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
Long-time collaborators Hiroshi Toki Research Center for Nuclear Physics Osaka University, Japan Jie Meng School of physics, Beijing University China 2
Outline 1 A brief review of relevant experimental quantities a. Nuclear masses b. Charge radii c. 2 + energies d. Deformations e. Odd-even effects 2 Theoretical framework a. The relativistic mean field (RMF) model b. The BCS method c. Model parameters 3The first systematic study of over 7000 nuclei 4 Further applications a. Proton drip line and proton emission b. Proton and neutron skins in light nuclei 5Summary and perspective 3
Introduction I: Nuclear masses Up to 1940! 4
Introduction I: Nuclear masses Up to 1948! 5
Introduction I: Nuclear masses Up to 1958! 6
Introduction I: Nuclear masses Up to 1968! 7
Introduction I: Nuclear masses Up to 1978! 8
Introduction I: Nuclear masses Up to 1988! 9
Introduction I: Nuclear masses Up to 1994! 10
Introduction I: Nuclear masses Up to 2004! 11
Introduction II: Charge radius E. G. Nadjakov, At. Data Nucl. Data Tables 56 (1994)133-157 523 12
Introduction II: Charge radius I. Angeli, At. Data Nucl. Data Tables 87 (2004)185-206 798 13
Introduction II: Charge radius 14
Introduction III: The energy of the first excited 2+ state S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128 15
Introduction III: The energy of the first excited 2+ state S. Raman, At. Data Nucl. Data Tables 78 (2001)1-128 16
Introduction IV: Nuclear deformation 17
Introduction V: The odd-even effect and pairing correlation 18
Introduction V: The odd-even effect and pairing correlation 19
The essential ingredients of a successful nuclear model 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 20
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 21
The relativistic mean field (RMF) model The RMF model starts from the following Lagrangian density: Dirac equation Klein-Gordon equation Scalar and Vector potentials 22
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! 23
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 herefore, solving the Dirac equation is transformed to diagonalizing the following matrix The meson fields can be treated similarly 24
The effect of deformation and pairing Binding energy per nucleon of Zirconium isotopes pairing deformation 25
Extending RMF to incorporate the pairing correlation From RMF to RMF+BCS Total energy: BCS equations: Or gap equation Occupation probability RMF RMF +BCS 26
The pairing correlation in weakly bound nuclei The constant-gap BCS method: very successful for stable 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! 27
The pairing correlation in weakly bound nuclei S.P.E [MeV] 0.59 3p3/2 0.57 4s1/2 0.55 1g7/2-0.56 2d 3/2 The state dependent BCS method can describe weakly bound nuclei properly Yadav and Toki, Mod. Phys. Lett A 17 (2002) 2523 28
Resonant states exist due to centrifugal barriers. 1g7/2 (0.55 MeV) this barrier traps 1g7/2 29
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 30
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 31
Model parameters of the pairing channel The pairing strength and the cutoff energy are determined by fitting experimental one- and two-nucleon separation energies! 32
Quaqrupole-constrained calculation and the true ground-state Z=58 Z=64 Z=71 The potential energy surfaces of 14 N=116 isotones 33
Model predictions: Binding energy per nucleon Z=64 Z=66 Z=68 Z=70 34
Model predictions: Two neutron separation energy Z=64 Z=66 Z=68 Z=70 35
Model predictions: Deformation Z=64 Z=66 Z=68 Z=70 36
Model predictions: Charge radius Z=64 Z=66 Z=68 Z=70 37
The first systematic study: Motivation First, the RMF model is just an effective field theory; its predictive power must and can only be checked by comparison with experiment. Second, any deficiency, if exists, can only be understood through an extensive study of all the nuclei throughout the periodic table. current RMF? current RMF? Nature! 38
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 number of nucleons properly treated 39
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. 40
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 oneneutron separation energies, which are more important in studies of nuclear structure 41
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. 42
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 43
Nuclear deformation: Theoretical predictions Z=92 Z=58 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 44 spherical
Nuclear charge radii: theory vs. experiment (I) Z RMF FRDM HFB The rms deviation for 523 nuclei over 42 isotopic chains is only 0.037 fm! 523 0.037 0.045 0.028 45
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. 46
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 47
The not well constrained isovector channel in the RMF model The shell closure at Z=92 Z=92 Overestimated neutron shell closure, underestimated proton shell closure Z=58 The shell closure at Z=58 Overestimated neutron shell closure, underestimated proton shell closure The present RMF model does not constrain the isovector channel very well! 48
Underbindings: Missing triaxial degree of freedom? H. Toki and A. Faessler, Nucl. Phys. A 253(1975)231-152 49
Proton drip line: theory vs. experiment Proton drip line nucleus: the heaviest nucleus with negative one-proton separation energy at each isotopic chain! N=Z 50
Proton emission beyond the proton drip line The decay probability depends strongly on the energy of the unbound proton and its angular momentum Proton separation energy [MeV] E=1.7 E=0.8 E=0.0 proton emitter 51
Studies of the rare-earth proton emission in RMF+BCS model proton-neurton pairing? odd-odd nuclei 52
Proton skin in Ar isotopes A. Ozawa et al., NPA709(2002)60 The difference between the proton Fermi surface and neutron Fermi surface is the driving force behind the skin phenomena! 53
Neutron skin in Na isotopes T. Suzuki et al., PRL75(1995)3241 Proton and neutron skins will always develop towards the drip lines 54
Summary The RMF model is very good in the whole region, but can be further improved! Subjects covered in this talk: The first systematic study of over 7000 nuclei Proton drip line and proton emission Proton and neutron skins Subjects not covered in this talk but covered in the thesis: Magic nuclei Rare-earth nuclei and actinide nuclei Superheavy nuclei Nuclear sizes Particle number conservation 55
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 56