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##### Lisheng Geng

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**The first systematic studyof the ground-state properties of**finite nucleiin the relativistic mean field model Lisheng Geng Research Center for Nuclear Physics, Osaka University School of Physics, Beijing University**Long-time collaborators**Hiroshi Toki Research Center for Nuclear Physics Osaka University, Japan Jie Meng School of physics, Beijing University China**Outline**• A brief review of relevant experimental quantities: nuclear masses, charge radii, 2+ energies, deformations, odd-even effects • Theoretical framework • The relativistic mean field (RMF) model • The BCS method • Model parameters • The first systematic study of over 7000 nuclei • Comparison with experimental data and other theoretical predictions • The causes of some discrepancies: the not-well-constrained isovector channel • Summary and perspective**Introduction I: Nuclear masses**Up to 1940!**Introduction I: Nuclear masses**Up to 1948!**Introduction I: Nuclear masses**Up to 1958!**Introduction I: Nuclear masses**Up to 1968!**Introduction I: Nuclear masses**Up to 1978!**Introduction I: Nuclear masses**Up to 1988!**Introduction I: Nuclear masses**Up to 1994!**Introduction I: Nuclear masses**Up to 2004!**Introduction II: Charge radius**E. G. Nadjakov, At. Data Nucl. Data Tables 56 (1994)133-157 523**Introduction II: Charge radius**I. Angeli, At. Data Nucl. Data Tables 87 (2004)185-206 798**Introduction III: The energy of the first excited 2+ state**S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128**Introduction III: The energy of the first excited 2+ state**S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128**The essential ingredients to build a nuclear structure**model • The spin-oribit interaction:i.e. the magic number effect • The deformation effect:most nuclei are deformed except a few magic nuclei • The pairing correlation:important to describe open-shell nuclei and responsible for the very existences of drip line nuclei**Spin-orbit interaction in non-relativistic models**no spin-orbit spin-orbit 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 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 Elementary theory of nuclear shell model, M. G. Mayer and J. Hans D. Jensen, 1956**The relativistic mean field (RMF) model**The RMF model starts from the following Lagrangian density: Dirac equation Klein-Gordon equation Scalar and Vector potentials**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!**The necessity of a relativistic model**• Spin-orbit interaction • Nuclear matter saturation • Polarization (spin) observables in nuclear reaction • Study of high density and high temperature nuclear matter • Connection to QCD • Pseudospin symmetry “The atomic nucleus as a Relativistic system”, L. N. Savushkin and H. Toki, springer, 2005**Nuclear matter saturation**non-relativistic relativistic Bruckner Hartree-Fock Coester band Empirical saturation relativistic mean field theory Non-relativistic calculations: not successful. Relativistic Bruckner Hartree-Fock calculations: encouraging! The RMF model: parameterized to describe the nuclear matter saturation.**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**The effect of deformation and pairing**Binding energy per nucleon of Zirconium isotopes pairing deformation**Extending RMF to incorporate the pairing correlation**From RMF to RMF+BCS Total energy: BCS equations: Or gap equation Occupation probability RMF RMF +BCS**The pairing correlation in weakly bound nuclei**The constant-gap BCS method: very successful forstable nuclei But for weakly bound nuclei, which are the subjects of present research, it fails. A zero-range delta force in the particle-particle channel is found to be useful!**The pairing correlation in weakly bound nuclei**S.P.E [MeV] 0.59 0.57 0.55 3p3/2 4s1/2 1g7/2 2d3/2 -0.56 The state dependent BCS method can describe weakly bound nuclei properly Yadav and Toki, Mod. Phys. Lett A 17 (2002) 2523**Resonant states exist due to centrifugal barriers.**1g7/2 (0.55 MeV) this barrier traps 1g7/2**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: Effective: valid for all nuclei Numerically simple: systematic study possible**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. • Nuclear matter: • saturation density, • binding energy per nucleon • symmetry energy • compression modulus • Finite nuclei: • Binding energy • Charge radius Stars: twelve nuclei used in DD-ME2**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**Model parameters of the pairing channel**The pairing strength and the cutoff energy are determined by fitting experimental one- and two-nucleon separation energies of a large number of nuclei!**Quaqrupole-constrained calculation and the true ground-state**Z=58 Z=71 Z=64 The potential energy surfaces of 14 N=116 isotones**Model predictions: Binding energy per nucleon**Z=64 Z=66 Z=68 Z=70**Model predictions: Two neutron separation energy**Z=64 Z=66 Z=68 Z=70**Model predictions: Deformation**Z=64 Z=66 Z=68 Z=70**Model predictions: Charge radius**Z=64 Z=66 Z=68 Z=70**The first systematic study: Motivation**current RMF? current RMF? Nature! • 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: Statistics**• 6969 nuclei, even and odd, compared to two previous works Hirata@1997, about 2000 even-even, no pairing; Lalazissis@1999, about 1000 even-even, the constant gap BCS method • 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**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 • 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-2 2000). • In this sense, the predictions of FRDM and HFB-2 are not really predictions.**One-neutron separation energy: theory vs. experiment**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 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**Nuclear mass: theory vs. experiment**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: How about other effective forces? NL3**TMA NL3 G. A. Lalazissis, S. Raman, and P. Ring, At. Data Nucl. Data Tables 71 (1999)1-40**Nuclear deformation: Theoretical predictions**Z=92 Z=58 Strongly deformed prolate and oblate shapes coexistin 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 charge radii: theory vs. experiment (I)**Z RMF FRDM HFB 523 0.037 0.045 0.028 The rms deviation for 523 nuclei over 42 isotopic chains is only 0.037 fm!**Nuclear charge radii: theory vs. experiment (II)**• 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.**Discrepancies at (82,58) and (92,126): Spurious shell**closures? 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