The Nuclear Symmetry Energy and Neutron Skin Thickness of Finite Nuclei Lie-Wen Chen ( 陈列文 ) (INPAC and Department of Physics, Shanghai Jiao Tong University. 第十三届全国核结构研讨会暨第九次全国 “ 核结构与量子力学 ” 专题 讨论会,2010 年 7 月 日, 赤峰, 内蒙古 Collaborators ： Che Ming Ko and Jun Xu (TAMU) Bao-An Li (TAMU-Commerce) Xin Wang (SJTU) Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Zhen Zhang (SJTU)

Outline EOS of asymmetric nuclear matter and the symmetry energy Constraints on density dependence of symmetry energy from nuclear structure and reactions – Present status Constraining the symmetry energy with the neutron skin thickness of heavy nuclei in a novel correlation analysis Symmetry energy and nuclear effective interactions Summary and outlook Main References: B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, (2008) L.W. Chen, B.J. Cai, C.M. Ko, B.A. Li, C. Shen, and J. Xu, PRC80, (2009) L.W. Chen, C.M. Ko, B.A. Li and J. Xu, arXiv: , 2010

Density Dependence of the Nuclear Symmetry Energy Reactions & Structures of Neutron- Rich Nuclei (CSR/Lanzho u, FRIB, GSI, RIKEN……) Most uncertain property of an asymmetric nuclear matter Isospin Nuclear Physics What is the isospin dependence of the in-medium nuclear effective interactions??? Neutron Stars … Structures of Neutron-rich Nuclei, … Isospin Effects in HIC’s … Many-Body Theory Transport Theory General Relativity Nuclear Force EOS for Asymmetric Nuclear Matter On Earth!!! In Heaven!!! Isospin in Nuclear Physics EOS of asymmetric nuclear matter and the symmetry energy

EOS of Nuclear Matter

Liquid-drop model (Isospin) Symmetry energy term W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,…… Symmetry energy including surface diffusion effects (y s =S v /S s ) The Nuclear Symmetry Energy

EOS of Isospin Asymmetric Nuclear Matter (Parabolic law) The Nuclear Symmetry Energy The Nuclear Matter Symmetry Energy Symmetry energy term (poorly known) Symmetric Nuclear Matter (relatively well-determined)

The Symmetry Energy The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). The symmetry energy is also related to some issues of fundamental physics: 1. The precision tests of the SM through atomic parity violation observables (Sil et al., PRC05) 2. Possible time variation of the gravitational constant (Jofre etal. PRL06; Krastev/Li, PRC07) 3. Non-Newtonian gravity proposed in grand unification theories (Wen/Li/Chen, PRL10)

Nuclear Matter EOS: Many-Body Approaches Microscopic Many-Body Approaches Non-relativistic Brueckner-Bethe-Goldstone (BBG) Theory Relativistic Dirac-Brueckner-Hartree-Fock (DBHF) approach Self-consistent Green’s Function (SCGF) Theory Variational Many-Body (VMB) approach …… Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) …… Phenomenological Approaches Relativistic mean-field (RMF) theory Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations Phenomenological potential models ……

Chen/Ko/Li, PRC72, (2005) Chen/Ko/Li, PRC76, (2007) Z.H. Li et al., PRC74, (2006)Dieperink et al., PRC68, (2003) BHF E sym : Many-Body Approaches

Promising Probes of the E sym (ρ) (an incomplete list !) Symmetry energy around saturation density

Li/ Chen, PRC72, (2005) Symmetry energy, isospin diffusion, in-medium cross section Isospin Diffusion Data  E sym (ρ 0 )=31.6 MeV L=88±25 MeV Chen/Ko/Li, PRC72, (2005) E sym : Isospin Diffusion in HIC’s Chen/Ko/Li, PRL94, (2005) Isospin dependent BUU transport model

Consistent with isospin diffusion data! Constraining Symmetry Energy by Isocaling: TAMU Data Shetty/Yennello/ Souliotis, PRC75,034602(2007); PRC76, (2007) E sym : Isoscaling in HIC’s Isoscaling Data  E sym (ρ 0 )=31.6 MeV L=65 MeV

