 The shell closure at Z=92 The shell closure at Z=58 The not well constrained isovector channel in the RMF model Z=92 Z=58 Overestimated neutron shell closure, underestimated proton shell closure The present RMF model does not constrain the isovector channel very well!

52 Underbindings: Missing triaxial degree of freedom? H. Toki and A. Faessler, Nucl. Phys. A 253(1975)

53 Summary 1.We have built a theoretical model which is valid for all nuclei. 2.Using this model, we have conducted the first systematic study of over 7000 nuclei from the proton drip line to the neutron drip line. 3.Extensive applications of our model to various regions demonstrate that our model is very good in all respects. 4.Further improvement of the current formulation is expected.

54 Future works:  Further improvements  New effective forces  Spurious shell closures at Z=58 and Z=92  Triaxial degree of freedom  Higher order correlations  Shell-model like approach treatment of the pairing correlation  Angular momentum projection  The residual proton and neutron pairing  Numerical methods  The basis expansion method with the woods-saxon basis  New mechanism  The effect of the Dirac sea  The contribution of pions