ImQMD: n/p ratios and two isospin diffusion measurements Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, (2009) E sym : Isospin diffusion and double n/p ratio in HIC’s ImQMD: Isospin Diffusion and double n/p ratio  E sym (ρ 0 )= MeV L= MeV

E sym : Nuclear Mass in Thomas-Fermi Model Thomas-Fermi Model + Nuclear Mass  E sym (ρ 0 )=32.65 MeV L=49.9 MeV Myers/Swiatecki, NPA 601, 141 (1996) Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8

E sym : Pygmy Dipole Resonances Pygmy Dipole Resonances of 130,132 Sn  E sym (ρ 0 )=32 ± 1.8 MeV L= ± 15 MeV Pygmy Dipole Resonances of 68 Ni and 132 Sn  E sym (ρ 0 )=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV

E sym from Isobaric Analog States + Liquid Drop model with surface symmetry energy IAS+Liquid Drop Model with Surface Esym  E sym (ρ 0 )=32.5 ± 1 MeV L=94.5 ± 16.5 MeV Danielewicz/Lee, NPA 818, 36 (2009) E sym : IAS+LDM

E sym : Droplet Model Analysis on Neutron Skin Droplet Model + N-skin  E sym (ρ 0 )=31.6 MeV, L=66.5 ± 36.5 MeV

Droplet Model + N-skin  E sym (ρ 0 )= MeV, L=55 ± 25 MeV E sym : Droplet Model Analysis on Neutron Skin

E sym around normal density E sym (ρ 0 )= MeV L= MeV 9 constraints on E sym (ρ 0 ) and L from nuclear reactions and structures Still within large uncertain region !!

The Nuclear Neutron Skin For heavier stable nuclei: N>Z Neutron Skin Thickness: Bodmer, Nucl. Phys. 17, 388 (1960) Sprung/Vallieres/Campi/Ko, NPA253, 1 (1975) Shlomo/Friedman, PRL39, 1180 (1977) ……

The E sym vs. Nuclear Neutron Skin Good linear Correlation: S-L Chen/Ko/Li, PRC72, (2005)

The E sym vs. Nuclear Neutron Skin Chen/Ko/Li, PRC72, (2005) For heavier nuclei: Still good linear correlation between S-L B.A. Brown, PRL85,5296 (2000)

The Skyrme HF Energy Density Functional Standard Skyrme Interaction: _________ 9 Skyrme parameters: 9 macroscopic nuclear properties: There are more than 120 sets of Skyrme- like Interactions in the literature Chen/Ko/Li/Xu arXiv: Agrawal/Shlomo/Kim Au PRC72, (2005) Yoshida/Sagawa PRC73, (2006)

The Skyrme HF Energy Density Functional Chen/Cai/Ko/Li/Shen/Xu, PRC80, (2009): Modified Skyrme-Like (MSL) Model Chen/Ko/Li/Xu, arXiv:

The Skyrme HF with MSL0 Chen/Ko/Li/Xu, arXiv:

Correlations between Nuetron-Skin thickness and macroscopic Nuclear Properties For heavy nuclei 208 Pb and 120 Sn: Δr np is strongly correlated with L, moderately with E sym (ρ 0 ), a little bit with m* s,0 For medium-heavy nucleus 48 Ca: Δr np correlation with E sym is much weaker; It further depends on G V and W 0 Important Terms

Constraining E sym with Neutron Skin Data Neutron skin constraints on L and E sym (ρ 0 ) are insensitive to the variations of other macroscopic quantities.

(~independent of E sym (ρ 0 )) A quite stringent constraint on Δr np of 208 Pb: Constraining E sym with Neutron Skin Data and Heavy-Ion Reactions N-Skin + HIC Core-Crust transition density in Neutron stars:

Global nucleon optical potential  E sym (ρ 0 )=31.3 ± 4.5 MeV, L=52.7 ± 22.5 MeV Xu/Li/Chen, arXiv: v1, 2010 E sym : Global nucleon optical potential Consistent with Sn neutron skin data!

Symmetry energy and Nuclear Effective Interaction Chen/Ko/Li, PRC72, (2005) Chen/Ko/Li, PRC76, (2007) L=58 ± 18 MeV: only 32/118L=58 ± 18 MeV: only 8/23

We have proposed a novel method to explore transparently the correlation between observables of finite nuclei and nuclear matter properties. The neutron skin thickness of heavy nuclei provides reliable information on the symmetry energy. The existing neutron skin data of Sn isotopes give important constraints on the symmetry energy and the neutron skin of 208 Pb Combining the constraints on E sym from neutron skin with that from isospin diffusion and double n/p ratios in HIC’s impose quite accurate constraint of L=58±18 MeV approximately independent of Esym Our correlation analysis method can be generalized to other mean- field models (e.g., RMF) or density functional theories and a number of other correlation analyses are being performed (giant resonance, shell structure,,…… ) IV. Summary and Outlook

谢谢！

(1) EOS of symmetric matter around the saturation density ρ 0 Giant Monopole Resonance K 0 =231±5 MeV PRL82, 691 (1999) Recent results: K 0 =240±10 MeV G. Colo et al. U. Garg et al. S. Shlomo et al. __ EOS of Symmetric Nuclear Matter

(2) EOS of symmetric matter for 1ρ 0 < ρ < 3ρ 0 from K + production in HIC’s J. Aichelin and C.M. Ko, PRL55, (1985) 2661 C. Fuchs, Prog. Part. Nucl. Phys. 56, (2006) 1 Transport calculations indicate that “results for the K + excitation function in Au + Au over C + C reactions as measured by the KaoS Collaboration strongly support the scenario with a soft EOS.” C. Fuchs et al, PRL86, (2001) 1974 See also: C. Hartnack, H. Oeschler, and J. Aichelin, PRL96, (2006) EOS of Symmetric Nuclear Matter

(3) Present constraints on the EOS of symmetric nuclear matter for 2ρ 0 < ρ < 5ρ 0 using flow data from BEVALAC, SIS/GSI and AGS Use constrained mean fields to predict the EOS for symmetric matter Width of pressure domain reflects uncertainties in comparison and of assumed momentum dependence. P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002) The highest pressure recorded under laboratory controlled conditions in nucleus-nucleus collisions High density nuclear matter 2 to 5ρ 0 EOS of Symmetric Nuclear Matter

Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σ exp b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σ in-medium c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking Isospin-dependent BUU (IBUU) model Transport model for HIC’s EOS

Isospin- and momentum-dependent potential (MDI) Chen/Ko/Li, PRL94, (2005) Li/Chen, PRC72, (2005) Das/Das Gupta/Gale/Li, PRC67, (2003) Transport model: IBUU04

E sym : Isospin Diffusion in HIC’s How to measure Isospin Diffusion? PRL84, 1120 (2000) ______________________________________ A+A,B+B,A+B X: isospin tracer Isospin Diffusion/Transport

Isoscaling in HIC’s Isoscaling observed in many reactions M.B. Tsang et al. PRL86, 5023 (2001) E sym : Isoscaling in HIC’s

High density behaviors of E sym n/p ratio of the high density region Li/Yong/Zuo, PRC 71, (2005) Isospin fractionation! Heavy-Ion Collisions at Higher Energies

Subthreshold K 0 /K + yield may be a sensitive probe of the symmetry energy at high densities Aichelin/Ko, PRL55, 2661 (1985): Subthreshold kaon yield is a sensitive probe of the EOS of nuclear matter at high densities Theory: Ferini et al., PRL97, (2006)Exp.: Lopez et al. FOPI, PRC75, (R) (2007) K 0 /K + yield is not so sensitive to the symmetry energy! Lower energy and more neutron-rich system??? High density behaviors of E sym : kaon ratio

IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102,062502(2009) High density behaviors of E sym : pion ratio A Quite Soft Esym at supra-saturation densities ??? Zhang et al.,PRC80,034616(2009) IDQMD, Feng/Jin, PLB683, 140(2010) Pion Medium Effects? Xu/Ko/Oh PRC81, (2010) Threshold effects? ……

High density behaviors of E sym : n/p v2 A Stiff Esym at supra-saturation densities ??? W. Trauntmann et al., arXiv:

E sym at very low densities: Clustering effects S. Kowalski, et al., PRC 75 (2007) Horowitz and Schwenk, Nucl. Phys. A 776 (2006) 55

E sym at very low densities: Clustering effects J. B. Natowitz et al., arXiv: PRL, 2010